Overview of Reduction Methods and Their Implementation 

Into Finite-Element Local-to-Global Techniques 

by Michael A. Minnicino II and David A. Hopkins 

ARL-TR-3340 September 2004

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Army Research Laboratory 

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ARL-TR-3340 September 2004

Overview of Reduction Methods and Their Implementation 

Into Finite-Element Local-to-Global Techniques 

Michael A. Minnicino II and David A. Hopkins 

Weapons and Materials Research Directorate, ARL 

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Overview of Reduction Methods and Their Implementation Into Finite-Element

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13. SUPPLEMENTARY NOTES 

14. ABSTRACT 

Reduction methods are used to reduce the number of degrees of freedom in a finite-element (FE) model, typically at the expense 

of high-fidelity solutions. The reduced solution’s fidelity depends on the reduction method implemented. Most of the reduction 

methods described in this report were developed to efficiently determine the natural frequencies and mode shapes of large FE 

models. It is the goal of this report to briefly review these reduction methods and describe their implementation into feasible FE 

local-to-global analysis techniques. Local-to-global techniques are also part of the family of model reduction methods, but are 

normally used for structural analysis purposes. 

15. SUBJECT TERMS

finite element, reduction, local-to-global 

16. SECURITY CLASSIFICATION OF: 19a. NAME OF RESPONSIBLE PERSON

Michael A. Minnicino a. REPORT 

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ii 
Contents 

List of Figures iv

1. Introduction 1

2. Motivation 1

3. Traditional Global/Local Process 3

4. Static, Quasi-Static, Dynamic, and Transient Domains 4

4.1 Static Domain ....................................................................................................................4 

4.2 Quasi-Static .......................................................................................................................4 

4.3 Dynamic Domain...............................................................................................................4 

4.4 Transient Domain ..............................................................................................................5 

5. Qualitative vs. Quantitative Analyses 5

6. Reduction Methods Overview and Local-to-Global Implementation 5

6.1 Overview of Nonmathematical Methods...........................................................................5 

6.1.1 Uncoupled Global/Local Analysis...........................................................................6 

6.1.2 Mass Element Approximation ................................................................................6 

6.1.3 Submodeling ...........................................................................................................7 

6.2 Overview of Mathematical Methods.................................................................................9 

6.3 Superelements/Substructuring and Static Reduction.......................................................11 

6.4 Dynamic Reduction.........................................................................................................12 

6.5 SWAT..............................................................................................................................13 

6.6 SEREP .............................................................................................................................16

7. Summary 18

8. References 19

Distribution List 20

iii 
List of Figures 

Figure 1. Depiction of global/local process. ...................................................................................3 

Figure 2. Comparison of axial displacement of baseline and lumped mass for a simple chip 

with components. .......................................................................................................................8 

Figure 3. Comparison of von Mises stress of baseline and lumped mass for a simple chip 

with components. .......................................................................................................................8 

Figure 4. Global model with three possible local-global boundaries. The “x” symbols denote 

the nodes from which results will be interpolated for boundary 2 and subsequently 

prescribed on the boundary of the local model to effectively drive the local solution..............9 

Figure 5. (a) Global model with three boundaries, (b) local model results corresponding to 

boundary 1, (c) local model results corresponding to boundary 2, and (d) local model 

results corresponding to boundary 3........................................................................................10 

Figure 6. Local-to-global model reduction process for SEREP. ..................................................16 

iv 
1. Introduction 

Reduction methods are used to reduce the number of degrees of freedom (DOF) in a 

finite-element (FE) model, typically at the expense of high-fidelity solutions. The reduced 

solution’s fidelity depends on the reduction method implemented. Most of the reduction 

methods described in this report were developed to efficiently determine the natural frequencies 

and mode shapes of large FE models. It is the goal of this report to briefly review these 

reduction methods and describe their implementation into feasible FE local-to-global analysis 

techniques. Local-to-global techniques are also part of the family of model reduction methods, 

but are normally used for structural analysis purposes. 

2. Motivation 

Explicit time domain FE solution techniques are attractive for simulations of many high-g loaded 

structures because of the increased computational speed and reduced memory requirements of 

these methods compared to implicit solution techniques. However, explicit methods require 

using small time-steps, compared to implicit methods, in order to solve the governing equations 

because explicit solvers are only conditionally stable. Each element’s time-step is determined by 

its size and its material properties. The conditional stability requirement is dependent on the 

explicit solver employed in the solution. The maximum value of this stable time-step is 

proportional to the size of the smallest element used in the model. This time-step restriction is 

commonly known as the Courant condition. Conversely, most implicit methods are 

unconditionally stable, thereby allowing an arbitrarily large time-step, but implicit methods also 

requires matrix inversion, which is computationally expensive. Accuracy considerations further 

limit the size of the allowable time-step in implicit methods. Finally, explicit techniques are 

faster and more reliable in obtaining results for most high-g FE simulation models. However, FE 

models of these structures can require hundreds of thousands of elements to accurately capture 

the desired structural response. This is especially true when modeling electronic packages to 

determine operational failure. If solutions are desired in a timeframe where the FE method can 

be used as a design tool for electronic packages, a significant reduction of the computational 

effort for a given CPU architecture (i.e., a set number of processors, processor speed, and amount 

of memory) is required. 

An FE model, whether of a cell phone or a smart projectile, can have large differences in the 

scales of the geometry that must be modeled. For instance, structural features in high-g launch 

applications may have characteristic dimensions of meters, while the electronic components 

and/or electronic packages can have characteristic dimensions of millimeters. Such large 

differences in scale lengths present significant issues for generating appropriate FE models of 

1
these structures. Some possible meshing choices include using a uniform mesh, using disparate 

meshes with constraints, and using global/local modeling techniques. 

First, a uniform mesh for the entire FE model with the characteristic element size based on the 

electronic component’s geometry can be generated. While the resultant mesh has element sizing 

that can accurately capture the electronic component’s response, larger features of the geometry 

may have an overly refined mesh. This approach can yield accurate response data for the entire 

model, but at the expense of solution time due to having more elements and therefore more DOF 

than needed. The solution time is further increased because of the size restriction of the solution 

time-step by the inclusion of the small structural features into the FE model. This reduction in 

the time-step size can also result in inaccurate results if numerical round-off errors accumulate in 

the solution.

A second approach is to mesh the individual components of an FE model separately. This 

approach reduces the total number of elements, but the resultant mesh has inconsistent element 

sizing. Subsequently, interface constraints, such as contact, are defined at the interfaces between 

the noncongruent meshes. Material properties excluded, the solution time-step is controlled by 

the smallest element in the model and thus long analysis times can still be issue. Also, 

depending on the type of interface used to couple the disparate meshes, results near the interfaces 

may be unsatisfactory. For very complex structures, it is possible that properly defining the 

interface can be a serious issue as well, since multiple interfaces may share some nodes or 

element faces (1). 

Finally, a third approach is to perform separate local and global analyses. The global/local 

approach is advantageous when disparate length scales must be modeled, such as when modeling 

the response on electronics that are embedded in projectiles. The local model normally consists 

of all those components whose inclusion would either lead to an excessive number of elements 

and/or a significant reduction in the time-step used in the solution. Determination of the local 

model is thus a subjective process. The global model then consists of all the remaining 

components. The goal in the selection of the global and local models is to allow larger elements 

to be employed in the global model, thus increasing the size of the stable time-step and 

consequently reducing the computational time needed. Also, because of the reduction in the 

number of elements and nodes there are fewer DOF to solve. The correct selection of the local 

and global models therefore reduces the computational effort in two ways: (1) increasing the 

characteristic time-step and (2) reducing the total number of elements in the model. 

A global/local approach is sometimes referred to as a submodeling or substructuring approach, 

depending upon how the model is formulated and solved. However, in the context of this report, 

we will reserve these terms for specific modeling approaches discussed later and use global/local 

to refer to the more general case where disparate length scales and corresponding mesh scales are 

required to obtain a representative model of the structure. The global/local approach can also 

allow the use of multiple local models, depending on the global-local modeling method used. 

2
However, the correct representation of the local model’s structural response or the proper 

representation of the effects of the local model upon the global model’s response can be difficult 

to determine and verify. This report reviews several possible approaches to global/local 

modeling and discusses their applicability to modeling high-g environments. 

