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Design Optimization: InternationalJournal for Product & ProcessImprovement, Vol. 1 No. 4, 1999,pp. 374-387. # MCB UniversityPress, 1463-5801

Received July 1999Accepted July 1999

Structural optimizationin occupant safety and

crash analysisUwe Schramm and Harold Thomas

Altair Engineering, Inc., USA, and

Detlef SchneiderAltair Engineering GmbH, Germany

Keywords Safety, Structural optimisation, Crash

Abstract In occupant safety and crash analysis, non-linear structural finite element codesbased on explicit integration schemes are used. Parameter changes are derived from theresults of the analysis to improve the design in order to meet the requirements of regulationsand design goals. Today, the common approach is to use a trial-and-error search. This is veryexpensive with respect to labor costs. Optimization methods can be used to automate thesearch for the optimum design. Closed loop software for the optimization of structures withnon-linear behavior such as is known for linear static analysis is not available currently.Usually add-ons to the non-linear codes based on response surface methods or finite differencesensitivity analysis are used. The paper discusses applications of optimization tools inoccupant safety and crash analysis using a very efficient response surface method. Someexamples are given.

IntroductionIn automotive design, the occupant and structural behavior in the event of acrash is of special interest. Non-linear finite element and rigid body analysis isapplied to predict the responses of the structure or of the occupant. Decisionsbased on these computations can lead to significant design modifications.Usually, intuition leads the iterative process of finding the best design. It isoften hard to determine necessary design modifications from the analysisresults only. In some cases many variations are tried before a satisfactorydesign is found.

Structural optimization, based on computational methods, is a useful tool tosupport the process of finding the right design. The design layout can bedescribed mathematically as an optimization problem. Using the results of acomputational optimization, the decision process can be improved.Optimization of structural elements can lead to significant improvements of thefinal design.

This paper was presented at the OptiCon '98 held in Newport Beach, CA, October 4-5, 1998. Theairbag and knee bolster examples appear courtesy of the BMW AG. The permission to publishthese results is gratefully acknowledged.

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The optimization problem appears as

Objective : W �p� ) min

Subject to Mt�d � t f �td; t _d;p�; 0 � t � te

with 0d � d0;0 _d � _d0

Constraints : g�d; _d; �d;p� � 0

Side constraints : pl � p � pu

�1�

Optimization of the structure can be performed to obtain a certain occupant orstructural response or to reduce the structural mass as much as possible. Theobjective function W can be any structural response, such as the total or a weightedmass of the structure. The constraint functions g� fgig; i � 1; . . . ; nc are occupantor structural responses. The state equation describes the dynamic behavior of thestructure that is to be optimized. Here, it is given in the form of an explicitintegration scheme, such as is usually applied in occupant and crash analysis. Thematrix M is the time independent mass matrix. The state vector td contains thegeneralized displacements. The load vector tf contains external and internal forces.The quantity p� fpjg; j � 1; . . . ;m is the vector of design variables.

Industrial application of structural optimization often depends on theavailability of software products. For linear static and dynamic problems suchsoftware is available and fairly well supported. Layout and shape optimizationcan be performed to design structural parts. In the design process based oncrashworthiness the trial and error approach is still the most commonly usedmethod to find a design that fits the needs. For analysis, explicit integrationschemes are mostly used for structural analysis. No commercial software isavailable to perform sensitivity analysis and optimization for crashworthinessproblems. A review of the optimization of structures under impact loading canbe found in Schramm and Pilkey (1996).

Design processViewing the design process as an optimization process to find structures andstructural parts that fulfill certain expectations toward their economy,functionality and appearance, structural optimization can be seen as the driver inthis process. It does not matter what functionality is expected from the design.

Figure 1 shows the design process in form of an optimization iteration goingthrough the following steps:

(1) Define an initial design p�s�0�;

(2) Analyze the behavior of the structure;

(3) Compare the results of the analysis with the requirements, such asallowable stresses, displacements, cost limits, injury criteria or energies;

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(4) If the requirements are not met, change the design such thatp�s�1� � p�2� � �p;

(5) Back to (2).

