Crashworthiness design methodology using a hybrid cellular automata algorithm for the synthesis of topologies for structures subject to nonlinear transient loading

Crashworthiness designing of a structure using a Hybrid Cellular Automata (HCA) algorithm where field states are computed using finite element analysis (FEA) and the material distribution of the structure is updated at each iteration using cellular automata method. The HCA algorithm optimizes the topology of the structures to achieve certain performance within the limits of various constraints applied to ensure crashworthiness of the structures. The HCA algorithm may also be applied to designing of structures to be fabricated by an extrusion method having the same cross section along the direction of extrusion or stamped structures having thickness varying across the structure.

FIELD OF THE INVENTION

This disclosure is related to a method and system for designing structures subject to nonlinear transient loads, more specifically to a method and system for crashworthiness designing structures using Hybrid Cellular Automata (HCA) algorithm.

BACKGROUND OF THE INVENTION

Crashworthiness design methodologies combine crash simulation and design methodology in order to design structures in vehicles. A crashworthiness design requires the modeling of complex interactions between components of the vehicles in a crash event. The design of the structures must also accommodate various conflicting criteria such as limiting the maximum acceleration to avoid injuries to and limiting the maximum deformation to avoid injuries attributable to deformed structures penetrating a passenger cabin. Further, the crashworthiness design often requires designing the structures so that the structures absorb as much energy as possible while deforming in a manner that does not cause injury the passengers.

Most commercial software products for designing structures assume that only elastic deformation takes place in the structures. Further, it is assumed that a load is applied to the structures over a long period of time, which results in a steady-state response from the structures. These assumptions, however, become unrealistic for crash events where loads to the structure and boundary conditions change as time progresses. Accordingly, most commercial software products are incapable of performing crashworthiness designing.

Gradient-based method is one of the techniques used in the crashworthiness design. The gradient-based method, however, requires a nonlinear dynamic finite element analysis that is computationally intensive. Therefore, the gradient-based method takes a very long time to design and evaluate the structures. One way to reduce the computation is to simplify the finite element analysis (FEA). But simplifying the finite element analysis will sacrifice the accuracy and reliability of the crashworthiness design.

SUMMARY OF THE INVENTION

Embodiments provide a method, a device and a computer readable storage medium for designing a structure representing a physical object by an optimization process using a Hybrid Cellular Automata (HCA) algorithm where the structure is segmented into a plurality of cells, analyzed by a finite element analysis (FEA), and updated by cellular automata computation. A design variable representing a material distribution in each cell is updated by a material distribution update rule to converge the field state of each cell towards a target field state of each cell. The process of analyzing the structure by the FEA and updating by the cellular automata computation is repeated until a termination condition is satisfied.

In one embodiment, a performance variable representing a property of the structure required to satisfy a performance constraint is calculated. The performance constraint may be associated with forces on the structure and displacements or deformation of the structure. The performance variable may be any property of the structure that is associated with the performance constraint.

In one embodiment, the design variable is a material density of the cell or a thickness of the cell. The material density or the thickness of the cell is updated by the cellular automata based on the field state of the cell to evolve the initial structure into an optimized structure.

In one embodiment, the field state of the cell is the internal energy density of the cell responsive to applying a load to the structure. Crashworthiness designing of a structure involves distributing the internal energy density across the cells as evenly as possible. By obtaining the design of the structure with an even distribution of the internal energy density, all of the cells contribute equally to the absorption of kinetic energy during a crash event. The even distribution of the internal energy density across the cells, therefore, may result in a more crashworthy design of the structure.

In one embodiment, the HCA algorithm is applied to designing a stamped structure with uneven cell thickness. The thickness is employed as a design variable, and is updated by the cellular automata computation.

DETAILED DESCRIPTION OF THE INVENTION

A preferred embodiment of the present invention is now described with reference to the figures where like reference numbers indicate identical or functionally similar elements.

Certain aspects of the present invention include process steps and instructions described herein in the form of an algorithm. It should be noted that the process steps and instructions of the present invention could be embodied in software, firmware or hardware, and when embodied in software, could be downloaded to reside on and be operated from different platforms used by a variety of operating systems.