3. Traditional Global/Local Process 

First, though, it is advantageous to review the typical global/local modeling process, which is 

shown in figure 1. Conceptually, this is very straightforward. As mentioned, models are 

typically constructed such that the smaller components of the model are modeled with high detail 

due to the need of hi-fidelity measurements/results. When this is done, the smaller elements of 

the highly detailed component are selected to form the local model and the larger elements form

the global model. The resulting FE models each retain consistent element sizes. In this 

approach, the highly detailed subscale local FE model and the larger scale global FE model can 

be developed simultaneously. The global model’s loading, both magnitude and location, is 

assumed known a priori. It is also assumed that the time histories for the global model, e.g., 

acceleration-time history, can be applied to the local structure at the local-to-global interfaces. 

The local model’s structural properties are homogenized via a qualitative local-to-global method, 

since the applied local load is not the true load seen by the local model. This homogenized local 

model is then incorporated into the global model. The response of the resulting model is then 

determined and a global solution is obtained. The global nodal solution at the local-to-global 

interfaces is then used as the applied load to the detailed local model for the local analysis. Also, 

for all reduction methods, the choice of the reduction method leads to different approximations 

to the true response of the structure. 

Local M odel G lobal M odel G lobal Solution

Local Solution

Local M odel 

D esign Change

Local Solution Process

G lobal Solution Process

5 

6 

4

1

3

2

Steps 

1) Homogenize Local Model and Incorporate into Global Model 

2) Solve Global Model 

3) Apply Local-to-Global Interface Forces to Local Model 

4) Solve Local Model 

5) Local Model Failure Analysis 

6) Update Local Model 

Figure 1. Depiction of global/local process.

3
The issue of whether the local response affects the global response can be investigated by 

repeating the global/local process, using various methods to homogenize the properties of the 

local structure, and then looking for differences in the global response. If the global model’s 

response does not change for the differently homogenized local structures, then it is normally 

assumed that the local response probably does not affect the global response. Alternatively, if it 

is found that the global response differs between the differently homogenized local structures, 

then it is assumed that the local response at least partially drives the global response. The 

question of how strongly one response affects the other needs to be determined in this latter case. 

If the dependence is weak, it can be concluded that the global response is largely unaffected by 

the local response except at or near the local-to-global interface. However, decisions concerning 

whether the solutions are accurate based on the qualitative response of the local and global 

models may not be sufficient for determining a viable design. It is precisely these uncertainties 

that this traditional approach to the analysis fails to answer. Quantitative measures of the 

accuracy of the analyses are needed if the designer is to make informed decisions regarding 

potential failure of a design. Several global/local techniques have been proposed which can 

address this concern. 

4. Static, Quasi-Static, Dynamic, and Transient Domains 

Before proceeding with discussion of the techniques, it is important to note that some of the 

discussed reduction methods were developed and implemented for specific time domains. For 

the remainder of this report, the following terms are defined: 

4.1 Static Domain 

The applied load is static, or changes so gradually that there are no rate effects and the entire 

system is in static equilibrium with the applied load at all times. Inertial effects are nonexistent 

and the model is completely defined by its stiffness matrix. 

4.2 Quasi-Static

The applied load is time dependent, yet rate effects are not present. The inertial response of the 

structure is restricted to the response where the entire structure is subjected to a constant 

acceleration field. The system deforms and may also translate/rotate in this constant acceleration 

field. Acceleration loads are normally applied as D’Alembert forces.

4.3 Dynamic Domain 

The period of load may be shorter than the quasi-static domain, and like the quasi-static domain, 

rate effects are not present. However, inertial effects are present such that different parts of the 

structure are accelerating at different rates. The system is in dynamic equilibrium. Vibratory 

motion exists, as well as deformation and translation/rotation in a variable acceleration field. 

4
4.4 Transient Domain 

The load is applied rapidly, as in a shock event. Rate effects are present. Steady state motion, 

such as vibratory motion, has not had time to become established. Wave propagation effects 

must be considered. 

5. Qualitative vs. Quantitative Analyses 

Currently, the simplest global/local methods to implement provide qualitative information in the 

sense that the true loading seen by the local structure is unknown and therefore only the trending 

of the response as a function of design iteration can be determined, similar to a parametric study. 

A typical qualitative example is applying the solution time-history results from a global model, 

e.g., an acceleration located at the global model’s CG, to the local structure at the global/local 

models’ interface. Thus, the designer can look at the differences in the local structure’s 

responses due to local structural changes. This approach can enable the designer to determine 

the important structural parameters at the expense of not knowing the true response. For initial 

design iterations, this level of knowledge is sufficient. These qualitative analyses can still 

require considerable computational time and effort. 

For dynamic events, these qualitative methods cannot be used to accurately predict mechanical 

failure of an electronics package or component, since they do not include the transmissibility 

effects that manifest as amplification or attenuation of the transmitted force to the local structure

from the global loading through the load path. This is because these qualitative analyses do not 

account for a coupled response between the local and global models. In a quantitative analysis, 

the amplification/attenuation of the transmitted force is captured and thus accurately represents 

the true load seen by the local model. This allows for the accurate determination of failure of an 

electronics package or component. 

6. Reduction Methods Overview and Local-to-Global Implementation 

6.1 Overview of Nonmathematical Methods 

The first class of global/local methods discussed are not mathematically based, but rather a 

“best-practices” approach to increase the computationally efficiency at the cost of quantitative 

results. 

The methods discussed in this section include the following: 

• uncoupled global/local analysis 

• lumped mass approximation 

• submodeling 

5
6.1.1 Uncoupled Global/Local Analysis 

The simplest reduction method, this qualitative method involves partitioning the original model 

into a global model and a local model. In this approach, the global model solution is obtained 

without feedback from the local model, i.e., it is assumed that local model has no effect on the 

global model’s response. Consequently, the local model isn’t even considered in obtaining the 

solution for the global model. The global solution, e.g., displacement or acceleration, at the 

global model’s local-to-global interface is then prescribed as the local model’s loading at the 

interface. The unconstrained explicit solution for the local model is then determined, and a 

qualitative failure analysis can be conducted as a function of design iteration. A slight 

refinement to this technique is to model the local model as a rigid body in the global model, thus 

capturing the rigid body inertial response of the local structure in the global solution. However, 

modeling the local model as a rigid body will stiffen the global model at the global/local 

interface. If the global model is actually compliant at this interface, then poor results will be 

obtained from this enhancement. 

As is typical for local/global approaches, this modeling technique reduces the FE computational 

cost in two ways. First, the total number of DOF is reduced for the solution by eliminating the 

elements associated with the local model. Secondly, the elimination of the smaller elements that 

constitute the local model increases the size of the characteristic time-step to be proportional to 

the smallest retained element of the global model. The computational savings of the global 

model is realized by taking larger time-steps and by solving fewer equations. 

One may consider the reduction in the number of DOF in the global model superficial because 

these DOF are subsequently solved for in the separate local model analysis. However, given the 

local excitation obtained from the global solution and the assumption that the local model 

response does not significantly affect the global model response, then, as a design tool, this FE 

approach is suitable because it allows for expedient local model solutions and analyses due to 

local model design modifications. Specifically, since the response of the global model is 

independent of the local model, different local models can be analyzed using the same loading as 

specified by the global model. This approach provides the fastest solution iterations, but at the 

expense of assuming there is no coupling between the global and local models’ responses. This 

approach is equivalent, both in spirit and practice, to the qualitative approach discussed earlier. 

6.1.2 Mass Element Approximation 

This method replaces the local model with equivalent point masses distributed at the local-toglobal interface in the global model. This method thus accounts for the local model’s 

translational inertial loading upon the global model’s response. The local model’s mass is 

distributed to the nodes on the interface surface of the global model’s elements. Using this 

method may allow the modeler to remove contact interfaces, thereby further decreasing the 

number of equations to solve. This effectively reduces the model’s computational complexity by 

eliminating/truncating the local model DOF, yet it includes the inertial feedback that may be 

6
important in high-g loading scenarios. Also, the use of mass elements means that the global 

solution’s time-step size is dictated by the smallest global model element for an explicit analysis. 