It is possible to determine the search direction �p directly from the results ofthe structural analysis. Then, the general formulation of the optimizationproblem (1) appears as:

Objective : 0�p� ) min

Constraints : i �p� � 0

Design space : pl � p � pu

�2�

Objective and constraints i�p�; i � 0; . . . ; nc can be approximated for eachdesign p�s� using the series expansion

i � i�p�s�� �Xm

j�1

d i

dpj

�pj �3�

The gradient d i=dpj can be obtained directly from the results of the numericalanalysis. If the gradient is known, the search direction �p can be obtained fromthe solution of an approximate optimization problem.

Optimization methodsFor the solution of structural optimization problems, approximation methodsare well established. Similar to the description in the previous section, thesequential solution of approximate optimization problems leads to the solutionof the optimization problem (1). Also in conjunction with crash analysis theseoptimization methods are applicable. In general two types of approximationmethods can be distinguished. First, the local approximation using gradientinformation, and second, global approximation methods that use anapproximation to the original structural optimization problem over a widerange of the design variables (Barthelemy and Haftka, 1993).

Product request

CADDesign

FEMDesign analysis

OptimizationDesign update

Figure 1.Design process

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Local approximationLocal approximation methods determine the solution of the optimizationproblem using the following steps:

(1) Analysis of the physical problem using finite elements;

(2) Convergence test, if the solution is found;

(3) Design sensitivity analysis;

(4) Solution of an approximate optimization problem formulated using thesensitivity information;

(5) Back to (1).

This approach is based on the assumption that only small changes of thedesign occur in each optimization step. The result is a local minimum. It shouldbe mentioned that the biggest changes occur in the first few optimization steps.Therefore, not too many system analyses are necessary in practicalapplications.

The design sensitivity analysis produces the gradient of the structuralresponses with respect to the design variables. It is one of the most importantingredients in local approximation methods.

Regarding implementation effort, the simplest method for design sensitivityanalysis is to use finite differences. In the form of central differences the designsensitivities for a function i appear as

d i

dpj

� i pj ��pj

ÿ �ÿ i pj ÿ�pj

ÿ �2�pj

�4�

The numerical effort for optimization using this method is quite large.Optimization for crash codes using finite difference sensitivity analysis isalready available.

Much more efficient with regard to the numerical effort would be ananalytical sensitivity analysis. In the case of using an explicit integrationscheme for the solution of the structural problem this sensitivity analysis canbe derived from the state equation. The state equation is usually solved usingcentral differences for the time domain.

Then, assuming a constant time step �t,

t�d �Mÿ1 tfÿ �

t��t=2 _d �tÿ�t=2 _d� t�d�t

t��td �td�t��t=2 _d�t

�5�

This notation follows that of Hallquist (1997).Differentiation of these equations with respect to the design variables pj

yields

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dt�d

dpj�Mÿ1 @tf

@pj� @tf

@td

dtd

dpj� @tf

@t _d

dt _d

dpjÿ dtM

dpj

�d

!dt��t=2 _d

dpj� dtÿ�t=2 _d

dpj� dt�d

dpj�t

dt��td

dpj� dtd

dpj� dt��t=2 _d

dpj�t

�6�

Using the derivatives of the state vector, the gradient of the objective andconstraint functions can be computed.

An adjoint variable method for sensitivity analysis and more detail can befound in Schramm and Pilkey (1996).

Global approximationThe implementation of a design sensitivity analysis for explicit integrationschemes is not a simple task. Analytical sensitivity analysis is not yetcommercially available. Therefore, it is common to introduce higher orderanalytical expressions to approximate the dependence between the objective orconstraint functions and the design variables. An approximate analyticalrelationship between structural responses and design variables can beestablished with only a few analyses.