Embodiments provide a method, a device and a computer readable storage medium for crashworthiness designing of a structure using a Hybrid Cellular Automata (HCA) algorithm where field states are computed using finite element analysis (FEA) and the material distribution of the structure is updated at each iteration using cellular automata method. The HCA algorithm optimizes the topology of the structures to achieve a target performance within the limits of various constraints applied to ensure crashworthiness of the structures. The HCA algorithm may also be applied to designing of structures to be fabricated by an extrusion method having the same cross section along the direction of extrusion or stamped structures having thickness varying across the structure.

The structures that can be designed using the embodiments include, among others, parts in various types of vehicles (e.g., automobiles, aircrafts, ships), construction elements, consumer products, furniture, construction machines, and safety equipments.

Architecture of Structure Designer

FIG. 1is a block diagram illustrating a structure designer110, according to one embodiment. The structure designer110may be a general purpose computer or a dedicated computer structured to design and evaluate structures using the method disclosed herein. In the embodiment ofFIG. 1, the structure designer110includes, among other components, a processor112, a memory114, an input module116, a display module118, and a bus130connecting these components. The structure designer110may include other components that are omitted herein to avoid obfuscation of the embodiment.

The processor112executes instructions stored in the memory114to generate the designs of structures and optimize the topologies of the structures. The processor112may comprise various computing architectures such as a complex instruction set computer (CISC) architecture, a reduced instruction set computer (RISC) architecture, or an architecture implementing a combination of instruction sets. Although only a single processor is shown inFIG. 1, multiple processors may be included. The processor112may comprise an arithmetic logic unit, a microprocessor, a general purpose computer, or some other information appliance equipped to transmit, receive and process data signals from the memory114.

The memory114stores various software components including, among others, a structure design storage210, a cell divider220, a finite element analysis (FEA) engine230, and a cellular automata engine240. The structure design storage210stores the parameters of the structures as they are generated in each iteration of the HCA algorithm processing. The parameters of the designed structures may be stored in the structure design storage210for display on the display module118or output via other output devices of the structure designer110. One or more of the components in the memory114may be embodied in hardware, software, firmware or any combinations thereof.

The cell divider220reads the data of the initial structure from the structure design storage210and segments the structure into multiple cells for processing by the FEA engine230and the cellular automata engine240. Specifically, the cells may have uniform size or have non-uniform size. In one embodiment, the FEA engine230and the cellular automata engine240use the same segmentation or configuration of cells. In another embodiment, the FEA engine230and the cellular automata engine240use different segmentation or configuration of cells.

The FEA engine230, in conjunction with the cellular automata engine240, performs the HCA algorithm. The FEA engine230performs numerical analysis on structures by dividing the structure in the current iteration of the HCA algorithm into cells (finite elements), and then computes equations representing the equilibrium of the cells to approximate one or more field states S (e.g., stress, strain and internal energy density) associated with the structure. In one embodiment, LS-Dyna available from Livermore Software Technology Corporation (LSTC) at Livermore, Calif. is used as the FEA engine230.

The cellular automata engine240updates a design variable x representing a material distribution for a cell in the structure based on a material distribution rule that takes into account the field state of neighborhood cells. The state αi(k)of ith cell in a kth iteration of the HCA algorithm may be expressed by the following equation:

αi(k)={Si(k)xi(k)}Equation⁢⁢(1)
where Si(k)represents a set of field states, and xirepresents the design variable of ith cell in the structure. The basic idea of the HCA algorithm is to compute the field state Si(k)using the FEA engine230while updating the design variable xiusing the cellular automata engine240. The HCA algorithm differs from a general cellular automata algorithm in that the information of the field state Si(k)is calculated from a global analysis (e.g., finite element analysis) and then used to adjust the design variable xi.

In one embodiment, the design variable xifor ith cell represents one of the following material properties associated with the structure: density (ρ), elastic modulus (E), yield stress (σY), strain-hardening, and modulus (Eh). The design variable ximay be scaled to have a value not less than 0 and not larger than 1 (0≦xi≦1). In this example, xi=0 means that ith cell is a void whereas xi=1 means that the cell is full of material. Any values between 0 and 1 means that the cell is partly filled with the material. In another embodiment, the design variable xifor ith cell represents the thickness of the cell, as described below in detail with reference to stamped structures.