The global solution is obtained and the nodal results corresponding to the local-to-global 

interface are then applied as the loading of the local model. Errors due to the misrepresentation 

of the moments of inertia may be significant at high-g loadings, thereby limiting its 

implementation into more complex loading scenarios. This approach assumes that the stiffness 

of the local model has no effect on the global response. This modeling technique is reasonable 

when the acceleration of the global system is dominated by a uniaxial response, and the stiffness 

of the local model is low compared to the global model. 

Figure 2 presents results for an axially accelerated disk with electronic components attached. 

The baseline response, figure 2a, is the computed response for the full model shown in figure 2d. 

Figure 2a shows contours of the axial displacement for the baseline model. Figure 2b shows the 

computed response when the components are replaced by lumped mass approximations. As can 

be seen, the axial response for the baseline model is very similar to the lumped mass 

approximation at the same solution time. This is more clearly shown in the plot in figure 2c, 

which plots the time history of the axial displacement for each model. However, while the axial 

displacement compares very well, the stress levels, as shown in figure 3, are very different. This 

difference is probably due to the stiffening effect of the actual components. Examining figure 3a 

closely, it is seen that the actual location of the components are clearly evident in the contour 

plots. The attachment points for the chips, capacitors, and resistors show up as either isolated 

high or low stress areas at these points. 

6.1.3 Submodeling 

Submodeling is used to analyze a local region of an FE model with a refined mesh based on the 

interpolated results of the coarsely meshed global solution. Submodeling makes use of St. 

Venant’s Principle; thus, the submodeling results are accurate if both St. Venant’s Principle is 

followed and if the homogenized properties for the local model are reasonable. Both of these 

conditions are vague; consequently, the validity of a submodel is highly dependent upon the skill 

of the modeler. 

In the submodeling approach, a global model is developed and its solution determined. The 

displacements at each node that enclose the local model’s boundary are determined from the 

global solution. These nodal displacements are then interpolated and applied to the local model 

at the local model’s boundary as the forced response boundary conditions (2). The accuracy of 

the local model depends on how closely the global model captures the solution at the local 

model’s boundary. The computational savings is achieved by forcing element size consistency 

and by reducing the total number of elements in the global model. The success of this approach 

is highly dependent upon the determination of the homogenized properties used in the global 

analysis. 

7
 

(a) (b) 

(c) (d)

Figure 2. Comparison of axial displacement of baseline and lumped mass for a simple chip with components. 

(a) (b) 

Figure 3. Comparison of von Mises stress of baseline and lumped mass for a simple chip with components.

8
Once the global solution is determined, the nodal displacements enclosing the local-global 

boundary are interpolated and are used as a prescribed displacement boundary condition of the 

local model. In figure 4, solid black lines depict three out of many possible local-global 

boundaries for a submodeling analysis. For local-global boundary 2, the “x” symbols denote the 

nodes of which the global solution nodal displacements are interpolated to effectively drive the 

local solution. Note that the local model mesh does not need to be congruent with the global 

mesh. The local results for the three boundaries are seen in figure 5. It is seen that the error 

increases as the boundary moves into the region affected by the local structure. This is due to 

St. Venant’s Principle. 

Figure 4. Global model with three possible local-global boundaries. The “x” 

symbols denote the nodes from which results will be interpolated for 

boundary 2 and subsequently prescribed on the boundary of the local 

model to effectively drive the local solution. 

6.2 Overview of Mathematical Methods 

In general, an FE analytical model leads to a system of equations consisting of mass, damping, 

and stiffness matrices M, C, and K, respectively. Usually, the model is considered to have 

proportional or zero damping if a linear analysis is conducted. In a nonlinear analysis, advanced 

constitutive models are typically employed which account for material damping directly in the 

constitutive law. For simplicity, in the following discussion, the damping is set to zero. The 

resulting equations of motion are given by 

M x Kx f(t) . (1) ..

+ =

The dimensionality of the mass and stiffness matrices is N × N, where N is the length of the 

coordinate vector x, which represents the DOF of the model. 

x x x x x x x x x

x

x 

x 

x 

x

x 

x

x 

x 

x

x 

x 

x 

x

x 

x

x x x x x x x

x x

x 

x x x x 

x 

x x x x x x x x x

x

x 

x x x x x x x 

x 

x 

x 

x 

x 

3 Local Model Boundaries

1

2

3

9
(a) Global

(b) Local 

(c) Local

(d) Local Model

1 

2 

3 

Figure 5. (a) Global model with three boundaries, (b) local model results corresponding to boundary 1, (c) local

model results corresponding to boundary 2, and (d) local model results corresponding to boundary 3. 

Reduction techniques often utilize a projection matrix T, sometimes referred to as a condensation 

or expansion matrix, depending on its usage, to relate the retained DOF and the truncated DOF 

of the original model. To clarify, the original full set of N DOF, xn, is partitioned into the 

retained “r” DOF and the truncated “t” DOF. Thus, xn can be expressed as 

r . (2) t

r

n Tx

x

x

x =

⎭

⎬

⎫

⎩

⎨

⎧ =

The form and calculation of the projection matrix, T, depends on the reduction method 

implemented. Note that, whereas the global/local methods described in subsection 6.1 do not 

have clearly defined projection matrices, the methods in this section are clearly defined. For 

10
these methods, the reduced mass and stiffness matrices are found by pre- and postmultiplying by 

the projection matrix T, 

(3)

K T KT,

M T MT, and

T

rr

T

rr

=

=

where the reduced mass and stiffness matrices, Mrr and Krr, are of size r x r. 

The following methods have been used to generate representations of the local model in the 

global model: 

• substructuring, superelements, and guyan reduction 

• dynamic reduction 

• summed of weighted accelerations technique (SWAT) 

• system equivalent reduction-expansion process (SEREP) 

6.3 Superelements/Substructuring and Static Reduction 

Generally, substructuring is the process of decomposing a large FE model into smaller, 

component-based models. These component models are the substructures of the full system. A 

definitive distinction between superelements and substructures is difficult to define. The 

ABAQUS/Standard User’s Manual (3) makes the following delineation between substructures 

and superelements: a substructure retains its response data at internal DOF locations within the 

local model; whereas, a superelement does not. Therefore, the implementation of substructures 

and superelements are equivalent with the recording of results being the difference. Substructure 

will refer to both substructure and superelement for the remainder of this report 

Static reduction, sometimes referred to as Guyan reduction or static condensation, is the most 

popular reduction method, and is the basis for several FE substructuring techniques. This 

reduction method applied in FE techniques reduces the FE model by condensing (truncating) 

internal DOF. Specifically, the technique removes the DOF not located at the substructure’s 

boundary, the local-to-global interface. These remaining DOF, located at the boundary, retain 

the stiffness of the local structure, but omit the inertial terms to create a more compact and thus 

more efficient representation, at the cost of accuracy for non-static loading conditions. This 

discussion is similar to another overview paper discussing Guyan reduction, as well as the 

discussion for dynamic reduction, improved reduced system (IRS), and SEREP (4, 5), where the 

focus is modal estimation. The difference here is the application of these reduction techniques 

for efficient structural analysis rather than efficient modal analysis. 

The static reduction method partitions the DOF, force vectors, and the mass and stiffness 

matrices into subvectors and submatrices relating to the retained DOF, xr, and the truncated 

DOF, xt. Ignoring the inertial terms of equation 1, the partitioned stiffness matrix K is 

. (4) ⎭

⎬

⎫

⎩

⎨

⎧ =

⎭

⎬

⎫

⎩

⎨

⎧

⎥

⎦

⎤ ⎢

⎣

⎡

f (t)

f (t)

x

x

K K

K K

t

r

t

r

tr tt

rr rt

11
The assumption is then made that no external forces act on the truncated DOF, thus 

 . (5) ⎭

⎬

⎫

⎩

⎨

⎧ =

⎭

⎬

⎫

⎩

⎨

⎧

⎥

⎦

⎤ ⎢

⎣

⎡

0

f (t)

x

x

K K

K K r

t

r

tr tt

rr rt

Equation 5 can be solved for the truncated DOF; thus, 

 (6) 

x K K x .

K x K x 0, and

tr r

1

t tt

tr r tt t

− = −

+ =

From equation 6, the relationship between the retained DOF and the truncated DOF is 

established as 

⎥ , (7) ⎦

⎤ ⎢

⎣

⎡

− ⎥ =

⎦

⎤ ⎢

⎣

⎡ = −

tr

1

s tt

s K K

I

t

I

T

where Ts is the static reduction projection matrix and the subscript “s” denotes static 

condensation. Therefore, the Guyan reduced DOF vector, xn, as well as the system’s reduced 

mass and stiffness matrices, is as given in equations 2 and 3, respectively, where T is defined as 

Ts. 

