Patent Publication Number: US-10315783-B2

Title: Ply blending and stacking sequence

Description:
CROSS-REFERENCE TO RELATED APPLICATION 
     This application is related to the following U.S. Pat. No. 10,195,812, entitled “Ply Blending and Stacking Sequence,” filed even date hereof, issued on Feb. 5, 2019, and is incorporated herein by reference in its entirety. 
     BACKGROUND INFORMATION 
     1. Field 
     The present disclosure relates generally to composite structures and, in particular, to layups for composite structures. More particularly, the present disclosure relates to a method and apparatus for determining a layup for a composite structure. 
     2. Background 
     Aircraft are being designed and manufactured with greater and greater percentages of composite materials. Composite materials are used in aircraft to decrease the weight of the aircraft. This decreased weight improves performance features such as payload capacity and fuel efficiency. Further, composite materials provide longer service life for various components in an aircraft. 
     Composite materials may be tough, light-weight materials created by combining two or more functional components. For example, a composite material may include reinforcing fibers bound in a polymer resin matrix. Resins used in composite materials may include thermoplastic or thermoset resins. A thermoplastic material may become soft upon heating and may harden upon cooling. A thermoplastic material may be able to be repeatedly heated and cooled. A thermoset material may become hard when heated. The fibers may be unidirectional or may take the form of a woven cloth or fabric. 
     In manufacturing composite structures, layers of composite material are typically laid up on a tool. The layers may be comprised of fibers in sheets. These sheets may take the form of fabrics, tape, tows, or other suitable forms. In some cases, resin may be infused or preimpregnated into the sheets. These types of sheets are commonly referred to as prepreg. 
     The different layers of prepreg may be laid up in different orientations, and different numbers of layers may be used depending on the thickness of the composite structure being manufactured. These layers may be laid up by hand or by using automated equipment such as a fiber placement system. After the layers of composite material have been laid up on the tool, the layers of composite material may be consolidated or cured by exposure to temperature and pressure, thus forming the final composite structure. 
     Traditional composite structures may include four types of composite layers: 0 degree layers, 45 degree layers, −45 degree layers, and 90 degree layers. These four angles may be conventionally and widely used to form composite structures such as composite skins. The behavior of a composite structure may be changed by varying the number of the four kinds of composite layers. 
     Advances in manufacturing equipment may now enable use of other fiber angles in composite layers. However, methods of designing composite structures with non-traditional angles of composite layers are needed. Further, methods of analysis for expected performance of composite structures with non-traditional angles of composite layers are needed. Therefore, it would be desirable to have a method and apparatus that take into account at least some of the issues discussed above, as well as other possible issues. 
     SUMMARY 
     In an illustrative embodiment, a method is presented. A desired thickness and a desired stiffness distribution for a composite structure are received. A model of the composite structure is split into a plurality of panels. A set of lamination parameters and a quantity of plies for each panel of the plurality of panels are determined using the desired thickness and the desired stiffness distribution for the composite structure. Ply shapes for each ply of the plurality of panels are determined using an objective function. A fiber angle value for each ply of the plurality of panels is determined such that the model of the composite structure has a stiffness equal to the desired stiffness. 
     In another illustrative embodiment, a method is presented. A desired thickness and a desired stiffness distribution for a composite structure are received. A model of the composite structure is split into a plurality of panels. A set of lamination parameters and a quantity of plies for each panel of the plurality of panels are determined using the desired thickness and the desired stiffness distribution for the composite structure. Ply shapes for each ply of the plurality of panels are determined using an objective function. Determining the ply shapes for each ply of the plurality of panels using the objective function comprises grouping panels of the plurality of panels with a same quantity of plies and same set of lamination parameters to form a plurality of groups and creating a plurality of ply sequences for the plurality of groups. Each panel of a group of the plurality of groups has a same layup. Each of the plurality of ply sequences represents a ply for at least one panel in the plurality of panels. A fiber angle for each ply of the plurality of panels is determined. Determining a fiber angle for each ply of the plurality of panels comprises generating equations for a plurality of lamination parameters for each group of the plurality of groups, combining the equations for each group of the plurality of groups into one system, and solving for a plurality of angle variables in the system. 
     A further illustrative embodiment of the present disclosure provides a method for forming a composite structure. A desired thickness and a desired stiffness distribution for a composite structure are received. A model of the composite structure is split into a plurality of panels. A set of lamination parameters and a quantity of plies for each panel of the plurality of panels are determined using the desired thickness and the desired stiffness distribution for the composite structure. Ply shapes for each ply of the plurality of panels are determined using an objective function. A fiber angle for each ply of the plurality of panels is determined. The plies are arranged according to stacking sequence rules to form a ply arrangement. The stacking sequence rules include at least one of symmetry, separation of plies with fiber angles with more than a 45 degrees difference, or avoidance of ply clusters. A composite material is laid up according to the ply arrangement to form the composite structure. The composite structure has a stiffness equal to the desired stiffness. 
     Yet another embodiment of the present disclosure provides a computer system. The computer system comprises a composite laminate designer, a ply shape optimizer, and a fiber angle selector. The composite laminate designer receives a desired thickness and a desired stiffness distribution for a composite structure, splits a model of the composite structure into a plurality of panels, and determines a set of lamination parameters and a quantity of plies for each panel of the plurality of panels using the desired thickness and the desired stiffness distribution for the composite structure. The ply shape optimizer determines ply shapes for each ply of the plurality of panels using an objective function. The fiber angle selector determines a fiber angle for each ply of the plurality of panels, such that the model of the composite structure has a stiffness equal to the desired stiffness. 
     A yet further embodiment of the present disclosure provides a computer program product for designing a composite structure. The computer program product comprises a computer readable storage media and first program code, second program code, third program code, fourth program code, and fifth program code each stored on the computer readable storage media. The first program code, stored on the computer readable storage media, receives a desired thickness and a desired stiffness distribution for a composite structure. The second program code, stored on the computer readable storage media, splits a model of the composite structure into a plurality of panels. The third program code, stored on the computer readable storage media, determines a set of lamination parameters and a quantity of plies for each panel of the plurality of panels using the desired thickness and the desired stiffness distribution for the composite structure. The fourth program code, stored on the computer readable storage media, determines ply shapes for each ply of the plurality of panels using an objective function. The fifth program code, stored on the computer readable storage media, determines a fiber angle for each ply of the plurality of panels, such that the model of the composite structure has a stiffness equal to the desired stiffness distribution. 
     The features and functions can be achieved independently in various embodiments of the present disclosure or may be combined in yet other embodiments in which further details can be seen with reference to the following description and drawings. 
    
    
     
       BRIEF DESCRIPTION OF THE DRAWINGS 
       The novel features believed to be characteristic of the illustrative embodiments are set forth in the appended claims. The illustrative embodiments, however, as well as a preferred mode of use, further objectives and features thereof, will best be understood by reference to the following detailed description of an illustrative embodiment of the present disclosure when read in conjunction with the accompanying drawings, wherein: 
         FIG. 1  is an illustration of an aircraft in which an illustrative embodiment may be implemented; 
         FIG. 2  is an illustration of a flowchart of a process for determining a layup for a composite structure in accordance with an illustrative embodiment; 
         FIG. 3  is an illustration of a flowchart of a process for determining a layup for a composite structure in accordance with an illustrative embodiment; 
         FIG. 4  is an illustration of a flowchart of a process for forming a composite structure in accordance with an illustrative embodiment; 
         FIGS. 5A and 5B  are illustrations of a block diagram of a manufacturing environment in accordance with an illustrative embodiment; 
         FIG. 6  is an illustration of a two-dimensional view of a model of a composite structure in accordance with an illustrative embodiment; 
         FIG. 7  is an illustration of a plurality of groups of panels in accordance with an illustrative embodiment; 
         FIG. 8  is an illustration of a plurality of groups of panels in accordance with an illustrative embodiment; 
         FIG. 9  is an illustration of ply sequences for a plurality of groups of panels in accordance with an illustrative embodiment; 
         FIG. 10  is an illustration of an exploded view of a ply arrangement in accordance with an illustrative embodiment; 
         FIG. 11  is an illustration of plies in ply sequences in accordance with an illustrative embodiment; 
         FIG. 12  is an illustration of a block diagram of a data processing system in accordance with an illustrative embodiment; 
         FIG. 13  is an illustration of a block diagram of a manufacturing environment in accordance with an illustrative embodiment; 
         FIG. 14  is an illustration of an aircraft manufacturing and service method in the form of a block diagram in accordance with an illustrative embodiment; and 
         FIG. 15  is an illustration of an aircraft in the form of a block diagram in which an illustrative embodiment may be implemented. 
     
    
    
     DETAILED DESCRIPTION 
     The different illustrative embodiments recognize and take into account one or more different considerations. For example, the different illustrative embodiments recognize and take into account that the shapes of composite layers in traditional composite structures may not take into account how the shapes of the layers may affect manufacturing rate. 
     The different illustrative embodiments recognize and take into account that it may be desirable to replace a traditional composite layup with a non-traditional layup. For example, a component typically formed of a traditional layup may have desirable characteristics if formed of a non-traditional layup having the same stiffness as the traditional layup. 
     The different illustrative embodiments recognize and take into account that load paths are driven by stiffness. Thus, it would be desirable for the non-traditional layup to have the same stiffness as the traditional layup it is replacing. Changing the stiffness of one component affects not only the behavior of that component, but also of the other components in the structure. 
     The different illustrative embodiments recognize and take into account that it may be desirable to replace a traditional composite layup with a non-traditional layup for many reasons. Changing from a traditional layup to a non-traditional layup may avoid undesirable failure modes by changing the fiber angle arrangement. 
     The different illustrative embodiments recognize and take into account that it may be desirable to replace a traditional laminate with a non-traditional laminate if the original thicknesses for the component were not specified in integer multiples of a ply thickness. The different illustrative embodiments further recognize and take into account that changing from a traditional layup to a non-traditional layup may create a composite component having the same stiffness properties but with a different ply thickness. 
     The different illustrative embodiments also recognize and take into account that certain ply angles may cause undesirable manufacturing results. Thus, the different illustrative embodiments recognize and take into account that replacing a traditional layup with a non-traditional layup may avoid using the certain ply angles that may cause undesirable manufacturing results. 
     The different illustrative embodiments also recognize and take into account that determining a sequence of composite layers for a single non-traditional composite layup may be performed using lamination parameters. Lamination parameters are a compact representation of a stacking sequence of layers. Lamination parameters may be non-dimensional parameters that together with thickness and material invariants describe a laminate stiffness matrix independent of the number of plies. 
     The different illustrative embodiments recognize and take into account that a composite may be an N ply layup with angles (θ 1 , θ 2 , . . . θ N ) and ply locations (z 0 =−t/2, z 0 &lt;z 1 &lt;z 2 , . . . z N =t/2) where t is the total thickness of the N plies. The composite can be described using 12 lamination parameters: 
     (V 1A , V 2A , V 3A , V 4A ), (V 1B , V 2B , V 3B , V 4B ), and (V 1D , V 2D , V 3D , V 4D ) which are defined by equations (1)-(3). 
     
       
         
           
             
               
                 
                   
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     For optimization, it is helpful to define normalized lamination parameters 
                     (         V   _       1   ⁢           ⁢   A       ,       V   _       2   ⁢           ⁢   A       ,       V   _       3   ⁢           ⁢   A       ,       V   _       4   ⁢           ⁢   A         )     =       1   t     ⁢     (       V     1   ⁢           ⁢   A       ,     V     2   ⁢           ⁢   A       ,     V     3   ⁢           ⁢   A       ,     V     4   ⁢           ⁢   A         )               (   4   )                 (         V   _       1   ⁢           ⁢   B       ,       V   _       2   ⁢           ⁢   B       ,       V   _       3   ⁢           ⁢   B       ,       V   _       4   ⁢           ⁢   B         )     =       4     t   2       ⁢     (       V     1   ⁢           ⁢   B       ,     V     2   ⁢           ⁢   B       ,     V     3   ⁢           ⁢   B       ,     V     4   ⁢           ⁢   B         )               (   5   )                 (         V   _       1   ⁢           ⁢   D       ,       V   _       2   ⁢           ⁢   D       ,       V   _       3   ⁢           ⁢   D       ,       V   _       4   ⁢           ⁢   D         )     =       12     t   3       ⁢     (       V     1   ⁢           ⁢   D       ,     V     2   ⁢           ⁢   D       ,     V     3   ⁢           ⁢   D       ,     V     4   ⁢           ⁢   D         )               (   6   )               
such that
 
−1≤(   V     1A   , V     2A   , V     3A   , V     4A )≤1
 
−1≤(   V     1B   , V     2B   , V     3B   , V     4B )≤1
 
−1≤(   V     1D   , V     2D   , V     3D   , V     4D )≤1  (7)
 
     Additional constraints on the lamination parameters exist to ensure the combination of lamination parameters is feasible. 
     The relationship between the normalized lamination parameters, thickness, and the stiffness matrices A, B, D is given by equations (8)-(10): 
                     (           A   11           A   12           A   16               A   21           A   22           A   26               A   61           A   62           A   66           )     =     t   (           ⁢             U   1     +       U   2     ⁢       V   _       1   ⁢           ⁢   A         +       U   3     ⁢       V   _       3   ⁢           ⁢   A                   U   4     -       U   3     ⁢       V   _       3   ⁢           ⁢   A                       U   2     2     ⁢       V   _       2   ⁢           ⁢   A         +       U   3     ⁢       V   _       4   ⁢           ⁢   A                       A   _     12             U   1     -       U   2     ⁢       V   _       1   ⁢           ⁢   A         +       U   3     ⁢       V   _       3   ⁢           ⁢   A                       U   2     2     ⁢       V   _       2   ⁢           ⁢   A         -       U   3     ⁢       V   _       4   ⁢           ⁢   A                       A   _     16             A   _     26             U   5     -       U   3     ⁢       V   _       3   ⁢           ⁢   A                 ⁢           )             (   8   )                 (           B   11           B   12           B   16               B   21           B   22           B   26               B   61           B   62           B   66           )     =         t   2     4     ⁢     (           ⁢               U   2     ⁢       V   _       1   ⁢           ⁢   B         +       U   3     ⁢       V   _       3   ⁢           ⁢   B                   -     U   3       ⁢       V   _       3   ⁢           ⁢   B                     U   2     2     ⁢       V   _       2   ⁢           ⁢   B         +       U   3     ⁢       V   _       4   ⁢           ⁢   B                       B   _     12               -     U   2       ⁢       V   _       1   ⁢           ⁢   B         +       U   3     ⁢       V   _       3   ⁢           ⁢   B                       U   2     2     ⁢       V   _       2   ⁢           ⁢   B         -       U   3     ⁢       V   _       4   ⁢           ⁢   B                       B   _     16             B   _     26             -     U   3       ⁢       V   _       3   ⁢           ⁢   B               ⁢           )               (   9   )                 (           D   11           D   12           D   16               D   21           D   22           D   26               D   61           D   62           D   66           )     =         t   3     12     ⁢     (           ⁢             U   1     +       U   2     ⁢       V   _       1   ⁢           ⁢   D         +       U   3     ⁢       V   _       3   ⁢           ⁢   D                   U   4     -       U   3     ⁢       V   _       3   ⁢           ⁢   D                       U   2     2     ⁢       V   _       2   ⁢           ⁢   D         +       U   3     ⁢       V   _       4   ⁢           ⁢   D                       D   _     12             U   1     -       U   2     ⁢       V   _       1   ⁢           ⁢   D         +       U   3     ⁢       V   _       3   ⁢           ⁢   D                       U   2     2     ⁢       V   _       2   ⁢           ⁢   D         -       U   3     ⁢       V   _       4   ⁢           ⁢   D                       D   _     16             D   _     26             U   5     -       U   3     ⁢       V   _       3   ⁢           ⁢   D                 ⁢           )               (   10   )               
where (U 1 , U 2 , U 3 , U 4 , and U 5 ) are material invariants that depend only on the ply material properties and are defined as follows:
 
     
       
         
           
             
               
                 
                   
                     
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                     = 
                     
                       G 
                       12 
                     
                   
                 
               
               
                 
                   ( 
                   12 
                   ) 
                 
               
             
           
         
       
     
     Equations (8)-(10) can be written in matrix notation by introducing normalized A, B, D matrices (Ā,  B ,  D ) as follows: 
     
       
         
           
             
               
                 
                   A 
                   = 
                   
                     t 
                     ⁢ 
                     
                         
                     
                     ⁢ 
                     
                       A 
                       _ 
                     
                   
                 
               
               
                 
                   ( 
                   13 
                   ) 
                 
               
             
             
               
                 
                   B 
                   = 
                   
                     
                       
                         t 
                         2 
                       
                       4 
                     
                     ⁢ 
                     
                       B 
                       _ 
                     
                   
                 
               
               
                 
                   ( 
                   14 
                   ) 
                 
               
             
             
               
                 
                   D 
                   = 
                   
                     
                       
                         t 
                         3 
                       
                       12 
                     
                     ⁢ 
                     
                       D 
                       _ 
                     
                   
                 
               
               
                 
                   ( 
                   15 
                   ) 
                 
               
             
           
         
       
     
     The different illustrative embodiments also recognize and take into account that determining a sequence of composite layers for adjacent non-traditional composite layups in a composite structure may be desirable. However, a method to provide continuity between layers of adjacent non-traditional composite layups that exactly match a prescribed thickness and stiffness for each layup is not known. For example, millions of stacking sequences may be possible for a given set of lamination parameters and a number of layers. Finding combinations of stacking sequences for adjacent non-traditional composite layups that have matching fiber angles is therefore unlikely, and it would be very time-consuming to search for matching stacking sequences. 
     Accordingly, it may be desirable to have a method and apparatus for determining a layup for a composite structure that exactly matches a desirable thickness and stiffness distribution. For example, it may be desirable to have a method and apparatus for determining a sequence and associated fiber angles for the plies of a non-traditional composite structure. 
     With reference now to the Figures, and in particular, with reference to  FIG. 1 , an illustration of an aircraft is depicted in which an illustrative embodiment may be implemented. In this illustrative example, aircraft  100  has wing  102  and wing  104  attached to body  106 . Aircraft  100  includes engine  108  attached to wing  102  and engine  110  attached to wing  104 . 
     Body  106  has tail section  112 . Horizontal stabilizer  114 , horizontal stabilizer  116 , and vertical stabilizer  118  are attached to tail section  112  of body  106 . 
     Aircraft  100  is an example of an aircraft having a composite structure whose layup may be determined in accordance with an illustrative embodiment. For example, a layup for composite skin  120  of wing  104  may be determined in accordance with an illustrative embodiment. As another example, a layup for composite skin  122  of wing  102  may be determined in accordance with an illustrative embodiment. As yet a further example, composite skin  124  of body  106  may be determined in accordance with an illustrative embodiment. 
     This illustration of aircraft  100  is provided for purposes of illustrating one environment in which the different illustrative embodiments may be implemented. The illustration of aircraft  100  in  FIG. 1  is not meant to imply architectural limitations as to the manner in which different illustrative embodiments may be implemented. For example, aircraft  100  is shown as a commercial passenger aircraft. The different illustrative embodiments may be applied to other types of aircraft, such as private passenger aircraft, a rotorcraft, and other suitable type of aircraft. 
     Although the illustrative examples for an illustrative embodiment are described with respect to an aircraft, an illustrative embodiment may be applied to other types of platforms. The platform may be, for example, a mobile platform, a stationary platform, a land-based structure, an aquatic-based structure, and a space-based structure. More specifically, the platform, may be a surface ship, a tank, a personnel carrier, a train, a spacecraft, a space station, a satellite, a submarine, an automobile, a manufacturing facility, a building, and other suitable platforms. 
     Turning now to  FIG. 2 , an illustration of a flowchart of a process for determining a layup for a composite structure is depicted in accordance with an illustrative embodiment. Process  200  may be a process for determining ply arrangement  509  of  FIG. 5A . In some illustrative examples, process  200  may be used in forming composite skin  120  of aircraft  100  of  FIG. 1 . 
     Process  200  may begin by receiving a desired thickness and a desired stiffness distribution for a composite structure (operation  202 ). In some illustrative examples, the composite structure may be a non-traditional composite laminate structure. In some illustrative examples, the desired stiffness may be presented in the form of lamination parameters. 
     A model of the composite structure may be split into a plurality of panels (operation  204 ). In some illustrative examples, the plurality of panels may be about the same size and shape. In some illustrative examples, the plurality of panels may include at least one of more than one size or more than one shape. In some illustrative examples, a first panel and a second panel of the plurality of panels have a different number of plies. 
     As used herein, the phrase “at least one of,” when used with a list of items, means different combinations of one or more of the listed items may be used and only one of each item in the list may be needed. In other words, “at least one of” means any combination of items and number of items may be used from the list, but not all of the items in the list are required. The item may be a particular object, thing, or a category. 
     For example, “at least one of item A, item B, or item C” may include, without limitation, item A, item A and item B, or item B. This example also may include item A, item B, and item C or item B and item C. Of course, any combinations of these items may be present. In other examples, “at least one of” may be, for example, without limitation, two of item A; one of item B; and ten of item C; four of item B and seven of item C; or other suitable combinations. 
     A set of lamination parameters and a quantity of plies may be determined for each panel of the plurality of panels using the desired thickness and the desired stiffness distribution for the composite structure (operation  206 ). Ply shapes for each ply of the plurality of panels may be determined using an objective function (operation  208 ). The objective function may be used along with constraints, such as desirable conditions  541  of  FIG. 5B . 
     There may be a number of desirable conditions for use with the objective function. In one illustrative example, in operation  208 , ply shapes for each ply of plurality of panels may be determined by maximizing objective function:
 
Φ obj =Σ s=1   S Σ p=1   P Σ q=p+1   P   Y   p,q,s   (16)
 
where s is a ply sequence variable, S is the quantity of the plurality of ply sequences, p, q are panel variables, P is the quantity of plurality of panels, and Y p,q,s  is a binary variable that represents connectivity between panels. Y p,q,s  is 1 if both panel p and q are present in ply sequence s and Y p,q,s  is 0 otherwise. In integer programming, this condition may be fulfilled by the following constraints: X s   λ(p)  
 
 Y   p,q,s ≤( X   s   λ(p)   +X   s   λ(q) )/2
 
 Y   p,q,s ≥( X   s   λ(p)   +X   s   λ(q) −1)/2
 
for all  Y  defined on (1≤ p≤P, p&lt;q≤P, s= 1 . . .  S )  (17)
 
In this equation, X s   λ(p)  is a binary variable that denotes if a ply is present (value=1) or absent (value=0) in panel p for ply sequence s. The function λ(p) provides the panel group index associated with panel p.
 
