Topology Optimization Based Lightweight Armor 

Development

by Dwight Hofstetter Jr., Rahul Gupta, and Robert Bitting

ARL-TR-5319 September 2010

Approved for public release; distribution is unlimited.
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Army Research Laboratory

Aberdeen Proving Ground, MD 21005-5066

ARL-TR-5319 September 2010

Topology Optimization Based Lightweight Armor 

Development

Dwight Hofstetter Jr.

Oak Ridge Institute for Science and Education (ORISE)

Rahul Gupta and Robert Bitting

Weapons and Materials Research Directorate, ARL

Approved for public release; distribution is unlimited.
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4. TITLE AND SUBTITLE

Topology Optimization Based Lightweight Armor Development

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6. AUTHOR(S)

Dwight Hofstetter Jr.*, Rahul Gupta, and Robert Bitting

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7. PERFORMING ORGANIZATION NAME(S) AND ADDRESS(ES)

Blast Protection Branch, Protection Division

U.S. Army Research Laboratory

ATTN: RDRL-WMP-F

Aberdeen Proving Ground, MD 21005-5066

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14. ABSTRACT

Improvised Explosive Devices (IEDs) present a constant and continually changing threat to the vehicles on the modern 

battlefield. Continual reengineering is needed to keep up with these threats and to safeguard the vehicles and the warfighters. 

To achieve this, protection kits need to be added to the vehicles. However, armor kits are a concession between maximum 

strength and minimum weight. Livermore Software Topology Optimization (LS-OPT)/Topology optimization software takes 

into account size, shape, and topology by designing for a uniform internal energy density in the structure while keeping the 

mass constrained. The user needs to develop a model matching the allowable limits of the desired part with a finite element 

analysis preprocessor. Then, the model is imported into LS-OPT/Topology and specific design limits are set. The program 

makes constant iterations until a tolerance with respect to the design limit (in this case mass fraction) is reached, or the amount 

of iterations have surpassed the set boundary. This paper explores the practical usage of LS-OPT/Topology that can be utilized 

by the engineers at U.S. Army Research Laboratory (ARL) to improve the modeling and simulation capabilities in the area of 

vehicle protection leading to Soldier survivability.

15. SUBJECT TERMS

summer student, topology, optimization, LS-OPT/topology, armor

16. SECURITY CLASSIFICATION OF: 

17. LIMITATION

OF ABSTRACT

UU

18. NUMBER

OF PAGES

34

19a. NAME OF RESPONSIBLE PERSON

Rahul Gupta

a. REPORT

Unclassified

b. ABSTRACT

Unclassified

c. THIS PAGE

Unclassified

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410-278-6035

Standard Form 298 (Rev. 8/98)

Prescribed by ANSI Std. Z39.18
iii

Contents

List of Figures v

List of Tables vi

Acknowledgments vii

1. Introduction/Background 1

2. Theory 1

2.1 Topology .........................................................................................................................1

2.2 Hybrid Cellular Automata (HCA)...................................................................................2

2.2.1 Internal Energy Density (6, 7).............................................................................3

2.3 Criteria for Evaluating Elements.....................................................................................3

2.3.1 Strain Hardening Modulus (6, 7).........................................................................5

3. Operation 5

3.1 LS-Prepost.......................................................................................................................5

3.2 Opening/Information.......................................................................................................5

3.3 Cases................................................................................................................................5

3.4 Problem ...........................................................................................................................6

3.5 Method.............................................................................................................................6

3.6 Solution ...........................................................................................................................6

3.7 View ................................................................................................................................6

4. Experiment 7

4.1 Observation .....................................................................................................................7

4.2 Hypothesis.......................................................................................................................7

4.3 Procedure.........................................................................................................................7

