Multi-material topology optimization with strength constraints

Struct Multidisc Optim (2011) 43:597–615DOI 10.1007/s00158-010-0581-z

RESEARCH PAPER

Multi-material topology optimization with strength constraints

Anand Ramani

Received: 30 November 2009 / Revised: 13 September 2010 / Accepted: 9 October 2010 / Published online: 20 November 2010c© Springer-Verlag 2010

Abstract A heuristic approach to handle strength con-straints based on material failure criteria in multi-materialtopology optimization is presented. This is particularlyadvantageous if different materials have different failurecriteria. The change in the material failure function in anelement due to a contemplated material change is estimatedwithout the need for expensive matrix factorizations. Thischange is used along with the changes to the objectiveand deflection-based constraint functions, computed usingpseudo-sensitivities, to determine a single aggregated rank-ing parameter for the element. Elements are ranked onthe basis of their ranking parameters and this rank is usedto modify the material ID-s of a few top-ranked elementsduring an optimization iteration. The working of the algo-rithm is demonstrated on a few example problems showingits effectiveness and utility in deriving optimal topolo-gies with multiple materials in the presence of stress andstrain-based failure criteria, in addition to the conventionalstiffness-based constraints.

Keywords Topology optimization · Multi-materialstructures · Stress constraints · Failure criteria

1 Introduction

Topology optimization methods for stiffness or compliance-driven synthesis of structures have evolved over the last

A. Ramani (B)India Science Lab, General Motors Global Researchand Development, GM Technical Centre India Pvt Ltd,Creator Building, International Tech Park Ltd.,Whitefield Road, Bangalore 560 066, Indiae-mail: [email protected]

couple of decades. Recent research attention has focusedon extending these methods for stress-constrained designs,nonlinear and dynamic problems, etc. Stress-constrained

topology optimization problems have been solved in liter-ature, but have been largely restricted to a single-materialsystem. In this paper, a methodology for topology optimiza-tion with multiple discrete materials, based on deflectionand strength constraints defined by general failure crite-ria, is presented, so that different materials with differentfailure criteria can be considered simultaneously in thetopology design process. Such problems are encountered inthe design of large structural systems with several candi-date materials. The pseudo-sensitivity approach developedearlier (Ramani 2009), in which the material is retained asa discrete variable, is utilized and a heuristic methodologyfor handling failure criteria is developed as an enhancementto the stiffness-based optimization. This is accomplishedby performing an efficient estimate of the change in anelement’s failure function due to a contemplated materialchange, without the need for expensive matrix factoriza-tions. The estimation uses the loading environment aroundthe element to compute the change in the failure func-tion. Following this computation, the strength constraintis treated in a similar manner as the deflection constraint,so that the pseudo-sensitivity approach and resubstitutionmethod developed by Ramani (2009) can be used withappropriate modifications. First, a short background onstress-constrained topology optimization is presented alongwith a review of literature on the subject, and a case ismade for developing an efficient method for incorporatingstrength-based constraints ab-initio in multi-material topol-ogy optimization. Following this, the optimization problemwith strength constraints is stated and the solution method-ology is described in detail along with the algorithm imple-mentation. Finally, several numerical case studies involving

598 A. Ramani

single material and multi-material optimization are providedto demonstrate the working of the methodology and theresults are discussed showing the efficacy of the method.

2 Background and literature review

Research in topology optimization in the initial stages in the1990’s focused on optimizing topologies for stiffness-baseddesigns. This, in itself posed several research challenges,which were overcome by the development of methods likethe homogenization method (Bendsøe et al. 1988), theSIMP method (Zhou and Rozvany 1991; Bendsøe 1989),evolutionary approaches (Xie and Steven 1997; Querinet al. 1998) and heuristic approaches (Goldberg 1989;Kirkpatrick 1984). During the development of these meth-ods, the ability to consider allowable stresses or other failurecriteria in topology optimization did not receive significantresearch attention, since it was difficult to incorporate suchlocal strength considerations in the optimization process. Atypical ‘work-around’ was to focus initially on stiffness con-siderations to arrive at load-paths and the general materiallayout to cater for them, followed by local shape and sizeoptimization exercises to account for the material strengthin these regions. However, in recent times, there is anincrease in research attention in the consideration of mate-rial strength in the topology design of structures, becauseof an increasing need to avoid modifications in later stagesof product development, thus needing consideration of sev-eral load cases upfront in the topology synthesis process. Itis realized that concurrent consideration of such constraintsas strength and failure at the initial stage itself can leadto better topologies. Additionally, advances in numericaltechniques and computational power have made the exten-sive computations needed for such concurrent considerationpotentially feasible. Also, with the availability of novelhigh-strength alloys and light-weight materials, there is aneed to consider multiple materials with different failurecriteria, simultaneously in the topology design process.

Many types of strength or failure criteria are used inpractice and are usually based on local stresses (such asvon Mises or maximum principal stress criterion), strainsor strain energy. The difficulties in the concurrent imple-mentation of strength or failure considerations in topologyoptimization are as follows:

Unlike global stiffness, strength considerations requirethe stress (or strain) field which varies over the structure.Thus, in a finite element (FE) context, this involves evaluat-ing such a field in every element of the FE mesh leadingto a considerably large number of stress (or strain) con-straints. Further, as shown by Kirsch (1990), the presenceof stress constraints can lead to singular optimum solu-tions and this implies that the optimum topology cannot be

achieved by a conventional optimizer using gradient basedmethods. Also, most of the commonly used topology opti-mization techniques employ the SIMP method (Zhou andRozvany 1991) which does not allow a straight-forwardinterpretation of “allowable stress” or “strength” for mate-rials with intermediate densities. Further, a stress constraintfor a specific material may become invalid when the elementis converted to a void. Also, in the multi-material sce-nario, different materials may have failure criteria definedin terms of different parameters (such as stress, strain, strainenergy, etc).

Literature on incorporating general failure criteria intopology optimization with multiple-materials is not avail-able. However, several approaches for incorporating stressconstraints in single-material topology optimization havebeen presented in the literature. Early attempts in address-ing stress constraints, such as Saxena and Ananthasuresh(2001), consider truss topologies and have limited utilityfor continuum applications. Amongst early approaches isthe fully-stressed design (FSD) approach for trusses (Makrisand Provatidis 2002; Patnaik and Hopkins 1998). The phi-losophy of FSD is that the optimum material distributionfor a structure is attained when every portion of the struc-ture is stressed to the maximum allowable limit. This can beextended to a multi-material scenario as well. The difficultywith the FSD approach arises when multiple load casesare considered and when stress constraints are combinedwith deflection or stiffness-based constraints. Under suchconditions, material removal or addition needs to accountfor the multiple loading conditions and also needs to con-sider the effect of the change in the deflections or stiffness.The fully-utilized design approach (Makris and Provatidis2002) addresses both deflection and stress constraints byscaling of the cross-sectional members, but extension ofthis to continuum problems with multiple materials is notstraight-forward.