     Desirable conditions may include the total number of ply sequences being equal to or larger than the largest quantity of plies for any of the panels. This may described by:
 
Σ s=1   S   X   s   l   ≥t   l    ∀l= 1 . . .  L   (18)
 
Where X x   l  is a binary variable that is 1 when a ply in panel group l is present in ply sequence s and that is 0 otherwise, and t l  is a quantity of plies for panel group l.
 
     All ply sequences should contain at least one ply. Without this desirable condition, the solution would use fewer ply sequences than provided, which could result in a larger number of equations to be solved than the number of variables available.
 
Σ l=1   L   X   s   l ≥1 ∀ s= 1 . . .  S   (19)
 
     Desirable conditions may include a duplicate ply limit. The duplicate ply limit may be represented as
 
Σ s=1   S   Z   s,k   ≤D  for each unique ply sequence  k   (20)
 
where Z s,k  is a binary variable that is 1 if ply sequence s matches unique ply sequence k, but is 0 otherwise, and D is the duplicate ply limit.
 
 Z   s,k ≤1−(Σ l=1   L   |X   s   l   −X   k   l |))/ L  
 
 Z   s,k ≥1−Σ l=1   L   |X   s   l   −X   k   l |  (21)
 
     Desirable conditions may also include measures of degree of freedom. Degree of freedom may include having at least a quantity of different layers between two groups equal to the quantity of the lamination parameters to be matched, when the layups of the two groups have a difference in quantity of plies less than the quantity of the lamination parameters:
 
Σ s=1   S   W   l,k,s ≤max( t   l   ,t   k )− V  
 
for all groups  l  and  k  with thickness difference&lt; V   (22)
 
where W l,k,s  is a binary variable that is 1 if both l and k are active on ply sequence s or 0 otherwise, where t l  is quantity of plies for layup l, where t k  is quantity of plies for layup k, and where V is a quantity of the lamination parameters to be matched per panel. In integer programming, the variable W l,k,s  is defined by the two constraints below:
 
 W   l,k,s ≤( X   s   l   +X   s   k )/2
 
 W   l,k,s ≥( X   s   l   +X   s   k −1)/2
 
for all  W  defined (all groups  l  and  k  with thickness difference&lt; V )  (23)
 
     In order to increase the chances of reducing the quantity of duplicate ply sequences without introducing duplicate ply constraints, noise may be introduced into the objective function. In some illustrative examples, the objective function with noise may be represented by:
 
Σ s=1   S Σ p=1   P Σ q=p+1   P (1+ε) Y   p,q,s   (24)
 
     where ε is generated by seeded random number generation and is small enough to not affect the optimization. Here, ε may serve as a small perturbation so that generating the same shaped plies for multiple ply sequences would be discouraged. 
     The fiber angle for each ply of the plurality of panels may be determined, such that the model of the composite structure has a stiffness equal to the desired stiffness (operation  210 ). Afterwards the process terminates. 
     In some illustrative examples, a fiber angle for each ply may be determined using a system of equations including a plurality of lamination parameters and a plurality of angle variables. The system includes a plurality of equations for the plurality of panels. In some illustrative examples, the plurality of panels may be placed into a plurality of groups. Each of the plurality of groups may have a quantity of equations equal to the quantity of lamination parameters in the set of lamination parameters for the group. A set of equations for first group  702  of  FIG. 7  may be: 
     
       
         
           
             
               
                 
                   
                     V 
                     1 
                     1 
                   
                   = 
                   
                     
                       1 
                       14 
                     
                     ⁢ 
                     
                       ( 
                       
                         
                           
                             
                               
                                 cos 
                                 ⁢ 
                                 
                                     
                                 
                                 ⁢ 
                                 2 
                                 ⁢ 
                                 
                                   θ 
                                   1 
                                 
                               
                               + 
                               
                                 cos 
                                 ⁢ 
                                 
                                     
                                 
                                 ⁢ 
                                 2 
                                 ⁢ 
                                 
                                   θ 
                                   2 
                                 
                               
                               + 
                               
                                 cos 
                                 ⁢ 
                                 
                                     
                                 
                                 ⁢ 
                                 2 
                                 ⁢ 
                                 
                                   θ 
                                   3 
                                 
                               
                               + 
                               
                                 cos 
                                 ⁢ 
                                 
                                     
                                 
                                 ⁢ 
                                 2 
                                 ⁢ 
                                 
                                   θ 
                                   4 
                                 
                               
                               + 
                               
                                 cos 
                                 ⁢ 
                                 
                                     
                                 
                                 ⁢ 
                                 2 
                                 ⁢ 
                                 
                                   θ 
                                   5 
                                 
                               
                               + 
                               
                                 cos 
                                 ⁢ 
                                 
                                     
                                 
                                 ⁢ 
                                 2 
                                 ⁢ 
                                 
                                   θ 
                                   6 
                                 
                               
                               + 
                             
                           
                         
                         
                           
                             
                               
                                 cos 
                                 ⁢ 
                                 
                                     
                                 
                                 ⁢ 
                                 2 
                                 ⁢ 
                                 
                                   θ 
                                   7 
                                 
                               
                               + 
                               
                                 cos 
                                 ⁢ 
                                 
                                     
                                 
                                 ⁢ 
                                 2 
                                 ⁢ 
                                 
                                   θ 
                                   8 
                                 
                               
                               + 
                               
                                 cos 
                                 ⁢ 
                                 
                                     
                                 
                                 ⁢ 
                                 2 
                                 ⁢ 
                                 
                                   θ 
                                   9 
                                 
                               
                               + 
                               
                                 cos 
                                 ⁢ 
                                 
                                     
                                 
                                 ⁢ 
                                 2 
                                 ⁢ 
                                 
                                   θ 
                                   10 
                                 
                               
                               + 
                               
                                 cos 
                                 ⁢ 
                                 
                                     
                                 
                                 ⁢ 
                                 2 
                                 ⁢ 
                                 
                                   θ 
                                   11 
                                 
                               
                               + 
                               
                                 cos 
                                 ⁢ 
                                 
                                     
                                 
                                 ⁢ 
                                 2 
                                 ⁢ 
                                 
                                   θ 
                                   12 
                                 
                               
                               + 
                             
                           
                         
                         
                           
                             
                               
                                 cos 
                                 ⁢ 
                                 
                                     
                                 
                                 ⁢ 
                                 2 
                                 ⁢ 
                                 
                                   θ 
                                   13 
                                 
                               
                               + 
                               
                                 cos 
                                 ⁢ 
                                 
                                     
                                 
                                 ⁢ 
                                 2 
                                 ⁢ 
                                 
                                   θ 
                                   14 
                                 
                               
                             
                           
                         
                       
                       ) 
                     
                   
                 
               
               
                 
                   ( 
                   25 
                   ) 
                 
               
             
             
               
                 
                   
                     V 
                     2 
                     1 
                   
                   = 
                   
                     
                       1 
                       14 
                     
                     ⁢ 
                     
                       ( 
                       
                         
                           
                             
                               
                                 sin 
                                 ⁢ 
                                 
                                     
                                 
                                 ⁢ 
                                 2 
                                 ⁢ 
                                 
                                   θ 
                                   1 
                                 
                               
                               + 
                               
                                 sin 
                                 ⁢ 
                                 
                                     
                                 
                                 ⁢ 
                                 2 
                                 ⁢ 
                                 
                                   θ 
                                   2 
                                 
                               
                               + 
                               
                                 sin 
                                 ⁢ 
                                 
                                     
                                 
                                 ⁢ 
                                 2 
                                 ⁢ 
                                 
                                   θ 
                                   3 
                                 
                               
                               + 
                               
                                 sin 
                                 ⁢ 
                                 
                                     
                                 
                                 ⁢ 
                                 2 
                                 ⁢ 
                                 
                                   θ 
                                   4 
                                 
                               
                               + 
                               
                                 sin 
                                 ⁢ 
                                 
                                     
                                 
                                 ⁢ 
                                 2 
                                 ⁢ 
                                 
                                   θ 
                                   5 
                                 
                               
                               + 
                               
                                 sin 
                                 ⁢ 
                                 
                                     
                                 
                                 ⁢ 
                                 2 
                                 ⁢ 
                                 
                                   θ 
                                   6 
                                 
                               
                               + 
                             
                           
                         
                         
                           
                             
                               
                                 sin 
                                 ⁢ 
                                 
                                     
                                 
                                 ⁢ 
                                 2 
                                 ⁢ 
                                 
                                   θ 
                                   7 
                                 
                               
                               + 
                               
                                 sin 
                                 ⁢ 
                                 
                                     
                                 
                                 ⁢ 
                                 2 
                                 ⁢ 
                                 
                                   θ 
                                   8 
                                 
                               
                               + 
                               
                                 sin 
                                 ⁢ 
                                 
                                     
                                 
                                 ⁢ 
                                 2 
                                 ⁢ 
                                 
                                   θ 
                                   9 
                                 
                               
                               + 
                               
                                 sin 
                                 ⁢ 
                                 
                                     
                                 
                                 ⁢ 
                                 2 
                                 ⁢ 
                                 
                                   θ 
                                   10 
                                 
                               
                               + 
                               
                                 sin 
                                 ⁢ 
                                 
                                     
                                 
                                 ⁢ 
                                 2 
                                 ⁢ 
                                 
                                   θ 
                                   11 
                                 
                               
                               + 
                               
                                 sin 
                                 ⁢ 
                                 
                                     
                                 
                                 ⁢ 
                                 2 
                                 ⁢ 
                                 
                                   θ 
                                   12 
                                 
                               
                               + 
                             
                           
                         
                         
                           
                             
                               
                                 sin 
                                 ⁢ 
                                 
                                     
                                 
                                 ⁢ 
                                 2 
                                 ⁢ 
                                 
                                   θ 
                                   13 
                                 
                               
                               + 
                               
                                 sin 
                                 ⁢ 
                                 
                                     
                                 
                                 ⁢ 
                                 2 
                                 ⁢ 
                                 
                                   θ 
                                   14 
                                 
                               
                             
                           
                         
                       
                       ) 
                     
                   
                 
               
               
                 
                   ( 
                   26 
                   ) 
                 
               
             
             
               
                 
                   
                     V 
                     3 
                     1 
                   
                   = 
                   
                     
                       1 
                       14 
                     
                     ⁢ 
                     
                       ( 
                       
                         
                           
                             
                               
                                 cos 
                                 ⁢ 
                                 
                                     
                                 
                                 ⁢ 
                                 4 
                                 ⁢ 
                                 
                                   θ 
                                   1 
                                 
                               
                               + 
                               
                                 cos 
                                 ⁢ 
                                 
                                     
                                 
                                 ⁢ 
                                 4 
                                 ⁢ 
                                 
                                   θ 
                                   2 
                                 
                               
                               + 
                               
                                 cos 
                                 ⁢ 
                                 
                                     
                                 
                                 ⁢ 
                                 4 
                                 ⁢ 
                                 
                                   θ 
                                   3 
                                 
                               
                               + 
                               
                                 cos 
                                 ⁢ 
                                 
                                     
                                 
                                 ⁢ 
                                 4 
                                 ⁢ 
                                 
                                   θ 
                                   4 
                                 
                               
                               + 
                               
                                 cos 
                                 ⁢ 
                                 
                                     
                                 
                                 ⁢ 
                                 4 
                                 ⁢ 
                                 
                                   θ 
                                   5 
                                 
                               
                               + 
                               
                                 cos 
                                 ⁢ 
                                 
                                     
                                 
                                 ⁢ 
                                 4 
                                 ⁢ 
                                 
                                   θ 
                                   6 
                                 
                               
                               + 
                             
                           
                         
                         
                           
                             
                               
                                 cos 
                                 ⁢ 
                                 
                                     
                                 
                                 ⁢ 
                                 4 
                                 ⁢ 
                                 
                                   θ 
                                   7 
                                 
                               
                               + 
                               
                                 cos 
                                 ⁢ 
                                 
                                     
                                 
                                 ⁢ 
                                 4 
                                 ⁢ 
                                 
                                   θ 
                                   8 
                                 
                               
                               + 
                               
                                 cos 
                                 ⁢ 
                                 
                                     
                                 
                                 ⁢ 
                                 4 
                                 ⁢ 
                                 
                                   θ 
                                   9 
                                 
                               
                               + 
                               
                                 cos 
                                 ⁢ 
                                 
                                     
                                 
                                 ⁢ 
                                 4 
                                 ⁢ 
                                 
                                   θ 
                                   10 
                                 
                               
                               + 
                               
                                 cos 
                                 ⁢ 
                                 
                                     
                                 
                                 ⁢ 
                                 4 
                                 ⁢ 
                                 
                                   θ 
                                   11 
                                 
                               
                               + 
                               
                                 cos 
                                 ⁢ 
                                 
                                     
                                 
                                 ⁢ 
                                 4 
                                 ⁢ 
                                 
                                   θ 
                                   12 
                                 
                               
                               + 
                             
                           
                         
                         
                           
                             
                               
                                 cos 
                                 ⁢ 
                                 
                                     
                                 
                                 ⁢ 
                                 4 
                                 ⁢ 
                                 
                                   θ 
                                   13 
                                 
                               
                               + 
                               
                                 cos 
                                 ⁢ 
                                 
                                     
                                 
                                 ⁢ 
                                 4 
                                 ⁢ 
                                 
                                   θ 
                                   14 
                                 
                               
                             
                           
                         
                       
                       ) 
                     
                   
                 
               
               
                 
                   ( 
                   27 
                   ) 
                 
               
             
             
               
                 
                   
                     V 
                     4 
                     1 
                   
                   = 
                   
                     
                       1 
                       14 
                     
                     ⁢ 
                     
                       ( 
                       
                         
                           
                             
                               
                                 sin 
                                 ⁢ 
                                 
                                     
                                 
                                 ⁢ 
                                 4 
                                 ⁢ 
                                 
                                   θ 
                                   1 
                                 
                               
                               + 
                               
                                 sin 
                                 ⁢ 
                                 
                                     
                                 
                                 ⁢ 
                                 4 
                                 ⁢ 
                                 
                                   θ 
                                   2 
                                 
                               
                               + 
                               
                                 sin 
                                 ⁢ 
                                 
                                     
                                 
                                 ⁢ 
                                 4 
                                 ⁢ 
                                 
                                   θ 
                                   3 
                                 
                               
                               + 
                               
                                 sin 
                                 ⁢ 
                                 
                                     
                                 
                                 ⁢ 
                                 4 
                                 ⁢ 
                                 
                                   θ 
                                   4 
                                 
                               
                               + 
                               
                                 sin 
                                 ⁢ 
                                 
                                     
                                 
                                 ⁢ 
                                 4 
                                 ⁢ 
                                 
                                   θ 
                                   5 
                                 
                               
                               + 
                               
                                 sin 
                                 ⁢ 
                                 
                                     
                                 
                                 ⁢ 
                                 4 
                                 ⁢ 
                                 
                                   θ 
                                   6 
                                 
                               
                               + 
                             
                           
                         
                         
                           
                             
                               
                                 sin 
                                 ⁢ 
                                 
                                     
                                 
                                 ⁢ 
                                 4 
                                 ⁢ 
                                 
                                   θ 
                                   7 
                                 
                               
                               + 
                               
                                 sin 
                                 ⁢ 
                                 
                                     
                                 
                                 ⁢ 
                                 4 
                                 ⁢ 
                                 
                                   θ 
                                   8 
                                 
                               
                               + 
                               
                                 sin 
                                 ⁢ 
                                 
                                     
                                 
                                 ⁢ 
                                 4 
                                 ⁢ 
                                 
                                   θ 
                                   9 
                                 
                               
                               + 
                               
                                 sin 
                                 ⁢ 
                                 
                                     
                                 
                                 ⁢ 
                                 4 
                                 ⁢ 
                                 
                                   θ 
                                   10 
                                 
                               
                               + 
                               
                                 sin 
                                 ⁢ 
                                 
                                     
                                 
                                 ⁢ 
                                 4 
                                 ⁢ 
                                 
                                   θ 
                                   11 
                                 
                               
                               + 
                               
                                 sin 
                                 ⁢ 
                                 
                                     
                                 
                                 ⁢ 
                                 4 
                                 ⁢ 
                                 
                                   θ 
                                   12 
                                 
                               
                               + 
                             
                           
                         
                         
                           
                             
                               
                                 sin 
                                 ⁢ 
                                 
                                     
                                 
                                 ⁢ 
                                 4 
                                 ⁢ 
                                 
                                   θ 
                                   13 
                                 
                               
                               + 
                               
                                 sin 
                                 ⁢ 
                                 
                                     
                                 
                                 ⁢ 
                                 4 
                                 ⁢ 
                                 
                                   θ 
                                   14 
                                 
                               
                             
                           
                         
                       
                       ) 
                     
                   
                 
               
               
                 
                   ( 
                   28 
                   ) 
                 
               
             
           
         
       
     
     In this illustrative example, the set of lamination parameters for first group  702  has four lamination parameters. Accordingly, four equations are present for first group  702 . As depicted, first group  702  has equation 25, equation 26, equation 27, and equation 28. 
     Each of equation 25, equation 26, equation 27, and equation 28 includes the fraction 1/14. The fraction 1/14 corresponds to a fraction of one over a quantity of plies in first group  702 . As can be seen in  FIGS. 7-9 , first group  702  has fourteen plies. 
     In each of equation 25, equation 26, equation 27, and equation 28, ply angles θ s  are present for only those ply sequences in which group  702  has a ply present. In other words, only ply sequences having plies in first group  702  will be present in equation 25, equation 26, equation 27, and equation 28. For example, each of equation 25, equation 26, equation 27, and equation 28 includes θ 1 . θ 1  is present in each of equation 25, equation 26, equation 27, and equation 28 because first group  702  has a “1” in first ply sequence  914  of  FIG. 9 . Likewise, θ 2  is present in each of equation 25, equation 26, equation 27, and equation 28 because column  906  representing first group  702  has a “1” in second ply sequence  916  of  FIG. 9 . Eleventh ply sequence  934  is the last ply sequence with a “1” for first group  702 , accordingly, θ 14  is the last ply angle present in equation 25, equation 26, equation 27, and equation 28. Ply sequence  15  of  FIG. 9  is not present in any of equation 25, equation 26, equation 27, and equation 28. Equation 25, equation 26, equation 27, and equation 28 will be combined with equations for second group  704 , third group  706 , and fourth group  708  to form a system of equations. 
     Second group  704  has four lamination parameters. Accordingly, second group  704  also has four equations. A set of equations for second group  704  may be: 
     