5. Results 8

5.1 The 0.75 Mass Fraction (Coarse Mesh) ..........................................................................9

5.2 The 0.50 Mass Fraction (Coarse Mesh) ........................................................................11
iv

5.3 The 0.4 Mass Fraction (Coarse Mesh) ..........................................................................13

5.4 The 0.75 Mass Fraction (Fine Mesh) ............................................................................15

5.5 The 0.50 Mass Fraction (Fine Mesh) ............................................................................17

6. Summary and Conclusion 18

7. References 19

List of Symbols, Abbreviations, and Acronyms 20

Distribution List 21
v

List of Figures

Figure 1. Hybrid cellular automata based topology optimization method......................................2

Figure 2. Hybrid cellular automata based topology optimization flowchart (6, 7).........................3

Figure 3. Topology optimization method simulation model. .........................................................8

Figure 4. Coarse mesh 0.75 mass fraction density redistribution.................................................10

Figure 5. Coarse mesh 0.75 mass fraction total internal energy density. .....................................10

Figure 6. Coarse mesh 0.75 mass fraction initial vs. optimized. ..................................................11

Figure 7. Coarse mesh 0.75 mass fraction density redistribution.................................................11

Figure 8. Coarse mesh 0.5 mass fraction density redistribution...................................................12

Figure 9. Coarse mesh 0.5 mass fraction total internal energy density. .......................................12

Figure 10. Coarse mesh 0.5 mass fraction initial vs. optimized. ..................................................13

Figure 11. Coarse mesh 0.5 mass fraction density redistribution.................................................13

Figure 12. Coarse mesh 0.4 mass fraction density redistribution.................................................14

Figure 13. Coarse mesh 0.4 mass fraction total internal energy density. .....................................14

Figure 14. Coarse mesh 0.4 mass fraction initial vs. optimized. ..................................................14

Figure 15. Coarse mesh 0.4 mass fraction density redistribution.................................................15

Figure 16. Fine mesh 0.75 mass fraction density redistribution...................................................15

Figure 17. Fine mesh 0.75 mass fraction total internal energy density. .......................................16

Figure 18. Fine Mesh 0.75 mass fraction initial vs. optimized.....................................................16

Figure 19. Fine mesh 0.75 mass fraction density redistribution...................................................16

Figure 20. Fine mesh 0.5 mass fraction initial vs. optimized .......................................................17

Figure 21. Fine mesh 0.5 mass fraction density redistribution.....................................................17
vi

List of Tables

Table 1. Coarse mesh (1 × 1 × 1 in, non-linear). ............................................................................9

Table 2. Fine mesh (0.5 × 0.5 × 0.5 in, non-linear). .......................................................................9
vii

Acknowledgments

The author wishes to acknowledge the mentorship of Dr. Rahul Gupta, Blast Protection Branch, 

U.S. Army Research Laboratory (ARL).
viii

INTENTIONALLY LEFT BLANK.
1

1. Introduction/Background

Today’s enemy has been relying heavily on Improvised Explosive Devices (IEDs) to defeat

vehicles used by the U.S. and its allies. Thus, there is a push to rush the immediate armoring of 

less protected vehicles. However, with added armor comes added weight, less maneuverability, 

and more frequent maintenance. It is the mission of U.S. Army Research Laboratory (ARL) to 

develop suitable designs that take into account the practicality of superior strength with the 

necessity of lighter weight. The problem that engineers face is determining how material can be 

removed without jeopardizing the integrity of the structure. Another aspect for the engineer to 

consider is that each physical test of a vehicle’s resistance to blast loading typically destroys the 

armor and/or the vehicle, making repeated tests both expensive and time consuming (1). In an 

effort to address the compromise between strength and weight while minimizing research 

expense, Livermore Software Topology Optimization (LS-OPT)/Topology appears to be a 

valuable tool for the development of lightweight armor kits. It is one of the first to explore the 

field of non-linear problems with dynamic loading. Physical nonlinearities are abundant in the 

process of a blast event (2). The topology optimization in the context of impact analysis is a 

very complex problem due to non-linear interactions among material non-linearities, geometry, 

and the transient nature of the boundary conditions. Conventional methods are not practical for 

solving these non-linear topology optimization problems due to the high computational cost and 

the lack of sensitivity information (3).