Attempts to use stress constraints in the more com-monly used density-based homogenization approach witha power-penalized stiffness model had to overcome thephenomenon of stress singularity (Duysinx and Bendsøe1998; Pereira et al. 2004; Bruggi and Venini 2008). Toquote from Pereira et al. (2004), “. . . if homogenized defor-mations remain finite, local stresses (used in the failurecriterion) also remain finite. On the other hand, when thedensity goes to zero, high deformations may occur due tothe low stiffness and, large, although finite, local stressesare obtained . . . As a consequence, local constraints satu-rate and lock the material-removal process. This is knownas the Stress Singularity Phenomenon . . . .” The interpreta-tion for failure stress in a homogenized material is madein Duysinx and Bendsøe (1998), in which a technique forovercoming the stress-singularity is proposed by studyingthe stress states of rank-2 microstructures and proposing an

Multi-material topology optimization with strength constraints 599

empirical model for relevant stress criteria for porous com-posite materials. This empirical model is a modified formof the density-based power-penalized stiffness model usedin the homogenization approach. Several other techniquesfor handling the stress singularity have been proposed. Forexample, Bruggi and Venini (2008) present a hybrid-FEhomogenization approach to handle the singularity by relax-ation. An alternative approach has been proposed by Leet al. (2010) in which the SIMP method has been extendedto also interpolate the stresses between minimum and maxi-mum values. In Navarrina et al. (2005), the stress constraintsingularity is overcome by rewriting the stress constraintinequality in terms of the effective stress and multiplyingthe inequality by the relative density.

Methods to circumvent the stress-singularity problemare also found in the literature. In Chiandussi et al. (2004),Chiandussi (2006), resizing rules to handle the stress con-straint based on strain energy ratio are formulated, thusavoiding the problem of stress singularity. In Chiandussi(2006), the stress constrained topology optimization prob-lem has been solved with the maximization of potentialenergy using optimality criteria with various allowablemean stresses. In Fancello (2006), another approach basedon homogenization and augmented Lagrangean techniquefor considering local stress constraints is presented. How-ever, it requires the minimization of a functional at everyiteration, which can be expensive when large problems areconsidered.

Methods for globalizing the stress constraints usingnorms and relaxation techniques (Duysinx and Sigmund1998), an independent continuous mapping (ICM) methodsuch as in Sui et al. (2006), Ye et al. (2006), etc. have beenproposed. Such a globalization allows many constraints tobe clubbed into a single constraint equation, which is thenused in lieu of the individual stress constraints. This oftencarries a penalty in terms of relaxation of the stress con-straints on the individual elements. A compromise approachhas been presented by Le et al. (2010), which combinesboth global and local stress constraints and also introducesa new concept of ‘regional constraints’ on stresses. Somestress constraint globalization functions have been proposedin Maute et al. (1999), Yang and Chen (1996). In Paytenand Law (1998), a globalization approach with a feedbackderivative (global maximum von Mises stress) analogous toa Lagrange multiplier, is used to keep the solution bounded.

Evolutionary approaches have also been considered forstress constraints in topology optimization. For a criticalreview of evolutionary structural optimization, the readeris referred to Rozvany (2009). In an evolutionary approachproposed by Miao and Bernitsas (2006), a nested algo-rithm is presented in which an outer loop deals with thetopology evolution and aims to redistribute material andredefine material properties, while an inner loop forces the

performance constraints. This requires an optimization sub-problem to be solved at each iteration. Another approachbased on evolution is discussed in Liang (1999) in whicha current design is scaled using a performance index cal-culated from the ratio of element stresses to the maximumallowable stress for the material. This is similar to the resiz-ing rules used in Chiandussi et al. (2004). Elements to beremoved are decided based on the value of the performanceindex. A metamorphic approach, which is in some sensea reverse of the evolutionary approach, is presented in Liuet al. (2000), but details of how the stress constraints arehandled are not provided.

Stress constraints have also been incorporated in inte-ger programming methods. In the approach presented inSvanberg and Werme (2007), a sequence of sub-problemsinvolving sequential linear integer programming andsequential quadratic integer programming methods havebeen used. This method requires every sub-problem to besolved as an optimization problem. Since the problem sizespresented are essentially small, it is not clear how the algo-rithm would work for large structural problems, if duringevery iteration an optimization sub-problem is to be solved.Moreover, the method starts with a coarse mesh which isgradually refined over the course of iterations. This maybe cumbersome to implement in a general finite elementanalysis.

Hierarchical methods have also been applied for topol-ogy optimization in general and in particular, to considerstress constraints. In Stolpe and Stidsen (2007), a hierarchi-cal optimization approach is presented using two meshes,one which defines the topology of the structure and isrefined when needed and the other, a fixed analysis meshon which the constraints are computed. Also, a branch andbound strategy is adopted, which can be very expensive forlarge meshes.

Phase-field approaches have been considered for topol-ogy optimization recently, but the accommodation of stressconstraints is very cumbersome with this method. In Burgerand Stainko (2006), a relaxation scheme based on phase-field approach has been presented, which yields a large-scale optimization problem with a large number of linearinequality constraints. Such schemes may become pro-hibitively expensive for large structural problems.

The issues mentioned above are compounded for thecase of multi-material topology optimization. Hardly anyapproach for handling stress constraints in a multi-materialtopology optimization is available in literature. There aremethods to perform stress constrained free-material or func-tionally graded material optimization such as in Stump et al.(2007), Kocvara and Stingl (2007). In Stump et al. (2007),the stress in the graded material is treated as the arith-metic mean of the stress in the two distinct phases, usingwhich a unique stress failure criterion is proposed to avoid

600 A. Ramani

the stress singularity phenomenon. In Kocvara and Stingl(2007), an algorithm based on a generalized augmentedLagrangean method is presented for free-material optimiza-tion. In another related work by Zhou and Li (2006), afiber-reinforced orthotropic composite is employed as thematerial model to simulate the constitutive relation oftruss-like continua, as an alternative to the density-homogenization technique. The fiber densities and orien-tations at the nodes are taken as design variables. First, foreach load case, the fiber orientations are aligned with theorientations of the principal stresses and the fiber densitiesare adjusted according to the strains along the fiber orien-tations. This involves the solution of non-linear equationsin each iteration, which may be computationally intensive.Further, its extension to multi-material topologies is notstraight-forward.

In the current approach, because of the need to considerdifferent failure criteria for different materials, the treatmentof stress constraints based on the homogenization techniqueis not directly applicable. Also, the problem of dealing withstress singularity is automatically avoided if the materialvariable is retained in the discrete form. Also, most meth-ods in the literature use the structural compliance as theobjective function to be minimized, subject to a volumefraction constraint. This way of posing the problem has aninherent difficulty in handling multiple load cases. In thepresent approach, this difficulty is circumvented by choos-ing the objective function as the structure mass and treatingthe compliances for multiple load cases as multiple con-straints. This way, multiple load cases can be easily handled.The problem statement and the solution methodology aredescribed in detail in the following sections.

3 Problem definition

The topology optimization problem with strength-basedconstraints defined by failure criteria for multiple materialsis stated as follows:

Given,

– a design space (or a packaging envelope) � for astructure, which may be in R2 or R3, dependingupon whether the problem is in 2-D or 3-D space,

– a choice of ‘m’ candidate materials, μk , k = 1,2, . . . m, with corresponding densities ρk , and withμ1 being the void material, and

– an objective function fo(μ) which defines the massof the structure

Find μk(P), for every point P e �, that minimizesthe objective function fo(μ) which, for the mass

minimization problem considered here, can be writ-ten as

fo(μ) =∫

�

ρk (P)d� (1)

Subject to Nc deflection-based constraints and Ns strength-based constraints defined by constraint func-tions f ’s as

r fc(μ) ≤ 0, r = 1, 2, . . . , Nc and

r fs(μ) ≤ 0, r = 1, 2, . . . , Ns ,

where subscripts ‘c’ and ‘s’ are used fordeflection-based and the strength-based con-straints respectively, and the left-superscript‘r ’ denotes the r th constraint.