Method of Designing a Structure Using HCA Algorithm

FIG. 3is a flow chart illustrating a method of designing a structure, according to one embodiment. First, an initial design of the structure is loaded310into the cell divider220. The structure is then segmented316into a plurality of cells by the cell divider220. The information about the structure and its segmentation is then fed into the FEA engine230to perform320a conventional structural analysis. In one embodiment, transient nonlinear finite element analysis is used to perform the structural analysis. The FEA engine230determines, among others, the field states Si(k)as a result of the structural analysis.

The result of the structural analysis is then provided to the cellular automata engine240together with information about the structure and its segmentation of the cells. The cellular automata engine240then updates330the material distribution of the structure according to a material distribution rule. Specifically, the design variable xiof each cell in the structure is updated according to the material distribution rule. In one embodiment, the distribution of material in the structure is updated to evolve the initial design of the structure over multiple iterations into a final structure with cells have approximately the same internal energy density. By obtaining the design of the structure with even distribution of the internal energy density, all of the cells contribute equally to absorption of kinetic energy during a crash event. The even distribution of the internal energy density across the cells, therefore, may result in a more crashworthy design of the structure.

In one embodiment, proportional, integral and derivative (PID) feedback is used to reduce the oscillatory behavior of the design variable xifrom one iteration of the HCA algorithm to the next. The change in the design variable xiof ith cell at kth iteration of the HCA algorithm using the PID feedback may be expressed in the following equation:
Δxi(k)=Kpei(k)+Ki∫k−2kei(k)dκ+KdΔei(k)equation (2)
where Kpis a proportional control gain, Kiis an integral control gain, Kdis the derivative gain, ei(k)is the error between the average field state of the cellSi(k)and a target field state of the cell S*i(k). The error ei(k)in equation (2) can be expressed as:
ei(k)=Si(k)−S*i(k)equation (4)

The target field state refers to the field state attempted to be reached in the current iteration. The target field state may be changed at each iteration based on, for example, the field states of the cells in the structure. The target field stat is caculated based on a target fractional mass provided by a user.

Although the integral error in equation (2) is limited to the error from the previous two iterations, the number of iterations for the integral error may be increased or decreased.

In one embodiment, equation (3) is simplified by setting Kiand Kdto zero, and combining equation (4). The resulting equation is as follows:
Δxi(k)=Kp(Si(k)−S*i(k))  equation (5)
The embodiments are described below using equation (5) for the sake of explanation.

In one embodiment, the field state of previous iterations may be used in conjunction with equation (5) to reduce oscillation between iterations. For example, the field state Si(k)of each cell may be represented by an exponential moving average of the internal energy density at the current iteration and n previous iterations, for example, as expressed in the following equation:

Si(k)=1∑k=0n⁢wik⁢∑j=0n⁢wj⁢Ui(k-j)equation⁢⁢(6)
where Uirepresents internal energy density of the ith cell, wirepresents weights given to the internal energy density computed at the current and previous iterations. Equation (6) takes into account the field states in previous iterations of the HCA algorithm and thereby gives each cell a memory of its previous states. The weights are expressed in the following equation:
wi=(xi−xmin)2equation (7)
where xminindicates the minimum density allowed. Ideally the xminis zero but this value is fixed to a small value (e.g., 0.05) to avoid computational issues.FIG. 4is a graph illustrating the relative weights

wj∑k=04⁢wik
given to the internal energy of ith cell at current (j=0) and previous iterations (j=1 to j=3) when n is set to 4.

After updating the distribution of material for all cells, the process proceeds to check whether a predetermined termination criteria is satisfied350. For example, the convergence of parameters in the design of the structure may be checked. That is, the termination criteria include the difference between a parameter in current iteration and a parameter in a next iteration falling below a threshold. If the criteria are not satisfied, the process returns to the step320of performing the structural analysis of the structure using the FEA engine320. If the criteria are satisfied, the final design is output360from the structure designer110.