2. Theory

2.1 Topology

LS-OPT/Topology is a novel optimization software that optimizes the size, shape, and topology 

of a model through reconstruction. Topology is an important factor in this process, because it 

deals with the material as well as the material properties. Figure 1 demonstrates an example of 

the topology optimization process. Topology is defined as a field in mathematics which focuses 

on an object’s spatial properties as they are preserved during deformations such as bending or 

twisting but not to include tearing (4). The topology aspect of LS-OPT/Topology essentially 

takes a raw or developed lump of clay then reshapes it to optimize the piece’s structural integrity 

while keeping it within its specific constraints. In the same manner, the program will also 

minimize the size and shape of the piece by removing cells from which it is made; yet the 

program will not cut the model into more than one piece. Design for structural topology 

optimization is a method of distributing material within a design domain of prescribed

dimensions. This domain is discretized into a large number of elements in which the 
2

Figure 1. Hybrid cellular automata based topology 

optimization method.

optimization algorithm removes, adds, or maintains the amount of material. The resulting 

structure maximizes a prescribed mechanical performance while satisfying functional and 

geometric constraints (2).

2.2 Hybrid Cellular Automata (HCA)

Over the years, several methods have been developed to analyze the mathematical field of 

topology. The LS-OPT/Topology developers believe that the most successful method for 

analyzing dynamically loaded problems is HCA. A cellular automaton (CA) is a collection of 

“colored” cells on a grid of a specified shape that evolves through a number of discrete time 

steps according to a set of rules based on the states of neighboring cells (5). However, for the 

structural optimization problems, the state of a cellular automaton is defined by rules that 

combine the local neighborhood information with the field variables that are calculated globally 

(6, 7), which makes the method a hybrid. The globally calculated field variables refer to those 

held within the entire model. Control rules drive a defined field variable to an optimum state or 

set point. The expression for the field variable and the value of the set point are derived from the 

optimality conditions of the structural design problem. Among different topology optimization 

algorithms, the hybrid cellular automaton HCA method has proven to be efficient and robust in 

problems involving large, plastic deformations (2). The idea of HCA can be seen in equation 1. 

(1)

(Ui) = energy density of ith cellular automata lattice

n= amount of neighbors/adjacent cells

(The number one is in the denominator so the denominator sums to n.)
3

2.2.1 Internal Energy Density (6, 7)

In the context of HCA, the field variables are subjected to the neighborhood update rule first and 

then the design variables are updated. This process is iterated until equilibrium is established 

among all state variables (6, 7). HCA is a simple yet effective method for analyzing the 

topology of a model as can be seen in figure 2.

Figure 2. Hybrid cellular automata based topology optimization flowchart (6, 7).

2.3 Criteria for Evaluating Elements

The basic principles behind how LS-OPT/Topology functions are easy to follow. The topology 

design problem is defined by the allowable geometric domain, how the part will be used, and 

properties of the part (8). These aspects are assigned in the initial LS-Prepost modeling. When 

the model is built within LS-Prepost, it is required that it be constructed using 

PIECEWISE_LINEAR_PLASTICITY using only eight-noded solid elements (8). A design 

variable is directly linked to each of these material elements such that each cell has its own

material model (8). The objective for optimization is to obtain a maximum uniform internal 

energy density among all of these elements (7, 9). The formulation of the optimization problem 

can be seen in equations 2 and 3. 

(2)

U = energy density of the i

th element

w= weighted parameter

U*= internal energy set point
4

L= amount of load cases

x = relative density/design variable

Optimization problem (6, 7)

(3)

Vi = volume of the ith element

ρ = density of the material

M*= updated mass of the structure

x = relative density/design variable

Optimization Problem Constraints (6, 7)

The solid isentropic material with a penalization model is used to interpolate the material 

properties as can be seen in the equations 4, 5, and 6.