Let i�p� be the response of interest. Then, a polynomial i�p�, calledresponse surface, of the degree n can be introduced such that

i�p� � i�p� � ai0 �Xm

j�1

aijpj�Xm

j�1

Xm

k�1

aijkpjpk � . . . �7�

The quantities ai0; aij; aijk are constants that are determined from the results ofthe system analyses by solving a linear system of equations. The number ofanalyses that are necessary depends on the polynomial degree n (1). If themixed terms are omitted, the computational effort can be reduced considerably.Then, the response surface appears as

i�p� � i�p� �Xn

l�0

Xm

j�1

alplj �8�

Using the equations above, an approximation to the optimization problem canbe established. It appears as

0�p� ) min

i�p� � 0; i � 1; . . . ; nc

�9�

The solution to this problem can be determined using mathematical programming.It yields an approximate solution to the structural optimization problem (1).

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The solution of the optimization problem (1) using response surfacesinvolves the following steps:

(1) Finite element solutions of the problem;

(2) Response surface computation for each response;

(3) Solution of the approximate optimization problem.

Problems of different physical content can be combined in one optimizationproblem easily.

Using the method above, a predefined number of designs is analyzedfollowed by the computation of the response surfaces and the optimizationsolution. The general idea of this method is shown in Figure 2 for a problemwith one design variable.

A different approach is to analyze the designs as the optimization proceeds.This method will be called sequential response surface method (AltairHyperOpt, User's Manual, 1997) (see Figure 3). Take a quadratic responsesurface

i�p� � i�p� � ai0 �Xm

j�1

aijpj�Xm

j�1

Xm

k�1

aijkpjpk �10�

Truemaxf

f(x)

RSmaxf Figure 2.

Ordinary responsesurface method

Truemaxf

f(x)RSmaxf

x

RS1

RS3

RS2

0 1 3 4 2

Figure 3.Sequential response

surface method

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The coefficients ai0; aij; aijk are determined using a least square fit of thefunction i�p� on the existing designs. In the first step a number of designss � 1�m is analyzed and the coefficients ai0; aij are determined. The next mdesigns are analyzed to determine the coefficients aijj which are the diagonalsof the Hessian. Additional designs are used to determine the remainingcoefficients, as it becomes necessary. After s � 1�m� �1�m�m=2 designs,an algorithm is used to weigh the designs.

The optimization procedure is as follows:

(1) Analyze the initial design and m perturbed designs;

(2) Use a least square fit to determine the polynomial coefficients for theobjective function 0�p� and each constraint i�p�;. If s � 1�m, then calculate ai0; aij;

. If 1�m < s < 2�1�m�, then compute ai0; aij; aijk; j � 1; sÿmÿ 1;

. If 1� 2m < s < 2�m� �1�m�m=2, then calculate ai0; aij; aijk andthe successive off-diagonal elements aijk;

. If s > 2�m� �1�m�m=2, then weigh the designs and calculateall ai0; aij; aijk;

(3) Solve for the approximate optimum design using mathematicalprogramming;

(4) Analyze the approximate optimum design;

(5) If the designs have converged, stop;

(6) Back to (2).

ApplicationsOptimization of a railThe design of a rail hit by a stone wall is used as the first example. Figure 4shows the deformed rail. The rail is clamped at the right end and loaded with astone wall of a mass of 1,000kg from the left side. The stone wall has an initialvelocity of 2m/s. The length of the rail is lx � 800mm.

The system needs to be designed such that it absorbs a maximum of energyduring the computation time of te = 0.03s. The displacement of the stone wall inx direction should be �lx = 50mm.

Problems of this kind are common in the case of the design of car bumpersand crash boxes. In many cases it is not possible to determine the optimumdesign using the trial and error approach. Optimization is the best approach toget a solution.

A sizing optimization has been performed using the thickness of six shellpatches as design variables (DVi, Figure 5). The initial thickness of eachelement was 2mm and could be modified from 0.2mm to 3mm.

