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Proceedings of TMCE 2006, April 18–22, 2006, Ljubljana, Slovenia, edited by I. Horvath and J. Duhovnikc© Organizing Committee of TMCE 2006, ISBN 961-6536-04-4

RAPID CONCEPT GENERATION AND EVALUATION BASED ON VAGUEDISCRETE INTERVAL MODEL AND VARIATIONAL ANALYSIS

Kazuhiro SaitouDept. of Mechanical Engineering

University of MichiganUSA

[email protected]

Zolt an Rus akImre Horv ath

Faculty of Industrial Design EngineeringDelft University of Technology

the Netherlands{z.rusak, i.horvath}@io.tudelft.nl

ABSTRACT

This paper presents a computational method forrapidly generating alternative product shapes andevaluating their structural responses during concep-tual design. It integrates the vague discrete intervalmodels (VDIM) as the representation of a set of al-ternative product shapes, and the variational analysis(VA) technique as the method for quickly approx-imating structural responses of alternative productshapes generated from the VDIM, based on the finiteelement analysis results of a nominal product shape.A simple case study on the two-dimensional crosssection of a chair demonstrates that the method is ca-pable of evaluating the structural responses of shapealternatives with a reasonable accuracy and negligi-ble computational overhead.

KEYWORDS

Conceptual design, concept evaluation, shape instan-tiation, structural analysis, approximation

1. INTRODUCTION

During the conceptual design phase, a designer ex-plores alternative design concepts and select one orseveral concept(s) to be examined further during thedetailed design phase (Ullman, 2003). This two-phase exploration process is widely accepted as astandard practice, since it facilitates the effective ex-ploration of a large design space where the close ex-aminations of all design alternatives are not practi-cal. It also naturally matches the thinking processof human designers, who tend to search for the em-

bodiments of design specifications by progressivelyrefining details.

For products whose primal functions are determinedby their shape, conceptual design requires rapid gen-eration of a large number of shape alternatives in acertain class of shapes, and their evaluations withrespect to desired design specifications. It poses aunique challenge to designers since they must eval-uate a shape alternative based on the incomplete de-scriptions of product geometry, which are being com-pleted during this very evaluation process.

As a solution to an aspect of this challenge, this paperpresents our first attempt towards a computer supportfor conceptual shape design of products with struc-tural specifications, such as stress and displacementunder certain loads. Considering that the completedetails of product shapes are yet to be finalized dur-ing detail design, it is hypothesized that the accu-racy of generated geometry is, if within a reasonablebound, is of less importance than the ease of gener-ating alternative shapes that allows interactive con-cept exploration. Similarly, the accuracy of struc-tural analysis, if within a reasonable bound, wouldbe of less importance than the speed of analysis thatallows interactive concept examination.

Based on this reasoning, the paper proposes a com-putational method for rapidly generating alternativeproduct shapes and evaluating their structural re-sponses during conceptual design. It integrates themethods of vague discrete interval modeling (VDIM)(Rusak, Z. and Horvath, I., 2005) to represent ofa set of alternative product shapes, and the varia-

149

tional analysis (VA) technique (CADOE, 2002) asthe method for quickly approximating structural re-sponses based on the finite element analysis resultsof a nominal product shape. A simple case study onthe two-dimensional cross section of a chair demon-strates that the method is capable of evaluating thestructural responses of shape alternatives with a rea-sonable accuracy and negligible computational over-head.

The rest of the paper is organized as follows: First,relevant literatures are reviewed. Next, the proposedmethod is described, followed by the results of a casestudy. Finally, the conclusion and future work aregiven.

2. RELATED WORK

2.1. Shape alternative generation forconceptual design

One approach to creating large number of shapealternatives is called evolutionary design. It sup-ports the designer by automatically generating a largenumber of shape mutations from an existing shapebased on genetic algorithms. The designer is not re-quired to understand the mathematical representationof the model, merely to use some fitness criterion ateach generation and guide the evolution of shapes.

Smyth and Wallace (2000) applied evolutionary de-sign approach to parametric surfaces, in which thestructure is defined by a skeleton model. To gener-ate shape alternatives, (a) first the designer chooses ahandle type, i.e. skeleton, lattice or bounding box ofthe shape, then (b) enters generic operation settingse.g. the mutation rate, finally (c) the algorithm gener-ates shape variances by alternating the parameters ofthe representation. Tauraet al. (1998) combined evo-lutionary design and procedural modeling, in whichshapes are represented by sets of rules instead of ge-ometrical data. When these rules are to be appliedto a given shape, an evolutionary approach is usedto determine a cell division model that best fits tocompute the results of shape manipulation. However,this cell division model is limited for well tessellatedshapes that do not contain any holes or incompleteshape components.