(4)

ρ = density of the material

0 = refers to the base material properties

x = relative density/design variable

2.3.2 Density (6, 7)

(5)

E = Young’s modulus

0 = refers to the base material properties

x = relative density/design variable

p= material linearity exponent

Young’s Modulus (6, 7)

(6)

σ = yield stress

0 = refers to the base material properties

x = relative density/design variable

q= material non-linearity exponent

Yield Stress (6, 7)

(7)

Eh = strain hardening modulus
5

0 = refers to the base material properties

x = relative density/design variable

q= material non-linearity exponent

2.3.1 Strain Hardening Modulus (6, 7)

Internal energy most often consists of elastic and plastic deformations. By decreasing the mass 

of the entire model, an increase in the total internal energy per total mass can be obtained (8).

When the blast pressure wave hits the targeted structure, the fluids kinetic energy is transformed 

into strain energy (SE) inside the solid medium. Maximum attenuation is reached when SE is 

maximized. Along with an optimum use of material, this condition is satisfied when SE is 

uniformly distributed in the design domain (2). When setting LS-OPT/Topology, several 

constraints can be set such as the number of design iterations, the desired mass fraction, and 

minimum density fraction before deleting an element. However, such constraints as maximum 

displacement are not accounted for within the current version of the LS-OPT/Topology software. 

3. Operation

3.1 LS-Prepost

Operating the LS-OPT/Topology software is quite simple once the user becomes familiar with 

the operational concepts. First, a user must create a model in LS-Prepost which will set the limit 

constraints for the final optimized model. The model must be created using the 

PEICEWISE_LINEAR_PLASTICITY material with eight-noded solid elements and it must also 

be defined using “*Part”. A user can include holes in the piece and if extrusions in the final 

model are desired the user can specify them in LS-Prepost. 

3.2 Opening/Information

As with other software, LS-OPT/Topology’s graphical user interface (GUI) can be opened from 

the file pull down menu. When the GUI window appears, the first item the user sees is the 

“information tab” with license and working directory information as well as a place to insert a 

problem description. The problem description is used for reference so anything can be written. 

3.3 Cases

The next tab titled “cases” is where all of the input file information will be inserted. First a user 

must select the “new” button at the bottom of the window. In the “edit cases” pop up window a 

“general” tab will be selected. The user must create a name for the case as well as specify the 

weight which is simply a scaling factor if multiple loads are used. One of two types of analysis 

need to be selected: linear or non-linear. At this point the model that was created in LS-PrePost 
6

can be imported into the program by browsing the input file name. LS-OPT/Topology requires a 

Livermore Software Dynamic Analysis (LS-DYNA) executable to solve which can be specified 

by browsing the execution command. Advanced users typically modify the “scheduling” tab by 

choosing to limit the number of concurring jobs, choosing a Queue system, and setting an 

environmental variable. The default settings on these options are sufficient for this experiment. 

When done, the user can select the “ok” button to finish the selection.

3.4 Problem

After selecting the “problem” tab a design-part id needs to be specified. The design-part 

identification is a means of referring to the place where all elements in the design domain have 

been placed in the LS-DYNA input deck. This reference number can be found in the model’s 

viewing window. If the LS-Prepost model has an extrusion specified, the extrusion set number 

can be inserted into the extrusion set box. On the bottom of the “problem” tab is a box labeled 

mass fraction. This refers to the amount of material that is desired to be eliminated where a 

value of 1 would conserve the original model and a value of 0 would remove the entire model. 

3.5 Method

Under the method tab is the termination criteria. The most often used termination criteria is the 

number of design iterations. LS-OPT/Topology will terminate when this number of iterations is 

reached. The convergence tolerance terminates the program when the topology begins making 

few changes. This is controlled by the density redistribution history variable. The proximity 

tolerance is based on the uniformity of neighboring elements. Elements with a density less than 

the minimum density fraction specified in the minimum density fraction box will be deleted on a 

given iteration.