       
         
           
             
               
                 
                   
                     V 
                     1 
                     2 
                   
                   = 
                   
                     
                       1 
                       18 
                     
                     ⁢ 
                     
                       ( 
                       
                         
                           
                             
                               
                                 cos 
                                 ⁢ 
                                 
                                     
                                 
                                 ⁢ 
                                 2 
                                 ⁢ 
                                 
                                   θ 
                                   1 
                                 
                               
                               + 
                               
                                 cos 
                                 ⁢ 
                                 
                                     
                                 
                                 ⁢ 
                                 2 
                                 ⁢ 
                                 
                                   θ 
                                   2 
                                 
                               
                               + 
                               
                                 cos 
                                 ⁢ 
                                 
                                     
                                 
                                 ⁢ 
                                 2 
                                 ⁢ 
                                 
                                   θ 
                                   3 
                                 
                               
                               + 
                               
                                 cos 
                                 ⁢ 
                                 
                                     
                                 
                                 ⁢ 
                                 2 
                                 ⁢ 
                                 
                                   θ 
                                   5 
                                 
                               
                               + 
                               
                                 cos 
                                 ⁢ 
                                 
                                     
                                 
                                 ⁢ 
                                 2 
                                 ⁢ 
                                 
                                   θ 
                                   6 
                                 
                               
                               + 
                               
                                 cos 
                                 ⁢ 
                                 
                                     
                                 
                                 ⁢ 
                                 2 
                                 ⁢ 
                                 
                                   θ 
                                   7 
                                 
                               
                               + 
                             
                           
                         
                         
                           
                             
                               
                                 cos 
                                 ⁢ 
                                 
                                     
                                 
                                 ⁢ 
                                 2 
                                 ⁢ 
                                 
                                   θ 
                                   8 
                                 
                               
                               + 
                               
                                 cos 
                                 ⁢ 
                                 
                                     
                                 
                                 ⁢ 
                                 2 
                                 ⁢ 
                                 
                                   θ 
                                   9 
                                 
                               
                               + 
                               
                                 cos 
                                 ⁢ 
                                 
                                     
                                 
                                 ⁢ 
                                 2 
                                 ⁢ 
                                 
                                   θ 
                                   11 
                                 
                               
                               + 
                               
                                 cos 
                                 ⁢ 
                                 
                                     
                                 
                                 ⁢ 
                                 2 
                                 ⁢ 
                                 
                                   θ 
                                   12 
                                 
                               
                               + 
                               
                                 cos 
                                 ⁢ 
                                 
                                     
                                 
                                 ⁢ 
                                 2 
                                 ⁢ 
                                 
                                   θ 
                                   13 
                                 
                               
                               + 
                               
                                 cos 
                                 ⁢ 
                                 
                                     
                                 
                                 ⁢ 
                                 2 
                                 ⁢ 
                                 
                                   θ 
                                   14 
                                 
                               
                               + 
                             
                           
                         
                         
                           
                             
                               
                                 cos 
                                 ⁢ 
                                 
                                     
                                 
                                 ⁢ 
                                 2 
                                 ⁢ 
                                 
                                   θ 
                                   15 
                                 
                               
                               + 
                               
                                 cos 
                                 ⁢ 
                                 
                                     
                                 
                                 ⁢ 
                                 2 
                                 ⁢ 
                                 
                                   θ 
                                   17 
                                 
                               
                               + 
                               
                                 cos 
                                 ⁢ 
                                 
                                     
                                 
                                 ⁢ 
                                 2 
                                 ⁢ 
                                 
                                   θ 
                                   18 
                                 
                               
                               + 
                               
                                 cos 
                                 ⁢ 
                                 
                                     
                                 
                                 ⁢ 
                                 2 
                                 ⁢ 
                                 
                                   θ 
                                   19 
                                 
                               
                               + 
                               
                                 cos 
                                 ⁢ 
                                 
                                     
                                 
                                 ⁢ 
                                 2 
                                 ⁢ 
                                 
                                   θ 
                                   20 
                                 
                               
                               + 
                               
                                 cos 
                                 ⁢ 
                                 
                                     
                                 
                                 ⁢ 
                                 2 
                                 ⁢ 
                                 
                                   θ 
                                   21 
                                 
                               
                             
                           
                         
                       
                       ) 
                     
                   
                 
               
               
                 
                   ( 
                   29 
                   ) 
                 
               
             
             
               
                 
                   
                     V 
                     2 
                     2 
                   
                   = 
                   
                     
                       1 
                       18 
                     
                     ⁢ 
                     
                       ( 
                       
                         
                           
                             
                               
                                 sin 
                                 ⁢ 
                                 
                                     
                                 
                                 ⁢ 
                                 2 
                                 ⁢ 
                                 
                                   θ 
                                   1 
                                 
                               
                               + 
                               
                                 sin 
                                 ⁢ 
                                 
                                     
                                 
                                 ⁢ 
                                 2 
                                 ⁢ 
                                 
                                   θ 
                                   2 
                                 
                               
                               + 
                               
                                 sin 
                                 ⁢ 
                                 
                                     
                                 
                                 ⁢ 
                                 2 
                                 ⁢ 
                                 
                                   θ 
                                   3 
                                 
                               
                               + 
                               
                                 sin 
                                 ⁢ 
                                 
                                     
                                 
                                 ⁢ 
                                 2 
                                 ⁢ 
                                 
                                   θ 
                                   5 
                                 
                               
                               + 
                               
                                 sin 
                                 ⁢ 
                                 
                                     
                                 
                                 ⁢ 
                                 2 
                                 ⁢ 
                                 
                                   θ 
                                   6 
                                 
                               
                               + 
                               
                                 sin 
                                 ⁢ 
                                 
                                     
                                 
                                 ⁢ 
                                 2 
                                 ⁢ 
                                 
                                   θ 
                                   7 
                                 
                               
                               + 
                             
                           
                         
                         
                           
                             
                               
                                 sin 
                                 ⁢ 
                                 
                                     
                                 
                                 ⁢ 
                                 2 
                                 ⁢ 
                                 
                                   θ 
                                   8 
                                 
                               
                               + 
                               
                                 sin 
                                 ⁢ 
                                 
                                     
                                 
                                 ⁢ 
                                 2 
                                 ⁢ 
                                 
                                   θ 
                                   9 
                                 
                               
                               + 
                               
                                 sin 
                                 ⁢ 
                                 
                                     
                                 
                                 ⁢ 
                                 2 
                                 ⁢ 
                                 
                                   θ 
                                   11 
                                 
                               
                               + 
                               
                                 sin 
                                 ⁢ 
                                 
                                     
                                 
                                 ⁢ 
                                 2 
                                 ⁢ 
                                 
                                   θ 
                                   12 
                                 
                               
                               + 
                               
                                 sin 
                                 ⁢ 
                                 
                                     
                                 
                                 ⁢ 
                                 2 
                                 ⁢ 
                                 
                                   θ 
                                   13 
                                 
                               
                               + 
                               
                                 sin 
                                 ⁢ 
                                 
                                     
                                 
                                 ⁢ 
                                 2 
                                 ⁢ 
                                 
                                   θ 
                                   14 
                                 
                               
                               + 
                             
                           
                         
                         
                           
                             
                               
                                 sin 
                                 ⁢ 
                                 
                                     
                                 
                                 ⁢ 
                                 2 
                                 ⁢ 
                                 
                                   θ 
                                   15 
                                 
                               
                               + 
                               
                                 sin 
                                 ⁢ 
                                 
                                     
                                 
                                 ⁢ 
                                 2 
                                 ⁢ 
                                 
                                   θ 
                                   17 
                                 
                               
                               + 
                               
                                 sin 
                                 ⁢ 
                                 
                                     
                                 
                                 ⁢ 
                                 2 
                                 ⁢ 
                                 
                                   θ 
                                   18 
                                 
                               
                               + 
                               
                                 sin 
                                 ⁢ 
                                 
                                     
                                 
                                 ⁢ 
                                 2 
                                 ⁢ 
                                 
                                   θ 
                                   19 
                                 
                               
                               + 
                               
                                 sin 
                                 ⁢ 
                                 
                                     
                                 
                                 ⁢ 
                                 2 
                                 ⁢ 
                                 
                                   θ 
                                   20 
                                 
                               
                               + 
                               
                                 sin 
                                 ⁢ 
                                 
                                     
                                 
                                 ⁢ 
                                 2 
                                 ⁢ 
                                 
                                   θ 
                                   21 
                                 
                               
                             
                           
                         
                       
                       ) 
                     
                   
                 
               
               
                 
                   ( 
                   30 
                   ) 
                 
               
             
             
               
                 
                   
                     V 
                     3 
                     2 
                   
                   = 
                   
                     
                       1 
                       18 
                     
                     ⁢ 
                     
                       ( 
                       
                         
                           
                             
                               
                                 cos 
                                 ⁢ 
                                 
                                     
                                 
                                 ⁢ 
                                 4 
                                 ⁢ 
                                 
                                   θ 
                                   1 
                                 
                               
                               + 
                               
                                 cos 
                                 ⁢ 
                                 
                                     
                                 
                                 ⁢ 
                                 4 
                                 ⁢ 
                                 
                                   θ 
                                   2 
                                 
                               
                               + 
                               
                                 cos 
                                 ⁢ 
                                 
                                     
                                 
                                 ⁢ 
                                 4 
                                 ⁢ 
                                 
                                   θ 
                                   3 
                                 
                               
                               + 
                               
                                 cos 
                                 ⁢ 
                                 
                                     
                                 
                                 ⁢ 
                                 4 
                                 ⁢ 
                                 
                                   θ 
                                   5 
                                 
                               
                               + 
                               
                                 cos 
                                 ⁢ 
                                 
                                     
                                 
                                 ⁢ 
                                 4 
                                 ⁢ 
                                 
                                   θ 
                                   6 
                                 
                               
                               + 
                               
                                 cos 
                                 ⁢ 
                                 
                                     
                                 
                                 ⁢ 
                                 4 
                                 ⁢ 
                                 
                                   θ 
                                   7 
                                 
                               
                               + 
                             
                           
                         
                         
                           
                             
                               
                                 cos 
                                 ⁢ 
                                 
                                     
                                 
                                 ⁢ 
                                 4 
                                 ⁢ 
                                 
                                   θ 
                                   8 
                                 
                               
                               + 
                               
                                 cos 
                                 ⁢ 
                                 
                                     
                                 
                                 ⁢ 
                                 4 
                                 ⁢ 
                                 
                                   θ 
                                   9 
                                 
                               
                               + 
                               
                                 cos 
                                 ⁢ 
                                 
                                     
                                 
                                 ⁢ 
                                 4 
                                 ⁢ 
                                 
                                   θ 
                                   11 
                                 
                               
                               + 
                               
                                 cos 
                                 ⁢ 
                                 
                                     
                                 
                                 ⁢ 
                                 4 
                                 ⁢ 
                                 
                                   θ 
                                   12 
                                 
                               
                               + 
                               
                                 cos 
                                 ⁢ 
                                 
                                     
                                 
                                 ⁢ 
                                 4 
                                 ⁢ 
                                 
                                   θ 
                                   13 
                                 
                               
                               + 
                               
                                 cos 
                                 ⁢ 
                                 
                                     
                                 
                                 ⁢ 
                                 4 
                                 ⁢ 
                                 
                                   θ 
                                   14 
                                 
                               
                               + 
                             
                           
                         
                         
                           
                             
                               
                                 cos 
                                 ⁢ 
                                 
                                     
                                 
                                 ⁢ 
                                 4 
                                 ⁢ 
                                 
                                   θ 
                                   15 
                                 
                               
                               + 
                               
                                 cos 
                                 ⁢ 
                                 
                                     
                                 
                                 ⁢ 
                                 4 
                                 ⁢ 
                                 
                                   θ 
                                   17 
                                 
                               
                               + 
                               
                                 cos 
                                 ⁢ 
                                 
                                     
                                 
                                 ⁢ 
                                 4 
                                 ⁢ 
                                 
                                   θ 
                                   18 
                                 
                               
                               + 
                               
                                 cos 
                                 ⁢ 
                                 
                                     
                                 
                                 ⁢ 
                                 4 
                                 ⁢ 
                                 
                                   θ 
                                   19 
                                 
                               
                               + 
                               
                                 cos 
                                 ⁢ 
                                 
                                     
                                 
                                 ⁢ 
                                 4 
                                 ⁢ 
                                 
                                   θ 
                                   20 
                                 
                               
                               + 
                               
                                 cos 
                                 ⁢ 
                                 
                                     
                                 
                                 ⁢ 
                                 4 
                                 ⁢ 
                                 
                                   θ 
                                   21 
                                 
                               
                             
                           
                         
                       
                       ) 
                     
                   
                 
               
               
                 
                   ( 
                   31 
                   ) 
                 
               
             
             
               
                 
                   
                     V 
                     4 
                     2 
                   
                   = 
                   
                     
                       1 
                       18 
                     
                     ⁢ 
                     
                       ( 
                       
                         
                           
                             
                               
                                 sin 
                                 ⁢ 
                                 
                                     
                                 
                                 ⁢ 
                                 4 
                                 ⁢ 
                                 
                                   θ 
                                   1 
                                 
                               
                               + 
                               
                                 sin 
                                 ⁢ 
                                 
                                     
                                 
                                 ⁢ 
                                 4 
                                 ⁢ 
                                 
                                   θ 
                                   2 
                                 
                               
                               + 
                               
                                 sin 
                                 ⁢ 
                                 
                                     
                                 
                                 ⁢ 
                                 4 
                                 ⁢ 
                                 
                                   θ 
                                   3 
                                 
                               
                               + 
                               
                                 sin 
                                 ⁢ 
                                 
                                     
                                 
                                 ⁢ 
                                 4 
                                 ⁢ 
                                 
                                   θ 
                                   5 
                                 
                               
                               + 
                               
                                 sin 
                                 ⁢ 
                                 
                                     
                                 
                                 ⁢ 
                                 4 
                                 ⁢ 
                                 
                                   θ 
                                   6 
                                 
                               
                               + 
                               
                                 sin 
                                 ⁢ 
                                 
                                     
                                 
                                 ⁢ 
                                 4 
                                 ⁢ 
                                 
                                   θ 
                                   7 
                                 
                               
                               + 
                             
                           
                         
                         
                           
                             
                               
                                 sin 
                                 ⁢ 
                                 
                                     
                                 
                                 ⁢ 
                                 4 
                                 ⁢ 
                                 
                                   θ 
                                   8 
                                 
                               
                               + 
                               
                                 sin 
                                 ⁢ 
                                 
                                     
                                 
                                 ⁢ 
                                 4 
                                 ⁢ 
                                 
                                   θ 
                                   9 
                                 
                               
                               + 
                               
                                 sin 
                                 ⁢ 
                                 
                                     
                                 
                                 ⁢ 
                                 4 
                                 ⁢ 
                                 
                                   θ 
                                   11 
                                 
                               
                               + 
                               
                                 sin 
                                 ⁢ 
                                 
                                     
                                 
                                 ⁢ 
                                 4 
                                 ⁢ 
                                 
                                   θ 
                                   12 
                                 
                               
                               + 
                               
                                 sin 
                                 ⁢ 
                                 
                                     
                                 
                                 ⁢ 
                                 4 
                                 ⁢ 
                                 
                                   θ 
                                   13 
                                 
                               
                               + 
                               
                                 sin 
                                 ⁢ 
                                 
                                     
                                 
                                 ⁢ 
                                 4 
                                 ⁢ 
                                 
                                   θ 
                                   14 
                                 
                               
                               + 
                             
                           
                         
                         
                           
                             
                               
                                 sin 
                                 ⁢ 
                                 
                                     
                                 
                                 ⁢ 
                                 4 
                                 ⁢ 
                                 
                                   θ 
                                   15 
                                 
                               
                               + 
                               
                                 sin 
                                 ⁢ 
                                 
                                     
                                 
                                 ⁢ 
                                 4 
                                 ⁢ 
                                 
                                   θ 
                                   17 
                                 
                               
                               + 
                               
                                 sin 
                                 ⁢ 
                                 
                                     
                                 
                                 ⁢ 
                                 4 
                                 ⁢ 
                                 
                                   θ 
                                   18 
                                 
                               
                               + 
                               
                                 sin 
                                 ⁢ 
                                 
                                     
                                 
                                 ⁢ 
                                 4 
                                 ⁢ 
                                 
                                   θ 
                                   19 
                                 
                               
                               + 
                               
                                 sin 
                                 ⁢ 
                                 
                                     
                                 
                                 ⁢ 
                                 4 
                                 ⁢ 
                                 
                                   θ 
                                   20 
                                 
                               
                               + 
                               
                                 sin 
                                 ⁢ 
                                 
                                     
                                 
                                 ⁢ 
                                 4 
                                 ⁢ 
                                 
                                   θ 
                                   21 
                                 
                               
                             
                           
                         
                       
                       ) 
                     
                   
                 
               
               
                 
                   ( 
                   32 
                   ) 
                 
               
             
           
         
       
     
     As depicted, second group  704  has equation 29, equation 30, equation 31, and equation 32. Each of equation 29, equation 30, equation 31, and equation 32 includes the fraction 1/18. The fraction 1/18 corresponds to a fraction of one over a quantity of plies in second group  704 . As can be seen in  FIGS. 7-9 , second group  704  has eighteen plies. 
     In each of equation 29, equation 30, equation 31, and equation 32, ply angles θ k  are present for only those ply sequences that have plies that are present for second group  704 . In other words, only ply sequences having plies of second group  704  will be present in equation 29, equation 30, equation 31, and equation 32. For example, each of equation 29, equation 30, equation 31, and equation 32 includes θ 1 . θ 1  is present in each of equation 29, equation 30, equation 31, and equation 32 because column  910  representing second group  704  has a “1” in first ply sequence  914  of  FIG. 9 . Likewise, θ 2  is present in each of equation 29, equation 30, equation 31, and equation 32 because second group  704  has a “1” in second ply sequence  916  of  FIG. 9 . Second group  704  does not have θ 4  and θ 10  because second group  704  has a “0” in both ply sequence  4  and ply sequence  10  of  FIG. 9 . Equation 29, equation 30, equation 31, and equation 32 will be combined with equations for first group  702 , third group  706 , and fourth group  708  to form a system of equations. 
     Third group  706  has four lamination parameters. Accordingly, third group  706  also has four equations. A set of equations for third group  706  may be: 
     
       
         
           
             
               
                 
                   
                     V 
                     1 
                     3 
                   
                   = 
                   
                     
                       1 
                       20 
                     
                     ⁢ 
                     
                       ( 
                       
                         
                           
                             
                               
                                 cos 
                                 ⁢ 
                                 
                                     
                                 
                                 ⁢ 
                                 2 
                                 ⁢ 
                                 
                                   θ 
                                   1 
                                 
                               
                               + 
                               
                                 cos 
                                 ⁢ 
                                 
                                     
                                 
                                 ⁢ 
                                 2 
                                 ⁢ 
                                 
                                   θ 
                                   3 
                                 
                               
                               + 
                               
                                 cos 
                                 ⁢ 
                                 
                                     
                                 
                                 ⁢ 
                                 2 
                                 ⁢ 
                                 
                                   θ 
                                   4 
                                 
                               
                               + 
                               
                                 cos 
                                 ⁢ 
                                 
                                     
                                 
                                 ⁢ 
                                 2 
                                 ⁢ 
                                 
                                   θ 
                                   5 
                                 
                               
                               + 
                               
                                 cos 
                                 ⁢ 
                                 
                                     
                                 
                                 ⁢ 
                                 2 
                                 ⁢ 
                                 
                                   θ 
                                   7 
                                 
                               
                               + 
                               
                                 cos 
                                 ⁢ 
                                 
                                     
                                 