In one embodiment, the HCA algorithm designs structures within the constraints of the total mass of the structure. The total mass of the structure is correlated with the production cost of the structure. Therefore, the total mass of the structure may be constrained to generate the design of the structure that satisfies cost constraints. The internal energy stored in the structure, however, is directly correlated with the mass of the structure. The higher the target internal energy, the lower the mass of the structure and vice versa. Therefore, to determine an acceptable final mass of the structure, a target internal energy must be determined.

One way of finding this target internal energy is to iterate the HCA algorithm and adjust the target internal energy until the mass constraint is satisfied. The target internal energy for the (k+1)th iteration of the HCA algorithm can be expressed as:

S*(j+1)=S*(j)⁡(Mf(k+1)Mf*)equation⁢⁢(8)
where M*fis a target fractional mass. The initial target field state S*(0)is the average field state of all cells in the kth iteration. The fractional mass Mfof the structure is defined as follows:

Mf=∑i=1N⁢xi∑i=1N⁢1equation⁢⁢(9)
where N represents the total number of cells in the structure. The fractional mass Mfindicates the fraction of mass relative to the mass of initial design having full density cells and is set by the user. The fractional mass Mfis proportional to the actual mass of the structure. When the fractional mass Mfof the structure at the kth iteration of the HCA algorithm satisfies the target mass M*fof the structure, the HCA algorithm is terminated. In this way, the final design of the structure satisfies the mass constraint.
Crashworthiness Design Based on Performance Constraints

Crashworthiness design of a structure may be limited by one or more factors. For example, the crashworthiness design may involve limiting of the displacement of the structure, reaction forces of the structure or the strain in the structure. If the displacement of the structure is excessive, passengers may experience acceleration that may injure the passengers or result in failure or buckling of the structure during the collusion. The reaction forces generated during the collision may also need to be constrained because high reaction forces may cause permanent damage to a vehicle. The strain may also need to be limited because excessive strain will cause the material to crack or fail and no longer withstand the load. The HCA algorithm may be applied to generate the designs of structures satisfying such predetermined performance constraints.

FIG. 5is a flow chart illustrating a method of designing a structure to satisfy the predetermined performance constraints, according to one embodiment. The steps510,516,520,530,550and560inFIG. 5are substantially the same as the steps310,316,320,530,550and560inFIG. 3, respectively. Therefore, the description of these steps is omitted herein. The only step added inFIG. 5compared toFIG. 3is the step524of updating the target fractional mass. Specifically, by updating the target fractional mass, the performance constraints of the structure may be adjusted while achieving a target optimization objective (e.g., even distribution of the internal energy density across the structure). In the following, the structures are designed to satisfy one of the three performance constraints: (i) a maximum displacement constraint, (ii) a maximum reaction force constraint, and (iii) a maximum strain constraint.

In one embodiment, the maximum displacement constraint is imposed as a performance constraint on the design of the structure. In this embodiment, a control-based strategy is used to determine the amount of change in the target fractional mass. Specifically, an error eDCis defined as follows:

eD⁢⁢C=dm⁢⁢ax-dm⁢⁢ax*dm⁢⁢ax*equation⁢⁢(10)
where dmaxis the maximum displacement of the structure, and d*maxis a target displacement set to satisfy design requirements. The reduction of the target fractional mass in step524in current iteration of the HCA algorithm is expressed as follows:
ΔM*f=(KDCx)eDCequation (11)
where KDCis a proportionality constant for displacement constraint, andxis a density scaling parameter indicating the portions of cells that is either void of filled as described below in detail.

The global stiffness of the structure in terms of maximum displacement is difficult to determine when the cells are partly filled with material (that is, 0<xi<1). Therefore, in one embodiment, larger changes in the fractional mass are allowed as the design converges using the HCA algorithm, and each cell moves toward being a void (xi=0) or being fully filled with material (xi=1). The scaling parameterxis defined as follows:

x_=∑i=1n_⁢xx>xm⁢⁢i⁢⁢nn_equation⁢⁢(12)
wherenrepresents the number of cells that have design variable xigreater than xmin.