3.6 Solution

After selecting the “run” tab, the only visible button is “run”. Upon clicking on this button, the 

user will be prompted to save the solution. A storage location and a name are given to the run, 

“ok” is selected, and the solution begins. Each iteration has a percent completion bar displayed

at the top of the page and an engine output display is visible at the bottom. At any time the run 

can be stopped and deleted or stopped to be restarted later.

3.7 View

The “view” tab allows a user to see a graph of the density redistribution as well as a graph of the 

total internal energy density. As the number of iterations increases the graphs readjust to show 

the new data. From the “view” tab a user can also open LS-Prepost to post process the iterations 

displaying the changes that have been made to the model. The iterations can be viewed 

individually or simultaneously in a window with regards to density or internal energy density. 
7

4. Experiment

4.1 Observation

After seeing LS-OPT/Topology presented, or solving example problems from the user’s manual, 

it can be concluded that this technology could be used in the design and analysis of blast 

protection. The examples within the LS-OPT/Topology user’s manual exhibited examples

solved with beams but plates are also feasible models. 

4.2 Hypothesis

It is hypothesized that in the utilization of LS-OPT/Topology for the designing and testing of 

armor plates, a model containing more initial elements will be developed into a more accurate 

optimized piece compared to a model containing fewer but larger initial elements. Since LSOPT/Topology removes whole elements, removing several smaller elements would be more 

precise and thorough than removing larger elements.

4.3 Procedure

To test the hypothesis, it was decided that a plate under pressure loading would be tested. Figure

3 demonstrates the model with dimension, loading, boundary conditions and material properties. 

The plate was 48 × 48 × 6 in with boundaries set along opposite edges with a bottom pressure 

loading in order to imitate a blast load from an underbody IED. The plate was assigned a mass 

density of 7850 kg/m3

, a Young’s modulus of 2.068 ×1011 Pa and a yield stress of 0.27 MPa. 

The first, second, third, and fourth effective strain values were 0.0, 0.0055, 0.0152, and 0.0248 

respectively. The fifth, sixth, seventh, and eighth effective strain values were 0.0627, 0.16, 0.28, 

and 0.42, respectively. The first, second, third, and fourth corresponding yield stress value to 

effective strain values were 7.922 × 108 Pa, 9.23 × 108 Pa, 9.610 × 108 Pa, and 9.830 × 108 Pa,

respectively. The fifth, sixth, seventh, and eighth corresponding yield stress value to effective 

strain values were 1.029 × 108 Pa, 1.077 × 109 Pa, 1.098 × 109 Pa, and 1.104 × 109 Pa,

respectively.
8

Figure 3. Topology optimization method simulation model.

The only difference between the two plates was that the course mesh plate was created with 1 × 1

× 1 in elements where the fine mesh plate was created with 0.5 × 0.5 × 0.5 in elements (finer 

mesh). In the experiment, the only variable that was changed was the mass fraction. Each plate 

would be optimized for: (1) 0.75 mass fraction, (2) 0.5 mass fraction, and (3) 0.25 mass fraction. 

However, the solutions ended up including 0.4 mass fraction, 0.5 mass fraction, and 0.75 mass 

fraction. Preliminary solutions showed that under the operating system used, an initial iteration 

could not be completed for a 0.25, 0.3, or a 0.35 mass fraction on the coarse mesh plate;

therefore, a 0.4 mass fraction was used. Each solution consisted of a non-linear analysis with the 

number of iterations set to 100. Other values were left as defaults. Most solutions were made 

overnight due to the length of time required to make all of the iterations.

5. Results

Each solution ended with a different number of iterations and all ended with error terminations 

due to negative volume. Negative volume is often the result of deforming an element so 

extensively that it is inverted inside out. Tables 1 and 2 summarize the iteration results.
9

Table 1. Coarse mesh (1 × 1 × 1 in, non-linear).