                                 ⁢ 
                                 2 
                                 ⁢ 
                                 
                                   θ 
                                   8 
                                 
                               
                               + 
                             
                           
                         
                         
                           
                             
                               
                                 cos 
                                 ⁢ 
                                 
                                     
                                 
                                 ⁢ 
                                 2 
                                 ⁢ 
                                 
                                   θ 
                                   9 
                                 
                               
                               + 
                               
                                 cos 
                                 ⁢ 
                                 
                                     
                                 
                                 ⁢ 
                                 2 
                                 ⁢ 
                                 
                                   θ 
                                   10 
                                 
                               
                               + 
                               
                                 cos 
                                 ⁢ 
                                 
                                     
                                 
                                 ⁢ 
                                 2 
                                 ⁢ 
                                 
                                   θ 
                                   11 
                                 
                               
                               + 
                               
                                 cos 
                                 ⁢ 
                                 
                                     
                                 
                                 ⁢ 
                                 2 
                                 ⁢ 
                                 
                                   θ 
                                   12 
                                 
                               
                               + 
                               
                                 cos 
                                 ⁢ 
                                 
                                     
                                 
                                 ⁢ 
                                 2 
                                 ⁢ 
                                 
                                   θ 
                                   13 
                                 
                               
                               + 
                             
                           
                         
                         
                           
                             
                               
                                 cos 
                                 ⁢ 
                                 
                                     
                                 
                                 ⁢ 
                                 2 
                                 ⁢ 
                                 
                                   θ 
                                   14 
                                 
                               
                               + 
                               
                                 cos 
                                 ⁢ 
                                 
                                     
                                 
                                 ⁢ 
                                 2 
                                 ⁢ 
                                 
                                   θ 
                                   15 
                                 
                               
                               + 
                               
                                 cos 
                                 ⁢ 
                                 
                                     
                                 
                                 ⁢ 
                                 2 
                                 ⁢ 
                                 
                                   θ 
                                   16 
                                 
                               
                               + 
                               
                                 cos 
                                 ⁢ 
                                 
                                     
                                 
                                 ⁢ 
                                 2 
                                 ⁢ 
                                 
                                   θ 
                                   17 
                                 
                               
                               + 
                               
                                 cos 
                                 ⁢ 
                                 
                                     
                                 
                                 ⁢ 
                                 2 
                                 ⁢ 
                                 
                                   θ 
                                   18 
                                 
                               
                               + 
                             
                           
                         
                         
                           
                             
                               
                                 cos 
                                 ⁢ 
                                 
                                     
                                 
                                 ⁢ 
                                 2 
                                 ⁢ 
                                 
                                   θ 
                                   19 
                                 
                               
                               + 
                               
                                 cos 
                                 ⁢ 
                                 
                                     
                                 
                                 ⁢ 
                                 2 
                                 ⁢ 
                                 
                                   θ 
                                   20 
                                 
                               
                               + 
                               
                                 cos 
                                 ⁢ 
                                 
                                     
                                 
                                 ⁢ 
                                 2 
                                 ⁢ 
                                 
                                   θ 
                                   21 
                                 
                               
                               + 
                               
                                 cos 
                                 ⁢ 
                                 
                                     
                                 
                                 ⁢ 
                                 2 
                                 ⁢ 
                                 
                                   θ 
                                   22 
                                 
                               
                             
                           
                         
                       
                       ) 
                     
                   
                 
               
               
                 
                   ( 
                   33 
                   ) 
                 
               
             
             
               
                 
                   
                     V 
                     2 
                     3 
                   
                   = 
                   
                     
                       1 
                       20 
                     
                     ⁢ 
                     
                       ( 
                       
                         
                           
                             
                               
                                 sin 
                                 ⁢ 
                                 
                                     
                                 
                                 ⁢ 
                                 2 
                                 ⁢ 
                                 
                                   θ 
                                   1 
                                 
                               
                               + 
                               
                                 sin 
                                 ⁢ 
                                 
                                     
                                 
                                 ⁢ 
                                 2 
                                 ⁢ 
                                 
                                   θ 
                                   3 
                                 
                               
                               + 
                               
                                 sin 
                                 ⁢ 
                                 
                                     
                                 
                                 ⁢ 
                                 2 
                                 ⁢ 
                                 
                                   θ 
                                   4 
                                 
                               
                               + 
                               
                                 sin 
                                 ⁢ 
                                 
                                     
                                 
                                 ⁢ 
                                 2 
                                 ⁢ 
                                 
                                   θ 
                                   5 
                                 
                               
                               + 
                               
                                 sin 
                                 ⁢ 
                                 
                                     
                                 
                                 ⁢ 
                                 2 
                                 ⁢ 
                                 
                                   θ 
                                   7 
                                 
                               
                               + 
                               
                                 sin 
                                 ⁢ 
                                 
                                     
                                 
                                 ⁢ 
                                 2 
                                 ⁢ 
                                 
                                   θ 
                                   8 
                                 
                               
                               + 
                             
                           
                         
                         
                           
                             
                               
                                 sin 
                                 ⁢ 
                                 
                                     
                                 
                                 ⁢ 
                                 2 
                                 ⁢ 
                                 
                                   θ 
                                   9 
                                 
                               
                               + 
                               
                                 sin 
                                 ⁢ 
                                 
                                     
                                 
                                 ⁢ 
                                 2 
                                 ⁢ 
                                 
                                   θ 
                                   10 
                                 
                               
                               + 
                               
                                 sin 
                                 ⁢ 
                                 
                                     
                                 
                                 ⁢ 
                                 2 
                                 ⁢ 
                                 
                                   θ 
                                   11 
                                 
                               
                               + 
                               
                                 sin 
                                 ⁢ 
                                 
                                     
                                 
                                 ⁢ 
                                 2 
                                 ⁢ 
                                 
                                   θ 
                                   12 
                                 
                               
                               + 
                               
                                 sin 
                                 ⁢ 
                                 
                                     
                                 
                                 ⁢ 
                                 2 
                                 ⁢ 
                                 
                                   θ 
                                   13 
                                 
                               
                               + 
                               
                                 sin 
                                 ⁢ 
                                 
                                     
                                 
                                 ⁢ 
                                 2 
                                 ⁢ 
                                 
                                   θ 
                                   14 
                                 
                               
                               + 
                             
                           
                         
                         
                           
                             
                               
                                 sin 
                                 ⁢ 
                                 
                                     
                                 
                                 ⁢ 
                                 2 
                                 ⁢ 
                                 
                                   θ 
                                   15 
                                 
                               
                               + 
                               
                                 sin 
                                 ⁢ 
                                 
                                     
                                 
                                 ⁢ 
                                 2 
                                 ⁢ 
                                 
                                   θ 
                                   16 
                                 
                               
                               + 
                               
                                 sin 
                                 ⁢ 
                                 
                                     
                                 
                                 ⁢ 
                                 2 
                                 ⁢ 
                                 
                                   θ 
                                   17 
                                 
                               
                               + 
                               
                                 sin 
                                 ⁢ 
                                 
                                     
                                 
                                 ⁢ 
                                 2 
                                 ⁢ 
                                 
                                   θ 
                                   18 
                                 
                               
                               + 
                               
                                 sin 
                                 ⁢ 
                                 
                                     
                                 
                                 ⁢ 
                                 2 
                                 ⁢ 
                                 
                                   θ 
                                   19 
                                 
                               
                               + 
                               
                                 sin 
                                 ⁢ 
                                 
                                     
                                 
                                 ⁢ 
                                 2 
                                 ⁢ 
                                 
                                   θ 
                                   20 
                                 
                               
                               + 
                             
                           
                         
                         
                           
                             
                               
                                 sin 
                                 ⁢ 
                                 
                                     
                                 
                                 ⁢ 
                                 2 
                                 ⁢ 
                                 
                                   θ 
                                   21 
                                 
                               
                               + 
                               
                                 sin 
                                 ⁢ 
                                 
                                     
                                 
                                 ⁢ 
                                 2 
                                 ⁢ 
                                 
                                   θ 
                                   22 
                                 
                               
                             
                           
                         
                       
                       ) 
                     
                   
                 
               
               
                 
                   ( 
                   34 
                   ) 
                 
               
             
             
               
                 
                   
                     V 
                     3 
                     3 
                   
                   = 
                   
                     
                       1 
                       20 
                     
                     ⁢ 
                     
                       ( 
                       
                         
                           
                             
                               
                                 cos 
                                 ⁢ 
                                 
                                     
                                 
                                 ⁢ 
                                 4 
                                 ⁢ 
                                 
                                   θ 
                                   1 
                                 
                               
                               + 
                               
                                 cos 
                                 ⁢ 
                                 
                                     
                                 
                                 ⁢ 
                                 4 
                                 ⁢ 
                                 
                                   θ 
                                   3 
                                 
                               
                               + 
                               
                                 cos 
                                 ⁢ 
                                 
                                     
                                 
                                 ⁢ 
                                 4 
                                 ⁢ 
                                 
                                   θ 
                                   4 
                                 
                               
                               + 
                               
                                 cos 
                                 ⁢ 
                                 
                                     
                                 
                                 ⁢ 
                                 4 
                                 ⁢ 
                                 
                                   θ 
                                   5 
                                 
                               
                               + 
                               
                                 cos 
                                 ⁢ 
                                 
                                     
                                 
                                 ⁢ 
                                 4 
                                 ⁢ 
                                 
                                   θ 
                                   7 
                                 
                               
                               + 
                               
                                 cos 
                                 ⁢ 
                                 
                                     
                                 
                                 ⁢ 
                                 4 
                                 ⁢ 
                                 
                                   θ 
                                   8 
                                 
                               
                               + 
                             
                           
                         
                         
                           
                             
                               
                                 cos 
                                 ⁢ 
                                 
                                     
                                 
                                 ⁢ 
                                 4 
                                 ⁢ 
                                 
                                   θ 
                                   9 
                                 
                               
                               + 
                               
                                 cos 
                                 ⁢ 
                                 
                                     
                                 
                                 ⁢ 
                                 4 
                                 ⁢ 
                                 
                                   θ 
                                   10 
                                 
                               
                               + 
                               
                                 cos 
                                 ⁢ 
                                 
                                     
                                 
                                 ⁢ 
                                 4 
                                 ⁢ 
                                 
                                   θ 
                                   11 
                                 
                               
                               + 
                               
                                 cos 
                                 ⁢ 
                                 
                                     
                                 
                                 ⁢ 
                                 4 
                                 ⁢ 
                                 
                                   θ 
                                   12 
                                 
                               
                               + 
                               
                                 cos 
                                 ⁢ 
                                 
                                     
                                 
                                 ⁢ 
                                 4 
                                 ⁢ 
                                 
                                   θ 
                                   13 
                                 
                               
                               + 
                             
                           
                         
                         
                           
                             
                               
                                 cos 
                                 ⁢ 
                                 
                                     
                                 
                                 ⁢ 
                                 4 
                                 ⁢ 
                                 
                                   θ 
                                   14 
                                 
                               
                               + 
                               
                                 cos 
                                 ⁢ 
                                 
                                     
                                 
                                 ⁢ 
                                 4 
                                 ⁢ 
                                 
                                   θ 
                                   15 
                                 
                               
                               + 
                               
                                 cos 
                                 ⁢ 
                                 
                                     
                                 
                                 ⁢ 
                                 4 
                                 ⁢ 
                                 
                                   θ 
                                   16 
                                 
                               
                               + 
                               
                                 cos 
                                 ⁢ 
                                 
                                     
                                 
                                 ⁢ 
                                 4 
                                 ⁢ 
                                 
                                   θ 
                                   17 
                                 
                               
                               + 
                               
                                 cos 
                                 ⁢ 
                                 
                                     
                                 
                                 ⁢ 
                                 4 
                                 ⁢ 
                                 
                                   θ 
                                   18 
                                 
                               
                               + 
                             
                           
                         
                         
                           
                             
                               
                                 cos 
                                 ⁢ 
                                 
                                     
                                 
                                 ⁢ 
                                 4 
                                 ⁢ 
                                 
                                   θ 
                                   19 
                                 
                               
                               + 
                               
                                 cos 
                                 ⁢ 
                                 
                                     
                                 
                                 ⁢ 
                                 4 
                                 ⁢ 
                                 
                                   θ 
                                   20 
                                 
                               
                               + 
                               
                                 cos 
                                 ⁢ 
                                 
                                     
                                 
                                 ⁢ 
                                 4 
                                 ⁢ 
                                 
                                   θ 
                                   21 
                                 
                               
                               + 
                               
                                 cos 
                                 ⁢ 
                                 
                                     
                                 
                                 ⁢ 
                                 4 
                                 ⁢ 
                                 
                                   θ 
                                   22 
                                 
                               
                             
                           
                         
                       
                       ) 
                     
                   
                 
               
               
                 
                   ( 
                   35 
                   ) 
                 
               
             
             
               
                 
                   
                     V 
                     4 
                     3 
                   
                   = 
                   
                     
                       1 
                       20 
                     
                     ⁢ 
                     
                       ( 
                       
                         
                           
                             
                               
                                 sin 
                                 ⁢ 
                                 
                                     
                                 
                                 ⁢ 
                                 4 
                                 ⁢ 
                                 
                                   θ 
                                   1 
                                 
                               
                               + 
                               
                                 sin 
                                 ⁢ 
                                 
                                     
                                 
                                 ⁢ 
                                 4 
                                 ⁢ 
                                 
                                   θ 
                                   3 
                                 
                               
                               + 
                               
                                 sin 
                                 ⁢ 
                                 
                                     
                                 
                                 ⁢ 
                                 4 
                                 ⁢ 
                                 
                                   θ 
                                   4 
                                 
                               
                               + 
                               
                                 sin 
                                 ⁢ 
                                 
                                     
                                 
                                 ⁢ 
                                 4 
                                 ⁢ 
                                 
                                   θ 
                                   5 
                                 
                               
                               + 
                               
                                 sin 
                                 ⁢ 
                                 
                                     
                                 
                                 ⁢ 
                                 4 
                                 ⁢ 
                                 
                                   θ 
                                   7 
                                 
                               
                               + 
                               
                                 sin 
                                 ⁢ 
                                 
                                     
                                 
                                 ⁢ 
                                 4 
                                 ⁢ 
                                 
                                   θ 
                                   8 
                                 
                               
                               + 
                             
                           
                         
                         
                           
                             
                               
                                 sin 
                                 ⁢ 
                                 
                                     
                                 
                                 ⁢ 
                                 4 
                                 ⁢ 
                                 
                                   θ 
                                   9 
                                 
                               
                               + 
                               
                                 sin 
                                 ⁢ 
                                 
                                     
                                 
                                 ⁢ 
                                 4 
                                 ⁢ 
                                 
                                   θ 
                                   10 
                                 
                               
                               + 
                               
                                 sin 
                                 ⁢ 
                                 
                                     
                                 
                                 ⁢ 
                                 4 
                                 ⁢ 
                                 
                                   θ 
                                   11 
                                 
                               
                               + 
                               
                                 sin 
                                 ⁢ 
                                 
                                     
                                 
                                 ⁢ 
                                 4 
                                 ⁢ 
                                 
                                   θ 
                                   12 
                                 
                               
                               + 
                               
                                 sin 
                                 ⁢ 
                                 
                                     
                                 
                                 ⁢ 
                                 4 
                                 ⁢ 
                                 
                                   θ 
                                   13 
                                 
                               
                               + 
                               
                                 sin 
                                 ⁢ 
                                 
                                     
                                 
                                 ⁢ 
                                 4 
                                 ⁢ 
                                 
                                   θ 
                                   14 
                                 
                               
                               + 
                             
                           
                         
                         
                           
                             
                               
                                 sin 
                                 ⁢ 
                                 
                                     
                                 
                                 ⁢ 
                                 4 
                                 ⁢ 
                                 
                                   θ 
                                   15 
                                 
                               
                               + 
                               
                                 sin 
                                 ⁢ 
                                 
                                     
                                 
                                 ⁢ 
                                 4 
                                 ⁢ 
                                 
                                   θ 
                                   16 
                                 
                               
                               + 
                               
                                 sin 
                                 ⁢ 
                                 
                                     
                                 
                                 ⁢ 
                                 4 
                                 ⁢ 
                                 
                                   θ 
                                   17 
                                 
                               
                               + 
                               
                                 sin 
                                 ⁢ 
                                 
                                     
                                 
                                 ⁢ 
                                 4 
                                 ⁢ 
                                 
                                   θ 
                                   18 
                                 
                               
                               + 
                               
                                 sin 
                                 ⁢ 
                                 
                                     
                                 
                                 ⁢ 
                                 4 
                                 ⁢ 
                                 
                                   θ 
                                   19 
                                 
                               
                               + 
                               
                                 sin 
                                 ⁢ 
                                 
                                     
                                 
                                 ⁢ 
                                 4 
                                 ⁢ 
                                 
                                   θ 
                                   20 
                                 
                               
                               + 
                             
                           
                         
                         
                           
                             
                               
                                 sin 
                                 ⁢ 
                                 
                                     
                                 
                                 ⁢ 
                                 4 
                                 ⁢ 
                                 
                                   θ 
                                   21 
                                 
                               
                               + 
                               
                                 sin 
                                 ⁢ 
                                 
                                     
                                 
                                 ⁢ 
                                 4 
                                 ⁢ 
                                 
                                   θ 
                                   22 
                                 
                               
                             
                           
                         
                       
                       ) 
                     
                   
                 
               
               
                 
                   ( 
                   36 
                   ) 
                 
               
             
           
         
       
     
     As depicted, column  912  representing third group  706  has equation 33, equation 34, equation 35, and equation 36. Each of equation 33, equation 34, equation 35, and equation 36 includes the fraction 1/20. The fraction 1/20 corresponds to a fraction of one over a quantity of plies in third group  706 . As can be seen in  FIGS. 7-9 , third group  706  has twenty plies. 
     In each of equation 33, equation 34, equation 35, and equation 36, ply angles θ k  are present for only those ply sequences that have a ply present for third group  706 . In other words, only ply sequences having plies of third group  706  will be present in equation 33, equation 34, equation 35, and equation 36. For example, each of equation 33, equation 34, equation 35, and equation 36 includes θ 1 . θ 1  is present in each of equation 33, equation 34, equation 35, and equation 36 because third group  706  has a “1” in first ply sequence  914  of  FIG. 9 . Likewise, θ 2  is not present in any of equation 33, equation 34, equation 35, and equation 36 because third group  706  has a “0” in second ply sequence  916  of  FIG. 9 . Third group  706  also does not have θ 6  because third group  706  has a “0” in both ply sequence  6  of  FIG. 9 . Equation 33, equation 34, equation 35, and equation 36 will be combined with equations for first group  702 , second group  704 , and fourth group  708  to form a system of equations. 
     Fourth group  708  has four lamination parameters. Accordingly, fourth group  708  also has four equations. A set of equations for fourth group  708  may be: 
     
       
         
           
             
               
                 
                   
                     V 
                     1 
                     4 
                   
                   = 
                   
                     
                       1 
                       16 
                     
                     ⁢ 
                     
                       ( 
                       
                         
                           
                             
                               
                                 cos 
                                 ⁢ 
                                 
                                     
                                 
                                 ⁢ 
                                 2 
                                 ⁢ 
                                 
                                   θ 
                                   5 
                                 
                               
                               + 
                               
                                 cos 
                                 ⁢ 
                                 
                                     
                                 
                                 ⁢ 
                                 2 
                                 ⁢ 
                                 
                                   θ 
                                   6 
                                 
                               
                               + 
                               
                                 cos 
                                 ⁢ 
                                 
                                     
                                 
                                 ⁢ 
                                 2 
                                 ⁢ 
                                 
                                   θ 
                                   7 
                                 
                               
                               + 
                               
                                 cos 
                                 ⁢ 
                                 
                                     
                                 
                                 ⁢ 
                                 2 
                                 ⁢ 
                                 
                                   θ 
                                   8 
                                 
                               
                               + 
                               
                                 cos 
                                 ⁢ 
                                 
                                     
                                 
                                 ⁢ 
                                 2 
                                 ⁢ 
                                 
                                   θ 
                                   9 
                                 
                               
                               + 
                               
                                 cos 
                                 ⁢ 
                                 
                                     
                                 