To eliminate an all-void topology and non-structuralsolutions, it is assumed that at least one deflection con-straint is specified so that Nc is at least 1. Thus, the probleminvolves determining the optimum topology of the structure,along with the optimum material distribution in the topologyunder both the deflection and strength-based constraints.

For FE-based solutions, the package space is meshedwith 2-D or 3-D finite elements, as appropriate. The struc-ture equilibrium equations are written as

K D = F, (2)

where K is the global structural stiffness matrix, D, theglobal structure displacement vector and F, the vector ofapplied forces.

In general, constraint equations involving displacementsmay be linear or non-linear; however, in the presentwork, only linear constraint equations involving linear staticdeflections are considered. Displacement-based constraintfunctions r fc are written in the form

r fc = r a0 +Ndof∑j=1

r a j d j (3)

where ra0 and ra j are constants, d j are the degrees of free-dom (dofs) of the FE mesh and Ndof is the total numberof dofs of the FE mesh. Not all the dofs may be involvedin specifying a deflection constraint, in which case thecorresponding ra j are equal to zero. For strength-basedconstraints, the failure criteria of the materials can be uti-lized. Generally, such failure criteria are defined in terms ofparameters such as stresses, strains, strain-energy, etc. andhave the form

S (p) ≤ pall, or[S (p)

/pall

] ≤ 1


where the failure function S(p) is a function of the param-eter p which may represent local stresses (σ ), strains (ε) orstrain energy (U ), and pall(μk) is the allowable value of theparameter for the chosen material μk . Denoting by S(p;P)

the value of S(p) at point P and pall(P), the allowable valueof p at P , the strength constraint can be written as[{S (p; P)/pall(P)} − 1

] ≤ 0, for all Pe �

In the FE scenario, this constraint needs to be applied forevery element ‘i’ and will result in Nelm constraints of theform

fsi ≡ [{Si (p;P)/pall(P)} −1] ≤ 0, for all P e �i ,

i = 1, 2, . . . , Nelm

where �i is the design space spanned by the i th elementand Nelm is the total number elements. In practice, one maychoose to apply each of these constraints over only a fewpoints in the element, such as the element centroid or Gausspoints. Also, note that pall(P) will correspond to the cho-sen material μk for the element and thus can be written aspall(μk). Choosing the application of the constraint at thecentroid, the strength constraint can be written as

fsi ≡ [{Si (p)/pall(μ)} − 1] ≤ 0,i= 1, 2, . . . ,Nelm

where Si (p) denotes the value of S(p) at the centroid ofthe i th element, which has material μ. For notational con-venience, the strength-based constraint for each element forone load case is represented by a single constraint equationof the form

r fs ≡ [{S(p)/pall} − 1] ≤ 0, r = 1, 2, . . . ,Ns (4)

where the left superscript ‘r ’ refers to the r th failure func-tion set r fs , which consists of the failure functions of all theelements evaluated at their respective centroids.

Only discrete materials (vis-à-vis graded materials) areconsidered in this study. Thus, the materials have uniqueand distinct modulus values. Two materials, say μa andμb, with the same modulus but different maximum allow-able values p a

all and p ball , (p a

all ≤ p ball ) are treated as the

same material with the higher allowable p ball . After the opti-

mal topology is obtained, the actual material assignment ismade based on the distribution of S(p) in the material sub-domain. For the sub-domain regions where S(p) < p a

all ,the material assignment will be μa and for the remainingsub-domain, the material assignment will be μb.

4 Solution methodology

The solution approach developed earlier (Ramani 2009) forstiffness-based topology optimization in a multi-material

framework is adopted here and modified to incorporate thestrength-based constraints. The main feature of the method-ology of Ramani (2009) is to use the potential changein objective and constraint functions due to a contem-plated change of material in a given element as a crite-rion for the choice of elements for change of material. Inthis respect, the approach presented differs from heuris-tic approaches such as ESO (Querin et al. 1998). In theESO-type approaches, the choice is based on a parameterlike stress or strain energy, which is only indirectly linkedto the structural compliance, which is usually chosen asthe performance index. In the present approach, the choiceof the element material change is made directly based onthe expected change in the performance indices (objec-tive and constraint functions). The elements are ranked onthe basis of this potential and only a fraction of the totalelements are modified to the new material in a single iter-ation. The optimal topology is obtained over several suchiterations.

In order to extend the above methodology for strength-based constraints, each such constraint is defined in termsof a material failure function as described in the last section.Every load case that imposes a strength-based constraintis treated as a separate load case. In each iteration, for acontemplated material change to an element, the changein the failure function �r fs is estimated by subjecting theelement to the loading environment around it at the currentiteration. This is used along with the estimated changes inthe objective function and the other (displacement-based)constraint functions in determining an aggregated elementranking parameter, using which optimization iterations areperformed. The estimation of the change in the failurefunctions (due to the material change in an element) ateach iteration, is carried out under the loading environmentaround the element at that iteration. This avoids refactoringof the stiffness matrix and makes the entire process compu-tationally efficient and economical. The description of theapproach follows.

Using Si (p) and Si ′(p) to denote the failure in element‘i’ before and after a contemplated material change fromμi to μi ′ respectively, under the same element loadingenvironment as at the current iteration, the failure ratiosin the element before and after the material change areSi (p)/pall(μi ) and Si ′(p)/pall(μi ′) respectively. (Note thatsuperscripts for μ are used to denote the material at the cur-rent iteration, while subscripts for μ used earlier refer toa specific material.) Thus, for element ‘i’, the change inthe failure function � fsi due to the contemplated materialchange is

� fsi=Si ′(p)/pall(μi ′)−Si (p)/pall(μ

i ) (5)

602 A. Ramani

The loading environment needs to be suitably defined basedon the choice of the parameter p. Thus,

1. If the parameters p refers to stress, the loading on theelement is defined in terms of the nodal displacements.

2. If the parameter p refers to strain, the loading on theelement is defined in terms of the nodal forces.

3. If the parameter p refers to strain-energy, the loading onthe element can be defined in terms of either the nodaldisplacements or the nodal forces.

4.1 Stress-based failure

For stress-based failure, the stress ratio Si (σ )/σall(μi ) isthe same as the stress ratio used in the resizing rules inChiandussi et al. (2004) and Chiandussi (2006). In the caseof multi-material optimization, the resizing rule needs to beappropriately modified to consider the stress ratios beforeand after the material change.

To calculate � fsi , first the change in the element stresstensor due to the contemplated material change is written as

�σ i=�(Eiεi )=�Eiεi + Ei�εi (6)

where εi is the element strain tensor, Ei is the elastic-ity matrix for the element material and �Ei represents thedifference in the elasticity matrix between the new andthe original material of the element. Equation (6) cap-tures the effect of the material change only for the elementconsidered; it does not capture stress changes due to mate-rial changes anywhere in the structure, and therefore it isnot equivalent to the true stress sensitivity. For optimiza-tion with discrete materials whose modulus values are not‘very close’ to each other, terms in the matrix �Ei are ofthe same order as the terms in the matrix Ei . The strainfield around the element is the loading environment forwhich, the change in the element stress tensor is evaluated.Therefore, setting �εi = 0, �σ i can be approximated as

�σ i ≈ �Eiεi = �Ei Bi Di (7)

where Bi is the matrix of shape function derivative operatorsand Di is the displacement vector for the dofs associatedwith the element.

4.2 Strain-based failure

For strain-based failure, the change in the element straintensor is written as

�εi=�(Ciσ i ) = �Ciσ i + Ci�σ i (8)

where Ci = E−1i . In this case, the stress-field around the

element is the loading environment for which, the changein the element strain tensor is evaluated. Therefore, setting�σ i = 0, �εi can be approximated as �εi ≈ � Ci σ i .