This reduced representation is only exact for static events and the error increases as the effects of 

the inertial terms become more significant, e.g., as the event transitions from the static to the 

quasi-static and the dynamic time domains. 

By reducing the total number of elements, the FE model is more efficiently solved, but at the 

cost of losing high-fidelity data of nonstatic loading conditions. The solution’s fidelity decreases 

with decreasing loading periods, i.e., higher frequency content in the loading time history. 

Implementation of the static reduction method into a local-to-global technique thus leads to a 

qualitative analysis due to the reduction process that retains the stiffness of the local model and 

neglects the inertial terms.

6.4 Dynamic Reduction 

Dynamic reduction attempts to accurately represent the system for specific frequencies 

corresponding to the system’s eigenvalues (4, 6). For the undamped system equation, equation 

1, the dynamic reduction method uses the eigenvalue problem associated with free vibrations at 

frequency ω, so that 

[K ω M]x 0 . (8) 2 − =

Forming the dynamic stiffness matrix B(ω2

), 

B(ω ) [K ω M], (9) 2 2 = −

12
such that B(ω2

) x = 0. If B(ω2) is partitioned in the same manner as in static reduction, with 

retained DOF and truncated DOF, the characteristic equation is 

0. (10) 

x

x

B (ω ) B (ω )

B (ω ) B (ω )

t

r

2

tt

2

tr

2

rt

2

rr =

⎭

⎬

⎫

⎩

⎨

⎧

⎥

⎦

⎤ ⎢

⎣

⎡

The relationship between the retained and truncated DOF is given by 

. (11) B (ω ) B (ω )

I

t (ω )

I

T (ω ) 2

tr

2 1

tt

2

d

2

d ⎥

⎦

⎤ ⎢

⎣

⎡

− =⎥

⎦

⎤ ⎢

⎣

⎡ = −

The parameter ω2

 explicitly states that the projection, or transformation, matrix Td, and hence B, 

depend on ω2. The dynamic reduction process is only exact when ω2 equals one of the systems 

eigenvalues, Ω2

, and, consequently, is only exact at that specific eigenvalue. This is evident by 

realizing that equation 10 is an eigenvalue problem expressed in a partitioned form. Therefore, if 

ω2 ≠ Ω, then B(ω2

) x ≠ 0, and equation 10 is not satisfied. Additionally, notice that the dynamic 

reduction method maps itself into the static reduction method when ω2

 = 0. Lastly, the 

dynamically reduced DOF vector, xn, as well as the system’s reduced mass and stiffness 

matrices, is as given in equations 2 and 3, respectively, where T is defined as Td. 

In addition to the dynamic reduction method described above, O’Callaghan introduced the IRS 

method (7) as an approach for improving the static reduction method by including dynamic 

effects. This method considers the inertial terms as so called quasi-static forces. Thus, the IRS 

method is an attempt to represent the low frequency response of the full system more accurately 

than the static reduction method. Accuracy of this approach decreases as the excitation 

frequency increases beyond the first few structural resonances. The IRS method works well for 

low frequency response, but loses accuracy as the inertial effects become more significant. This 

is equivalent to assuming that the higher frequency response is unimportant (5). This is not the 

case for gun dynamics and other high-g events; therefore, the IRS method will not be considered 

further. 

6.5 SWAT 

Developed at Sandia National Laboratory, SWAT (8, 9) is a force reconstruction technique that 

utilizes the sums of the mass-scaled acceleration data. Although SWAT was not developed for 

FE model reduction, it can be implemented as such by replacing the local model with a SWATdefined force distributed over the local-to-global interface. The SWAT weighting coefficients 

are determined from the inverse modal matrix of the system with free boundary conditions, a 

least squares solution, or from measured frequency response function data. This technique 

calculates the net force and moment, effectively acting at the selected generalized DOF’s center 

of mass. The spatial distribution of the forces and moments over the structure is not known and 

is therefore not unique. Although this technique yields non-unique results, for a model 

13
containing a single local-to-global interface, this procedure can be used to determine 

approximate load transmissibility, including dynamic effects. As with the other reduction 

methods, SWAT starts from the system’s equations of motion in a generalized coordinate 

system. For an undamped system, the equations of motion are again given by equation 1. The 

modal matrix U and natural frequencies ω are determined from modal analysis. The coordinate 

transformation 

x(t) = Uq(t) (12)

defines the mapping of the generalized coordinates to physical coordinates. U includes both 

rigid body modes and elastic modes. Substitution of this coordinate transform into equation 1 

with the premultiplication by a single rigid body mode Urigid yields 

U MU q U KUq U f(t). (13) T

rigid

T

rigid

.. T

rigid + =

In rigid body modes, no elastic deformation exists; therefore,

KUrigid = 0 or U K 0. (14) T

rigid =

Substituting equation 15 into equation 14 yields 

m q (t) U f(t), (15) T

rigid rigid

..

rigid =

where mrigid is 1 x 1 due to orthogonality of eigenvectors and mrigid represents the mass 

associated with the rigid modal coordinate qrigid. From equation 15, it is apparent that the 

external net force applied to the system can be determined by the rigid body modal coordinate 

acceleration-time histories, which is simply Newton’s Law. The goal is therefore to determine a 

specific modal coordinate acceleration-time history from the sum of the generalized coordinate 

acceleration-time histories. Equation 12 and equation 15 provide insight for the sequence of 

steps required to reach this goal. To determine the SWAT force, the rigid modal coordinate time 

history needs to be developed from the vector of generalized acceleration-time histories. Modal 

analysis states that the acceleration-time response, a(t), can be estimated by a sum of vectors. If 

the free modes are used as the vectors, then the approximation will be accurate up to a limiting 

frequency dependent on the number of modes used in the estimation, hence the mode’s 

eigenvalues. The acceleration can be expressed in modal coordinates as

a(t) Ψq(t), (16) ..

=

where ψ is the retained free modes used in the approximation of a(t) using the modal coordinate 

acceleration vector q(t)

..

. A SWAT DOF is defined as 

s(t) w a(t), (17) T =

14
where w is a vector of to be defined weights and a(t) is the measured set of accelerations. Thus, 

equation 17 becomes 

s(t) w Ψq(t). (18) .. T =

Next, the weights are chosen so that s(t) will be a modal coordinate, e.g., wT

ψ = [1 0 ... 0], 

resulting in s(t) = . Similarly, the weight vector needed to extract any modal coordinate 

from acceleration-time histories can be determined by solving 

q (t) 1

..

 or (19) W Ψ I T = Ψ W I, T =

where I is the identity matrix and W is a matrix of which its jth column is the jth weight vector, 

wj, for the jth modal coordinate. The weight vectors computed in equation 19 are referred to as

the reciprocal modal vectors. In summary, once the desired modal coordinate response has been 

determined, multiplication by the modal mass for the corresponding modal coordinate yields the 

net externally applied force. 

SWAT, if implemented as a global-to-local FE reduction method, has several advantageous 

features: 

• allows use of a SWAT reconstructed force to determine an accurate approximation of the 

local loading, 

• can be used to check an alternate method’s loading on local structure, 

• can be used to determine whether or not local response effects global response, and 

• can convert experimental acceleration data to numerical force data. 

SWAT would be most useful in a single local-to-global interface case for which the local loading 

is applied from a single point, line, or plane. The SWAT reconstructed force based on the global 

nodal accelerations would be able to determine the exact local loading force acting through a 

point, and the net local loading force acting through a line or a plane. If the loading interface 

were a line or plane, then the net SWAT force could be partitioned by observing the weighting 

factors of the elastic modes (7) or by dividing the net force equally by the number of loading 

points contained in the line or plane. Although not exact, this method does capture the dynamics 

of the global structure. Thus, the accuracy of the local loading magnitude is adequate, but the 

distribution of this local force is not. SWAT is not a reduction technique by condensation, but 

rather a reduction technique by substitution. SWAT substitutes the local model with an 

approximately equivalent, time-dependant loading at the local-to-global interface, thereby 

simultaneously reducing the number of DOF in the FE model and increasing the characteristic 

time-step. The disadvantage to SWAT is that accurate acceleration-time histories need to exist a 

priori for SWAT to be utilized. Additionally, for small electronic packages, these accelerations 

may be difficult or impossible to measure accurately. 