Mass 

Fraction

Vertical 

Deformation 

(in)

Initial 

Mass 

(kg)

Final 

Mass 

(kg)

No. of 

Iterations

Density 

Redistribution

Internal 

Energy Density 

(KN/m2)

0.75 0.35 1778.42 1333.84 11 Initial 0.072 4.6E+011

Final 0.015 3.1E+011

0.5 0.63 1778.42 889.149 7

Initial 0.062 9E+011

Final 0.055 5.5 E+011

0.4 0.87 1778.42 711.319 5 Initial 0.06 3.5E+012

Final 0.055 0.75E+012

Table 2. Fine mesh (0.5 × 0.5 × 0.5 in, non-linear).

Mass 

Fraction

Vertical 

Deformation

(in)

Initial 

Mass 

(kg)

Final 

Mass 

(kg)

No. of 

Iterations

Density 

Redistribution

Internal Energy 

Density 

(KN/m2

)

0.75 0.5118 1778.42 1333.84 4 Initial 0.072 4.52E+012

Final 0.072 3.69E+012

0.5 1.142 1778.42 889.149 2 Initial N/A N/A

Final N/A N/A

0.4 N/A 1778.42 N/A 0 Initial N/A N/A

Final N/A N/A

Note: N/A = not applicable.

5.1 The 0.75 Mass Fraction (Coarse Mesh)

The 0.75 mass fraction solution of the coarse mesh plate ended in an error termination due to a 

negative volume on its eleventh iteration. The density distribution increased to 0.072 then 

leveled off to around 0.015 (figure 4). The total internal energy density dropped from 4.6 × 1011

KN/m2 and leveled off at 3.1 × 1011 KN/m2 (figure 5). Figure 6 shows that the final iteration 

(right) had symmetrical crescent shapes removed from the blue surface of the plate. Figure 7

demonstrates the mass fraction density redistribution.
10

Figure 4. Coarse mesh 0.75 mass fraction density 

redistribution.

Figure 5. Coarse mesh 0.75 mass fraction total 

internal energy density.
11

 

 Initial Mass = 1778.42 kg Optimized Mass = 1333.84 kg

Mass reduction = 25%

Figure 6. Coarse mesh 0.75 mass fraction initial vs. optimized.

Figure 7. Coarse mesh 0.75 mass fraction density redistribution.

5.2 The 0.50 Mass Fraction (Coarse Mesh)

The 0.5 mass fraction solution of the coarse mesh plate ended in an error termination due to a 

negative volume on its seventh iteration. The density distribution increased to 0.062 then leveled 

off to around 0.055 (figure 8). The total internal energy density dropped from 9 ×1011 KN/m2

and leveled off at 5.5 ×1011 KN/m2 (figure 9). Figure 10 demonstrates the final iteration (right) 

had symmetrical crescent shapes removed from the blue surface of the plate. Figure 11

demonstrates the mass fraction density redistribution.
12

Figure 8. Coarse mesh 0.5 mass fraction density 

redistribution.

Figure 9. Coarse mesh 0.5 mass fraction total 

internal energy density.
13

 

Initial Mass = 1778.42 

 

Optimized Mass = 889.149 kg 

Mass reduction = 50% 

Figure 10. Coarse mesh 0.5 mass fraction initial vs. optimized.

Figure 11. Coarse mesh 0.5 mass fraction density redistribution.

5.3 The 0.4 Mass Fraction (Coarse Mesh)

The 0.4 mass fraction solution for the coarse mesh plate ended in an error termination due to a 

negative volume on its fifth iteration. The density distribution increased to 0.06 then leveled off 

to around 0.055 (figure 12). The total internal energy density dropped from 3.5 ×10 12 KN/m2

and leveled off at 0.75 × 1012 KN/m2 (figure 13). Figure 14 demonstrates that the final iteration 

(right) had symmetrical crescent shapes removed from the blue surface of the plate. Figure 15

demonstrates the mass fraction density redistribution.
14

Figure 12. Coarse mesh 0.4 mass fraction density 

redistribution.

