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Topology optimization of cellular materials with pe riodic microstructure under stress constraints Pedro G. Coelho *, José M. Guedes ** and João B. Cardoso * * UNIDEMI, Faculty of Sciences and Technology Universidade Nova de Lisboa FCT, 2829-516 Caparica, Portugal {pgc@fct.unl.pt, jbc@fct.unl.pt} **IDMEC, Instituto Superior Técnico Universidade de Lisboa Av. Rovisco Pais 1, 1049-001 Lisboa, Portugal {jmguedes@tecnico.ulisboa.pt} Abstract Material design is a critical development area for industries dealing with lightweight construction. T rying to respond to these industrial needs topology optimiza tion has been extended from structural optimization to the design of material microstructures to improve overa ll structural performance. Traditional formulations based on compliance and volume control result in stiffnes s-oriented optimal designs. However, strength-orien ted designs are crucial in engineering practice. Topolo gy optimization with stress control has been applie d mainly to (macro) structures, but here it is applie d to material microstructure design. Here, in the c ontext of density based topology optimization, well-establish ed techniques and analyses are used to address know n difficulties of stress control in optimization prob lems. A convergence analysis is performed and a den sity filtering technique is used to minimize the risk of results inaccuracy due to coarser finite element m eshes associated with highly nonlinear stress behaviour. A stress-constraint relaxation technique ( qp -approach) is applied to overcome the singularity phenomenon. Par allel computing is used to minimize the impact of t he local nature of the stress constraints and the fini te difference design sensitivities on the overall c omputational cost of the problem. Finally, several examples test the developed model showing its inherent difficult ies. Keywords Topology, Optimization, Microstructures, Convergen ce, Stress, Homogenization Acknowledgements This work was partially supported by Fundação para a Ciência e a Tecnologia (Portugal) through the projects UID/EMS/00667/2013, UID/EMS/50022/2013 and PTDC/EMS- PRO/4732/2014. Authors wish to thank Professor Krister Svanberg (R oyal Institute of Technology, Stockholm, Sweden) fo r the MMA optimization code and Professor Hélder C. R odrigues (IDMEC, Instituto Superior Técnico, Portugal) for all the discussions on this work.
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1. Introduction Significant improvements in industrial products hav e been achieved through the use of new structural materials that exhibit extreme properties found by optimal design techniques. For example, optimized composite materials have seen increased interest fo r lightweight construction purposes namely in autom otive and aerospace industries (see, e.g., Gürdal et al. 1999). To fully grasp the overall response of a composite one needs a greater understanding of its microstruc ture behaviour. A micromechanical analysis based on deta iled modelling of the mixture of base constituents, often relying on homogenization techniques, is cruc ial to characterize its mechanical performance and to transfer the respective data across the different m aterial length scales (see, e.g., Zohdi and Wrigger s, 2005). The macroscopic behaviour of a composite is thus me chanically linked to its microstructure. To efficiently explore the potential of composite m aterials, one searches for the optimal layout of it s constituents within a design domain adequately para meterized and representative of the material heterogeneities (e.g., RVE, representative volume e lement or UC, material unit-cell). A material distr ibution optimization problem is then solved, targeting pres cribed behaviours at the macroscopic level as desig n objectives or constraints. This procedure, also kno wn as inverse homogenization, often uses density-ba sed topology optimization (see e.g. Sigmund, 1994). Topology optimization has been applied to design st ructure and material separately (single-scale model s, see e.g. Sigmund and Maute 2013) as well as concurr ently (multi-scale models, see e.g. Theocaris and Stavroulaki 1999, Rodrigues et al. 2002, Deng et al . 2013, Nakshatrala et al. 2013, Xia and Breitkopf 2014, Coelho and Rodrigues 2015). The basic idea is to fi nd an optimal layout, within a particular domain, determining which spatial points should have materi al and which should be void (no material). The most common approach is to use density-based design vari ables to relax the integer nature of the original t opology optimization problem transforming it into a continu ous and differentiable problem solvable by gradient -based optimization methods (Bendsøe and Sigmund 200 3). Although the present paper is focused on the mixture of solid and void phases (cellular material s), the formulation is readily extendable to bi-mat erial solutions (composites). A cellular material with periodic microstructure is studied in this work. The UC represents its smalle st periodic material heterogeneity and asymptotic homo genization is used to extract the