When interactive approach is applied to generateshape concepts, communication between the de-signer and the modeling tool need to be amplified.To help the designer to easily express his ideas, sev-eral attempts have been made that focused on imple-menting shape grammars for product design. First

efforts concentrated on only specific shape grammarsthat can be used to generate products, e.g. build-ings (Heisserman, 1994), and coffeemakers (Cagan,1998). By limiting the application field to specificproducts provided a better background for the re-searchers to develop highly sophisticated grammars.A general shape grammar based approach has beenproposed by Hsiao and Chen (1997), which facili-tates the designer to create shape variances in threesteps. Although, they addressed the issue of mappinglinguistic rules to shape manipulation commands,their solution was limited to a few rules.

Most of the above discussed methods are efficient toquickly generate alternative shapes, but their resultscannot be directly used for physically based evalu-ation due to the mismatch in their geometric repre-sentation for shape conceptualization and for finiteelement analysis. To address this issue, Rusak andHorvath (2005) proposed a particles system repre-sentation as a part of vague discrete interval model-ing (VDIM), which is summarized in the followingsection. This particle system representation inher-ently ensures that the same finite element mesh couldbe used to evaluate any shape instance that is derivedfrom the same vague discrete interval model.

2.2. Shape alternative evaluation forconceptual design

Even with today’s fast PCs, the finite element analy-sis of products with reasonable complexity can takeseveral minutes to hours. It is undesirable, there-fore, to conduct the finite element analyses of eachshape alternative generated during the conceptual de-sign. To facilitate interactive concept exploration, ap-proximation methods can be applied to rapidly esti-mate the structural responses of an alternative shapebased on the analysis results of one or several nomi-nal shapes. Major classes of approximation methodsfor structural analyses are surrogate models and re-analysis.

Surrogate models estimate the structural responses ofshape alternatives within certain ranges, by interpo-lating the analysis results of several nominal shapessampled within the range. Prior to the evaluation ofshape alternatives, the multiple sample shapes mustbe analyzed first, and then the parameters of the in-terpolation functions must be adjusted for the mini-mum estimation error of the sampled responses. De-pending on the type of interpolation functions, Arti-ficial Neural Network (ANN) (Cheng and Tittering-ton, 1994), Polynomial Regression, Kriging Method

150 Kazuhiro Saitou, Zoltan Rusak, Imre Horv ath

(Forsberg and Nilsson, 2005; Sakata,et al., 2003)and Radial Basis Function (Mechesheimer, 2001;Jin, et al., 2001) are popularly used. However, itcan potentially take a long time to construct an ac-curate surrogate model, since the overall accuracy ofthe estimation is affected by the number of samples,Therefore, these methods may not be suitable for theuse with concept exploration.

Reanalysis is a class of methods for approximatingthe structural responses of modified designs, basedon the analysis results of a single original design.In contrast to surrogate models that require multi-ple analyses (multi-point approximation), reanalysismethods only utilize a single analysis result (singlepoint approximation). Due to this character, reanaly-sis methods are more suitable for the concept explo-ration of shape alternatives within a neighborhood ofa nominal design.

Of most notable in structural reanalysis is CombinedApproximations (CA) method (Levy, 2000; Kirschand Papalambros, 2001a; Kirsch and Papalambros,2001b), which is based on the reduced-basis approx-imation of the nodal displacement vector in terms ofa binomial series. Initially developed for static lin-ear analyses, the method is later extended to nonlin-ear static analyses, eigenvalue analyses, and also ap-plied to calculate design sensitivities (Kirsch, 2002;Kirsch, 2003). Another class of reanalysis methodsis based on Taylor series approximation (Choi, andChang, 1994; Yeh and Vance, 1998; CADOE, 2002),where the derivatives of desired structural responseswith respect to design variables are calculated by us-ing Adjoint Method (Choi, and Chang, 1994; Yehand Vance, 1998) or Direct Method (CADOE, 2002).The variational analysis (VA) adopted in this paper isof this category.