4.3 Strain-energy-based failure

For strain-energy-based failure, using Ki to denote the ele-ment stiffness matrix before the material change, the changein the element strain energy due to a contemplated materialchange can be estimated as

�Ui = 1

2(Di + �Di )

T(Ki + �Ki ) (Di+�Di )

− 1

2DT

i Ki Di = 1

2DT

i �Ki Di (9)

ignoring the higher order terms and assuming that �Di isnegligible for a material change to one element, as explainedbefore. Thus the change in the element strain energy is thedifference between the strain energies before and after thecontemplated material change, assuming that the displace-ment field around the element is unaltered by the materialchange.

Once the change in stress, strain or strain energy is esti-mated, the change in the failure function due to the materialchange can be calculated using (5). For illustration, if vonMises stress-based failure is to be considered, then using(7), the change in the von Mises stress at the elementcentroid �σ i , can be calculated as

�σi = fVM (σ i + �σ i ) − fVM (σ i ) (10)

where fVM() is the function that calculates the von Misesstress from the element stress tensor σ i . From (10), the vonMises stress in the element after the material change can becalculated as

σ ′i = σ i + �σ i (11)

In an optimization iteration, three kinds of changes arepossible to an element material:

1. change from one material to another2. change from a material to void, and3. change from void to a material.

For case (1), the change in failure parameters can be cal-culated as explained above. For case (2), � fsi is undefinedsince failure parameters are not defined for a void materialand therefore � fsi is not computed. For case (3), the com-putation of � fsi requires the specification of the maximumallowable value of the failure parameter for the void mate-rial. A low elastic modulus is used for the void material,using which the stress and strain tensors can be calculated. Ifstress-based failure is to be considered, the maximum allow-able stress in the void material μvoid, may be specified as

σall (μvoid) = cσ σall (μ2) E (μvoid)/E (μ2) (12)


where μ2 is the ID of the material with the lowest modulus(excluding the void material), E is the material modulus andcσ is a non-dimensional parameter. A low allowable stressfor the void material may be specified by choosing a lowvalue for cσ . In a similar manner, for strain-based failure, alow allowable strain for the void material can be specified.

4.4 Algorithm implementation

The methodology developed above was implemented in thetopology optimization code developed earlier for stiffness-driven optimization (Ramani 2009). Every load case that

(i) Start with an initial feasible design.Initialize s to zero.Evaluate objective and constraint functions fo, rfc and rfs.

(ii) Set material ID change to lower density material µi-.

Evaluate foi-, r fci

- and r fsi-.

Evaluate aggregated values fci*- and fsi

*- .

(iii) Calculate ranking parameters Rci , Rsi and Ri .

(iv) Rank elements based on Ri .

(v) Assign new material ID-s to s fraction of elements.

(vi) Re-solve FE model.Evaluate objective function fo, constraint functions rfc and rfs.

(vii) Is design feasible?

Yes

No

(viii) Set material ID change to higher modulus material µi .Evaluate foi

+, r fci+ and r fsi

+

+

.Evaluate aggregated values fci

*+ and fsi*+.

(ix) Calculate ranking parameters Rci+ , Rsi

+ and Ri+.

(x) Rank elements based on Ri+.

(xi) Assign new material ID-s to s fraction of elements.

(xii) Re-solve FE model.Evaluate objective function fo, constraint functions rfc and rfs.

(xiii) Isdesign feasible?

No

Yes

Termination criteria met?

Yes

End

(xiv) Increment cycle counter s.Update best feasible design.

No

Fig. 1 Optimization process flow

imposes a material failure constraint is treated as a sepa-rate load case. In each iteration, for a contemplated materialchange to the element, the change in the failure func-tion is calculated using the appropriate equations. This isused along with the computed values for the changes inthe objective and displacement-based constraint functionsin determining an aggregated element ranking parameter.Elements are ranked on the basis of their ranking param-eters and this rank is used to modify the material ID-s ofa fraction of elements during the iteration process. There-after, the resubstitution scheme (Ramani 2009) is used forperforming optimization iterations until user-specified ter-mination criteria are attained. Figure 1 provides an overviewof the entire optimization process. The detailed algorithm isexplained below.

1. The algorithm is begun with an initial material assign-ment to all the elements in the package space. The FEmodel is solved and the objective function fo, and allthe constraint functions, r fc and r fs are evaluated. Acycle counter s is initialized to one.

2. Let the initial design be a feasible design satisfyingall the constraints. Therefore, in order to decrease theobjective function fo, a material change �μ−

i to theimmediately lower density material is considered. Let� f −

oi be the change in the objective function corre-sponding to this material change. The change in thedisplacement �Dq for a dof ‘q’ is calculated as

�Dq = − (�KD)T q (13)

where q is the qth column of K−1 (Ramani 2009).Equation (13) is used to calculate the change in thedisplacements for all the dofs involved in the r th

displacement-based constraint equation, using whichthe change in the r th displacement-based constraintfunction r� f −

ci , is evaluated. The change in the r th fail-ure function r� f −

si can be calculated using (5). For amaterial change to the void material, r� f −

si is set to0. For multiple load cases, � f ∗−

ci and � f ∗−si are cal-

culated from the values of r� f +ci and r� f +

si using (14)and (15) below.

� f ∗−ci =

Nc∑r=1

r� f −ci

|r fc| (14)

� f ∗−si =

Ns∑r=1

r� f −si

∣∣r fsi + 1∣∣ (15)

The term |r fsi + 1| in (15) is chosen so as to nor-malize the change in the failure function such thatequal changes in the failure function from a materialstate close to and far exceeding the allowable value

604 A. Ramani

for the material are weighted unequally, with the latterweighted more heavily compared to the former.

3. Using the values of � f −oi , � f ∗−

ci and � f ∗−si , element

ranking parameters R−ci and R−

si are computed as

R−ci = � f −

oi

{−sgn(� f ∗−

ci

)}� f ∗−

ci

{−sgn(� f −

oi

)}(16)

R−si = � f −

oi

{−sgn(� f ∗−

si

)}� f ∗−

si

{−sgn(� f −

oi

)}(17)

For an element with a material ID change to the voidmaterial, the ranking parameter R−

si is not computed.4. Elements are ranked in the descending order of their

element ranking parameter values. Two rank vectorsare obtained, one each corresponding to R−

ci and R−si .

Using these, the average rank R−i is obtained for each

element. For an element with a material ID change tothe void material, its average rank is set to R−

ci .5. Proceeding in the order of the average rank vector,

the material ID-s of a fraction αs of elements arechanged to the new material as determined by thecorresponding values of �μ−

i .6. The finite element model is re-solved and the objective

function fo, and all the constraint functions r fc and r fs

are re-evaluated.7. If the resulting design remains feasible, steps (2)

through (6) above are repeated till an infeasible designis obtained.

8. With an infeasible design, a step similar to step (2) isexecuted. For element i , let r� f +

ci denote the changein the r th displacement-based constraint function dueto a contemplated material change, and r� f +

si denotethe change in the r th failure function for this load case.Let � f +

oi be the change in the objective function cor-responding to this material change. The current designbeing infeasible, only a material change �μ+

i to theimmediately higher modulus material is considered.For multiple load cases, � f ∗+

ci and � f ∗+si are calcu-

lated from the values of r� f +ci and r� f +

si using (18)and (19) below.

� f ∗+ci =

Nc∑r=1

r Cr � f +

ci|r fc|Nc∑

r=1

r C

(18)

� f ∗+si =

Ns∑r=1

r C r� f +si |r fsi + 1|

Ns∑r=1

r C

(19)

The constant rC = 1 if either the r th displacement-based or strength-based constraint is violated, elserC = 0.