15
6.6 SEREP 

SEREP is an attractive approach for FE global/local system reduction because only the DOF 

located at the local-to-global interface need to be retained, and the method will maintain an 

accurate representation of the local system. This is advantageous since, like SWAT, the local 

structure in the global model can then be replaced by an equivalent set of force-time history 

functions obtained by the reduction process. Also, this method can be modified so that the local 

model can be completely replaced with a simpler, equivalent representation. Additionally, the 

local FE model does not need to be developed to get the equivalent set of force-time histories, 

since a modal model can be experimentally measured to determine the set of equivalent forces. 

The process is depicted in figure 6. 

Experimental Modal

Analysis

Finite Element Modal 

Analysis

Analytical Modal Analysis

Prototype Local

Structure 

Set of Equivalent

Forces

Reduction Sequence in SEREP

SEREP

Figure 6. Local-to-global model reduction process for SEREP. 

Like the prior reduction techniques, excluding SWAT (which was not initially formulated as a 

reduction technique), SEREP partitions its mass, damping, and stiffness matrices into retained 

and truncated DOF. In an analytical analysis, the number of modes that can be determined 

equals the number of DOF. Conversely, in FE analysis, it is typical to have many more DOF 

than captured modes and thus n>>m, where n is the number of DOF and m is the number of 

modes. In both cases, however, the desire is to reduce the size of the system to include only the 

i modes and the j DOF that are necessary to describe the system in the frequency regime of 

interest. 

The derivation of SEREP is fully developed in references (4, 10). Once again, the generalized 

coordinate vector x is partitioned into two parts, xr and xt, the retained and truncated coordinates, 

respectively. Neglecting damping, the mass and stiffness matrices are then reorganized such that 

 . (20) ⎭

⎬

⎫

⎩

⎨

⎧ =

⎭

⎬

⎫

⎩

⎨

⎧

⎥

⎦

⎤ ⎢

⎣

⎡

+

⎪⎭

⎪

⎬

⎫

⎪⎩

⎪

⎨

⎧

⎥

⎦

⎤ ⎢

⎣

⎡

f (t)

f (t)

x

x

K K

K K

x

x

M M

M M

t

r

t

r

tr tt

rr rt

t

..

r

..

tr tt

rr rt

16
The eigensolution of this system generates the eigenvalues Ω2

 and the corresponding 

eigenvectors U. The coordinate transformation from physical coordinates to modal coordinates 

is defined as 

x = Uq. (21)

Typically, the calculated modal matrix is a subset of the theoretical full modal matrix, and the 

coordinate transform becomes 

 , x = Uacqc (22)

where the Uac, the reduced modal matrix, denotes the ‘c’ captured modes for ‘a’ all generalized 

coordinate DOF, and qc is c × 1. The modal response of the reduced system is directly related to 

which modes are included. For high-g loads, it is normally desirable to generate a reduced 

model that is representative up to some upper-frequency limit determined by the frequency 

content of the anticipated loading. Using the reduced modal matrix, the physical coordinate 

vector is partitioned as 

q . (23) U

U

x

x

x c

tc

rc

t

r

⎥

⎦

⎤ ⎢

⎣

⎡ =

⎭

⎬

⎫

⎩

⎨

⎧ =

Expanding the equation regarding the retained DOF, 

x U q . r = rc c (24) 

Solving for qc requires the generalized inverse of the possibly nonsquare matrix Urc; 

U (U U ) U , (25) T

rc

1

rc

T

rc rc

+ − =

where + represents that the generalized inverse is taken. Solving for qc, 

q U x . (26) c rc r

+ =

Substitution of equation 26 into equation 23 gives the relationship between the retained DOF and 

the full set of DOF and also provides the definition of the SEREP projection matrix, Tu, via 

 . x U U x T x (27) = ac rc r = u r

+

The eigensolution of the reduced mass and stiffness matrices, Mrr and Krr, respectively, produce 

the reduced eigenvalues R

Ω2 and eigenvectors RU, where the presuperscript “R” denotes reduced. 

The modal transformation of equation 24 is still valid while 

 . U U (28) rc

R =

The modal matrix, , of the full system is obtainable from Un

U T U . (29) R

n = u

17
Equation 29 is important because it implies that the SEREP is reversible and exact in the sense 

that the full eigensolution for all the DOF can be obtained by solving the reduced eigenproblem. 

Using the transformation given by equation 27, the SEREP-reduced mass and stiffness matrices 

can be written as 

 (30) 

K U U ,

M U U , and

rr rc rc

rr rc rc

+ +

+ +

= Λ

=

T

T

where Λ = Ω2

. 

These reduced matrices can be used directly in an FE analysis since they are in terms of the 

physical coordinates. With this method, the dynamic information which is omitted in the FE 

analysis depends on the number of modes captured in the modal analysis and the number of 

interface DOF. Usually, the limiting parameter, especially for a distributed system, will be the 

number of captured modes during the modal analysis of the local structure. Also, the reduced 

mass matrix is typically a full matrix requiring a matrix inversion technique; consequently, the 

solution is no longer explicit. However, the method allows such a significant reduction in the 

number of DOF that must be retained for accurately representing the dynamic response that the 

increase in computational costs associated with the implicit solution technique may be 

acceptable. The SEREP reduction technique has several features that are beneficial to an FE 

analysis: 

• the mass and stiffness matrices are in physical coordinates, 

• the expansion to the full matrices (the inverse mapping) is exact, 

• the resultant equations capture the exact response of the full system for every natural 

frequency that is retained in the reduction process, and 

• the method allows the use of either analytical or experimentally derived modal 

information. 

However, it is worth noting that, because of the reduction process, equation 30a is generally 

singular. This presents some numerical issues that must be addressed in any implementation of 

the SEREP method as a general reduction method. 

7. Summary 

Several model reduction techniques have been introduced and briefly discussed along with their 

advantages and disadvantages. It has been determined that SEREP is the appropriate reduction 

techniques to be implemented in the FE method in order to reduce complex models subjected to 

dynamic loads to smaller, simpler representations while maintaining increased accuracy, as 

compared with other reduction techniques. Also SEREP, due to its modal based implementation, 

has the potential of being the most accurate if all modes of the local system are determined. 

18
8. References 

1. Wilkerson, S.; Hopkins, D.; Gazonas, G.; Berman, M. Developing a Transient Finite Element 

Model to Simulate the Launch Environment of the 155-mm SADARM Projectile; ARL-TR2341; U. S. Army Research Laboratory: Aberdeen Proving Ground, MD, September 2000. 

2. Submodeling. ABAQUS/Standard User’s Manual, Volume I, Version 6.1; Hibbit, Karlsson & 

Sorensen, Inc., 2000; 7.3.1. 

3. Using Substructuring/Superelements. ABAQUS/Standard User’s Manual, Vol. I, Version 6.1; 

Hibbit, Karlsson & Sorensen, Inc., 2000; 7.2.1. 

4. Avitabile, P.; O’Callahan, J.; Milani, J. Comparison of System Characteristics Using Various 

Model Reduction Techniques. Seventh International Modal Analysis Conference, Las Vegas, 

NV, February 1989. 

5. Avitabile, P.; O’Callahan, J. Effects of Various Model Reduction Techniques on Computed 

System Response. Seventh International Modal Analysis Conference, Las Vegas, NV, 

February 1989. 

6. Friswell, M. I.; Garvey, S. D.; Penny, J. E. T. Model Reduction Using Dynamic and Iterated 

IRS Techniques. Journal of Sound and Vibration 1995, 186 (2), 311-323. 

7. O’Callahan, J. A Procedure for an Improved Reduced System (IRS) Model. Seventh 

International Modal Analysis Conference, Las Vegas, NV, February 1989. 

8. Carne, T. G.; Bateman, V. I.; Dohrmann, C. R. Force Reconstruction Using the Inverse of the 

Mode-Shape Matrix. Thirteenth Biennial ASME Vibration Conference, Miami, FL, 

September 1991. 