3. PROPOSED METHOD

3.1. Overview

VDIM represents a class of non-parameterized ge-ometry by a set of distribution trajectories alongwhich the position of the points of an instance prod-uct geometry can be defined by the designer by ap-plying linguistic rules. This facilitates fast and inter-active modeling of variances of product geometries.VA technique approximates the displacements of avariant shape using the Taylor series expansion of thedisplacement vectors around the ones of a nominalshape computed by the conventional finite elementanalysis, The discrete nature of VDIM allows theseamless integration of the generation of alternative

shapes and their rapid evaluation using VA, withoutthe need of re-meshing product geometry. VA allowsvery quick estimation of structural performances ofvariant shapes without additional finite element anal-yses. The integration of these two techniques enablesthe interactive design modification and evaluation bythe human designer, greatly facilitating the conceptexploration for a class of products where both prod-uct geometry and structural responses are importantdesign criteria.

3.2. Vague discrete interval models(VDIM)

For the representation of the variances of individ-ual shapes as well as for modeling a cluster ofshapes in one single geometric model, the theoryand methodology ofvague discrete interval model-ing (VDIM) was developed. VDIM is vague in thesense that it models (i) a cluster of shapes with asingle representation allowing the integration of anominal shape with its domain of variance, (ii) de-scribes the structural relations between shape com-ponents, which represent the shape either completelyor incompletely, and (iii) supports manipulation ofan evolving shape by means of dedicated modelingmethods and tools. It is discrete since the represen-tation of the geometry is composed from discrete en-tities. Finally, VDIM is an interval modeling tech-nique, since it describes the domain of variance ofthe shape by a finite interval around a hypothesizednominal geometry.

The fundamental modeling entity of VDIM is theparticle,π, or more precisely, coupled pairs of par-ticles, c(π1, π2). Coupling of particles makes it pos-sible to introduce various physical relationships andconstraints in order to provide the means for a phys-ically based manipulation of shapes and for the in-vestigation of their behavior as physical objects. Ina general form, a particle is represented by its refer-ence vector,

:

r , and its metric occurrence,E. Refer-ence vectors provide positional information and met-ric occurrence and couplings supply morphologicalinformation for the geometric representation. Fig-ure 1 shows the fundamental entities of VDIM. Avague discrete interval model is vague since the ref-erence points of the particles are uncertainly speci-fied. In practice it means, that a metric occurrenceis defined as a finite distribution space and mergedin the shape model represented as a particle system.Note that a particle system contains boundary andinternal particles, but internal particles do not have

RAPID CONCEPT GENERATION AND EVALUATION 151

metric occurrences. Internal particles are not used inthe vague representation for the reason that they onlyplay a role in physically based modeling.

Figure 1 Fundamental entities of VDIM

Having uncertain positions for particles, introducesweakness in the topological relationships betweenthe neighboring elements, since they can have dif-ferent neighbors depending on their actual position.In practice to use the same finite element mesh foreach instance shape generated from the same VDIMmodel, the topological relationships of any neighbor-ing particle has to be the same. To resolve this is-sue a characteristic discreteness,ε, is established inthe vague discrete model, which is the distance be-tween the neighboring particles. Discreteness of themodel comes from the fact that the particles them-selves never coincide. Nonetheless, the condition ofdiscreteness does not assure topological robustness,since the actual discreteness depends on the actualpositions of the particles so that a set of particlescan be neighbors with a characteristic discretenessin a given position, but in another position they arenot. Figure 2 illustrates the problem. To resolve thisissue, the metric occurrence of the particles is gen-erated in a way that avoids these situations. Hav-ing topologically robust representation assures thatself-intersecting shapes are not instantiated from thevague model, and coupling relations of particles arethe same for each instance.

The minimal and maximal overlaying surfaces aregenerated on the extremes of distribution specified bythe boundary particles of the vague discrete model.The distribution interval is a subspace that is in be-tween the minimal and maximal overlaying surfaces.Thus, it represents a cluster or a family of possibleshapes rather than a single nominal shape. The im-ages (actually, image points) of the particles on the

two overlaying surfaces of the distribution intervalare connected by so-called distribution trajectories,

:

τ . This is a sort of simplification of the represen-tation, that enables to have 1D metric occurrences.Having a 1D metric occurrence reduces the requiredinformation over a vague shape as well as it regulatesthe discreteness of each shape that is represented inthe vague interval model.