9. Using the values of � f +oi , � f ∗+

ci and � f ∗+si for each

element, ranking parameters R+ci and R+

si are computedas

R+ci = � f +

oi

{−sgn(� f ∗+

ci

)}� f ∗+

ci

{−sgn(� f +

oi

)}(20)

R+si = � f +

oi

{−sgn(� f ∗+

si

)}� f ∗+

si

{−sgn(� f +

oi

)}(21)

If max(� f ∗+si ) > 0, the values of � f ∗+

si of all theelements are offset by −(1 + ε)max(� f ∗+

si ), where0<ε<<1. This results in max(� f ∗+

si ) < 0 so that R+si

can be calculated using the offset values of � f ∗+si .

10. Elements are ranked in the descending order of theirelement ranking parameter values. Two ranks areobtained, one each corresponding to R+

ci and R+si .

Using these two ranks, the average rank R+i is obtained

for each element.11. Proceeding rank-wise, a user-specified fraction of

the elements αs , are assigned the new material asindicated by the corresponding �μ+

i values.12. The FE model is re-solved and the objective function

fo, and all the constraint functions r fc and r fs , are re-evaluated.

13. A check is made on design feasibility. If infeasible,steps (8) through (12) are repeated until a feasiblesolution is found.

14. When a feasible design is obtained, one re-substitutioncycle is said to be completed and the cycle counters is incremented by one. Thus, steps (2) through(13) constitute one re-substitution cycle. In this pro-cess, the best feasible design-to-date is updated. Thenext re-substitution cycle is begun from the latestfeasible design of the previous re-substitution cycle.During each re-substitution cycle, the best feasibledesign-to-date is updated.

Iterations are repeated until either the improvement in theobjective function over successive feasible designs is lessthan a user-specified tolerance or the user-specified maxi-mum number of re-substitution cycles smax are performedor the maximum number of iterations are reached.

Fraction αs (for the number of elements to change mate-rial) is chosen to be initially high, gradually reducing to asmaller value over successive re-substitution cycles. For thecurrent research, this is achieved by using the relations

αs = αmax(

αmax

αmin

) −ssmax

(22)

where αmax and αmin are user-specified values and s is thenumber of the current re-substitution cycle, incremented by


VerticalRestraint

SymmetryBoundaryCondition

Force

0.1 m

0.1 m

Fig. 2 Short MBB beam problem set-up

one for every completed re-substitution cycle. Other tech-niques such as a linear decay to gradually reduce αs from itsmaximum to minimum values are possible (Ramani 2009),but were not considered here.

Instead of starting from a feasible design, the resubstitu-tion algorithm can be started from an infeasible design aswell. In such a scenario, the first iteration performed willcorrespond to step (viii) of the algorithm. Given that topol-ogy optimization problems have multiple optima, it may bebeneficial to obtain optimal topologies with various startingdesigns to arrive at a topology closer to the global optimum.

The effectiveness of the present implementation isbrought out in the case studies in the following section.

5 Examples, results and discussion

The methodology developed for handling stress constraintsand implemented in a computer code was used to findthe optimal topologies for the problem of a short MBB(Messerschmitt-Bolkow-Blohm GmbH) beam, long MBBbeam, an L-domain, a plate with a stiffener, and a solid L-bracket with graded material. Some of these problems havebeen studied as benchmark problems in the topology opti-mization literature (Duysinx and Bendsøe 1998; Svanbergand Werme 2007; Kocvara and Stingl 2007).

5.1 Short MBB beam

The first problem involves the determination of optimaltopologies with deflection and stress constraints for a shortMBB beam package space. The package space has a lengthof 0.2 m, depth of 0.1 m and a thickness of 2 mm. A verticalforce of 20 N is applied at the mid-span of the lower edge,with the lower corners restrained from vertical motion (sim-ply supported). Because of the symmetry of the packagespace, only the left half of the design domain is modeled.The FE mesh along with the applied load and boundary con-ditions is shown in Fig. 2. The model has 2,500 elementsand 2,601 nodes, each with two translational dofs in themesh plane. A vertical deflection constraint is specified atthe lower right corner node in Fig. 2.

First, solutions were obtained with steel as the chosenmaterial (elastic modulus of 2.1 × 1011 N/m2, Poisson’sratio of 0.3 and density of 7,890 kg/m3) for a maximumallowable stress of 400 MPa and with various values of the

Fig. 3 Short MBB beamoptimal topologies with400 MPa stress constraint

Steel

(f) Deflection < 0.04 mm Mass 123.3 g

(e) Deflection < 0.05 mm Mass 90.89 g

(d) Deflection < 0.1 mm Mass 46.33 g

(c) Deflection < 0.2 mm Mass 25.94 g

(b) Deflection < 0.3 mm Mass 22.66 g

(a) Deflection < 1 mm Mass 22.66 g

606 A. Ramani

Fig. 4 Convergence plot

Infeasible Design

Feasible Design

Feasible Design with a

Lower Mass than a

Previous Feasible

DesignIteration Number

Mas

s (k

g)

maximum allowable vertical deflection ranging from 1.0 to0.04 mm. Results are shown in Fig. 3. The jagged mate-rial boundaries in Fig. 3 are a result of treating the materialas a discrete variable so that each element as a whole canhave only one material phase. It is possible that the pres-ence of jagged edges may cause stress raisers at the corners.As explained before, the optimization process is based oncentroidal stresses and does not take into consideration theeffect of stress raisers at the corners. In order to obtain anengineering structure from the derived topology, the jaggededges would need to be smoothened, so that the stress rais-ers will not occur in practice. On further examination, it isobserved that the same topologies are obtained in Fig. 3aand b implying that the stress constraint is not active untila deflection constraint of about 0.3 mm. Thereafter, Fig. 3cthrough f show that the mass increases when the maximumallowable deflection is decreased. Figure 4 shows a sam-ple convergence plot corresponding to the result obtainedin Fig. 3d. It can be observed that the algorithm alter-nately finds feasible and infeasible designs in its progressiontowards the final topology.

The second set of results was obtained with a 1 mmmaximum allowable vertical deflection and various values

of the maximum allowable stress, ranging from 200 to500 MPa. The topologies obtained for this set of simulationsare shown in Fig. 5. The general trend shows that the massdecreases when the maximum allowable stress is relaxed.In Fig. 5e, the topology without any stress constraint (i.e.,deflection constraint only) is plotted. It is interesting tosee that the mass obtained is slightly higher than the onewith the 500 MPa maximum allowable stress, indicatingthat the algorithm converged to a different topology witha slightly higher mass than the one in Fig. 5d. This showsthe influence of the search path on the solution for a prob-lem such as this, where multiple optima are present. Thepresence and absence of the stress constraint causes thealgorithm to follow slightly different search paths leadingto different topologies.