9. Carne, T. G.; Bateman, V. I.; Mayes, R. L. Force Reconstruction Using a Sum of Weighted 

Accelerations Technique. Proceedings of the 10th International Modal Analysis Conference, 

February 1992. 

10. O’Callahan, J.; Avitable, P.; Riemer, R. System Equivalent Reduction Exapansion Process 

(SEREP). Seventh International Modal Analysis Conference, Las Vegas, NV, February 

1989.

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SOLDIER SYSTEMS CENTER 

 BALLISTICS TEAM 

 J WARD 

 W ZUKAS 

 P CUNNIFF 

 J SONG

MARINE CORPS TEAM 

 J MACKIEWICZ 

BUS AREA ADVOCACY TEAM

 W HASKELL 

 AMSSB RCP SS 

 W NYKVIST 

 S BEAUDOIN 

KANSAS ST 

 NATICK MA 01760-5019

7 US ARMY RESEARCH OFC

 A CROWSON 

 H EVERETT 

 J PRATER 

 G ANDERSON

 D STEPP 

 D KISEROW 

 J CHANG 

PO BOX 12211

RESEARCH TRIANGLE PARK NC 

 27709-2211

 1 AFRL MLBC 

2941 P ST RM 136

WRIGHT PATTERSON AFB OH

 45433-7750

 1 DIRECTOR 

LOS ALAMOS NATL LAB 

F L ADDESSIO T 3 MS 5000

PO BOX 1633 

LOS ALAMOS NM 87545

 8 NSWC

J FRANCIS CODE G30 

D WILSON CODE G32 

R D COOPER CODE G32 

J FRAYSSE CODE G33

E ROWE CODE G33 

T DURAN CODE G33 

L DE SIMONE CODE G33 

R HUBBARD CODE G33 

 DAHLGREN VA 22448

 1 NSWC

 CARDEROCK DIVISION 

R CRANE CODE 6553

9500 MACARTHUR BLVD 

WEST BETHESDA MD 20817-5700

 1 AFRL MLSS 

 R THOMSON 

2179 12TH ST RM 122 

WRIGHT PATTERSON AFB OH

 45433-7718

 2 AFRL 

 F ABRAMS 

 J BROWN 

 BLDG 653

2977 P ST STE 6 

WRIGHT PATTERSON AFB OH

 45433-7739

 5 DIRECTOR 

 LLNL 

 R CHRISTENSEN 

 S DETERESA 

 F MAGNESS 

M FINGER MS 313

M MURPHY L 282

PO BOX 808

 LIVERMORE CA 94550

 1 AFRL MLS OL 

 L COULTER 

5851 F AVE 

BLDG 849 RM AD1A

HILL AFB UT 84056-5713 

 1 OSD 

JOINT CCD TEST FORCE 

 OSD JCCD 

 R WILLIAMS

3909 HALLS FERRY RD

 VICKSBURG MS 29180-6199 

25
NO. OF NO. OF

COPIES ORGANIZATION COPIES ORGANIZATION

 3 DARPA

 M VANFOSSEN 

 S WAX 

 L CHRISTODOULOU

3701 N FAIRFAX DR

 ARLINGTON VA 22203-1714 

2 SERDP PROGRAM OFC 

 PM P2 

 C PELLERIN 

 B SMITH 

901 N STUART ST STE 303 

 ARLINGTON VA 22203

1 OAK RIDGE NATL LAB 

R M DAVIS 

PO BOX 2008 

OAK RIDGE TN 37831-6195

1 OAK RIDGE NATL LAB 

C EBERLE MS 8048 

PO BOX 2008 

OAK RIDGE TN 37831

 3 DIRECTOR 

SANDIA NATL LABS

 APPLIED MECHS DEPT 

 MS 9042

 J HANDROCK

 Y R KAN 

 J LAUFFER 

PO BOX 969

 LIVERMORE CA 94551-0969

1 OAK RIDGE NATL LAB 

C D WARREN MS 8039

PO BOX 2008 

OAK RIDGE TN 37831

 4 NIST 

M VANLANDINGHAM MS 8621

J CHIN MS 8621 

J MARTIN MS 8621 

D DUTHINH MS 8611

 100 BUREAU DR 

 GAITHERSBURG MD 20899 

 1 HYDROGEOLOGIC INC 

SERDP ESTCP SPT OFC 

 S WALSH 

1155 HERNDON PKWY STE 900

 HERNDON VA 20170

3 NASA LANGLEY RESEARCH CTR 

 AMSRD ARL VS 

W ELBER MS 266 

F BARTLETT JR MS 266

G FARLEY MS 266

HAMPTON VA 23681-0001 

1 NASA LANGLEY RESEARCH CTR 

T GATES MS 188E 

HAMPTON VA 23661-3400 

 1 FHWA

 E MUNLEY

 6300 GEORGETOWN PIKE

MCLEAN VA 22101

1 USDOT FEDERAL RAILROAD 

M FATEH RDV 31 

 WASHINGTON DC 20590

 3 CYTEC FIBERITE 

 R DUNNE 

 D KOHLI 

 R MAYHEW 

1300 REVOLUTION ST 

HAVRE DE GRACE MD 21078

 1 DIRECTOR 

 NGIC 

 IANG TMT 

2055 BOULDERS RD 

 CHARLOTTESVILLE VA 

 22911-8318

 1 SIOUX MFG

 B KRIEL 

PO BOX 400

FT TOTTEN ND 58335

 2 3TEX CORP 

 A BOGDANOVICH

 J SINGLETARY

109 MACKENAN DR

CARY NC 27511 

 1 3M CORP 

 J SKILDUM

3M CENTER BLDG 60 IN 01

ST PAUL MN 55144-1000

26
NO. OF NO. OF

COPIES ORGANIZATION COPIES ORGANIZATION

 1 DIRECTOR 

 DEFENSE INTLLGNC AGNCY 

 TA 5 

 K CRELLING 

 WASHINGTON DC 20310

1 ADVANCED GLASS FIBER YARNS 

 T COLLINS 

281 SPRING RUN LANE STE A 

 DOWNINGTON PA 19335 

1 COMPOSITE MATERIALS INC 

 D SHORTT 

19105 63 AVE NE

PO BOX 25 

 ARLINGTON WA 98223

 1 JPS GLASS 

 L CARTER 

PO BOX 260

 SLATER RD 

 SLATER SC 29683

1 COMPOSITE MATERIALS INC 

 R HOLLAND 

11 JEWEL CT

 ORINDA CA 94563 

1 COMPOSITE MATERIALS INC 

 C RILEY 

14530 S ANSON AVE 

SANTA FE SPRINGS CA 90670 

 2 SIMULA

 J COLTMAN 

 R HUYETT

10016 S 51ST ST 

 PHOENIX AZ 85044

2 PROTECTION MATERIALS INC 

 M MILLER 

 F CRILLEY 

14000 NW 58 CT 

MIAMI LAKES FL 33014

 2 FOSTER MILLER 

 M ROYLANCE

 W ZUKAS 

195 BEAR HILL RD

 WALTHAM MA 02354-1196

1 ROM DEVELOPMENT CORP 

R O MEARA 

 136 SWINEBURNE ROW 

 BRICK MARKET PLACE 

NEWPORT RI 02840

 2 TEXTRON SYSTEMS 

 T FOLTZ 

 M TREASURE

 1449 MIDDLESEX ST 

LOWELL MA 01851 

1 O GARA HESS & EISENHARDT 

 M GILLESPIE

9113 LESAINT DR 

FAIRFIELD OH 45014 

2 MILLIKEN RESEARCH CORP 

 H KUHN

 M MACLEOD 

PO BOX 1926 

 SPARTANBURG SC 29303

1 CONNEAUGHT INDUSTRIES INC 

 J SANTOS 

PO BOX 1425 

 COVENTRY RI 02816

 1 ARMTEC DEFENSE PRODUCTS 

 S DYER 

85 901 AVE 53

PO BOX 848

 COACHELLA CA 92236