Figure 2 Topologically weak vague discrete shape

To generate interval models with robust topologicalrelationships, we have developed an algorithm that isshown in Algorithm 1. The input of this algorithm istwo point sets, which define the minimal and maxi-mal overlaying surfaces of a vague discrete intervalmodel. In the first step of the algorithm, the closestpoints of the two point sets are looked for, and an ini-tial distribution trajectory is generated between them.Then ascending order of the points is computed foreach point set based on the shortest path from the endpoints of this initial distribution trajectory. Follow-ing this ascending order, distribution trajectories aregenerated one by one. In case a newly generated dis-tribution trajectory causes topological anomaly in thestructure of the interval model, then the next point isselected as a candidate for generating a distributiontrajectory. The algorithm stops when each point ineach point-set has been assigned to at least one dis-tribution trajectory.

To identify topological anomalies in a particle cloud,Π, the geometric relation of the distribution trajecto-ries is investigated between particles being closer toeach other than 2|

::

τ |. We can distinguish three rela-tions of distribution trajectories,

:

τ1 and:

τ2, namely,that they: (a) are parallel, (b) intersect, and (c) areskewed. In case

:

τ1 and::

τ2 are parallel, the topo-logical relation between each instance particle is thesame. If they intersect, the topological relationshipamong the particles in the neighborhood of

:

τ1 and

:

τ2, is influenced by the position of the selected in-stance particle(s), depending whether it is above or


below the point of intersection. For skewed:

τ1 and

:

τ2, the topological relation to the neighboring parti-cles is influenced by the position of the instance parti-cle relative to the joining point of the normal traversalbetween

::

τ1 and::

τ2. The problem is illustrated in Fig-ure 3. We have to investigate whether the intersectingpoint or the normal traversal of

::

τ1 and:

τ 2 falls intothe distribution interval. Topological anomalies areremoved from the model, and new distribution trajec-tories are searched for those points that are not partof any other distribution trajectory.

Figure 3 Occurrence of topological anomalies (pointsconnected by the dark triangle are in differ-ent neighboring relations below and above thenormal traversal)

3.3. Variational Analysis (VA)

Variational Analysis (VA) (CADOE, 2002) in lin-ear static analysis1 is a type of reanalysis methodbased on Talylor series approximation of the dis-placement vector with respect to key design vari-ables. The uniqueness of the VA over the otherTaylor-series based reanalysis methods is the useof symbolic derivatives of the element stiffness ma-trices in computing the derivatives of the displace-ment vector. This allows scalable computational ef-ficiency, approximation accuracy, and numerical sta-bility, compared to the other methods that numeri-cally evaluate derivatives such as finite differencing.

Let u(x) be the nodal displacement vector of the finiteelement model of a product geometry represented bydesign variablex, andx0 be the value of design vari-able representing the nominal product geometry. VAestimates the nodal displacement vectoru(x0 + ∆x)of a modified geometryx0 + ∆x using Taylor series

1While the description and case study in this paper assumeslinear static structural analysis, VT can generalize to other typeof analyses (CADOE, 2003).

Algorithm 1: Generating interval from two point-sets (P1and P2)based on topologically closest points

Match Basedon TopologicalRelations(P1, P2, Π)If(Π empty) πc=Find GeometricallyClosestPoints(P1, P2)Elseπc=Find ShortestDistribution Trajectory(Π)P12= P1+ P2

RemovePointsIf In ParticleCloud(P12, Π)Sort PointsAscendingBy ShortestPath(P12, from πc)For p=eachpoint of P12

Pn=CollectCloseNeighbors(p, P1 (if p ∈ P1), P2 (if p ∈ P2))Πn= CollectParticlesThatContainPoints(Πsgn)Pm=CollectMatchedPointsOfCloseNeighbors(If pn=rn return

pm=rn + τn; If pn=rn + τn return pm=rn)Pm=CollectCloseNeighbors(pm, P1 (if pm ∈P1),

P2 (if pm ∈P2))Sort PointsAscendingBy Distance(Pm, from p)For pm=eachpoint of Pm

πm =GenerateDistributionTrajectory (pm, p)addπm to Πn

Πanomalies =CheckTopologicalAnomalies(Πn)If (Πanomalies <>0)removeπm from Πn

go to next point in Pm

Elseaddπm to Πgo to next point in P12

end forend for

approximation:

u(x0 + ∆x) = u(x0) +n

X

i=1

1

i!u

(i)(x0)∆xi + O(∆x

n+1) (1)

where u(i)(x0) is the i-th order derivative ofu(x)evaluated atx = x0. The displacement vectoru(x0)of the nominal designx0 is obtained by solving thestatic equilibrium equation:

K(x0)u(x0) = f(x0) (2)

where K(x0) andf (x0) are the global stiffness matrixand the global load vector, respectively, of the nomi-nal designx0.Note that bothK andf are the functionsof design variablex.