The next set of results was obtained with both magne-sium (elastic modulus of 0.44 × 1011 N/m2, Poisson’s ratioof 0.3 and density of 1,740 kg/m3) and steel as the twoavailable materials. Various combinations of deflection andstress constraints were studied for the two-material problem.Topology results for all these cases are shown in Fig. 6. Ingeneral, it is observed that the topologies show greater useof steel at lower allowable deflections. Minor changes in

Fig. 5 Short MBB beamoptimal topologies with 1 mmdeflection constraint

Steel

(e) No Stress Constraint Mass 22.09 g

(d) Stress < 500 MPa Mass 20.76 g

(c) Stress < 400 MPa Mass 22.66 g

(b) Stress < 300 MPa Mass 23.29 g

(a) Stress < 200 MPa Mass 47.21 g


Fig. 6 Short MBB beamoptimal topologies with twomaterials

Magnesium Steel

(h) Deflection < 0.5 mm Stress < 400 MPa (Mg) Stress < 400 MPa (Steel) Mass 9.828 g

(i) Deflection < 0.5 mm Stress < 200 MPa (Mg) Stress < 600 MPa (Steel) Mass 9.869 g

(g) Deflection < 0.5 mm Stress < 200 MPa (Mg) Stress < 200 MPa (Steel) Mass 9.856 g

(e) Deflection < 0.25 mm Stress < 400 MPa (Mg) Stress < 400 MPa (Steel) Mass 17.12 g

(f) Deflection < 0.25 mm Stress < 200 MPa (Mg) Stress < 600 MPa (Steel) Mass 16.57 g

(d) Deflection < 0.25 mm Stress < 200 MPa (Mg) Stress < 200 MPa (Steel) Mass 18.73 g

(a) Deflection < 0.1 mm Stress < 200 MPa (Mg) Stress < 200 MPa (Steel) Mass 38.46 g

(b) Deflection < 0.1 mm Stress < 400 MPa (Mg) Stress < 400 MPa (Steel) Mass 37.70 g

(c) Deflection < 0.1 mm Stress < 200 MPa (Mg) Stress < 600 MPa (Steel) Mass 37.65 g

the topology can be observed for various combinations ofstress constraints for a 0.1 mm deflection constraint (Fig. 6athrough c). When the stress constraint on steel is relaxed,

a lighter topology is obtained. A similar observation canbe made regarding Fig. 6d through f, although the topol-ogy in Fig. 6e is very different from the other two. For a

Fig. 7 Long MBB beamoptimal topologies with400 MPa stress constraint

(c) Deflection < 0.5 mmMass 66.90 g

Steel

(b) Deflection < 0.6 mmMass 49.80 g

(a) Deflection < 1.6 mmMass 49.80 g

(f) Deflection < 0.2 mmMass 132.6 g

(e) Deflection < 0.3 mmMass 93.99 g

(d) Deflection < 0.4 mmMass 73.47 g

608 A. Ramani

Fig. 8 Long MBB beamoptimal topologies with 1 mmdeflection constraint

Steel

(f) No Stress ConstraintMass 45.38 g

(e) Stress < 600 MPaMass 42.86 g

(d) Stress < 500 MPaMass 42.86 g

(c) Stress < 400 MPaMass 49.80 g

(b) Stress < 300 MPaMass 78.84 g

(a) Stress < 200 MPaMass 136.9 g

deflection constraint of 0.5 mm, an all-magnesium topol-ogy was obtained. The masses obtained in Fig. 6g and iare very close indicating that since steel is not used in thetopology, the stress constraint on steel does not influencethe solution significantly. Relaxing the stress constraint onmagnesium to 400 MPa results in hardly any change in themass; however, the topology in Fig. 6h with a 400 MPa max-imum allowable stress for magnesium is different from theones in Fig. 6g and i.

5.2 Long MBB beam

In the second example, a longer MBB beam was considered.The length of the beam package space was 0.4 m, with the

other dimensions being the same as those for the shorterbeam, i.e., depth of 0.1 m and thickness of 2 mm. Theloads and boundary conditions were also the same as before.Again, only one half of the beam was modeled because ofsymmetry. With the same element size as for the shorterbeam, the FE mesh for this problem had 5,000 elementsand 5,151 nodes. As for the previous problem, a verticaldeflection constraint was specified at the bottom-most nodeon the symmetry plane.

Four sets of results are presented for this example. Thefirst set of results shown in Fig. 7a is for steel-only optimiza-tion with a maximum allowable stress of 400 MPa and withvarious deflection constraints ranging from 1.6 to 0.2 mm.The same topologies obtained in Fig. 7a and b imply that thestress constraint is not active until a deflection constraint

Fig. 9 Long MBB beamoptimal topologies with twomaterials—same stress butvarying deflection constraints

Magnesium Steel

(a) Stress < 200 MPa (Mg)Stress < 200 MPa (Steel)Mass 41.26 g

(d) Stress < 400 MPa (Mg)Stress < 200 MPa (Steel)Mass 69.60 g

(b) Stress < 200 MPa (Mg)Stress < 400 MPa (Steel)Mass 33.00 g

(c) Stress < 200 MPa (Mg)Stress < 600 MPa (Steel)Mass 31.18 g

(e) Stress < 400 MPa (Mg)Stress < 400 MPa (Steel)Mass 31.77 g

(f) Stress < 400 MPa (Mg)Stress < 600 MPa (Steel)Mass 30.51 g


of about 0.6 mm. As for the shorter beam, the topologiesobtained in Fig. 7c through f show the influence of boththe constraints, with increase in mass for lower allowabledeflections.

The second set of results was obtained keeping the max-imum vertical deflection at 1 mm and varying the allowablestress from 200 to 600 MPa. The topologies for this set ofsimulations are shown in Fig. 8a. As in the previous exam-ple, the mass decreases when the maximum allowable stressis relaxed. Comparing the topologies in Fig. 8c and d, it isobserved that the upper horizontal ‘truss’ member is low-ered in Fig. 8d to lighten the topology because of the higherallowable stress. The same topology is obtained for max-imum allowable stresses of 500 and 600 MPa. In Fig. 8f,the topology without any stress constraint (i.e., deflectionconstraint only) is plotted. The topology is essentially thesame as in Fig. 8d and f, except that since there was no stressconstraint, the upper horizontal ‘truss’ member occupied theupper extreme position of the package space, resulting in aslightly higher mass.

The next set of results was obtained with both magne-sium and steel as the two available materials. The maximumallowable stress for both the materials was specified as400 MPa and the deflection constraint was varied to obtaindifferent optimal topologies. These results are shown inFig. 9. With a 2 mm allowable vertical deflection, an all-magnesium solution was obtained. The results in Fig. 9bthrough f show that increasing use of steel is made at lowerallowable deflections, resulting in higher masses.

Finally, results are shown in Fig. 10 for a verticaldeflection constraint of 1 mm and a few combinations ofmaximum allowable stresses for magnesium and steel. It

Magnesium Steel

(a) Stress < 200 MPa (Mg)Stress < 400 MPa (Steel)Mass 33.00 g

(b) Stress < 200 MPa (Mg)Stress < 600 MPa (Steel)Mass 31.18 g

(c) Stress < 400 MPa (Mg)Stress < 400 MPa (Steel)Mass 31.77 g

(d) Stress < 400 MPa (Mg)Stress < 600 MPa (Steel)Mass 30.51 g

Fig. 10 Long MBB beam optimal topologies with two materials—same deflection but varying stress constraints

0.15 m

0.15 m

0.06 m

0.06 m

A

B

Restraints

1 mm thickD

C

Fig. 11 L-domain problem set-up

is observed the mass values are almost the same, but thetopologies are very different. An all-magnesium solution isobtained in Fig. 10d, while some steel was used for the othercombinations of stress constraints.