                                 ⁢ 
                                 2 
                                 ⁢ 
                                 
                                   θ 
                                   10 
                                 
                               
                               + 
                             
                           
                         
                         
                           
                             
                               
                                 cos 
                                 ⁢ 
                                 
                                     
                                 
                                 ⁢ 
                                 2 
                                 ⁢ 
                                 
                                   θ 
                                   11 
                                 
                               
                               + 
                               
                                 cos 
                                 ⁢ 
                                 
                                     
                                 
                                 ⁢ 
                                 2 
                                 ⁢ 
                                 
                                   θ 
                                   12 
                                 
                               
                               + 
                               
                                 cos 
                                 ⁢ 
                                 
                                     
                                 
                                 ⁢ 
                                 2 
                                 ⁢ 
                                 
                                   θ 
                                   13 
                                 
                               
                               + 
                               
                                 cos 
                                 ⁢ 
                                 
                                     
                                 
                                 ⁢ 
                                 2 
                                 ⁢ 
                                 
                                   θ 
                                   14 
                                 
                               
                               + 
                               
                                 cos 
                                 ⁢ 
                                 
                                     
                                 
                                 ⁢ 
                                 2 
                                 ⁢ 
                                 
                                   θ 
                                   15 
                                 
                               
                               + 
                             
                           
                         
                         
                           
                             
                               
                                 cos 
                                 ⁢ 
                                 
                                     
                                 
                                 ⁢ 
                                 2 
                                 ⁢ 
                                 
                                   θ 
                                   16 
                                 
                               
                               + 
                               
                                 cos 
                                 ⁢ 
                                 
                                     
                                 
                                 ⁢ 
                                 2 
                                 ⁢ 
                                 
                                   θ 
                                   17 
                                 
                               
                               + 
                               
                                 cos 
                                 ⁢ 
                                 
                                     
                                 
                                 ⁢ 
                                 2 
                                 ⁢ 
                                 
                                   θ 
                                   18 
                                 
                               
                               + 
                               
                                 cos 
                                 ⁢ 
                                 
                                     
                                 
                                 ⁢ 
                                 2 
                                 ⁢ 
                                 
                                   θ 
                                   19 
                                 
                               
                               + 
                               
                                 cos 
                                 ⁢ 
                                 
                                     
                                 
                                 ⁢ 
                                 2 
                                 ⁢ 
                                 
                                   θ 
                                   20 
                                 
                               
                             
                           
                         
                       
                       ) 
                     
                   
                 
               
               
                 
                   ( 
                   37 
                   ) 
                 
               
             
             
               
                 
                   
                     V 
                     2 
                     4 
                   
                   = 
                   
                     
                       1 
                       16 
                     
                     ⁢ 
                     
                       ( 
                       
                         
                           
                             
                               
                                 sin 
                                 ⁢ 
                                 
                                     
                                 
                                 ⁢ 
                                 2 
                                 ⁢ 
                                 
                                   θ 
                                   5 
                                 
                               
                               + 
                               
                                 sin 
                                 ⁢ 
                                 
                                     
                                 
                                 ⁢ 
                                 2 
                                 ⁢ 
                                 
                                   θ 
                                   6 
                                 
                               
                               + 
                               
                                 sin 
                                 ⁢ 
                                 
                                     
                                 
                                 ⁢ 
                                 2 
                                 ⁢ 
                                 
                                   θ 
                                   7 
                                 
                               
                               + 
                               
                                 sin 
                                 ⁢ 
                                 
                                     
                                 
                                 ⁢ 
                                 2 
                                 ⁢ 
                                 
                                   θ 
                                   8 
                                 
                               
                               + 
                             
                           
                         
                         
                           
                             
                               
                                 sin 
                                 ⁢ 
                                 
                                     
                                 
                                 ⁢ 
                                 2 
                                 ⁢ 
                                 
                                   θ 
                                   9 
                                 
                               
                               + 
                               
                                 sin 
                                 ⁢ 
                                 
                                     
                                 
                                 ⁢ 
                                 2 
                                 ⁢ 
                                 
                                   θ 
                                   10 
                                 
                               
                               + 
                               
                                 sin 
                                 ⁢ 
                                 
                                     
                                 
                                 ⁢ 
                                 2 
                                 ⁢ 
                                 
                                   θ 
                                   11 
                                 
                               
                               + 
                               
                                 sin 
                                 ⁢ 
                                 
                                     
                                 
                                 ⁢ 
                                 2 
                                 ⁢ 
                                 
                                   θ 
                                   12 
                                 
                               
                               + 
                               
                                 sin 
                                 ⁢ 
                                 
                                     
                                 
                                 ⁢ 
                                 2 
                                 ⁢ 
                                 
                                   θ 
                                   13 
                                 
                               
                               + 
                               
                                 sin 
                                 ⁢ 
                                 
                                     
                                 
                                 ⁢ 
                                 2 
                                 ⁢ 
                                 
                                   θ 
                                   14 
                                 
                               
                               + 
                             
                           
                         
                         
                           
                             
                               
                                 sin 
                                 ⁢ 
                                 
                                     
                                 
                                 ⁢ 
                                 2 
                                 ⁢ 
                                 
                                   θ 
                                   15 
                                 
                               
                               + 
                               
                                 sin 
                                 ⁢ 
                                 
                                     
                                 
                                 ⁢ 
                                 2 
                                 ⁢ 
                                 
                                   θ 
                                   16 
                                 
                               
                               + 
                               
                                 sin 
                                 ⁢ 
                                 
                                     
                                 
                                 ⁢ 
                                 2 
                                 ⁢ 
                                 
                                   θ 
                                   17 
                                 
                               
                               + 
                               
                                 sin 
                                 ⁢ 
                                 
                                     
                                 
                                 ⁢ 
                                 2 
                                 ⁢ 
                                 
                                   θ 
                                   18 
                                 
                               
                               + 
                               
                                 sin 
                                 ⁢ 
                                 
                                     
                                 
                                 ⁢ 
                                 2 
                                 ⁢ 
                                 
                                   θ 
                                   19 
                                 
                               
                               + 
                               
                                 sin 
                                 ⁢ 
                                 
                                     
                                 
                                 ⁢ 
                                 2 
                                 ⁢ 
                                 
                                   θ 
                                   20 
                                 
                               
                             
                           
                         
                       
                       ) 
                     
                   
                 
               
               
                 
                   ( 
                   38 
                   ) 
                 
               
             
             
               
                 
                   
                     V 
                     3 
                     4 
                   
                   = 
                   
                     
                       1 
                       16 
                     
                     ⁢ 
                     
                       ( 
                       
                         
                           
                             
                               
                                 cos 
                                 ⁢ 
                                 
                                     
                                 
                                 ⁢ 
                                 4 
                                 ⁢ 
                                 
                                   θ 
                                   5 
                                 
                               
                               + 
                               
                                 cos 
                                 ⁢ 
                                 
                                     
                                 
                                 ⁢ 
                                 4 
                                 ⁢ 
                                 
                                   θ 
                                   6 
                                 
                               
                               + 
                               
                                 cos 
                                 ⁢ 
                                 
                                     
                                 
                                 ⁢ 
                                 4 
                                 ⁢ 
                                 
                                   θ 
                                   7 
                                 
                               
                               + 
                               
                                 cos 
                                 ⁢ 
                                 
                                     
                                 
                                 ⁢ 
                                 4 
                                 ⁢ 
                                 
                                   θ 
                                   8 
                                 
                               
                               + 
                               
                                 cos 
                                 ⁢ 
                                 
                                     
                                 
                                 ⁢ 
                                 4 
                                 ⁢ 
                                 
                                   θ 
                                   9 
                                 
                               
                               + 
                               
                                 cos 
                                 ⁢ 
                                 
                                     
                                 
                                 ⁢ 
                                 4 
                                 ⁢ 
                                 
                                   θ 
                                   10 
                                 
                               
                               + 
                             
                           
                         
                         
                           
                             
                               
                                 cos 
                                 ⁢ 
                                 
                                     
                                 
                                 ⁢ 
                                 4 
                                 ⁢ 
                                 
                                   θ 
                                   11 
                                 
                               
                               + 
                               
                                 cos 
                                 ⁢ 
                                 
                                     
                                 
                                 ⁢ 
                                 4 
                                 ⁢ 
                                 
                                   θ 
                                   12 
                                 
                               
                               + 
                               
                                 cos 
                                 ⁢ 
                                 
                                     
                                 
                                 ⁢ 
                                 4 
                                 ⁢ 
                                 
                                   θ 
                                   13 
                                 
                               
                               + 
                               
                                 cos 
                                 ⁢ 
                                 
                                     
                                 
                                 ⁢ 
                                 4 
                                 ⁢ 
                                 
                                   θ 
                                   14 
                                 
                               
                               + 
                               
                                 cos 
                                 ⁢ 
                                 
                                     
                                 
                                 ⁢ 
                                 4 
                                 ⁢ 
                                 
                                   θ 
                                   15 
                                 
                               
                               + 
                             
                           
                         
                         
                           
                             
                               
                                 cos 
                                 ⁢ 
                                 
                                     
                                 
                                 ⁢ 
                                 4 
                                 ⁢ 
                                 
                                   θ 
                                   16 
                                 
                               
                               + 
                               
                                 cos 
                                 ⁢ 
                                 
                                     
                                 
                                 ⁢ 
                                 4 
                                 ⁢ 
                                 
                                   θ 
                                   17 
                                 
                               
                               + 
                               
                                 cos 
                                 ⁢ 
                                 
                                     
                                 
                                 ⁢ 
                                 4 
                                 ⁢ 
                                 
                                   θ 
                                   18 
                                 
                               
                               + 
                               
                                 cos 
                                 ⁢ 
                                 
                                     
                                 
                                 ⁢ 
                                 4 
                                 ⁢ 
                                 
                                   θ 
                                   19 
                                 
                               
                               + 
                               
                                 cos 
                                 ⁢ 
                                 
                                     
                                 
                                 ⁢ 
                                 4 
                                 ⁢ 
                                 
                                   θ 
                                   20 
                                 
                               
                             
                           
                         
                       
                       ) 
                     
                   
                 
               
               
                 
                   ( 
                   39 
                   ) 
                 
               
             
             
               
                 
                   
                     V 
                     4 
                     4 
                   
                   = 
                   
                     
                       1 
                       16 
                     
                     ⁢ 
                     
                       ( 
                       
                         
                           
                             
                               
                                 sin 
                                 ⁢ 
                                 
                                     
                                 
                                 ⁢ 
                                 4 
                                 ⁢ 
                                 
                                   θ 
                                   5 
                                 
                               
                               + 
                               
                                 sin 
                                 ⁢ 
                                 
                                     
                                 
                                 ⁢ 
                                 4 
                                 ⁢ 
                                 
                                   θ 
                                   6 
                                 
                               
                               + 
                               
                                 sin 
                                 ⁢ 
                                 
                                     
                                 
                                 ⁢ 
                                 4 
                                 ⁢ 
                                 
                                   θ 
                                   7 
                                 
                               
                               + 
                               
                                 sin 
                                 ⁢ 
                                 
                                     
                                 
                                 ⁢ 
                                 4 
                                 ⁢ 
                                 
                                   θ 
                                   8 
                                 
                               
                               + 
                             
                           
                         
                         
                           
                             
                               
                                 sin 
                                 ⁢ 
                                 
                                     
                                 
                                 ⁢ 
                                 4 
                                 ⁢ 
                                 
                                   θ 
                                   9 
                                 
                               
                               + 
                               
                                 sin 
                                 ⁢ 
                                 
                                     
                                 
                                 ⁢ 
                                 4 
                                 ⁢ 
                                 
                                   θ 
                                   10 
                                 
                               
                               + 
                               
                                 sin 
                                 ⁢ 
                                 
                                     
                                 
                                 ⁢ 
                                 4 
                                 ⁢ 
                                 
                                   θ 
                                   11 
                                 
                               
                               + 
                               
                                 sin 
                                 ⁢ 
                                 
                                     
                                 
                                 ⁢ 
                                 4 
                                 ⁢ 
                                 
                                   θ 
                                   12 
                                 
                               
                               + 
                               
                                 sin 
                                 ⁢ 
                                 
                                     
                                 
                                 ⁢ 
                                 4 
                                 ⁢ 
                                 
                                   θ 
                                   13 
                                 
                               
                               + 
                               
                                 sin 
                                 ⁢ 
                                 
                                     
                                 
                                 ⁢ 
                                 4 
                                 ⁢ 
                                 
                                   θ 
                                   14 
                                 
                               
                               + 
                             
                           
                         
                         
                           
                             
                               
                                 sin 
                                 ⁢ 
                                 
                                     
                                 
                                 ⁢ 
                                 4 
                                 ⁢ 
                                 
                                   θ 
                                   15 
                                 
                               
                               + 
                               
                                 sin 
                                 ⁢ 
                                 
                                     
                                 
                                 ⁢ 
                                 4 
                                 ⁢ 
                                 
                                   θ 
                                   16 
                                 
                               
                               + 
                               
                                 sin 
                                 ⁢ 
                                 
                                     
                                 
                                 ⁢ 
                                 4 
                                 ⁢ 
                                 
                                   θ 
                                   17 
                                 
                               
                               + 
                               
                                 sin 
                                 ⁢ 
                                 
                                     
                                 
                                 ⁢ 
                                 4 
                                 ⁢ 
                                 
                                   θ 
                                   18 
                                 
                               
                               + 
                               
                                 sin 
                                 ⁢ 
                                 
                                     
                                 
                                 ⁢ 
                                 4 
                                 ⁢ 
                                 
                                   θ 
                                   19 
                                 
                               
                               + 
                               
                                 sin 
                                 ⁢ 
                                 
                                     
                                 
                                 ⁢ 
                                 4 
                                 ⁢ 
                                 
                                   θ 
                                   20 
                                 
                               
                             
                           
                         
                       
                       ) 
                     
                   
                 
               
               
                 
                   ( 
                   40 
                   ) 
                 
               
             
           
         
       
     
     As depicted, fourth group  708  has equation 37, equation 38, equation 39, and equation 40. Each of equation 37, equation 38, equation 39, and equation 40 includes the fraction 1/16. The fraction 1/16 corresponds to a fraction of one over a quantity of plies in fourth group  708 . As can be seen in  FIGS. 7-9 , fourth group  708  has sixteen plies. 
     In each of equation 37, equation 38, equation 39, and equation 40, ply angles θ k  are present for only those ply sequences that have a ply present for fourth group  708 . In other words, only ply sequences having plies of fourth group  708  will be present in equation 37, equation 38, equation 39, and equation 40. For example, each of equation 37, equation 38, equation 39, and equation 40 includes θ 1 . θ 1  is not present in any of equation 37, equation 38, equation 39, and equation 40 because fourth group  708  has a “0” in first ply sequence  914  of  FIG. 9 . Likewise, θ 2  is not present in any of equation 37, equation 38, equation 39, and equation 40 because fourth group  708  has a “0” in second ply sequence  916  of  FIG. 9 . Fourth group  708  does have a “1” in fourth ply sequence  920 , fifth ply sequence  922 , and sixth ply sequence  924 . As a result, θ 5 , θ 7 , and θ 8  are present in each of equation 37, equation 38, equation 39, and equation 40. 
     Each of equation 37, equation 38, equation 39, and equation 40 will be combined with equations for first group  702 , second group  704 , and third group  706  to form a system of equations. In general, the system of equations may be expressed as: 
                   [             V   1   1     =       ∑     s   =   1     S     ⁢       X   s   1     ⁢   cos   ⁢           ⁢   2   ⁢       θ   s     /     t   1                         V   2   1     =       ∑     s   =   1     S     ⁢       X   s   1     ⁢   sin   ⁢           ⁢   2   ⁢       θ   s     /     t   1                         V   3   1     =       ∑     s   =   1     S     ⁢       X   s   1     ⁢   cos   ⁢           ⁢   4   ⁢       θ   s     /     t   1                         V   4   1     =       ∑     s   =   1     S     ⁢       X   s   1     ⁢   sin   ⁢           ⁢   4   ⁢       θ   s     /     t   1                     …               V   1   l     =       ∑     s   =   1     S     ⁢       X   s   l     ⁢   cos   ⁢           ⁢   2   ⁢       θ   s     /     t   l                         V   2   l     =       ∑     s   =   1     S     ⁢       X   s   l     ⁢   sin   ⁢           ⁢   2   ⁢       θ   s     /     t   l                         V   3   l     =       ∑     s   =   1     S     ⁢       X   s   l     ⁢   cos   ⁢           ⁢   4   ⁢       θ   s     /     t   l                         V   4   l     =       ∑     s   =   1     S     ⁢       X   s   l     ⁢   sin   ⁢           ⁢   4   ⁢       θ   s     /     t   l                     …               V   1   L     =       ∑     s   =   1     S     ⁢       X   s   L     ⁢   cos   ⁢           ⁢   2   ⁢       θ   s     /     t   L                         V   2   L     =       ∑     s   =   1     S     ⁢       X   s   L     ⁢   sin   ⁢           ⁢   2   ⁢       θ   s     /     t   L                         V   3   L     =       ∑     s   =   1     S     ⁢       X   s   L     ⁢   cos   ⁢           ⁢   4   ⁢       θ   s     /     t   L                         V   4   L     =       ∑     s   =   1     S     ⁢       X   s   L     ⁢   sin   ⁢           ⁢   4   ⁢       θ   s     /     t   L                   ]           (   41   )               
where V i   l  is a lamination parameter in which i is a lamination parameter index and l is the layup index (panel group index), while θ s  is a ply angle in which s is a ply sequence number, and where X s   l  is a ply index that is 1 if the ply in layup l is present in sequence s, and 0 if that ply is absent in sequence s.
 