Figure 13. Coarse mesh 0.4 mass fraction total internal 

energy density.

 Initial Mass = 1778.42 kg Optimized Mass= 711.319 kg 

Mass reduction = 60%

Figure 14. Coarse mesh 0.4 mass fraction initial vs. optimized.
15

Figure 15. Coarse mesh 0.4 mass fraction density redistribution.

5.4 The 0.75 Mass Fraction (Fine Mesh)

The 0.75 mass fraction solution of the fine mesh plate ended in an error termination due to a 

negative volume on its fourth iteration. The density distribution increased to 0.072 then leveled 

off as shown in figure 16. The total internal energy density dropped from 4.52 × 1012 KN/m2 and

steadily declined as shown in figure 17. Figure 18 demonstrates that the final iteration (right) did 

not have any material removed at the time of testing. Figure 19 demonstrates the mass fraction 

density redistribution. Solving the 0.75 mass fraction with the fine mesh took roughly 2 days 

(46 h) to complete. 

Figure 16. Fine mesh 0.75 mass fraction density 

redistribution.
16

Figure 17. Fine mesh 0.75 mass fraction total internal 

energy density.

 Initial Mass = 1778.42 kg Optimized Mass = 1333.84 kg 

Mass reduction = 25%

Figure 18. Fine Mesh 0.75 mass fraction initial vs. optimized.

Figure 19. Fine mesh 0.75 mass fraction density redistribution.
17

5.5 The 0.50 Mass Fraction (Fine Mesh)

The 0.5 mass fraction solution of the fine mesh plate ended in an error termination due to a 

negative volume on its second iteration. Since the program was unable to begin to remove 

material as of the second iteration, neither the density redistribution nor the total internal energy 

density graphs could be drawn. However, the initial iteration was able to produce an analyzed 

plate as shown in figure 20. Figure 21 demonstrates the mass fraction density redistribution.

 Initial Mass = 1778.42 kg Optimized Mass= 889.149 kg 

Mass reduction = 50%

Figure 20. Fine mesh 0.5 mass fraction initial vs. optimized

Figure 21. Fine mesh 0.5 mass fraction density redistribution.
18

The 0.4 mass fraction solution of the fine mesh plate ended in an error termination due to a 

negative volume on its first iteration. Since the first iteration was not completed, no data can be 

viewed concerning its potential form.

6. Summary and Conclusion

Initially, it was believed that an error termination was acceptable since most solutions displayed 

an optimized model before they were terminated. However, it was discovered subsequently that

the coarse mesh plate used for this experiment can be solved to completion using a mass fraction 

of 0.25 since LSTC was using massively parallel processing (MPP) on a Linux operating system

while the ARL simulation experiment was conducted using symmetric multiprocessing (SMP) 

on a Windows operating system. If the experiment had been successful and every case solved 

through completion, each resulting model would show a plate that would progressively reduce

mass in order to reflect the required mass fraction. A model with a finer mesh should portray a

more accurate depiction of the optimized form. In the experiment, the solutions with lower mass 

fractions ended with less number of iterations, most likely because the program had more mass 

that was required to be removed so that the initial removal could be substantial. The solutions of 

the model containing the finer mesh also ended with a less number of iterations. However, the 

iterations themselves took much longer to complete than those of the coarser mesh. This is 

possibly due to the fact that more elements had to be removed to equal the amount of larger 

elements removed from the coarser meshed model. Upon completion of the experimental study, 

further analysis will be presented in future publications.
19

7. References

1. Optimization of an Armored Plate for Light Armored Vehicles Subjected to Blast Loading. 

Red Ceder Technologies, Inc. East Lansing, Michigan, 2009.

2. Goetz, J. C.; Tan, H.; Renaud, J. E.; Tovar, A. Structural Topology Optimization for Blast 

Mitigation Using Hybrid Cellular Automata. 2009 Ground Vehicle Systems Engineering and 

Technology Symposium (GVSETS). 2009.