The structural analysis to obtain the displacement and energy absorbed wasperformed using LS-Dyna (Hallquist, 1997). Table I summarizes the results.The code DoE (1997) provided the ordinary response surface solution using a

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Time step 31, t=3.000e–02

Displacements

>4.72e+01<4.72e+01<3.92e+01<3.13e+01<2.33e+01<1.53e+01<7.39e+00<–5.63e–01

max = 5.51e+01min =–5.63e–01

Figure 4.Rail: deformed initial

design

DV4

DV3

DV6 DV2

DV1

DV5

Figure 5.Rail: sizing optimization;

design variables

Initial design Ordinary RS Sequential RSLocal

approximation

DV1 [mm] 2.00 1.70 2.02 2.25

DV2 [mm] 2.00 1.92 1.57 2.95

DV3 [mm] 2.00 0.20 1.56 2.11

DV4 [mm] 2.00 1.95 1.65 1.94

DV5 [mm] 2.00 2.63 3.00 2.28

DV6 [mm] 2.00 3.00 2.48 2.28

Emax [Nm] 1010 1177 1254 1170

�lx [mm] 52.8 49.95 49.88 49.85

Table I.Rail: sizing

optimization results

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cubic response surface as (1). Altair HyperOpt (1997) was applied to obtain theoptimization solution using the sequential response surface method. Further,another optimum design using a local approximation approach with finitedifference sensitivity analysis has been obtained.

All three methods did yield approximately the same results for the structuralbehavior. The thickness distribution is somewhat different. This shows thatdifferent solutions to the problem exist.

In another investigation the same rail was designed using shape optimization.The optimization problem was the same as stated above. Again six designvariables are used. Figure 6 shows the iteration history, and Figure 7 shows theinitial and final shape. One can see that the target value of the displacement couldbe reached within a certain bound and the strain energy could be maximized.

AirbagIn another application, an airbag impactor test was simulated. The goal was tofind a parameter combination that would minimize the Head Injury Criterion(HIC), which is an occupant injury criterion that provides a measurement of theload on the occupant's head. The HIC is computed from the deceleration of theimpactor such that

HIC � maxt2ÿt1�36ms

1

t2 ÿ t1

Z t2

t1

�dImpactordt

� �2:5

t2 ÿ t1� � �11�

Its value should never exceed 1,000. The constraint was that the impactorshould not move more than 260mm. The design variables are typical for airbagdesign such as the vent orifice area, the permeability of the fabric and theignition time. Also, the mass flow of the gas has been modified using fivepoints on the mass flow time curve as design variables. Figure 8 shows the testsetup. The iteration history and the structural behavior is shown in Figures 9

EnergyDeflection

Figure 6.Rail: shape optimization.Iteration history

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and 10, respectively. The result was a parameter combination that reduces theHIC considerably by letting the displacement be far below the limit.

The dynamic analysis was performed using PAM-CRASH (1997). Theoptimum design has been obtained using the sequential response surfacemethod implemented in Altair HyperOpt (1997).

Roof crush layoutIn automotive design certain government regulations have to be fulfilled by thedesign. Numerical analysis is used to check if a design complies with these

Final Design

Initial Design

z

x

y

Figure 7.Rail: shape optimization

initial and final shape

ν0

Figure 8.Airbag: impactor setup

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rules to avoid expensive testing. Here the US standard FMVSS216 is of interest.This standard requires a test that simulates a roll-over of the car. The goal isthat the A-pillar withstands a certain load. Figure 11 shows the test as it isgiven by the standard.

HICDeflection

Figure 9.Airbag: iteration history

Iteration 22

Iteration 1

Iteration 1HIC = 870max a = 9.1 g

Iteration 22HIC = 605max a = 8.0 g

Figure 10.Airbag: responses,initial and final design

F

Plate

Figure 11.Roof crush testFMVSS216: test setup

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In the test setup a plate is pushed on the A-pillar with constant velocity. The load-displacement curve is measured. In a simulation the test setup is modeled usingfinite elements. Just the region of interest is modeled and not the whole structure.