To evaluate the right hand side of Equation (1), thederivativeu(i)(x0) needs be computed. In VA, thisis done by exploiting the mathematical properties ofthe equilibrium equation, instead of numerically esti-mation. By differentiating the both sides of the equi-librium equation for arbitraryx, one can obtain:

K(1)(x)u(x) + K(x)u(1)(x) = f (1)(x)

K(2)(x)u(x) + 2K(1)(x)u(1)(x) + K(x)u(2)(x) = f (2)(x)

...n−1P

i=0

„

ni

«

K(n−i)(x)u(i)(x) + K(x)u(n)(x) = f (n)(x)

(3)


or equivalently:

K(x)u(1)(x) = f (1)(x) − K(1)(x)u(x)

K(x)u(2)(x) = f (2)(x) − K(2)(x)u(x) − 2K(1)(x)u(1)(x)

...

K(x)u(n)(x) = f (n)(x) −n−1P

i=0

„

ni

«

K(n−i)(x)u(i)(x)

(4)

It can be seen from Equation (4) that:1. The left hand sides of all equations are in the form

Kv (a vectorv multiplied by the stiffness matrix),same as the original equilibrium equation (2).

2. The right hand side of thei-th equation (the equa-tion with K (x)u(i)(x) in the left hand side) de-pends only onf (i)(x), K (i)(x), and u(j)(x), j=1,. . . ,i-1.

The first observation leads to the efficient solutionof Equations (4) by utilizing the decomposed stiff-ness matrix obtained while solving Equation (2). Thesecond observation leads to the efficient evaluationof the right hand sides of Equations (4), by recur-sively substitutingu(j)(x0), j = 1, 2, . . . ,i-1 to thei-th equation starting withi = 1, oncef (i)(x0) andK (i)(x0) are obtained.

The calculations off (i)(x0) andK (i)(x0) are done byevaluating the analytic equations of thei-th deriva-tives of the boundary and loading conditions and theelement stiffness matrix, respectively, with respect todesign variablex. As an illustration, let us considera two-dimensional constant strain triangle (CST) el-ement and suppose the design variablex is the nodalcoordinates:

x = (x1, y1, x2, y2, x3, y3)T (4)

The element stiffness matrix of a CST element isgiven by (Chandrupatla and Belegundu, 1997):

K e(x) = tA(x) BT (x) D B(x) (5)

wheret andA(x) are the thickness and area of theelement, respectively,B(x) is the strain-displacementmatrix, andD is the stress-strain matrix. They aregiven as:

A(x) =1

2|∆| (5)

B(x) =1

∆

y23 0 y31 0 y12 00 x32 0 x13 0 x21

x32 y23 x13 y31 x21 y12

(6)

D =E

1 − ν2

1 ν 0ν 1 00 0 (1 − ν)/2

(7)

where:

∆ = x13y23 − x23y13

xij = xi − xj

yij = yi − yj

(8)

Using Equations (5)-(8), the first derivative of the el-ement stiffness matrixK e(x) with respect tox1, forexample, is given as:

∂Ke(x)

∂x1= t

∂A(x)

∂x1B

T (x)DB(x)

+ tA(x)∂B

T (x)

∂x1DB(x)

+ tA(x)BT (x)D∂B(x)

∂x1

(9)

where

∂A(x)

∂x1=

{

y23 if ∆ > 0−y23 otherwise

(10)

∂B(x)

∂x1=

1

∆2

0

B

B

B

B

B

@

−y23y23 0 −x32y23

0 −x32y23 −y23y23

−y31y23 0 ∆ − x13y23

0 ∆ − x13y23 −y31y23

−y12y23 0 −∆ − x21y23

0 −∆ − x21y23 −y12y23

1

C

C

C

C

C

A

T

(11)

Basic steps for computingu(x0 + ∆x) for given∆xcan be summarized as follows:1. Solve Equation (2) to obtain the displacement

vectoru(x0) for the nominal product shape.2. For i = 1, 2, . . . ,n, compute thei-th derivative

∂iK e(x)/gxiof the element stiffness matrix andthe i-th derivative∂if(x)/gxi of the load vector,using analytical equations.