5.3 L-shaped domain with stress and strain basedfailure criteria

An L-shaped domain was considered in the third case study.The dimensions and FE mesh are shown in Fig. 11. The FEmodel had 3,600 elements and 3,751 nodes, each with twotranslational dofs in the mesh plane. The first set of opti-mization studies are presented for a single material. For this,aluminum with a modulus of 7.0 × 1010 N/m2, Poisson’sratio of 0.3, density of 2,630 kg/m3 and a maximum allow-able stress of 50 MPa was chosen as the design material.With all dofs on the bottom edge restrained, three differentcases of loading and constraints were studied: (1) a hor-izontal load of 200 N along with a horizontal deflectionconstraint of 3 mm specified at the mid-point C, of edgeAB, (2) a vertical load of 100 N along with a verticaldeflection constraint of 3 mm specified at Point C and(3) both the horizontal and vertical load cases along withtheir corresponding constraints in (1) and (2) above. Resultsfor all these cases are shown in Fig. 12. Figure 12a showsthe topology obtained for the horizontal load case withonly the deflection constraint. This topology had a mass of5.281 g. The Von Mises stress distribution in the topologyis shown in Fig. 12b. The maximum stress in the topologywas 327 MPa, which is considerably higher than the max-imum allowable stress and occurs in the transition regionbetween the horizontal and vertical portions of the structure.Now, with the stress and the deflection constraint, the topol-ogy obtained is shown in Fig. 12c and had a higher massof 18.78 g. The stress distribution in this topology is shownin Fig. 12d and reveals that stress constrained optimization

610 A. Ramani

(a) Aluminum (c) AluminumHorizontal Load CaseDeflection and StressMass 18.78 g

(d) AluminumHorizontal Load CaseDeflection and StressMaximum Stress 49.9 MPa

(b) AluminumHorizontal Load CaseDeflection onlyMaximum Stress 327 MPa

Horizontal Load CaseDeflection onlyMass 5.281 g

(e) Aluminum (g) AluminumVertical Load CaseDeflection and StressMass 17.42 g

(h) AluminumVertical Load CaseDeflection and StressMaximum Stress 50.0 MPa

(f) AluminumVertical Load CaseDeflection onlyMaximum Stress 139 MPa

Vertical Load CaseDeflection onlyMass 7.859 g

(i) Aluminum (k) Aluminum and PlasticTwo Load CasesDeflection onlyMass 10.37 g

(l) Aluminum and PlasticTwo Load CasesDeflection and StressMass 29.52 g

(j) AluminumTwo Load CasesDeflection and StressMass 29.92 g

Two Load CasesDeflection onlyMass 9.510 g

Aluminum

Plastic

Fig. 12 L-domain optimum topologies

is able to bring down the stresses to within the allowablelimit, by the use of additional material. The location ofmaximum stress shifts to the corners where the restraintsare applied. The corresponding results for the vertical loadcase are shown in Fig. 12e through h. For this load case, thetopology with only the deflection constraint yielded a massof 7.859 g as shown in Fig. 12e. The stress distribution inthis topology shown in Fig. 12f indicates that high stressesoccur at the connection regions of the truss-like structure.The maximum stress in this topology was 139 MPa andoccurred at Point D shown in Fig. 11. When the stress con-straint for this load case was included, the topology obtainedis shown in Fig. 12g and has a higher mass of 17.42 g. In

this new topology the stress at the inner corner of the ‘L’shaped region is reduced from 139 to 50 MPa by suitabletopology design, in order to meet the stress constraint, asseen in the plot of stresses in Fig. 12h. Next, topology opti-mization with both the load cases and their correspondingdeflection constraints yielded the structure shown in Fig. 12iwith a mass of 9.510 g. The maximum stresses in this struc-ture for the horizontal and vertical load cases were 342and 375 MPa respectively. With the addition of the stressconstraint for both the load cases, the topology obtained isshown in Fig. 12j. The new topology had a mass of 29.92 gand had stresses and deflections within the allowable limitfor both the load cases.


Thus, although the re-entrant corner is not removed, thealgorithm is able to arrive at a suitable topology in thepresence of stress constraints. Earlier attempts (Bruggi andVenini 2008; Svanberg and Werme 2005; Pereira et al. 2004;Le et al. 2010) to solve the L-shaped geometry problem havehighlighted the difficulties at the re-entrant corner. The solu-tion in Bruggi and Venini (2008) shows that the re-entrantcorner is not completely removed, but the stresses are withinthe constraint as obtained in the present method. Topologieswithout re-entrant corners have been obtained by Svanbergand Werme (2005), through a series of mesh refinementsstarting with a coarse mesh. Solutions in Pereira et al. (2004)do not show the corner but a smoothly-radiused geometry.The method is SIMP-based and uses penalization and relax-ation parameters which may need to be tuned for a particularproblem. Recently, Le et al. (2010) have developed anotherSIMP-based approach using global and regional stressconstraint definitions. Using this approach, the removal ofthe re-entrant corner has been demonstrated without theneed for tuning parameters.

Results for a two-material problem with both the horizon-tal and vertical load cases are next presented. The secondmaterial considered was a plastic material with an elasticmodulus of 3.1 × 109 N/m2, Poisson’s ratio of 0.4 anddensity of 1,110 kg/m3. The failure criterion for the plas-tic material was a 1% maximum allowable major principalstrain. It may be noted that for the two-load case problem,no feasible solution was found to exist if the plastic mate-rial alone was considered for topology design. The optimumtopology for the two-material problem with only the 3 mmdeflection constraints for both the horizontal and verticalload cases is shown in Fig. 12k. It can be observed that thealgorithm resulted in a topology which utilized the lighterplastic material for the most part and makes minimal useof the heavier aluminum at the regions of maximum stress(or strain). The mass of this topology was 10.37 g and isslightly higher than that obtained with aluminum only as

1 m

X

Y

1 m

120 kN vertical load appliedat this central patch

All edges simply supported

0.05 m

0.175 m

0.075 m

Fig. 13 Plate problem details

0.05 m

0.5 m

Shell elements

X

Y

Solid elements

0.5 m

30 kN vertical load

Simply supported edge

Symmetr

ic re

strain

ts

Symmetric restraints

Simply

supporte

d edge

Fig. 14 Plate model details

the design material. While a lower mass structure is gen-erally expected with two materials, it is possible that theintroduction of the additional plastic material with a muchlower stiffness-to-mass ratio caused the algorithm to con-verge to a local optimum with a slightly higher mass. Thisis another result which brings out the influence of the searchpath on the obtained solution for a multiple-optima prob-lem. In the topology in Fig. 12k, the maximum stresses inaluminum for the horizontal and vertical load cases were305 and 170 MPa respectively. The maximum major princi-pal strain values for the plastic material for these load caseswere 1.35% and 0.838% respectively. Next, with the inclu-sion of the failure constraints, i.e., maximum allowable VonMises stress constraint for aluminum and maximum allow-able major principal strain constraint for the plastic material,along with the deflection constraints, a topology with a

X

X

Z

Y

Z

Y Magnesium

Aluminum

Top View

Bottom View

Center

Center

Fig. 15 Optimized stiffener topology

612 A. Ramani

Fig. 16 Solid frame problemset-up

45360 Elements54229 Nodes

Constraint: Load Point Deflection < 3 mm

Fixed dofs on this face

1000 N Force

60 mm

60m

m

6 mm

60 mm

6 mm

mass of 29.52 g as shown in Fig. 12l was obtained. Thismass is only slightly lower than that for the single materialcase. The maximum Von Mises stresses in aluminum were28.8 and 28.6 MPa respectively for the horizontal and ver-tical load cases. The maximum major principal strains inthe plastic material were 0.119% and 0.154% for the hor-izontal and vertical load cases. Thus, the results for thetwo-material problem indicate that the consideration of anadditional plastic material with a very low stiffness-to-massratio in comparison to aluminum is not of significant advan-tage with respect to mass savings, as evidenced in the resultswith and without the failure constraints.