The objective in this optimization is to minimize the mass of the design andstill hold the required load level. The load-displacement curve was discretizedin five points to define the constraints. The curves for the initial and finaldesign are shown in Figure 12. Design variables have been the shell thicknessof five material properties in the A-pillar and the roof frame.

It can be seen that the initial design was well below the bound for most of thesimulation time. The optimization did yield a design that would exceed the loadlimit. Usually a large number of design variations is used before a result isobtained. For the example problem 13 designs have been tried automatically.

The analysis has been performed using LS-Dyna (Hallquist, 1997) and theoptimization using Altair HyperOpt (1997).

Knee bolster designIn this example, rigid body simulation is used to analyze the occupant behaviorand to make initial package studies for a knee bolster. The use of optimizationwill be demonstrated (Schramm et al., 1997). For the investigation, just thelower extremities of the dummy are modeled (Figure 13). The dummy is loadedwith an acceleration pulse ax�t� resulting from a crash test according to the USgovernment regulation FMVSS208. The model of the initial design wasvalidated against a sled test.

The objective of the optimization is to minimize the femur load with a givenlimit on the knee penetration into the knee bolster. The optimization problemappears as

FFemur ��;�kx;�lx;�lz;��;� � ) min

� ��;�kx;�lx;�lz;��;� � � 0:07m�12�

Initial Design

Final Design (iteration 13)

Lower Bound Figure 12.Roof crush test: load-

displacement (time)curves for initial and

final design

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The design variables are defined in Figure 14. The optimization was performedusing the sequential response surface method. The results are summarized inTable II.

Figure 13.Knee bolster designmodel

KneebolsterZ

P

H-Point

fix

Xax

∆θ

∆kx

∆lx

∆lz ∆α

Figure 14.Knee bolster design:design variables

FV Lower bound Upper bound Optimal value

����� ±10.00 10.00 ±10.00�kx[mm] ±0.10 0.07 ±0.10�lx[mm] ±0.05 0.03 0.03�lz[mm] ±0.05 0.05 0.05����� ±5.00 5.00 ±4.65FFemur [N] Minimize 5332� [mm] 0.07 0.0617

Table II.Knee bolster design:results

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The analysis has been performed using MADYMO (1997) and the optimizationusing Altair HyperOpt (1997).

Concluding remarksThe results from computational analyses can be employed effectively for thedesign of structural systems and parts if structural optimization technology isused. Such methods reduce the effort for expensive design variations andcomparisons considerably. An objective oriented search for the best design ispossible if the design problem is formulated as a structural optimizationproblem. Of course, such a formulation requires the accurate analysis of thephysics of the problem and the sufficient determination of the input data suchas loading, boundary conditions, material data, and so forth.

Today, the application of structural optimization is not just limited to linearproblems, but it can also be applied to complex physical behavior, such asanalyzed in crash and occupant simulations. Therefore such methods are ofgrowing interest in the automotive industry. The examples in this paper showthat structural optimization is a valuable tool in structural design even ofstructures with complex nonlinear dynamic behavior.

References

Altair HyperOpt, User's Manual (1997), V4.0, Altair Computing Inc., Troy, MI.

Barthelemy, J.-F.M. and Haftka, R.T. (1993), ` Approximation concepts for optimum structuraldesign ± a review'', Structural Optimization, Vol. 5 , pp. 129-44.

DoE, User's Manual (1997), Altair Computing Inc., Troy, MI.

Hallquist, J.O. (1997), LS-DYNA, Theoretical Manual, Livermore Software TechnologyCorporation, Livermore, CA.

MADYMO, Theory Manual (1997), TNO Road Vehicle Research Institute.

PAM-CRASH, Theory Manual (1997), ESI Inc.

Schramm, U. and Pilkey, W.D. (1996), ` Review: optimal design of structures under impactloading'', Shock and Vibration, Vol. 3, pp. 69-81.

Schramm, U., SchoÈnfeld, V. and Baldauf, H. (1997), ` Package layout using MADYMO and adesign of experiments approach'', Proceedings of the 1st European MADYMO Users'Meeting, Heidelberg, pp. 121-30.