3. Fori = 1, 2, . . . ,n, assemble∂iK e(x)/gxi to thei-th derivative∂iK (x)/gxiof the global stiffness ma-trix.

4. Fromi = 1 throughi = n, recursively solve Equa-tions (4) by using the decomposed global stiffnessmatrix K (x) obtained in step 1, and by substitut-ing u(j)(x0), j = 1, 2, . . . ,i-1 to the right handside of thei-th equation.

5. Computeu(x0 + ∆x) by substitutingu(x0) ob-tained in step 1 andu(i)(x0), i = 1, 2, . . . ,n, ob-tained in step 4, into Equation (1).


Since the global matrix is solved only once at step1 and the subsequent steps are only assembling ma-trices and evaluating polynomials,u(x0 + ∆x) canbe obtained with far smaller computational cost thansolving Equation (1) forx = x0 + ∆x.

The choice ofn depends on the desired accuracy inEquation (1), whichn = 2 to 4 is reported as suffi-cient in most cases. Since the derivation of the an-alytical equations up to then-th derivative can beextremely tedious and complex (especially for ele-ments with large DOF), higher order derivatives canbe approximated by a numerical method such as fi-nite differencing.

4. EXAMPLES

VDIM and VA have been implemented as computersoftware, and a case study with a two-dimensionalcross section of the seats of chairs is conducted todemonstrate the feasibility of the method. The finiteelement analysis and VA are implemented for CSTelements withn = 2. CST elements are chosen asa first attempt since they are relatively simple to im-plement VT due to the small DOF and compatibleto the triangulated surfaces in the shape instances ofVDIM.

4.1. VDIM

As an application of VDIM, three popular designs ofthe seats of chairs have been reproduced, and their2D cross sections have been tested. The photos ofFigure 5 illustrate the targeted seats. To produce thetargeted shapes with VDIM, first a vague model wascreated as a composition of planar point-sets. Thevague model had to be carefully chosen to be ableto derive each targeted shape from it. This requiredgenerating a large interval. The left top image of Fig-ure 5 presents the vague model of a seat. In the nextstep a number of regions have been selected by usinga so called 3D cursor. Figure 4 shows the 3D cursorin action. Particles, that are in contact with the the 3Dcursor during the selection procedure are assigned toa region. The selected regions represent various fea-tures of the targeted shapes. To derive the instanceshapes, shape formation rules of curving, offsetting,and tilting were applied to the regions. Interestedreaders are referred to (Rusak and Horvath 2005) fordetails on shape instantiation based on shapes forma-tion rules. The derived instance seats are presentedon Figure 5.

Figure 4 Region selection with a 3D cursor

Figure 5 Vague shape model of a seat and some in-stance shape variances

4.2. VA

The structural performances the shape alternativesinstantiated from VDIM are examined by using VA.As an initial implementation, 2D cross sections ofthe seats are used for estimation and the results byVA are compared with the finite element analysis interms of accuracy and speed.


Figure 6a shows the cross-section of the VDIM ofseats in Figures 4 and 5, and its nominal shape.The dimension of the seats is modeled as approxi-mately 700 mm in length and height, and 500 mmin width. As seen in Figure 6a, the shape intervalsare defined only to the points on the seating surface,with the range between 15 mm and 30 mm. Figure 6bshows the boundary and loading conditions used forthe evaluation of structural responses. These condi-tions intend to mimic the loads exerted by the personwith 70 kg of weight sitting on the chair. For sim-plicity, the loads are assumed constant during the in-stantiation of different shape alternatives,i.e., f (i)(x)≡ 0 in Equation (4).

(a) (b)

Figure 6 (a) cross-section of the VDIM of seats and (b)boundary and loading conditions

Figure 7 shows a snapshot of the developed softwareimplementing VA. The upper left window shows thedisplacement of the nominal shape obtained by theconventional finite element analysis, and the rest ofthe windows show the displacements of the shape al-ternatives derived from the VDIM in Figure 6a, esti-mated by VA utilizing up to the second-order termsin the Taylor series expansion. For clarification, dis-placements are magnified to 5 times. Although notshown in the figure, stress values estimated by VAcan also be visualized. Using this user interface, adesigner can quickly examine the structural perfor-mances of alternative shapes at virtually no compu-tational overhead.