Although the maximum displacement and the target fractional mass have approximately quadratic relationship for a structure where most of its cells have the design variable xiof 0 or 1, a linear relationship is approximated for small changes in the fractional mass during the updating of the target fractional mass (e.g., ΔM*f=0.1) because the maximum allowable change of the design variable Δxmaxis 0.1 for numerical stability in the FEA. By defining the reduction of the target fractional mass as a function of the scaling parameterxthat indicates the portion of cells in the structure that is either void of fully filled, an efficient convergence to optimized design of the structure can be achieved. As the iteration of the HCA algorithm proceeds and approaches the final structure exhibiting the desired force-displacement behavior, the permitted change in the target fractional mass may be increased. By changing the target fractional mass (and hence the total mass), the force and displacement constraints can be accommodated.

In one embodiment, the maximum reaction force constraint is imposed as a performance constraint on the design of the structure. The reaction force generated during a contact between two structures is also directly correlated with the mass of the structure and the global stiffness of the structure. Therefore, the methodology for controlling the reaction force is again based on finding the appropriate mass of the structure. Similar to the embodiment of imposing the maximum displacement constraint as set forth above, the reaction force constraint is applied globally by penalizing the amount of material needed for the structure. Specifically, the threshold mass of the structure is lowered until the maximum reaction force is reached. A control-based strategy is also used to adjust the target fractional mass M*fto control the maximum reaction force in the structure.

An error for maximum reaction force eFCis quantified by integrating the normalized error between the actual force-displacement Fr(d) of the structure and a target set of forces applied to the structure Fr*(d) as expressed by the following equation:

In one embodiment, the maximum strain constraint is imposed as a performance constraint on the design of the structure. The structure designer110may generate the design of a structure that maximizes the energy absorption of the structure while not exceeding a strain level in any parts of the structure during a collision. If a cell is determined as having largest strain that exceeds predetermine maximum strain, the mass of the material in the cell is increased. By increasing the material in the cell, the overall level of strain in the cells is decreased.

FIG. 6is a diagram illustrating stress-strain curves for varying design variables xi(relative density in this case) of the material in a cell, according to one embodiment. At kth iteration of the HCA algorithm, the density variable is xi* and the maximum strain in the ith cell is determined as EK. By increasing the design variable from xik=x*i* to xik+1=xi*+Δx, the maximum strain in the ith cell is reduced to a target strain Ek+1=E*. Vx may be determined by computing a design variable xik+1where the amount of additional internal energy (represented by hashed area610) stored in the cell by increasing the design variable to xi*+Vx corresponds to the internal energy (represented by hashed area620) stored in the cell for undergoing excessive strain at the current density variable xi*. All design variables of the cells in the structure is updated in this manner to reduce the maximum strain to a target strain E*.

The adjustment of the design variable ximay be performed in the step530of updating the material distribution. Specifically, the material distribution rule is expressed by the following equation to achieve (i) uniform distribution of the internal energy, and (ii) the strain constraint:
Δxi=ω1Δx1i+ω2Δx2iequation (15)
where Δx1(k)=KP(Si(k)−S*i(k)) represents material distribution update for uniform internal energy distribution,

Δ⁢⁢x2(k)=Ks⁡(ɛ*-ɛi*ɛ*)
represents material distribution update for the strain constraint, Ksis a constant representing a proportionality constant for strain, ω1and ω2represent weighted values represent how much weight should be given to uniform distribution of the internal energy and the strain constraint, respectfully. ε* represents a target strain value or a maximum allowable strain in the structure, and εi* represents the strain value at the present iteration.

The weighted values ω1and ω2are fixed during the iterations. If the final design of the structure is not satisfactory, however, a new design may be obtained by performing another set of iterations based on different weights ω1and ω2. The user may change the weights ω1and ω2based on experience to obtain a desired final design.