3. LSTC, Goel, T. Topology Optimization using LS-OPT/Topology. http://www.lsoptsupport.

com/news/topology-optimization-using-ls-optae-topology (accessed 2010).

4. Wikipedia, Topology, http://en.wikipedia.org/wiki/Topology (accessed 2010).

5. Weisstein, E. W. Mathworld, Cellular Automaton. 

http://mathworld.wolfram.com/CellularAutomaton.html (accessed 2010).

6. Goel, T.; Roux, W. LS-OPT/Topology A Topology Optimization Tool for LS-DYNA: User’s 

Manual: Livermore Software Technology Corporation, Livermore, CA, 2009.

7. Goel, T.; Roux, W.; Stander, N. A Topology Optimization Tool for LS-DYNA Users: LSOPT/Topology. 7th European LS-DYNA Conference. 2009.

8. Roux, W.; Goel, T.; Bjӧrkevik, D. LS-OPT/Topology Version 1. 11th International LSDYNA Users Conference. 2010.

9. Mayer, R. R.; Kikuchi, N.; Scott, R. A. Application of Topological Optimization 

Techniques to Structural Crashworthiness. International Journal for Numerical Methods in 

Engineering 1996, 39, 1383–1403.

10. LS-DYNA Keyword User’s Manual: Volume 1: Livermore Software Technology 

Corporation, Livermore, CA, 2007.
20

List of Symbols, Abbreviations, and Acronyms

ARL U.S. Army Research Laboratory

CA cellular automaton

GUI graphical user interface

HCA Hybrid Cellular Automata

IEDs Improvised Explosive Devices

LS-DYNA Livermore Software Dynamic Analysis

LS-OPT Livermore Software Topology Optimization

MPP massively parallel processing

N/A not applicable

SE strain energy

SMP symmetric multiprocessing
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LANL

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2 THE BOEING CO

J CHILDRESS

N GERKEN

MS 84-69

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6 BAE SYSTEMS

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EMERGING TECHNOLOGIES

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SYSTEMS

STERLING HEIGHTS COMPLEX

JIM ERIDON

TOM ZELEZNIK

38500 MOUND ROAD

STERLING HEIGHTS MI 

48310-3200

2 MARINE CORPS SCIENCE AND 

TECHNOLOGY

OFFICE OF NAVAL RESEARCH

LEE MASTROIANNI

JEFF BRADEL

ONE LIBERTY CENTER

875 NORTH RANDOLPH STREET, 

SUITE 1156B CODE 30

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ELECTRICAL S&T DIV 

OFFICE OF NAVAL RESEARCH

ONR334 

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PE 

800 N QUINCY ST

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ABERDEEN PROVING GROUND

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RDRL CI

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M ADAMSON

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R COATES

RDRL SLA B

P KUSS

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E HUNT

B WARD

RDRL SLB D

R GROTE

J POLESNE

RDRL SLB E

D LYNCH

RDRL VTA

J BORNSTEIN

RDRL WM

J SMITH

L BURTON

RDRL WML

M ZOLTOSKI

J NEWILL

RDRL WML G

W DRYSDALE

S KUKUCK

RDRL WML H

B SORENSEN

T FARRAND

E KENNEDY

L MAGNESS

RDRL WMM A

J WOLBERT

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C YEN

R DOOLEY

C FOUNTZOULAS

RDRL WMM C

R JENSEN

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RDRL WMM D

E CHIN

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K DOHERTY

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P BAKER

RDRL WMP

S SCHOENFELD

M BURKINS

RDRL WMP A

B RINGERS

RDRL WMP C

T BJERKE

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T HAVEL

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M KLUSEWITZ

W GOOCH

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E HORWATH

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M LOVE

RDRL WMP F

N GNIAZDOWSKI

E FIORAVANTE

A FRYDMAN

R GUPTA

X HUANG

D FOX

R BITTING

RDRL WMP G

N ELDREDGE