1 NATL COMPOSITE CTR 

 T CORDELL 

2000 COMPOSITE DR

 KETTERING OH 45420 

3 PACIFIC NORTHWEST LAB 

 M SMITH

G VAN ARSDALE 

 R SHIPPELL 

PO BOX 999

 RICHLAND WA 99352

 1 SAIC

 M PALMER 

1410 SPRING HILL RD STE 400 

 MS SH4 5 

MCLEAN VA 22102 

27
NO. OF NO. OF

COPIES ORGANIZATION COPIES ORGANIZATION

1 ALLIANT TECHSYSTEMS INC 

4700 NATHAN LN N 

 PLYMOUTH MN 55442-2512

 1 APPLIED COMPOSITES 

 W GRISCH

333 NORTH SIXTH ST 

ST CHARLES IL 60174

 1 CUSTOM ANALYTICAL 

ENG SYS INC 

 A ALEXANDER 

13000 TENSOR LANE NE 

 FLINTSTONE MD 21530 

 1 AAI CORP 

DR N B MCNELLIS 

PO BOX 126

HUNT VALLEY MD 21030-0126

1 OFC DEPUTY UNDER SEC DEFNS 

 J THOMPSON 

1745 JEFFERSON DAVIS HWY 

CRYSTAL SQ 4 STE 501 

 ARLINGTON VA 22202

3 ALLIANT TECHSYSTEMS INC 

 J CONDON

 E LYNAM 

 J GERHARD 

WV01 16 STATE RT 956 

PO BOX 210

ROCKET CENTER WV 

 26726-0210

 1 PROJECTILE TECHNOLOGY INC 

 515 GILES ST

HAVRE DE GRACE MD 21078

 1 HEXCEL INC

 R BOE 

PO BOX 18748

SALT LAKE CITY UT 84118 

 1 PRATT & WHITNEY 

 C WATSON 

400 MAIN ST MS 114 37

EAST HARTFORD CT 06108

 5 NORTHROP GRUMMAN 

 B IRWIN 

 K EVANS 

 D EWART 

 A SHREKENHAMER 

 J MCGLYNN

BLDG 160 DEPT 3700

1100 WEST HOLLYVALE ST

AZUSA CA 91701

 1 HERCULES INC 

 HERCULES PLAZA 

WILMINGTON DE 19894

 1 BRIGS COMPANY 

 J BACKOFEN 

2668 PETERBOROUGH ST 

 HERNDON VA 22071-2443 

1 ZERNOW TECHNICAL SERVICES 

 L ZERNOW 

425 W BONITA AVE STE 208 

SAN DIMAS CA 91773

1 GENERAL DYNAMICS OTS 

 L WHITMORE

10101 NINTH ST NORTH

ST PETERSBURG FL 33702 

2 GENERAL DYNAMICS OTS 

 FLINCHBAUGH DIV

 K LINDE 

 T LYNCH 

PO BOX 127

RED LION PA 17356 

1 GKN WESTLAND AEROSPACE 

 D OLDS

 450 MURDOCK AVE 

MERIDEN CT 06450-8324

 2 BOEING ROTORCRAFT 

 P MINGURT 

 P HANDEL 

800 B PUTNAM BLVD 

 WALLINGFORD PA 19086 

28
NO. OF NO. OF

COPIES ORGANIZATION COPIES ORGANIZATION

 5 SIKORSKY AIRCRAFT 

 G JACARUSO

 T CARSTENSAN 

 B KAY 

S GARBO MS S330A 

 J ADELMANN 

6900 MAIN ST 

PO BOX 9729 

 STRATFORD CT 06497-9729

 1 AEROSPACE CORP 

G HAWKINS M4 945

2350 E EL SEGUNDO BLVD

EL SEGUNDO CA 90245

 2 CYTEC FIBERITE 

 M LIN

 W WEB 

1440 N KRAEMER BLVD

 ANAHEIM CA 92806 

 2 UDLP 

 G THOMAS 

 M MACLEAN 

PO BOX 58123

SANTA CLARA CA 95052 

1 UDLP WARREN OFC 

A LEE 

31201 CHICAGO RD SOUTH

 SUITE B102 

 WARREN MI 48093 

 2 UDLP 

 R BRYNSVOLD 

P JANKE MS 170 

4800 EAST RIVER RD 

 MINNEAPOLIS MN 55421-1498

 1 LOCKHEED MARTIN 

 SKUNK WORKS 

 D FORTNEY 

 1011 LOCKHEED WAY

 PALMDALE CA 93599-2502

 1 LOCKHEED MARTIN 

 R FIELDS 

5537 PGA BLVD 

 SUITE 4516

ORLANDO FL 32839

1 NORTHRUP GRUMMAN CORP 

 ELECTRONIC SENSORS 

 & SYSTEMS DIV

E SCHOCH MS V 16

1745A W NURSERY RD

 LINTHICUM MD 21090

 1 GDLS DIVISION 

 D BARTLE 

PO BOX 1901 

 WARREN MI 48090 

 2 GDLS 

 D REES

 M PASIK

PO BOX 2074 

 WARREN MI 48090-2074

 1 GDLS 

 MUSKEGON OPER 

 M SOIMAR

76 GETTY ST

 MUSKEGON MI 49442

 1 GENERAL DYNAMICS 

 AMPHIBIOUS SYS 

 SURVIVABILITY LEAD 

 G WALKER 

991 ANNAPOLIS WAY 

 WOODBRIDGE VA 22191 

 6 INST FOR ADVANCED

 TECH 

 H FAIR 

 I MCNAB 

 P SULLIVAN 

 S BLESS 

 W REINECKE

 C PERSAD 

3925 W BRAKER LN STE 400

 AUSTIN TX 78759-5316

 1 ARROW TECH ASSOC 

1233 SHELBURNE RD STE D8 

SOUTH BURLINGTON VT

 05403-7700

 1 R EICHELBERGER 

 CONSULTANT

409 W CATHERINE ST 

BEL AIR MD 21014-3613

29
NO. OF NO. OF

COPIES ORGANIZATION COPIES ORGANIZATION

1 SAIC

 G CHRYSSOMALLIS 

8500 NORMANDALE LAKE BLVD

 SUITE 1610

 BLOOMINGTON MN 55437-3828

1 UCLA MANE DEPT ENGR IV 

H T HAHN

LOS ANGELES CA 90024-1597 

 2 UNIV OF DAYTON

 RESEARCH INST 

 R Y KIM

 A K ROY 

300 COLLEGE PARK AVE

DAYTON OH 45469-0168 

1 UMASS LOWELL 

 PLASTICS DEPT 

 N SCHOTT 

1 UNIVERSITY AVE 

LOWELL MA 01854 

1 IIT RESEARCH CTR 

D ROSE 

 201 MILL ST 

 ROME NY 13440-6916

1 GA TECH RESEARCH INST 

GA INST OF TCHNLGY 

 P FRIEDERICH

 ATLANTA GA 30392

1 MICHIGAN ST UNIV

 MSM DEPT 

 R AVERILL 

 3515 EB 

EAST LANSING MI 48824-1226

 1 UNIV OF WYOMING 

 D ADAMS 

PO BOX 3295 

LARAMIE WY 82071 

1 PENN STATE UNIV

R S ENGEL 

 245 HAMMOND BLDG

UNIVERSITY PARK PA 16801

2 PENN STATE UNIV

 R MCNITT 

 C BAKIS 

 212 EARTH ENGR 

 SCIENCES BLDG 

UNIVERSITY PARK PA 16802

 1 PURDUE UNIV

SCHOOL OF AERO & ASTRO 

C T SUN 

W LAFAYETTE IN 47907-1282

 1 STANFORD UNIV

 DEPT OF AERONAUTICS 

 & AEROBALLISTICS

 S TSAI 

 DURANT BLDG 

 STANFORD CA 94305

1 UNIV OF MAINE 

ADV STR & COMP LAB 

 R LOPEZ ANIDO 

5793 AEWC BLDG 

 ORONO ME 04469-5793

1 JOHNS HOPKINS UNIV

APPLIED PHYSICS LAB 

 P WIENHOLD 

11100 JOHNS HOPKINS RD

 LAUREL MD 20723-6099

 1 UNIV OF DAYTON

 J M WHITNEY 

 COLLEGE PARK AVE 

DAYTON OH 45469-0240 