In order to statistically access the accuracy and speedof VA, samples of 100 shape variants are randomlyinstantiated from the VDIM in Figure 6a, and the es-timation by VA are compared to the results of the fi-nite element analyses. The comparisons are made bycalculating the displacement errorseu and von Misesstress errorseσ averaged over all nodes in a sampleshape variant:

eu = 16m

6m∑

i=1

|ui−vi|vi

eσ = 1m

m∑

i=1

|σi−si|si

(12)

wherem is is the number of elements in the FE mesh,uiand vi are thei-th element of the (global) nodaldisplacement vectors obtained by VA and FEM, re-spectively, andσi andsi are von Mises stresses ofthe i-th CST element obtained by VA and FEM, re-spectively. Table 3.3 showseu andeσ averaged overthe 100 samples. It also shows the CPU time of the100 runs for VA and FEA on a Windows laptop.

Figure 7 Developed software visualizing the displace-ments of the nominal shape (upper left), andshape variants (rest).

The statistics show that von Mises stresses have 22%error on average, whereas the nodal displacementshave error as large as 45%. While the error in vonMises stress is within an acceptable range (as stan-dard FE is known to have an error as large as 10-20%), the displacement error is fairly large, with thenotably large standard deviation (250%!).

Figures 8 and 9 show the displacement errors andvon Mises stress errors of the 100 samples, plottedagainst the average and maximum coordinate devia-tions from the nominal shape. In these figures, thesquare and triangle points indicate the average coor-dinate deviation and the maximum coordinate devia-tion, respectively. The plots show no obvious corre-lation between the errors in structural responses and


Table 1 Comparison of VA and FEA for 100 randomsamples

quantities avg of 100samples

stddev of100 samples

Average displacementerror [%] 45 250

Average von Misesstress error [%] 22 34

CPU time for VT [ms]560 35

CPU time for FEA [ms]3900 97

the deviations from the nominal shape. On the otherhand, Figure 8 shows the several (6) samples withextremely large errors, with the rest of samples inone small cluster. These extreme samples are a likelycause of very large standard deviation (250%) of thedisplacement errors.

Figure 8 Average displacement errors vs. average co-ordinate deviations (squares) and vs. max-imum coordinate deviations (triangles) fromthe nominal shape

The close examination of each sample shape revealsthat that the large errors in the displacements occurnear the horizontal seat of the chair where the mag-nitude of displacements is almost zero compared tothe size of the chair. For the sample shapes withrelatively large shape variations at the seat, the dis-placement errors of the nodes on the seat can be verylarge sincevi ≈ 0 in Equation (12). The errors in vonMises stresses in the element on the seat, on the otherhand, are not as large, since the stress is constant in a

Figure 9 Average von Mises stress errors vs. averagecoordinate deviations (squares) and vs. max-imum coordinate deviations (triangles) fromthe nominal shape

CST element, which seems to average out the effectsof errors in nodal displacements.

In terms of computational speed, VA clearly showsthe advantage over FEA, with 7-fold improvement inthe average CPU time. Since VA requires only oneFE analysis of the nominal shape and the estimationsfor the shape variants are done by evaluating poly-nomials, this speed gain is expected to scale up wellwith the number of nodes in the finite element model.

5. CONCLUSION AND FUTURE WORK

The paper proposed the integration of the vague dis-crete interval models (VDIM) and the variationalanalysis (VA) technique, in order to realize a designenvironment for rapidly generating alternative shapesand evaluating their structural responses. While theaccuracy of VA has some concern in the presentedexamples, it is likely due to the naıve implemen-tation by the authors, considering the reported ac-curacy of the highly-tuned, commercial 3D imple-mentation of the technique (CADOE, 2002). SinceVDIM was originally developed for 3D (Rusak andHorvath, 2005), its integration with the 3D versionof VA, possibly using a commercial code, will proveto be a very effective tool for concept exploration forpractical designs.

ACKNOWLEDGMENTS

The first author acknowledges the University ofMichigan for partially supporting his sabbatical leaveat the Delft University of Technology.


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