     In these equations, the ply angle for each ply may be determined by solving for θ s  in each equation of set of equations 41. In set of equations 26, V i   l  are the lamination parameters determined using the desired thickness and the desired stiffness distribution in operation  206 . The ply index X s   l  in each equation is determined in operation  208 . 
     Turning now to  FIG. 3 , an illustration of a flowchart of a process for determining a layup for a composite structure is depicted in accordance with an illustrative embodiment. Process  300  may be a process for determining ply arrangement  509  of  FIG. 5A . In some illustrative examples, process  300  may be used in forming composite skin  120  of aircraft  100  of  FIG. 1 . 
     Process  300  may begin by receiving a desired thickness and a desired stiffness distribution for a composite structure (operation  302 ). A model of the composite structure may be split into a plurality of panels (operation  304 ). 
     A set of lamination parameters and a quantity of plies for each panel of the plurality of panels are determined using the desired thickness and the desired stiffness distribution for the composite structure (operation  306 ). Ply shapes for each ply of the plurality of panels are determined using an objective function. Determining the ply shapes for each ply of the plurality of panels using the objective function comprises grouping panels of the plurality of panels with a same quantity of plies and same set of lamination parameters to form a plurality of groups and creating a plurality of ply sequences for the plurality of groups. Each panel of a group of the plurality of groups has a same layup. Each of the plurality of ply sequences represents a ply for at least one panel in the plurality of panels (operation  308 ). A layup may represent a collection of plies and respective ply angles through the thickness of the panel. 
     A fiber angle for each ply of the plurality of panels is determined. Determining a fiber angle for each ply of the plurality of panels comprises generating equations for a plurality of lamination parameters for each group of the plurality of groups, combining the equations for each group of the plurality of groups into a system, and solving for a plurality of angle variables in the system (operation  310 ). Afterwards the process terminates. 
     Turning now to  FIG. 4 , an illustration of a flowchart of a process for forming a composite structure is depicted in accordance with an illustrative embodiment. Process  400  may be a process for determining ply arrangement  509  of  FIG. 5A . In some illustrative examples, process  400  may be used in forming composite skin  120  of aircraft  100  of  FIG. 1 . 
     Process  400  may begin by receiving a desired thickness and a desired stiffness distribution for a composite structure (operation  402 ). In some illustrative examples, the composite structure may be a non-traditional composite laminate structure. 
     A model of the composite structure may be split into a plurality of panels (operation  404 ). A set of lamination parameters and a quantity of plies for each panel of the plurality of panels may be determined using the desired thickness and the desired stiffness distribution for the composite structure (operation  406 ). Ply shapes for each ply of the plurality of panels may be determined using an objective function (operation  408 ). A fiber angle for each ply of the plurality of panels may be determined (operation  410 ). The plies may be arranged according to stacking sequence rules to form a ply arrangement, wherein the stacking sequence rules include at least one of symmetry, separation of plies with fiber angles with more than a 45 degrees difference, or avoidance of ply clusters (operation  412 ). Composite material may be laid up according to the ply arrangement to form the composite structure, wherein the composite structure has a stiffness equal to the desired stiffness (operation  414 ). Afterwards, the process terminates. 
     In some illustrative examples, determining ply shapes for each ply of the plurality of panels using an objective function comprises grouping panels of the plurality of panels with a same quantity of plies and same lamination parameters to form a plurality of groups; creating a plurality of ply sequences for the plurality of groups; and determining a ply shape for each ply sequence of the plurality of ply sequences. In these examples, each panel of a group of the plurality of groups has a same layup. Further, in these examples, each of the plurality of ply sequences represents a ply in the plurality of panels. 
     In some illustrative examples, determining the fiber angle for each ply of the plurality of panels comprises generating equations for lamination parameters for each group of the plurality of groups; combining the equations for each group of the plurality of groups into one system; and solving for a plurality of angle variables in the system. 
     The flowcharts and block diagrams in the different depicted embodiments illustrate the architecture, functionality, and operation of some possible implementations of apparatuses and methods in an illustrative embodiment. In this regard, each block in the flowcharts or block diagrams may represent a module, a segment, a function, and/or a portion of an operation or step. 
     In some alternative implementations of an illustrative embodiment, the function or functions noted in the blocks may occur out of the order noted in the figures. For example, in some cases, two blocks shown in succession may be executed substantially concurrently, or the blocks may sometimes be performed in the reverse order, depending upon the functionality involved. Also, other blocks may be added in addition to the illustrated blocks in a flowchart or block diagram. 
     For example, in some illustrative examples, process  200  may thereafter arrange the plies according to stacking sequence rules to form a ply arrangement. In some illustrative examples, the stacking sequence rules include at least one of symmetry, manufacturing constraints, separation of plies with fiber angles with more than a 45 degree difference, or avoidance of ply clusters. 
     In some illustrative examples, determining ply shapes for each ply of the plurality of panels comprises maximizing a connection quantity over all of the plurality of panels, wherein a connection exists when there is a common ply between two adjacent panels of the plurality of panels, and wherein a connection quantity is a sum of connections. In some illustrative examples, determining ply shapes for each ply of the plurality of panels comprises limiting a quantity of plies with a same geometry to be less than a quantity of lamination parameters. 
     In some illustrative examples, process  200  may also include grouping panels of the plurality of panels with a same quantity of plies and same in-plane lamination parameters to form a plurality of groups, wherein each panel of a group of the plurality of groups has a same layup. In one illustrative example, process  300  may further include creating a plurality of ply sequences for the plurality of groups, wherein each of the plurality of ply sequences represents a ply in the composite structure. In some examples, determining ply shapes for each ply of the plurality of panels may comprise minimizing the total length of the ply boundary over all of the plurality of groups. In some examples, the system is underdetermined and multiple solutions are possible. In these examples, the method may further comprise selecting one of the multiple solutions based on at least one of a desirable stacking sequence or matching fiber angles with the ply shapes. 
     In some examples, determining a fiber angle for each ply of the plurality of panels comprises: generating equations for a plurality of lamination parameters for each group of the plurality of groups; combining the equations for each group of the plurality of groups into one system; and solving for a plurality of angle variables in the system. In one example, a quantity of the plurality of ply sequences is equal to at least a quantity of the plurality of groups times a quantity of the plurality of lamination parameters. In an example, a quantity of the plurality of ply sequences is chosen such that for any two groups in the plurality of groups with a thickness difference of fewer plies than a quantity of the plurality of lamination parameters, there are at least a quantity of ply sequences equal to a quantity of the plurality of lamination parameters in which the two groups have different ply present or absent states. 
     Turning now to  FIGS. 5A and 5B , illustrations of a block diagram of a manufacturing environment are depicted in accordance with an illustrative embodiment. Manufacturing environment  500  may be an environment for manufacturing at least a portion of aircraft  501 . For example, manufacturing environment  500  may be an environment for at least one of design or manufacture of composite structure  502  of aircraft  501 . 
     In some illustrative examples, aircraft  100  of  FIG. 1  may be a physical implementation of aircraft  501  of  FIG. 5B . In some illustrative examples, composite skin  120  of  FIG. 1  may be a physical implementation of composite structure  502  of  FIG. 5B . 
     Composite design system  503  of manufacturing environment  500  may be used to design composite structure  502 . In this illustrative example, composite design system  503  may be implemented using software, hardware, firmware, or a combination thereof. 
     When software is used, the operations performed by composite design system  503  may be implemented using, for example, without limitation, program code configured to run on a processor unit. When firmware is used, the operations performed by composite design system  503  may be implemented using, for example, without limitation, program code and data and stored in persistent memory to run on a processor unit. 
     When hardware is employed, the hardware may include one or more circuits that operate to perform the operations performed by composite design system  503 . Depending on the implementation, the hardware may take the form of a circuit system, an integrated circuit, an application-specific integrated circuit (ASIC), a programmable logic device, or some other suitable type of hardware device configured to perform any number of operations. 
     A programmable logic device may be configured to perform certain operations. The device may be permanently configured to perform these operations or may be reconfigurable. A programmable logic device may take the form of, for example, without limitation, a programmable logic array, programmable array logic, a field programmable logic array, a field programmable gate array, or some other type of programmable hardware device. 
     In one illustrative example, composite design system  503  may be implemented in computer system  504 . Composite design system  503  may be used to design composite structure  502 . 
     Composite structure  502  may be non-traditional laminate  505 . In these illustrative examples, non-traditional laminate  505  may be a laminate that does not have a traditional layup. When composite structure  502  is non-traditional laminate  505 , plurality of plies  506  may have non-traditional fiber angles, such as fiber angles other than 0°, 90° or ±45°. Composite structure  502  may have desired thickness  507  and desired stiffness distribution  508 . Desired stiffness distribution  508  may be provided in any suitable form. In some illustrative examples, desired stiffness distribution  508  may be provided as a total thickness and in-plane lamination parameters. In another illustrative example, desired stiffness distribution  508  may be provided as a total thickness and ply percentages for given ply angles of a traditional laminate. Non-traditional laminate  505  may be designed in accordance with an illustrative embodiment to have the same stiffness distribution  508  as the traditional laminate having the total thickness and ply percentages for the given ply angles. 
     In another illustrative example, desired stiffness distribution  508  may be provided as a number of plies of given fiber angles and ply thicknesses. In another illustrative example, desired stiffness distribution  508  may be provided as a specific stacking sequence including not only the ply count and angles, but also the order of the plies through the thickness. In another illustrative example, desired stiffness distribution  508  may be provided as a laminate stiffness matrix, also known as an ABD matrix, generated from one of the previous examples of thickness, lamination parameter, or ply number data. The laminate stiffness matrix relates in-plane and bending loads in the laminate to in-plane strains and curvature in the laminate according to the Classical Laminate Theory. In some illustrative examples, desired thickness  507  and desired stiffness distribution  508  may be used in combination to determine a set of lamination parameters. The set of lamination parameters may be at least one of in-plane lamination parameters (A), out-of-plane lamination parameters (D), or coupling lamination parameters (B). Coupling lamination parameters are used to define the B stiffness matrix, which is 0 if the layup is defined to be symmetric. The set of lamination parameters may be a subset of in-plane lamination parameters. 
     In some illustrative examples, desired thickness  507  and desired stiffness distribution  508  may be used to determine ply arrangement  509  of model  510  of composite structure  502 . Model  510  may be a virtual representation of composite structure  502 . Model  510  may be split into plurality of panels  511 . 
     Plurality of panels  511  may be selected based on desired thickness  507  and desired stiffness distribution  508 . In some illustrative examples, each panel in plurality of panels  511  may have the same size and shape. In some illustrative examples, plurality of panels  511  may have at least one of a plurality of shapes or a plurality of sizes. In some illustrative examples, each panel in plurality of panels  511  may be rectangular. 
     Plurality of panels  511  is formed of plurality of plies  512 . Plurality of plies  512  is a representation of plurality of layers of composite material. Plurality of plies  512  may be a representation of plurality of plies  506  of composite structure  502 . 
     Plurality of panels  511  may be grouped into plurality of groups  513 . Each group in plurality of groups  513  may include a number of panels of plurality of panels  511  having the same quantity of plies and same set of lamination parameters. In some illustrative examples, a group of plurality of groups  513  may only include one panel. In some illustrative examples, a group of plurality of groups  513  may include more than one panel. In some illustrative examples, a number of panels of plurality of panels  511  may only be grouped if each of the number of panels are adjacent. In some illustrative examples, a number of panels of plurality of panels  511  may be grouped even if the number of panels are not adjacent. An illustrative example is provided in  FIG. 7 . 
     As depicted, plurality of groups  513  includes first group  514  and second group  515 . First group  514  includes first panel  516  and second panel  517 . First panel  516  has quantity of plies  518  and set of lamination parameters  519 . In some illustrative examples, quantity of plies  518  may represent the quantity of plies in a respective portion of composite structure  502 . In some illustrative examples, quantity of plies  518  may be only half of the number of plies that will form the respective portion of composite structure  502 . When only half of the number of plies is modeled, the composite structure  502  is assumed to be symmetric about the laminate mid-plane and both halves are defined simultaneously. 
     Second panel  517  has quantity of plies  520  and set of lamination parameters  521 . In some illustrative examples, quantity of plies  520  may represent the quantity of plies in a respective portion of composite structure  502 . In some illustrative examples, quantity of plies  520  may be only half of the number of plies that will form the respective portion of composite structure  502 . 
     As depicted, quantity of plies  518  of first panel  516  may be the same as quantity of plies  520  of second panel  517 . Set of lamination parameters  519  of first panel  516  may be the same as set of lamination parameters  521  of second panel  517 . As a result, first panel  516  and second panel  517  will both have layup  522 . Layup  522  may describe at least one of the order, fiber angles, ply shapes, or other characteristics of a plurality of plies forming first group  514 . 
     As depicted, second group  515  includes number of panels  523 . Number of panels  523  may include any desirable quantity of panels. In some illustrative examples, number of panels  523  may include one panel. In some illustrative examples, number of panels  523  may include more than one panel. Each panel in number of panels  523  has the same quantity of plies and set of lamination parameters as each other panel in number of panels  523 . As a result, each panel in number of panels  523  has layup  524 . Layup  524  may describe at least one of the order, fiber angles, ply shapes, or other characteristics of a plurality of plies forming second group  515 . 
     Quantity  525  of plurality of groups  513  represents the quantity of groups in plurality of groups  513 . As depicted, quantity  525  of plurality of groups  513  is two. However, quantity  525  of plurality of groups  513  may be any desirable value. In some illustrative examples, quantity  525  may be greater than two. 
     Plurality of ply sequences  526  may be created for plurality of groups  513 . Plurality of ply sequences  526  may be positions for plurality of plies  512  of plurality of groups  513 . Plurality of ply sequences  526  may be described as a plurality of “slots” which may be filled with plurality of plies  512 . Plies of plurality of plies  512  present in a ply sequence of plurality of ply sequences  526  are laid up at the same time during manufacturing of composite structure  502 . All plies present of plurality of plies  512  present in the same ply sequence of plurality of ply sequences  526  have the same fiber angle. Plies of plurality of plies  512  within the same ply sequence of plurality of ply sequences  526  need not be continuous. 
     Plurality of plies  512  may form plurality of groups  513 . Although plurality of plies  512  forms plurality of groups  513 , each ply of plurality of plies  512  may not be present in each of plurality of groups  513 . Some of plurality of plies  512  may be present in only first group  514 . Some of plurality of plies  512  may only be present in second group  515 . Some of plurality of plies  512  may be present in both first group  514  and second group  515 . Each ply sequence of the plurality of ply sequences  526  must contain at least one ply of the plurality of plies  512  in at least one group of the plurality of groups  515 . 
     First group  514  may only have plies in a subset of plurality of ply sequences  526 . Second group  515  may only have plies in a subset of plurality of ply sequences  526 . 
     Plurality of ply sequences  526  includes first ply sequence  527 , second ply sequence  528 , and third ply sequence  529 . First ply  530  of plurality of plies  512  is present in first ply sequence  527 . First ply  530  is present in layup  522  but not in layup  524 . In other words, first ply  530  is absent in layup  524 . 
     Second ply  533  of plurality of plies  512  is present in second ply sequence  528 . Second ply  533  is present in layup  524  but not in layup  522 . In other words, second ply  533  is absent in layup  522 . 
     Third ply  536  of plurality of plies  512  is present in third ply sequence  529 . Third ply  536  is present in both layup  522  and layup  524 . 
     Plurality of plies  512  in plurality of ply sequences  526  may have ply shapes  539 . Ply shapes  539  may be selected using objective function  540 . In some illustrative examples, ply shapes  539  may include more than one shape. Ply shapes  539  of respective plies of plurality of plies  512  may be influenced by which groups of plurality of groups  513  contain the respective plies. Ply shapes  539  may be selected such that model  510  has desired thickness  507  of composite structure  502 . Ply shapes  539  may be selected such that each group of plurality of groups  513  has a desired quantity of plies. 
     Objective function  540  may determine ply shapes  539  based on desirable conditions  541 . Desirable conditions  541  may be constraints for optimizing objective function  540 . Desirable conditions  541  may be set by a user. Desirable conditions  541  may include connectivity  542 , duplicate ply limit  543 , equation  544 , and difference  545 . Duplicate ply limit  543 , equation  544 , and difference  545  may affect degrees of freedom  546  of system  547  of equations  548 . System  547  of equations  548  may be used to determine fiber angles  549  of plurality of plies  512  after using objective function  540  to determine ply shapes  539 . Specifically, each ply sequence of plurality of ply sequences  526  may have a ply angle variable in system  547  of equations  548 . By solving for a value of the ply angle variable for a specific ply sequence in ply sequences  526 , the fiber angle for all plies in the respective ply sequence is determined. Fiber angles for the plies of plurality of plies  512  are inherited from the ply sequence of ply sequences  526  to which they belong. 
     In some illustrative examples, desirable conditions  541  may include maximizing connectivity  542 . In some illustrative examples, connectivity  542  may be described using connection quantity  550 . Connection quantity  550  is a sum of connections across all of plurality of panels  511 . A connection exists if plies within the same ply sequence are present for two adjacent panels of plurality of panels  511 . 
     In other illustrative examples, connectivity  542  may be described using ply boundary length  551 . A ply boundary is the boundary between two adjacent panels in one ply sequence, in which a ply is present in one panel and the ply is absent in the other panel in the same ply sequence. The ply boundary length is the sum of the lengths of all the boundaries of that ply. In some illustrative examples, maximizing connectivity  542  may be described by minimizing ply boundary length  551  over all of plurality of plies  512 . 
     In some illustrative examples, connectivity  542  may be described by a function of ply boundary length  551  and ply area  552 . Ply area  552  may be the sum of the area of all panels of plurality of panels  511  that are connected within a ply. In one illustrative example, the function may divide the ply boundary length  551  by ply area  552  for each ply, and maximizing connectivity  550  may be described by minimizing the sum of the function values over all plies of plurality of plies  518 . 
     In some illustrative examples, desirable conditions  541  may include controlling reductions  553  to degrees of freedom  546 . Degrees of freedom  546  may be affected by a quantity of duplicate ply sequences. Reductions  553  to degrees of freedom  546  may be controlled by keeping a quantity of duplicate ply sequences below duplicate ply limit  543 . Duplicate ply limit  543  may be a maximum number of ply sequences that cover the same subset of groups of plurality of groups  513 . These duplicate ply sequences may act as a sublaminate. A sublaminate can be represented by a set of lamination parameters. If the quantity of lamination parameters that describes the sublaminate is smaller than the number of ply sequences forming the sublaminate, a change of variables can be made, reducing degrees of freedom  546  in system  547 . System  547  might become overdetermined if the number of degrees of freedom  546  is smaller than the quantity of equations  554 . 
     In some illustrative examples, desirable conditions  541  may include having a desirable value for degrees of freedom  546 . Degrees of freedom  546  may be desirably affected by equation  544 . Degrees of freedom  546  may be influenced by difference  545  of thickness between any two groups of plurality of groups  513  and differences between the present and absent plies of those respective groups. 
     Although not listed in desirable conditions  541 , it may also be desirable for each ply sequence of plurality of ply sequences  526  to contain at least one ply in plurality of plies  512 . If a ply sequence in plurality of ply sequences  526  were to be empty, the number of variables in system  547  would be reduced. Reducing the number of variables in system  547  may result in a situation with more equations  548  than solution variables. Having a situation with more equations than plurality of angle variables  559  will result in system  547  being overdetermined. 
     System  547  may be used to determine fiber angles  549  of plurality of ply sequences  526 , resulting in fiber angles  549  for each ply of plurality of plies  512 . System  547  may include equations  548 . Quantity  554  of equations  548  may equal total quantity  555  of lamination parameters in plurality of lamination parameters  556  in system  547 . Total quantity  555  of plurality of lamination parameters  556  in system  547  is the product of quantity  525  of plurality of groups  513  and quantity  557  of lamination parameters per group of plurality of groups  513 . Equations  548  may be formed using plurality of lamination parameters  556 , fiber angles  549 , and ply shapes  539 . 
     In some illustrative examples, each group in plurality of groups  513  may have quantity  557  of lamination parameters. Quantity  557  of lamination parameters per group of plurality of groups  513  may be four. In these illustrative examples, quantity  554  of equations  548  and total quantity  555  of plurality of lamination parameters  556  may be four times quantity  525  of plurality of groups  513 . 
     Quantity  554  of equations  548  may affect the ability to solve system  547 . In some illustrative examples, desirable values for other features may be influenced by quantity  554  of equations  548 . 
     For example, quantity  558  of plurality of ply sequences  526  may be chosen to be larger than quantity  554  of equations  548 . The number of degrees of freedom  546  is necessarily equal to or smaller than quantity  558  of plurality of ply sequences  526 . System  547  of equations  548  will not be solvable if the number of degrees of freedom  546  is smaller than quantity  554  of equations  548 . When degrees of freedom  546  are smaller than quantity  554  of equations  548 , system  547  of equations  548  is overdetermined. 