5.4 Plate with a stiffener

The next example is that of a plate with a stiffener. A 1×1 msteel plate of 1 mm thickness was to be stiffened withina design space extending for a depth of 50 mm directlybeneath the plate. The plate was simply supported on itsedges and a central vertical load of 120 kN was appliedas shown in Fig. 13. For the stiffener, magnesium andaluminum were considered for topology optimization. Theproperties of these materials were the same as those in theprevious examples. For topology optimization, a constraintof 1.9 mm was specified on the vertical deflection at the cen-

Fig. 17 Solid frame optimumtopologies with deflectionconstraint a) Graded material Topology

Mass: 87.84 g

b) Steel-only TopologyMass: 102.3 g

View 1 View 2

View 1 View 2

Void

Steel


tral node of the plate mesh. In addition to this, all stresseswere to be within the allowable limits—300 MPa for steel,300 MPa for magnesium and 50 MPa for aluminum.

Because of geometry and loading symmetry, only onequarter of the plate was modeled. The plate was meshedwith 1,600 shell elements with a uniform mesh size of12.5 mm in the X–Y plane as shown in Fig. 14. These shellelements were non-design elements whose topology was notto be altered during optimization. The mesh of the topologydesign space shown in Fig. 14 consisted of 6,400 solid ele-ments. There were a total of 8,405 nodes in the model. Theload of one-quarter magnitude was applied on a group ofnodes in the center of the plate as shown in Fig. 14.

An interesting aspect of this problem is that neithermagnesium nor aluminum, if used alone can provide a fea-sible solution. If the entire stiffener topology is made ofmagnesium, then the stress constraint is satisfied, but thedeflection constraint is violated. On the other hand, for anall-aluminum package space mesh, the deflection is withinthe limit specified, but the stresses exceed the allowable lim-its. When topology optimization was performed with boththe materials, the algorithm determined an optimal topol-ogy, which is shown in Fig. 15. The topology had a massof 21.61 kg. It can be observed that there is a central patchof magnesium on the top surface to satisfy the stress con-straint; otherwise, aluminum is used in the central region toprovide higher stiffness. There is a predominant use of thelighter magnesium around the edges. A small region of thepackage space remains unutilized, as seen in Fig. 15. Thus,this problem demonstrates how the algorithm can be used

to determine multi-material topologies when single mate-rial solutions satisfying the deflection and stress constraintscannot be obtained.

5.5 Solid frame topology optimization with gradedmaterial

The next case study involves the topology optimization ofan L-frame for minimum mass when a graded material wasconsidered. The geometry and mesh details for the prob-lem are shown in Fig. 16. With about 45,360 elements and163,000 dofs, this is a reasonably large problem. A keyassumption in the method to handle failure criteria was thatthe materials considered were distinct and had unique mod-ulus values not close to each other. A continuously gradedmaterial does not satisfy this assumption. However, thesolution approach developed here can be used if the con-tinuously graded material is ‘discretized’ into a number ofdistinct materials, implying that the assumptions made inthe development of method are not violated. For the cur-rent problem, the modulus and density values of steel weregraded over ten intervals resulting in ten materials withmodulii 0.21 × 1011, 0.42 × 1011, . . . , 2.1 × 1011 N/m2

and corresponding densities 789, 1,578,. . . , 7,890 kg/m3.The design space mesh is shown in Fig. 16 along with thedimensional details. For a 1,000 N applied load as shownin Fig. 16, the specified load point deflection constraint was3 mm in the direction of the applied load. Without any stressconstraints, the topology obtained with the graded materialis shown in Fig. 17a. This topology is compared to the one

Fig. 18 Solid frame optimumtopologies with deflection andstress constraints a) Graded material Topology

Mass: 91.02 g

b) Steel-only TopologyMass: 112.6 g

View 1 View 2

View 1 View 2

Void

Steel

614 A. Ramani

obtained with only steel as the chosen material shown inFig. 17b. Compared to the optimum steel-only topology,about a 13% reduction in mass is obtained with a gradedmaterial. To solve the same problem with the inclusion ofstress constraints, a 300 MPa maximum allowable stresswas specified for steel. This maximum allowable stress wasalso graded equally from 30 to 300 MPa for the ten mate-rials. The optimum topology obtained with deflection andstress constraints is shown in Fig. 18a. For comparison, theoptimum steel-only topology obtained with deflection andstress constraints is also shown in Fig. 18b. In this case,about a 20% reduction in mass is obtained with a gradedmaterial. For the cases with stress constraints, on a desk-top computer with 2 GB RAM and 2.2 GHz CPU, the timetaken for FE analysis per iteration was about 200 s and thetime taken for pseudo-sensitivity computation and elementranking was about 100 s, resulting in a total iteration timeof about 300 s. The different examples presented as part ofthis case study took about 25 to 50 iterations, resulting in aturnaround time of about 3 to 4 h.

These results demonstrate that the method can also beapplied for the topology synthesis of structures with gradedmaterials, in the presence of stress constraints.

6 Conclusions

A heuristic but efficient approach to handle failure criteriain multi-material topology optimization has been developedand presented. The methodology developed to handle mul-tiple materials retains the material as a discrete variable andtherefore avoids the phenomenon of stress singularity. Inaddition, it accommodates different failure criteria neces-sary to represent strength constraints in a multiple-materialscenario. Further, the estimation of the changes in the fail-ure function due to a contemplated material change uses theloading environment around the element in a computation-ally effective calculation which avoids matrix factorizationsin the optimization iteration. This makes it possible to usethe methodology effectively for many stress-based designproblems. This has been demonstrated on a few bench-mark problems with various combinations of deflectionand stress constraints, both for single and multiple mate-rials. In addition, the versatility of the approach has beendemonstrated by solving different kinds of problems such as(1) topology synthesis using two materials with differentfailure criteria, (2) stiffener topology design using twomaterials, when either material used alone would resultin an infeasible design, and (3) synthesis of a structurewith graded material properties. The presence of multipleoptima in topology optimization problems, the ability of the

algorithm to attain some of the multiple optima and alsoits sensitivity to the search path have been demonstratedusing the case studies. Thus, the results obtained show theeffectiveness and utility of the method in deriving optimaltopologies with multiple materials in the presence of generalfailure criteria.

A significant limitation of the approach is that it breaksdown in design domains involving certain kinds of geome-tries like re-entrant corners, which may cause it to convergeto non-optimal designs. This is recognized as an issue forfurther research attention.

As a final point, in this paper, attention is focused onthe methodology for evolution of topology and associatedload-paths considering only stresses at the element level. Ina conventional displacement-based finite element formula-tion, stresses are either discontinuous or may have only C0

continuity across elements, requiring the use of only cen-troidal (or Gauss point) stresses for stress representation. Asa consequence, interface stresses are not accounted for in thepresent analysis. One approach to handle interface stresseswould be to carry out a separate analysis a-priori to arrive atconstraints on the use of neighboring materials (such as theratio of maximum modulii or maximum allowable stresses)or a-posteriori to design the interface properly.

Acknowledgments The author is thankful to Dr. Prakash D.Mangalgiri of General Motors, India Science Lab., Bangalore, India,for suggesting example problems, participation in discussions, helpin interpretation of results and reviewing the manuscript. Thanks arealso due to Dr. Prabhakar Marur of General Motors, India ScienceLab., for developing an automatic communication interface betweenthe algorithm and the FE solver, which enabled efficient solution of thecase studies. Many thanks to Narendran Balan and Srinivasan Surya-narayanan of General Motors, India Science Lab., for their commentsand suggestions during the reviews.

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Multi-material topology optimization with strength constraints