1 NORTH CAROLINA ST UNIV

 CIVIL ENGINEERING DEPT

 W RASDORF

PO BOX 7908 

 RALEIGH NC 27696-7908

 5 UNIV OF DELAWARE 

CTR FOR COMPOSITE MTRLS 

 J GILLESPIE 

 M SANTARE

 S YARLAGADDA

 S ADVANI

 D HEIDER 

 201 SPENCER LAB

 NEWARK DE 19716

30
NO. OF NO. OF

COPIES ORGANIZATION COPIES ORGANIZATION

1 DEPT OF MTRLS 

SCIENCE & ENGRG 

 UNIV OF ILLINOIS

AT URBANA CHAMPAIGN

 J ECONOMY

1304 WEST GREEN ST 115B 

 URBANA IL 61801 

1 UNIV OF MARYLAND

DEPT OF AEROSPACE ENGRG 

 A J VIZZINI

COLLEGE PARK MD 20742 

 1 DREXEL UNIV 

A S D WANG

3141 CHESTNUT ST 

 PHILADELPHIA PA 19104 

3 UNIV OF TEXAS AT AUSTIN 

 CTR FOR ELECTROMECHANICS

 J PRICE 

 A WALLS 

 J KITZMILLER 

10100 BURNET RD

 AUSTIN TX 78758-4497

 3 VA POLYTECHNICAL 

INST & STATE UNIV

DEPT OF ESM

M W HYER 

 K REIFSNIDER

 R JONES 

 BLACKSBURG VA 24061-0219 

 1 SOUTHWEST RESEARCH INST 

ENGR & MATL SCIENCES DIV 

 J RIEGEL 

6220 CULEBRA RD 

PO DRAWER 28510 

SAN ANTONIO TX 78228-0510

1 BATELLE NATICK OPERS

 B HALPIN 

313 SPEEN ST

 NATICK MA 01760 

 3 DIRECTOR 

US ARMY RESEARCH LAB 

AMSRD ARL WM MB

 A FRYDMAN

2800 POWDER MILL RD

ADELPHI MD 20783-1197 

ABERDEEN PROVING GROUND

 1 US ARMY ATC 

CSTE DTC AT AC I 

 W C FRAZER

400 COLLERAN RD 

APG MD 21005-5059

 91 DIR USARL 

 AMSRD ARL CI 

AMSRD ARL O AP EG

 M ADAMSON 

AMSRD ARL SL BA 

AMSRD ARL SL BB 

 D BELY

 AMSRD ARL WM

 J SMITH 

 H WALLACE

AMSRD ARL WM B 

 A HORST

 T KOGLER 

AMSRD ARL WM BA 

 D LYON

AMSRD ARL WM BC 

 J NEWILL 

 P PLOSTINS 

 A ZIELINSKI

AMSRD ARL WM BD 

 P CONROY 

 B FORCH 

 M LEADORE

 C LEVERITT

 R LIEB 

R PESCE RODRIGUEZ 

 B RICE 

AMSRD ARL WM BF 

 S WILKERSON

AMSRD ARL WM M

 B FINK 

 J MCCAULEY

AMSRD ARL WM MA

 L GHIORSE 

 S MCKNIGHT 

 E WETZEL 

AMSRD ARL WM MB

 J BENDER 

 T BOGETTI 

 L BURTON 

 R CARTER 

 K CHO 

 W DE ROSSET

 G DEWING 

 R DOWDING 

 W DRYSDALE

 R EMERSON 

 D HENRY 

31
NO. OF NO. OF

COPIES ORGANIZATION COPIES ORGANIZATION

 D HOPKINS 

 R KASTE 

 L KECSKES 

 M MINNICINO

 B POWERS 

 D SNOHA 

 J SOUTH 

 M STAKER 

 J SWAB 

 J TZENG 

AMSRD ARL WM MC

 J BEATTY 

 R BOSSOLI 

 E CHIN 

 S CORNELISON 

 D GRANVILLE 

 B HART 

 J LASALVIA 

 J MONTGOMERY 

 F PIERCE 

 E RIGAS 

 W SPURGEON

AMSRD ARL WM MD

 B CHEESEMAN 

 P DEHMER 

 R DOOLEY

 G GAZONAS

 S GHIORSE 

 C HOPPEL 

 M KLUSEWITZ 

 W ROY 

 J SANDS 

 D SPAGNUOLO

 S WALSH 

 S WOLF 

AMSRD ARL WM RP 

 J BORNSTEIN 

 C SHOEMAKER 

AMSRD ARL WM T 

 B BURNS 

AMSRD ARL WM TA

 W BRUCHEY 

 M BURKINS 

 W GILLICH 

 B GOOCH 

 T HAVEL 

 E HORWATH 

 M NORMANDIA

 J RUNYEON

 M ZOLTOSKI 

AMSRD ARL WM TB 

 P BAKER 

AMSRD ARL WM TC 

 R COATES 

AMSRD ARL WM TD

 D DANDEKAR 

 T HADUCH

 T MOYNIHAN

 M RAFTENBERG 

 S SCHOENFELD 

 T WEERASOORIYA 

AMSRD ARL WM TE 

 A NIILER 

 J POWELL 

32
NO. OF NO. OF

COPIES ORGANIZATION COPIES ORGANIZATION

1 LTD 

 R MARTIN 

 MERL 

 TAMWORTH RD 

HERTFORD SG13 7DG 

 UK 

 1 SMC SCOTLAND 

 P W LAY 

 DERA ROSYTH 

ROSYTH ROYAL DOCKYARD 

DUNFERMLINE FIFE KY 11 2XR 

 UK 

 1 CIVIL AVIATION

 ADMINSTRATION

 T GOTTESMAN 

PO BOX 8 

BEN GURION INTRNL AIRPORT 

 LOD 70150

 ISRAEL

 1 AEROSPATIALE 

 S ANDRE 

A BTE CC RTE MD132

316 ROUTE DE BAYONNE 

 TOULOUSE 31060

 FRANCE 

 1 DRA FORT HALSTEAD

P N JONES 

SEVEN OAKS KENT TN 147BP 

 UK 

1 SWISS FEDERAL ARMAMENTS 

 WKS 

 W LANZ 

 ALLMENDSTRASSE 86

 3602 THUN

 SWITZERLAND 

1 DYNAMEC RESEARCH LAB

 AKE PERSSON 

 BOX 201 

SE 151 23 SODERTALJE 

 SWEDEN

1 ISRAEL INST OF TECHLGY

 S BODNER 

FACULTY OF MECHANICAL 

 ENGR 

 HAIFA 3200

 ISRAEL

 1 DSTO

WEAPONS SYSTEMS DIVISION

 N BURMAN RLLWS 

 SALISBURY 

 SOUTH AUSTRALIA 5108 

 AUSTRALIA 

1 DEF RES ESTABLISHMENT 

 VALCARTIER 

 A DUPUIS 

2459 BLVD PIE XI NORTH

 VALCARTIER QUEBEC 

 CANADA

PO BOX 8800 COURCELETTE 

 GOA IRO QUEBEC 

 CANADA

 1 ECOLE POLYTECH 

 J MANSON 

 DMX LTC 

CH 1015 LAUSANNE

 SWITZERLAND 

1 TNO DEFENSE RESEARCH 

 R IJSSELSTEIN 

ACCOUNT DIRECTOR 

 R&D ARMEE

PO BOX 6006 

 2600 JA DELFT 

 THE NETHERLANDS 

 2 FOA NATL DEFENSE RESEARCH

 ESTAB 

DIR DEPT OF WEAPONS &

 PROTECTION 

 B JANZON 

 R HOLMLIN 

S 172 90 STOCKHOLM 

 SWEDEN

33
NO. OF

COPIES ORGANIZATION 

2 DEFENSE TECH & PROC 

 AGENCY GROUND

 I CREWTHER

GENERAL HERZOG HAUS 

 3602 THUN

 SWITZERLAND 

1 MINISTRY OF DEFENCE 

 RAFAEL 

 ARMAMENT DEVELOPMENT 

 AUTH 

 M MAYSELESS 

PO BOX 2250 

 HAIFA 31021 

 ISRAEL

1 TNO DEFENSE RESEARCH 

I H PASMAN

 POSTBUS 6006

 2600 JA DELFT 

 THE NETHERLANDS 

 1 B HIRSCH

 TACHKEMONY ST 6 

 NETAMUA 42611

 ISRAEL

1 DEUTSCHE AEROSPACE AG 

 DYNAMICS SYSTEMS 

 M HELD

PO BOX 1340 

 D 86523 SCHROBENHAUSEN 

 GERMANY 

34