     If system  547  of equations  548  is underdetermined, multiple solutions may be possible. If multiple solutions are possible, one of these solutions may be selected based on a desirable stacking sequence. If multiple solutions are possible, one of these solutions may be selected based on how well fiber angles match with the ply shapes, or both. For example, a 0° fiber angle would be better for a long, narrow ply shape that is aligned to the 0° direction than a 90° fiber angle. 
     As another example, the quantity of degrees of freedom  546  might be reduced by the presence of duplicate ply sequences  585 . Duplicate ply sequences  585  are ply sequences with plies of the same shape. If quantity of duplicates  586  of one type of duplicate ply sequences  585  is larger than quantity  557  of lamination parameters per group of plurality of groups  513 , reduction  553  in degrees of freedom  546  is equal to the difference between quantity of duplicates  586  and quantity  557  of lamination parameters per group of plurality of groups  513 . The total quantity of degrees of freedom  546  may then equal quantity  558  of plurality of ply sequences  526  minus reduction  553  of degrees of freedom  546  for each of duplicate ply sequences  585 . System  547  of equations  548  will not be solvable if the quantity of degrees of freedom  546  is smaller than quantity  554  of equations  548 . As yet a further example, system  547  of equations  548  may be set up for any subset of groups in plurality of groups  513  for which the number of degrees of freedom  546  in system  547  for that subset of groups has to be larger than quantity  554  of equations  548  for that subset of groups. 
     Equations  548  may be generated for plurality of lamination parameters  556 . Equations  548  may also include plurality of angle variables  559 . Equations  548  for plurality of groups  513  may be combined into system  547 . Afterwards, system  547  may be solved to determine the value of each angle in plurality of angle variables  559 . 
     In some illustrative examples, system  547  may be solved to determine angles for all plies in all groups in plurality of groups  513 . Layup  522  for first panel  516  in first group  514  with quantity of plies  518  may represent the entirety of a portion of composite structure  502 . Layup  522  may comprise plies with fiber angles corresponding to those plies in first group  514  that are present in ply sequences of plurality of ply sequences  526 . When quantity of plies  518  represents the entirety of a portion of composite structure  502 , each ply in first panel  516  may have a different fiber angle. 
     In another illustrative example, system  547  may be solved to determine angles for only half of each group of plurality of groups  513 . For example, system  547  may be solved to determine angles for first half  560  of first panel  516 . When system  547  is solved to determine angles for first half  560 , each ply in first half  560  of first panel  516  may have a different fiber angle. When system  547  is solved to determine angles for first half  560 , second half  561  of first panel  516  may include plies with the same angles as the fiber angles in first half  560 . For example, second half  561  may be a mirror image of first half  560  to form a symmetric laminate. 
     By solving objective function  540 , ply shapes  539  of plurality of plies  512  may be determined. By solving system  547  of equations  548 , fiber angles  549  of plurality of ply sequences  526  may be determined, consequently defining fiber angles of plies  512 . 
     As depicted, ply shapes  539  may include first ply shape  562 , second ply shape  563 , and third ply shape  564 . First ply shape  562  is a shape of first ply  530  in first ply sequence  527 . As depicted, first ply  530  is present in layup  522  and not in layup  524 . Accordingly, first ply shape  562  may be substantially the same as the shape of first group  514 . 
     Second ply shape  563  is a shape of second ply  533  in second ply sequence  528 . As depicted, second ply  533  is present in layup  524  and not in layup  522 . Accordingly, second ply shape  563  may be substantially the same as the shape of second group  515 . In this illustrative example, first ply shape  562  and second ply shape  563  are different. 
     Third ply shape  564  is a shape of third ply  536  in third ply sequence  529 . As depicted, third ply  536  is present in both layup  522  and layup  524 . Accordingly, third ply shape  564  may be substantially the same as the shape of first group  514  and second group  515  combined. In this illustrative example, third ply shape  563  may be different from both first ply shape  562  and second ply shape  563 . 
     First ply  530 , second ply  533 , and third ply  536  are only illustrative examples of plies in plurality of plies  512 . Although each of first ply shape  562 , second ply shape  563 , and third ply shape  564  are different from each other shape of first ply shape  562 , second ply shape  563 , and third ply shape  564 , in some illustrative examples, plies in plurality of plies  512  may have substantially the same shape. 
     For example, in other implementations, first ply  530  may be present in both layup  522  and layup  524 . In these implementations, first ply shape  562  would be substantially the same as third ply shape  564 . As another example, first ply  530  could instead be present in layup  524  and not layup  522 . In this example, first ply shape  562  would be substantially the same as second ply shape  563 . 
     After determining ply shapes  539  using objective function  540 , fiber angles  549  may be determined using system  547  of equations  548 . Fiber angles  549  may be determined by solving for plurality of angle variables  559 . In this illustrative example, fiber angles  549  include first fiber angle  565 , second fiber angle  566 , and third fiber angle  567 . First fiber angle  565  may be a fiber angle for first ply  530  in first ply sequence  527 . Second fiber angle  566  may be a fiber angle for second ply  533  in second ply sequence  528 . Third fiber angle  567  may be a fiber angle for third ply  536  in third ply sequence  529 . 
     In some illustrative examples, first fiber angle  565  may be different from each other fiber angle in fiber angles  549 . For example, first fiber angle  565  may be different from second fiber angle  566  and third fiber angle  567 . 
     After determining values for plurality of angle variables  559  using system  547 , stacking sequence rules  576  may be used to form ply arrangement  509 . In some illustrative examples, if plurality of lamination parameters  556  include in-plane and out-of-plane lamination parameters, stacking sequence rules  576  may not be used. In these illustrative examples, the order of plurality of ply sequences  526  may be the same as ply arrangement  509 . 
     In illustrative examples in which plurality of lamination parameters  556  are only in-plane lamination parameters, stacking sequence rules  576  may be used to form ply arrangement  509 . Ply arrangement  509  may reflect an order in which plurality of plies  512  are laid down during manufacturing. Stacking sequence rules  576  may include symmetry  578 , separation  580 , manufacturing constraints  582 , and avoidance of ply clusters  584  with similar fiber angles. 
     Symmetry  578  may be desirable in each panel of plurality of panels  511 . Specifically, it may be desirable for each panel of plurality of panels  511  to be symmetric about a center plane. For example, when symmetry  578  is present in first panel  516 , first half  560  may be a mirror image of second half  561  about the center plane. 
     Separation  580  may describe separation of plurality of plies  512  with fiber angles  549  with more than a specified number of degrees difference. For example, it may be desirable to separate plies of plurality of plies  512  that are adjacent in a layup that have fiber angles  549  that are more than threshold angle  581  apart. Threshold angle  581  have any desirable value. In some illustrative examples, threshold angle  581  may be 45 degrees. Separating plies of plurality of plies  512  in a layup that have fiber angles  549  that are more than 45 degrees apart may avoid high interlaminar stresses. Separation  580  may be desirable when each ply has a fiber angle that differs by more than threshold angle  581  of the fiber angle of each adjacent ply. Threshold angle  581  might be chosen to be other than 45 degrees. For example, threshold angle  581  may be 50 degrees, 55 degrees, 60 degrees, or any other desirable value. 
     Manufacturing constraints  582  may include desirable conditions for manufacturing. For example, it may be undesirable to lay down the same fiber angle across multiple ply sequences of plurality of ply sequences  526 . It may be desirable to combine ply sequences that have the same ply angles and whose ply shapes do not overlap to minimize the quantity of ply sequences to be programmed and inspected. 
     Avoidance of ply clusters  584  may describe the placement of plies having substantially similar fiber angles that are adjacent to each other in a layup. Clusters of plies with the same fiber angle may result in a laminate that is prone to delaminations. In some illustrative examples, plies having the same fiber angle may be desirably separated from each other. As a result, avoidance of ply clusters  584  may be desirable. 
     Ply arrangement  509  is formed by arranging plurality of plies  512  according to stacking sequence rules  576 . Composite structure  502  may be laid up according to ply arrangement  509 . As a result, plurality of plies  506  may be in the same order as plurality of plies  512  in ply arrangement  509  of model  510 . 
     By determining ply shapes  539  using objective function  540  and selecting fiber angles  549  using system  547 , resulting composite structure  502  may have a desired level of blending, desired thickness  507 , and desired stiffness distribution  508 . Further, by using stacking sequence rules  576  to form ply arrangement  509 , manufacturing composite structure  502  may not take an undesirable amount of time. Using stacking sequence rules  576  to form ply arrangement  509  may result in high quality laminates by avoiding undesired failure modes such as delamination. 
     Computer system  504  may include composite laminate designer  590 , ply shape optimizer  592 , fiber angle selector  594 , and ply arranger  596 . In this illustrative example, composite laminate designer  590  may be implemented in software, hardware, firmware or a combination thereof. When software is used, the operations performed by composite laminate designer  590  may be implemented in program code configured to run on hardware, such as a processor unit. When firmware is used, the operations performed by composite laminate designer  590  may be implemented in program code and data and stored in persistent memory to run on a processor unit. When hardware is employed, the hardware may include circuits that operate to perform the operations in composite laminate designer  590 . 
     Composite laminate designer  590  receives desired thickness  507  and desired stiffness distribution  508  for composite structure  502 , splits model  510  of composite structure  502  into plurality of panels  511 , and determines a set of lamination parameters and a quantity of plies for each panel of plurality of panels  511  using desired thickness  507  and desired stiffness distribution  508  for composite structure  502 . Composite laminate designer  590  may further group panels of plurality of panels  511  with a same quantity of plies and same set of lamination parameters to form plurality of groups  513 , wherein each panel of a group of plurality of groups  513  has a same layup. Composite laminate designer  590  may further create plurality of ply sequences  526  for plurality of groups  513 , wherein each of plurality of ply sequences  526  represents at least one ply in composite structure  502 . 
     In this illustrative example, ply shape optimizer  592  may be implemented in software, hardware, firmware or a combination thereof. When software is used, the operations performed by ply shape optimizer  592  may be implemented in program code configured to run on hardware, such as a processor unit. When firmware is used, the operations performed by ply shape optimizer  592  may be implemented in program code and data and stored in persistent memory to run on a processor unit. When hardware is employed, the hardware may include circuits that operate to perform the operations in ply shape optimizer  592 . 
     Ply shape optimizer  592  determines ply shapes  539  for each ply of plurality of panels  511  using objective function  540 . Ply shape optimizer  592  may minimize ply boundary length  551  over all of plurality of groups  513 . Ply shape optimizer  592  may maximize a connection quantity  550  over all of plurality of panels  511 . A connection may be a common ply between two adjacent panels of plurality of panels  511 . Connection quantity  550  may be a sum of connections. 
     In this illustrative example, fiber angle selector  594  may be implemented in software, hardware, firmware or a combination thereof. When software is used, the operations performed by fiber angle selector  594  may be implemented in program code configured to run on hardware, such as a processor unit. When firmware is used, the operations performed by fiber angle selector  594  may be implemented in program code and data and stored in persistent memory to run on a processor unit. When hardware is employed, the hardware may include circuits that operate to perform the operations in fiber angle selector  594 . 
     Fiber angle selector  594  determines a fiber angle for each ply of plurality of panels  511  such that model  510  of composite structure  502  has a stiffness equal to desired stiffness distribution  508 . Fiber angle selector  594  may generate equations  548  for set of lamination parameters  521  for each group of plurality of groups  513 , combine equations  548  for each group of plurality of groups  513  into one system  547 , and solve for plurality of angle variables  559  in system  547 . 
     In this illustrative example, ply arranger  596  may be implemented in software, hardware, firmware or a combination thereof. When software is used, the operations performed by ply arranger  596  may be implemented in program code configured to run on hardware, such as a processor unit. When firmware is used, the operations performed by ply arranger  596  may be implemented in program code and data and stored in persistent memory to run on a processor unit. When hardware is employed, the hardware may include circuits that operate to perform the operations in ply arranger  596 . 
     Ply arranger  596  may arrange plurality of plies  512  according to stacking sequence rules  576  to form ply arrangement  509 . Stacking sequence rules  576  may include at least one of symmetry  578 , separation  580  of plies with fiber angles  549  with more than a 45 degree difference, or avoidance of ply clusters  584 . 
     The illustration of manufacturing environment  500  in  FIGS. 5A and 5B  is not meant to imply physical or architectural limitations to the manner in which an illustrative embodiment may be implemented. Other components in addition to or in place of the ones illustrated may be used. Some components may be unnecessary. Also, the blocks are presented to illustrate some functional components. One or more of these blocks may be combined, divided, or combined and divided into different blocks when implemented in an illustrative embodiment. 
     For example, although plurality of groups  513  only depicts first group  514  and second group  515 , plurality of groups  513  may include any desirable number of groups. In some illustrative examples, plurality of groups  513  may include more than two groups. 
     As another example, although first group  514  is depicted as having only first panel  516  and second panel  517 , first group  514  may have any desirable number of panels. In some illustrative examples, first group  514  may have only one panel. In other illustrative examples, first group  514  may have more than two panels. As another example, although first ply sequence  527  is depicted as having only first ply  530 , first ply sequence  527  might have multiple discontinuous plies in plurality of plies  512  corresponding to groups that are not adjacent. 
     Turning now to  FIG. 6 , an illustration of a two-dimensional view of a model of a composite structure is depicted in accordance with an illustrative embodiment. View  600  may be a two-dimensional view of model  602  of a composite structure, such as composite skin  122  of wing  102  of  FIG. 1 . Model  602  is divided into plurality of panels  604 . Plurality of panels  604  may be an implementation of plurality of panels  511  of  FIG. 5A . Plurality of panels  604  has quantities of plies  606  indicated on each of plurality of panels  604 . In some illustrative examples, quantities of plies  606  may be an actual number of plies to form a desired thickness for each of plurality of panels  604 . For example, panel  608  of plurality of panels  604  may have 30 plies. When the composite structure is formed, the section of the composite structure represented by panel  608  may have 30 plies. In some other illustrative examples, quantities of plies  606  may be only half of the number of plies that will form the composite structure. For example, panel  608 , as depicted in view  600 , may have 30 plies. However, when the composite structure is formed, the section of the composite structure represented by panel  608  may have 60 plies. In these illustrative examples, panel  608  may represent the first half of a respective portion of the composite structure. The second half of the resulting composite structure may have the same types of plies as the first half. For example, the first half and the second half of the composite structure may be symmetric about a center plane. In some illustrative examples, symmetry for panel  608  may be taken into account when determining a stacking sequence for plurality of panels  604 . In some illustrative examples, only a first half of the composite structure may be represented by panel  608  to simplify ply shape optimization and calculations for the ply directions. 
     As depicted, each of plurality of panels  604  are rectangular. However, plurality of panels  604  may be any desirable shape. For example, in some illustrative examples, plurality of panels  604  may be triangular. In some illustrative examples, plurality of panels  604  may have more than one shape. 
     Further, as depicted, each of plurality of panels  604  are substantially the same size. In other illustrative examples, plurality of panels  604  may be more than one size. 
     Turning now to  FIG. 7 , an illustration of a plurality of groups of panels is depicted in accordance with an illustrative embodiment. Plurality of groups  700  may be a view of a subset of plurality of panels  604  of  FIG. 6  within box  7 . Plurality of groups  700  includes first group  702 , second group  704 , third group  706 , and fourth group  708 . First group  702  may have twelve adjacent panels. As depicted, each panel in first group  702  has fourteen plies. Each panel in first group  702  has substantially the same set of lamination parameters. 
     Second group  704  may have eight adjacent panels. As depicted, each panel in second group  704  may have eighteen plies. Each panel in second group  704  also has substantially the same set of lamination parameters. 
     Third group  706  may have seven panels. As depicted, six of the seven panels in third group  706  are adjacent. However, panel  710  does not contact any of the other panels of third group  706 . As depicted, each panel in third group  706  may have twenty plies. Each panel in third group  706  may has substantially the same set of lamination parameters. In this example, panel  710  is chosen to be grouped with the other seven panels in group  706 . In another example, panel  710  may be considered as a separate group. 
     Fourth group  708  may have three consecutive panels. As depicted, each panel in fourth group  708  may have sixteen plies. Each panel in fourth group  708  has substantially the same in-plane lamination parameters. 
     Turning now to  FIG. 8 , an illustration of a plurality of groups of panels is depicted in accordance with an illustrative embodiment. View  800  is a view of plurality of groups  700  with plurality of connected panels  802  indicated by plurality of connections  804 . In this illustrative example, plurality of connections  804  is represented by dots. Each connection of plurality of connections  804  may be a shared side of two panels of plurality of groups  700  covered by a same ply. For example, connection  806  is a shared side between panel  808  and panel  810  of first group  702 . As depicted, plurality of connected panels  802  covers multiple groups. For example, connection  812  is a shared side between panel  808  of first group and panel  814  of fourth group  708 . As depicted, plurality of connected panels  802  is connected panels for a ply covering first group  702 , second group  704 , and fourth group  708 . As depicted, plurality of connected panels  802  includes thirty-six connections  804 . 
     Plurality of connected panels  802  is only a single illustrative example of a quantity and location of connected panels for a single ply. However, each ply will have its own respective size and shape. The quantity of connections for a respective ply may be related to its size and shape. Although plurality of connected panels  802  is depicted as covering multiple groups, another illustrative example of a plurality of connected panels may only cover connected panels in a single group. 
     A total of all connections across all ply sequences in a composite structure may be referred to as a connection quantity. It may be desirable to maximize the connection quantity in a composite structure. By increasing the connection quantity in a composite structure, the resulting composite structure may have a higher degree of blending, which makes it easier to manufacture. 
     Turning now to  FIG. 9 , an illustration of ply sequences for a plurality of groups of panels is depicted in accordance with an illustrative embodiment. View  900  may be a representation of plurality of ply sequences in chart form. Chart  902  includes five columns. First column  904  may provide an indication of a ply sequence number. 
     Second column  906  represents the layup in the panels making up first group  702  of  FIG. 7 . A “1” in a row of second column  906  indicates that a ply in first group  702  of  FIG. 7  is present in the respective ply sequence indicated by first column  904 . When a “1” is present in a row of second column  906 , a ply is present in the respective ply sequence for first group  702 . A “0” in a row of second column  906  indicates that there are no plies in first group  702  of  FIG. 7  in a respective ply sequence indicated by first column  904 . When a “0” is present in a row of second column  906 , a ply is absent in the respective ply sequence for first group  702 . The total number of plies in second column  906  may equal the total number of plies for first group  702  of  FIG. 7 . In some illustrative examples, the total number of plies in second column  906  may equal half the total number of plies for first group  702  of  FIG. 7 . 
     Third column  908  represents fourth group  708  of  FIG. 7 . A “1” in a row of third column  908  indicates that a ply in fourth group  708  of  FIG. 7  is present in the respective ply sequence indicated by first column  904 . When a “1” is present in a row of third column  908 , a ply is present in the respective ply sequence for fourth group  708  of  FIG. 7 . A “0” in a row of third column  908  indicates that there are no plies in fourth group  708  of  FIG. 7  in a respective ply sequence indicated by first column  904 . When a “0” is present in a row of third column  908 , a ply is absent in the respective ply sequence for fourth group  708 . The total number of plies in third column  908  may equal the total number of plies for fourth group  708  of  FIG. 7 . In some illustrative examples, the total number of plies in third column  908  may equal half the total number of plies for fourth group  708  of  FIG. 7 . 
     Fourth column  910  represents second group  704  of  FIG. 7 . A “1” in a row of fourth column  910  indicates that a ply in second group  704  of  FIG. 7  is present in the respective ply sequence indicated by first column  904 . When a “1” is present in a row of fourth column  910 , a ply is present in the respective ply sequence for second group  704  of  FIG. 7 . A “0” in a row of fourth column  910  indicates that there are no plies in second group  704  of  FIG. 7  in a respective ply sequence indicated by first column  904 . When a “0” is present in a row of fourth column  910 , a ply is absent in the respective ply sequence for second group  704 . The total number of plies in fourth column  910  may equal the total number of plies for second group  704  of  FIG. 7 . In some illustrative examples, the total number of plies in fourth column  910  may equal half the total number of plies for second group  704  of  FIG. 7 . 
     Fifth column  912  represents third group  706  of  FIG. 7 . A “1” in a row of fifth column  912  indicates that a ply in third group  706  of  FIG. 7  is present in the respective ply sequence indicated by first column  904 . When a “1” is present in a row of fifth column  912 , a ply is present in the respective ply sequence for third group  706  of  FIG. 7 . A “0” in a row of fifth column  912  indicates that there are no plies in third group  706  of  FIG. 7  in a respective ply sequence indicated by first column  904 . When a “0” is present in a row of fifth column  912 , a ply is absent in the respective ply sequence for third group  706  of  FIG. 7 . The total number of plies in fifth column  912  may equal the total number of plies for third group  706  of  FIG. 7 . In some illustrative examples, the total number of plies in fifth column  912  may equal half the total number of plies for third group  706  of  FIG. 7 . 
     As depicted, a ply in first ply sequence  914  will be present in first group  702 , second group  704 , and third group  706 . In this illustrative example, a ply in second ply sequence  916  will be present in first group  702  and second group  704 . As a result, the shape of a ply in first ply sequence  914  would be different than the shape of a ply in second ply sequence  916 . 