Designing of Structures Fabricated by Extrusion Method

An extrusion manufacturing process is often employed as a method for fabricating a structure with reduced cost. In the extrusion manufacturing process, material is forced through a die using either a mechanical or hydraulic press. The structure fabricated by extrusion has a cross-section corresponding to the shape of the die. The structures fabricated by the extrusion process often require minimal secondary machining, which reduces the overall cost associated with fabricating the structures. The design of the structure manufactured by the extrusion process is different in that the two-dimensional cross-section of the structure must be defined instead of a three-dimensional topology of the structure.

In one embodiment, the HCA algorithm is modified to generate designs with a constant cross-section in a direction perpendicular to the direction of extrusion.FIG. 7is a diagram illustrating a discretized three-dimensional structure710for fabrication by an extrusion method, according to one embodiment. The structure710is fabricated by extruding the material in direction ξ. In the extrusion method, the cross-section of the structure710must be retained along the direction of extrusion ξ. Therefore, the process of designing the structure710is reduced to a two dimensional problem of assigning design variables xito the cross-sectional columns A1through D4.

FIG. 8is a flow chart illustrating a method of designing the structure710to be fabricated by the extrusion method, according to one embodiment. First, the initial design of the structure710is loaded810. The structure710is then segmented816into columns (for example, columns A1through D4) along the direction of extrusion ξ. Each column A1through D4is then segmented820into multiple cells (for example, column B2is segmented to cells702A through702N).

The FEA engine230performs826the structural analysis of the structure710applied with a load to compute the internal energy density in each cells of the structure710. Then, the internal energy densities in the cells of a column are added to obtain830a total internal energy density in each column for the structure710. Referring toFIG. 7, for example, the internal energy densities in the cells702A through720N is added to obtain the total internal energy density of column B2. The total internal energy densities in other columns are obtained in the same manner. The total internal energies of the columns A1through D4are illustrated inFIG. 9as blocks with different patterns.

The cellular automata engine240then updates836the distribution of materials in each block A1through D4to evenly distribute the total internal energy density across the columns A1through D4. The distribution of materials is performed in two-dimension.

Then the updated design is analyzed to confirm if predetermined termination criteria are satisfied840. In one embodiment, the updated design is checked against the design in a previous iteration for differences in a parameter of the designs after updating the material distribution. If the change in the parameter is below a threshold, it is determined that the termination criteria are satisfied. If the termination criteria are satisfied, the process proceeds to output846the final design of the structure710. If the termination criteria are not satisfied, the process returns to step826.

Designing of Stamped Structures

In one embodiment, the HCA algorithm is used for designing a stamped structure with varying structural thickness. Like other examples described above, the goal of the design may be to evenly distribute the internal energy density across the stamped structure as much as possible. If the stamped structure has a low internal energy density at one portion, the thickness is reduced to increase the internal energy density at the same portion. In contrast, if the stamped structure has a high internal energy density at a portion, the thickness is increased to reduce the internal density at the same portion. The HCA algorithm adjusts the thickness profile of the stamped structure to evenly distribute the internal energy density across the stamped structure.

The method of designing the stamped structure using the HCA algorithm is essentially the same as the method described above with reference toFIG. 3. The major difference in designing the stamped structure compared to the embodiment ofFIG. 3is that the thickness (t) of the cell is used as the design variable instead of density (ρ), elastic modulus (E), yield stress (σY) or strain-hardening modulus (Eh). Specifically, the thickness (t) is updated in step330using the following equation:
Δti(k)=KP(Ūi(k)−U*i(k))  equation (16)
where Δti(k)is the change of thickness in ith cell at kth iteration of the HCA algorithm, Kpis a proportional control gain, Ūi(k)is the internal energy density of ith cell, and U*i(k)is a target internal energy density of the cell. The target internal energy density U*i(k)may be updated at each iteration based on the internal energy density of each cell.

Various modifications to the equations and processes may be made. For example, although-above embodiments were described mainly with reference to a design variable xihaving values not below 0 and not above 1, other unscaled variables having values beyond 0 and 1 may also be used. Further, actual mass of the cell may be used instead of the factional mass when performing the HCA algorithm.

Although the present invention has been described above with respect to several embodiments, various modifications can be made within the scope of the present invention. Accordingly, the disclosure of the present invention is intended to be illustrative, but not limiting, of the scope of the invention, which is set forth in the following claims.