     As depicted, a ply in third ply sequence  918  will be present in first group  702 , second group  704 , and third group  706 . As a result, the shape of a ply in first ply sequence  914  and a ply in third ply sequence  918  would be substantially the same. 
     As depicted, fourth ply sequence  920 , fifth ply sequence  922 , sixth ply sequence  924 , seventh ply sequence  926 , eighth ply sequence  928 , ninth ply sequence  930 , tenth ply sequence  932 , and eleventh ply sequence  934  are duplicate plies. In this example, the duplicate quantity is eight. In this example, the duplicate quantity reduces the number of degrees of freedom of a system of equations by four if the quantity of lamination parameters in the set of lamination parameters is four. 
     As depicted, quantity  936  of plurality of ply sequences describes the quantity of ply sequences for plurality of groups  700  of  FIG. 7 . As depicted, quantity  936  is twenty-two ply sequences. Quantity  936  of plurality of ply sequences may be selected based on a quantity of lamination parameters for the composite structure to be formed. 
     In some illustrative examples, chart  902  may include more or fewer columns. For example, additional columns may be present in chart  902  when more groups are present in the model. For example, to calculate all ply angles for each panel of plurality of panels  604  of model  602 , chart  902  may also include at least a fifth group. Chart  902  may also include further groups such as a sixth group, seventh group, or any other desirable number of groups. 
     Additionally, although only twenty-two ply sequences are depicted, more or fewer ply sequences may be present in chart  902 . More ply sequences may be present with increased number of groups or increased number of plies in at least one group. For example, if first group  702  included thirty plies, there would necessarily be more than twenty-two ply sequences. Likewise, if chart  902  also included a fifth group, a sixth group, a seventh group, and an eighth group, there may be more than twenty-two ply sequences such that a required quantity of degrees of freedom is present for a system of equations. 
     Turning now to  FIG. 10 , an illustration of an exploded view of a ply arrangement is depicted in accordance with an illustrative embodiment. Ply arrangement  1000  may be an exploded three-dimensional representation of ply arrangement  509  of  FIG. 5A . Ply arrangement  1000  may have plurality of plies  1002  in plurality of ply sequences  1004 . Each panel of plurality of panels  604  of  FIG. 6  may contain at least one ply of plurality of plies  1002 . 
     Turning now to  FIG. 11 , an illustration of plies in ply sequences is depicted in accordance with an illustrative embodiment. View  1100  may be a view of a portion of plurality of plies  1002  in plurality of ply sequences  1004  of  FIG. 10 . In some illustrative examples, view  1100  may be a view within bracket  11 - 1  in  FIG. 10 . As depicted, view  1100  may include plurality of plies  1102  in plurality of ply sequences  1104 . Plurality of ply sequences  1104  includes first ply sequence  1106 , second ply sequence  1108 , third ply sequence  1110 , fourth ply sequence  1112 , fifth ply sequence  1114 , sixth ply sequence  1116 , seventh ply sequence  1118 , and eighth ply sequence  1120 . First ply sequence  1106  corresponds to ply sequence  936  in  FIG. 9 . As shown in  FIG. 9 , third group  706  of  FIG. 7  is present in first ply sequence  1106 . Ply  1121  and ply  1122  are present within first ply sequence  1106 . Ply  1122  has ply shape  1123  and markings  1124 . Markings  1124  may indicate which panels of the plurality of panels are included in ply  1122 . Markings  1124  are only for reference purposes. 
     Second ply sequence  1108  corresponds to ply sequence  21  in  FIG. 9 . As shown in  FIG. 9 , second group  704  and third group  706  of  FIG. 7  are present in second ply sequence  1108 . Ply  1125  and ply  1126  are present in second ply sequence  1108 . Ply  1126  has ply shape  1127  and markings  1128 . Markings  1128  may indicate which panels of the plurality of panels include ply  1126 . As depicted, ply  1126  may be included in more panels than ply  1122 . The difference in the number of panels may be seen by comparing markings  1124  and markings  1128 . Further, the difference in the number of panels may also be seen by comparing ply shape  1123  and ply shape  1127 . 
     Third ply sequence  1110  corresponds to ply sequence  20  in  FIG. 9 . As shown in  FIG. 9 , second group  704 , third group  706 , and fourth group  708  of  FIG. 7  are present in third ply sequence  1110 . Ply  1130  is present in third ply sequence  1110 . Ply  1130  has ply shape  1131  and markings  1132 . Markings  1132  may indicate which panels of the plurality of panels include ply  1130 . 
     Fourth ply sequence  1112  corresponds to ply sequence  19  in  FIG. 9 . As shown in  FIG. 9 , second group  704 , third group  706 , and fourth group  708  of  FIG. 7  are present in fourth ply sequence  1112 . As depicted, ply  1130  may be included in the same panels as ply  1136 . As a result, ply shape  1131  and ply shape  1134  may be substantially the same. 
     Fifth ply sequence  1114  corresponds to ply sequence  18  in  FIG. 9 . As shown in  FIG. 9 , second group  704 , third group  706 , and fourth group  708  of  FIG. 7  are present in fifth ply sequence  1110 . As depicted, ply shape  1134  of ply  1136  in fourth ply sequence  1112  and ply shape  1138  of ply  1140  in fifth ply sequence  1114  are substantially the same as ply shape  1131  of ply  1130 . As depicted, third ply sequence  1110 , fourth ply sequence  1112 , fifth ply sequence  1114 , and sixth ply sequence  1116  are duplicate ply sequences. Duplicate ply sequences can be identified by observing the ply shapes as in  FIG. 11 . Duplicate ply sequences may also be identified by observing the presence of ones and zeroes in  FIG. 9 . Duplicate ply sequences may be limited such that a resulting system of equations is solvable. 
     The number of ply sequences and the quantity of lamination parameters in the set of lamination parameters may limit the number of plies having the same ply shape in a model. Plies having the same ply shape may act as a sublaminate which may be undesirable. This may be undesirable if the number of plies in the sublaminate is larger than the quantity of lamination parameters to be matched. The number of plies in the sublaminate may reduce the number of degrees of freedom of a system of equations used to determine the ply angles. 
     Turning now to  FIG. 12 , an illustration of a block diagram of a data processing system is depicted in accordance with an illustrative embodiment. Data processing system  1200  may be used to implement computer system  504  in  FIG. 5B . As depicted, data processing system  1200  includes communications framework  1202 , which provides communications between processor unit  1204 , storage devices  1206 , communications unit  1208 , input/output unit  1210 , and display  1212 . In some cases, communications framework  1202  may be implemented as a bus system. 
     Processor unit  1204  is configured to execute instructions for software to perform a number of operations. Processor unit  1204  may comprise a number of processors, a multi-processor core, and/or some other type of processor, depending on the implementation. In some cases, processor unit  1204  may take the form of a hardware unit, such as a circuit system, an application-specific integrated circuit (ASIC), a programmable logic device, or some other suitable type of hardware unit. 
     Instructions for the operating system, applications, and/or programs run by processor unit  1204  may be located in storage devices  1206 . Storage devices  1206  may be in communication with processor unit  1204  through communications framework  1202 . As used herein, a storage device, also referred to as a computer readable storage device, is any piece of hardware capable of storing information on a temporary and/or permanent basis. This information may include, but is not limited to, data, program code, and/or other information. 
     Memory  1214  and persistent storage  1216  are examples of storage devices  1206 . Memory  1214  may take the form of, for example, a random access memory or some type of volatile or non-volatile storage device. Persistent storage  1216  may comprise any number of components or devices. For example, persistent storage  1216  may comprise a hard drive, a flash memory, a rewritable optical disk, a rewritable magnetic tape, or some combination of the above. The media used by persistent storage  1216  may or may not be removable. 
     Communications unit  1208  allows data processing system  1200  to communicate with other data processing systems and/or devices. Communications unit  1208  may provide communications using physical and/or wireless communications links. 
     Input/output unit  1210  allows input to be received from and output to be sent to other devices connected to data processing system  1200 . For example, input/output unit  1210  may allow user input to be received through a keyboard, a mouse, and/or some other type of input device. As another example, input/output unit  1210  may allow output to be sent to a printer connected to data processing system  1200 . 
     Display  1212  is configured to display information to a user. Display  1212  may comprise, for example, without limitation, a monitor, a touch screen, a laser display, a holographic display, a virtual display device, and/or some other type of display device. 
     In this illustrative example, the processes of the different illustrative embodiments may be performed by processor unit  1204  using computer-implemented instructions. These instructions may be referred to as program code, computer usable program code, or computer readable program code and may be read and executed by one or more processors in processor unit  1204 . 
     In these examples, program code  1218  is located in a functional form on computer readable media  1220 , which is selectively removable, and may be loaded onto or transferred to data processing system  1200  for execution by processor unit  1204 . Program code  1218  and computer readable media  1220  together form computer program product  1222 . In this illustrative example, computer readable media  1220  may be computer readable storage media  1224  or computer readable signal media  1226 . 
     Computer readable storage media  1224  is a physical or tangible storage device used to store program code  1218  rather than a medium that propagates or transmits program code  1218 . Computer readable storage media  1224  may be, for example, without limitation, an optical or magnetic disk or a persistent storage device that is connected to data processing system  1200 . 
     Alternatively, program code  1218  may be transferred to data processing system  1200  using computer readable signal media  1226 . Computer readable signal media  1226  may be, for example, a propagated data signal containing program code  1218 . This data signal may be an electromagnetic signal, an optical signal, and/or some other type of signal that can be transmitted over physical and/or wireless communications links. 
     The illustration of data processing system  1200  in  FIG. 12  is not meant to provide architectural limitations to the manner in which the illustrative embodiments may be implemented. The different illustrative embodiments may be implemented in a data processing system that includes components in addition to or in place of those illustrated for data processing system  1200 . Further, components shown in  FIG. 12  may be varied from the illustrative examples shown. 
     Turning now to  FIG. 13 , an illustration of a block diagram of a manufacturing environment is depicted in accordance with an illustrative embodiment. Manufacturing environment  1300  may be an implementation of part of manufacturing environment  500  of  FIGS. 5A and 5B . 
     Manufacturing environment  1300  includes composite structure  1302 . Composite structure  1302  may be composite laminate  1304  formed of plurality of plies  1306  having non-traditional angles  1308 . Each of plurality of plies  1306  has a respective shape of respective shapes  1310  configured to minimize total length of ply boundary  1312  over all of plurality of plies  1306 . Composite laminate  1304  has stiffness  1314  equal to desired stiffness  1316  of traditional composite laminate  1318 . 
     Traditional composite laminate  1318  may be conventionally used. However, it may be desirable to replace traditional composite laminate  1318  with composite laminate  1304 . For example, composite laminate has thickness  1320  less than thickness  1322  of traditional composite laminate  1318 . Having a reduced thickness may also reduce the weight of composite laminate  1304  compared to traditional composite laminate  1318 . In some illustrative examples, composite laminate  1304  may be wing skin  1324  of aircraft wing  1326  of aircraft  1328 . It may be desirable to have thickness  1320  to reduce weight  1330  of aircraft  1328 . 
     Traditional composite laminate  1318  may be formed of plurality of plies  1331  having traditional angles  1332 . In some illustrative examples, it may be desirable to replace traditional composite laminate  1318  with composite laminate  1304  such that one or more of traditional angles  1332 , such as at least one of 0, +45, −45, or 90, are not present in composite laminate  1304 . 
     Composite laminate  1304  may include plurality of panels  1334 . Plurality of panels  1334  may be formed of plurality of plies  1306 . Each panel of plurality of panels  1334  shares at least one ply with an adjacent panel. For example, if first panel  1336  and second panel  1338  are adjacent, first panel  1336  and second panel  1338  may share at least one ply of plurality of plies  1306 . 
     Plurality of panels  1334  may be grouped into plurality of groups  1340 . Each group of plurality of groups  1340  contains a respective portion of first surface  1342  of composite laminate  1304 . For example, first group  1344  of plurality of groups  1340  may contain respective portion  1346  of first surface  1342 . As another example, second group  1348  of plurality of groups  1340  may contain respective portion  1350  of first surface  1342 . In some illustrative examples,  FIG. 7  may be a depiction of a portion of plurality of groups  1340  and first surface  1342 . 
     A respective layup of each group of plurality of groups  1340  is different than a respective layup of each other group of plurality of groups  1340 . For example, a layup of first group  1344  may differ from a layup of second group  1348 . For example, for any two groups in plurality of groups  1340  having a thickness difference of fewer plies than a quantity of lamination parameters, the two groups have a quantity of plies that is present in only one of the two groups equal to at least the quantity of lamination parameters. The lamination parameters may be plurality of lamination parameters  556  of  FIG. 5B . In some illustrative examples, quantity of lamination parameters may be quantity  557  of  FIG. 5B . 
     For example, if thickness  1352  and thickness  1354  have a difference of fewer plies than the quantity of lamination parameters, first group  1344  and second group  1348  have at least a quantity of plies in only first group  1344  or second group  1348  equal to at least the quantity of lamination parameters. For example, if there are six lamination parameters and the difference between thickness  1352  and thickness  1354  is less than six plies, there are at least six plies that are found in only first group  1344  or only second group  1348 . As another example, if there are four lamination parameters and the difference between thickness  1352  and thickness  1354  is less than four plies, there are at least four plies that are found in only first group  1344  or only second group  1348 . 
     In some illustrative examples, each ply of a first half of a panel of the plurality of panels has a different fiber angle value than each other ply of the first half of the panel. For example, each ply of first half  1374  of first panel  1336  of plurality of panels  1334  has a different fiber angle value than each other ply of first half  1374  of first panel  1336 . 
     Composite laminate  1304  may have plurality of ply sequences  1356  containing plurality of plies  1306 . Quantity  1358  of plurality of ply sequences  1356  is equal to at least quantity  1360  of plurality of groups  1340  times a quantity of lamination parameters. 
     Plurality of plies  1306  may have respective shapes  1310  based upon which groups of plurality of groups  1340  contain each respective ply. Plurality of plies  1306  may also have respective shapes  1310  based on shapes of plurality of panels  1334 . In some illustrative examples, at least one panel of plurality of panels  1334  may be rectangular. For example, second panel  1338  may be rectangular  1362 . In some illustrative examples, at least one ply of plurality of plies  1306  may have rectangular edges  1364 . 
     When composite laminate  1304  is wing skin  1324  for aircraft wing  1326 , a width of a panel of plurality of panels  1334  may be a distance between two adjacent stringers of aircraft wing  1326 . For example, at least one of first panel  1336  or second panel  1338  may have a width equal to distance  1366  between first stringer  1368  and second stringer  1370  of plurality of stringers  1372 . 
     The different components shown in  FIGS. 1 and 6-13  may be combined with components in  FIGS. 5A and 5B , used with components in  FIGS. 5A and 5B , or a combination of the two. Additionally, some of the components in  FIGS. 1 and 6-13  may be illustrative examples of how components shown in block form in  FIGS. 5A and 5B  can be implemented as physical structures. 
     The illustrative embodiments of the present disclosure may be described in the context of aircraft manufacturing and service method  1400  as shown in  FIG. 14  and aircraft  1500  as shown in  FIG. 15 . Turning first to  FIG. 14 , an illustration of an aircraft manufacturing and service method is depicted in the form of a block diagram in accordance with an illustrative embodiment. During pre-production, aircraft manufacturing and service method  1400  may include specification and design  1402  of aircraft  1500  of  FIG. 15  and material procurement  1404 . 
     During production, component and subassembly manufacturing  1406  and system integration  1408  of aircraft  1500  of  FIG. 15  takes place. Thereafter, aircraft  1500  of  FIG. 15  may go through certification and delivery  1410  in order to be placed in service  1412 . While in service  1412  by a customer, aircraft  1500  of  FIG. 15  is scheduled for routine maintenance and service  1414 , which may include modification, reconfiguration, refurbishment, and other maintenance or service. 
     Each of the processes of aircraft manufacturing and service method  1400  may be performed or carried out by a system integrator, a third party, and/or an operator. In these examples, the operator may be a customer. For the purposes of this description, a system integrator may include, without limitation, any number of aircraft manufacturers and major-system subcontractors; a third party may include, without limitation, any number of vendors, subcontractors, and suppliers; and an operator may be an airline, a leasing company, a military entity, a service organization, and so on. 
     With reference now to  FIG. 15 , an illustration of an aircraft is depicted in the form of a block diagram in which an illustrative embodiment may be implemented. In this example, aircraft  1500  is produced by aircraft manufacturing and service method  1400  of  FIG. 14  and may include airframe  1502  with systems  1504  and interior  1506 . Examples of systems  1504  include one or more of propulsion system  1508 , electrical system  1510 , hydraulic system  1512 , and environmental system  1514 . Any number of other systems may be included. Although an aerospace example is shown, different illustrative embodiments may be applied to other industries, such as the automotive industry. 
     Apparatuses and methods embodied herein may be employed during at least one of the stages of aircraft manufacturing and service method  1400  of  FIG. 14 . One or more illustrative embodiments may be used during specification and design  1402  of aircraft  1500  as well as component and subassembly manufacturing  1406 . For example, a composite structure of aircraft  1500  may be designed using the illustrative embodiments during specification and design  1402 . Further, the composite structure may be formed using the resulting ply arrangement during component and subassembly manufacturing  1406 . 
     The illustrative embodiments provide a method and apparatus for determining a layup for a composite structure exactly matching a specified stiffness and thickness distribution. Specifically, the illustrative embodiments provide a method and apparatus for determining a layup for a nontraditional composite structure exactly matching a specified stiffness and thickness distribution. 
     In the illustrative examples, an objective function is used to determine ply shapes in the composite structure. The objective function may be solved within desirable conditions. The desirable conditions may include maximizing connectivity while filling all ply sequences. By having a desirable connectivity, loads may be transferred as desirable through a composite component. Further, by having a desirable connectivity, manufacturing time may be reduced. 
     The number of duplicate ply sequences may be limited. Degrees of freedom may be controlled by controlling a difference and following an equation for a number of ply sequences. 
     Equations with a plurality of lamination parameters and a plurality of angle variables form a system. The system may be solved to determine the fiber angles of each ply of the composite structure. 
     Using the illustrative embodiments, non-traditional laminates may be designed to have desirable connectivity. Further, by using the illustrative embodiments, design and layout of a composite structure may take less time. Design and layout using the illustrative examples may take less time than conventional design processes which continuously refine after testing. Yet further, the layup process for a composite structure designed by the illustrative examples may take less time due to at least one of less machine downtime, greater machine efficiency, or ply shape optimization. 
     The different illustrative embodiments can take the form of an entirely hardware embodiment, an entirely software embodiment, or an embodiment containing both hardware and software elements. Some embodiments are implemented in software, which includes, but is not limited to, forms such as, for example, firmware, resident software, and microcode. 
     Furthermore, the different embodiments can take the form of a computer program product accessible from a computer usable or computer readable medium providing program code for use by or in connection with a computer or any device or system that executes instructions. For the purposes of this disclosure, a computer usable or computer readable medium can generally be any tangible apparatus that can contain, store, communicate, propagate, or transport the program for use by or in connection with the instruction execution system, apparatus, or device. 
     The computer usable or computer readable medium can be, for example, without limitation an electronic, magnetic, optical, electromagnetic, infrared, or semiconductor system, or a propagation medium. Non-limiting examples of a computer readable medium include a semiconductor or solid state memory, magnetic tape, a removable computer diskette, a random access memory (RAM), a read-only memory (ROM), a rigid magnetic disk, and an optical disk. Optical disks may include a compact disk-read only memory (CD-ROM), a compact disk-read/write (CD-R/W), and a DVD. 
     Further, a computer usable or computer readable medium may contain or store a computer readable or computer usable program code such that when the computer readable or computer usable program code is executed on a computer, the execution of this computer readable or computer usable program code causes the computer to transmit another computer readable or computer usable program code over a communications link. This communications link may use a medium that is, for example, without limitation, physical or wireless. 
     A data processing system suitable for storing and/or executing computer readable or computer usable program code will include one or more processors coupled directly or indirectly to memory elements through a communications fabric, such as a system bus. The memory elements may include local memory employed during actual execution of the program code, bulk storage, and cache memories which provide temporary storage of at least some computer readable or computer usable program code to reduce the number of times code may be retrieved from bulk storage during execution of the code. 
     Input/output or I/O devices can be coupled to the system either directly or through intervening I/O controllers. These devices may include, for example, without limitation, keyboards, touch screen displays, and pointing devices. Different communications adapters may also be coupled to the system to enable the data processing system to become coupled to other data processing systems, remote printers, or storage devices through intervening private or public networks. Non-limiting examples of modems and network adapters are just a few of the currently available types of communications adapters. 
     The description of the different illustrative embodiments has been presented for purposes of illustration and description, and is not intended to be exhaustive or limited to the embodiments in the form disclosed. The different illustrative examples describe components that perform actions or operations. In an illustrative embodiment, a component may be configured to perform the action or operation described. For example, the component may have a configuration or design for a structure that provides the component an ability to perform the action or operation that is described in the illustrative examples as being performed by the component. 
     Many modifications and variations will be apparent to those of ordinary skill in the art. Further, different illustrative embodiments may provide different features as compared to other desirable embodiments. The embodiment or embodiments selected are chosen and described in order to best explain the principles of the embodiments, the practical application, and to enable others of ordinary skill in the art to understand the disclosure for various embodiments with various modifications as are suited to the particular use contemplated.