behaviour of the periodic material based on the analysis of this UC (see, e.g., Guedes and Kikuchi, 1990). Topology optimization problems with stress based cr iteria are especially interesting to engineering practice because they guarantee very efficient desi gns and directly address aspects of material failur e. However, dealing with stresses is quite challenging , mainly due to: (1) highly nonlinear stress behavi our with respect to (w.r.t.) design changes; (2) design singularity phenomenon; (3) local nature of the st ress constraint(s). Topology optimization of macro-struc tures has developed techniques to effectively tackl e these issues (see, e.g., Le et al. 2010, Deaton and Grand hi 2014). Topology optimization in material microst ructure design with stress constraints has only a few contr ibutions (see, e.g., the conference proceedings pap ers by Collet et al. 2016a,b). One should note the importa nt works by Lipton and Stuebner (2005, 2006) and al so recent research papers on the shape optimization of microstructures with stress control (see, e.g. Noë l and Duysinx, 2017). The primary goal of the present work is to study an d optimize the topology of a material UC with stres s constraints and compare (or validate) the obtained designs with literature known results. To meet this challenge, one applies current procedures, summariz ed below, to gain insight into the distribution of stresses within the UCs as well as to generate a well-posed topology optimization problem. Thus, given a UC domain and the applied macroscopic stress or strain field, one computes the micro- stress distribution across the two base constituent s (weaker and stronger), mixed in the UC domain, us ing an asymptotic based homogenization model. Typically, t he stress field is highly nonlinear with design
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sensitivities strongly dependent on design details. High stress gradient values are mainly located at boundary points with strong curvatures or re-entrant corners and they are very sensitive to the finite element (FE) discretization. If the accuracy of stress computati on is not guaranteed, reliable designs for stress-r elated optimization problems cannot be expected. To overco me this, a mesh convergence analysis is performed a s described in appendix A. This study further shows t he benefit of using density filters in connection w ith square grid meshes for layout optimization with str ess criteria (Sigmund 2007). The design singularity problem means that optimal p oints are singular, i.e., they are located in degen erate feasible domain subspaces, which are of a lower dim ension than the design space, and thus unreachable by gradient-based optimization algorithms (Cheng and J iang 1992, Kirsch 1990, Sved and Ginos 1968). Successful stress-constraint relaxation techniques to overcome singular optima are: (1) -relaxation (Cheng and Guo 1997, Duysinx and Bendsøe 1998); (2) qp -approach (Bruggi 2008); (3) relaxed stress (Le et al. 2010, Luo and Kang 2012); (4) damage-approach (Verb art et al. 2016). Here one uses the qp -approach due to its straightforward implementation. The local nature of the stress constraint implies t hat the optimization problem number of constraints can quickly increase with FE discretization. Thus a poi ntwise control of stresses based on the FE approach provides an accurate verification of stress constra ints admissibility but at the expense of increasing the computational cost. A possible way to mitigate this is the use of an active set strategy where only th e potentially active stress constraints are considere d in each design iteration (Bruggi and Duysinx 2012 ). Aggregation techniques can also be used to reduce t he number of local stress constraints by lumping th em into a single aggregation function (Duysinx and Sig mund 1998). Regional stress measures also reduce th at number (see Le et al. 2010). These later strategies decrease the computational cost but at the expense of not being able to control effectively the peak stress v alues. In the present work the stress constraint is imposed at each FE to avoid compromising that control. The optimal designs shown are obtained with three-d imensional meshes, but only 2-D layouts are shown since design uniformity in one (thickness) spatial direction is imposed. The applied loads studied lea d to symmetric density and stress distributions. This sy mmetry implies a lower number of stress-constraints since only a quarter of the UC domain is discretized. Som e reference solutions can be found in the literatur e and are revisited here for comparison purposes. This paper is organized as follows. In section 2 th e material model is presented. The stress-based optimization problem is discussed in section 3. In section 4, stiffness and strength-oriented designs are compared. The impact of stress control on the optim al material layout is summarized in the final secti on 5. 2. Material Model Consider a cellular material generated through the periodic repetition of a UC representing the smalle st periodic heterogeneity of the material domain W, see Fig. 1. A solid and fivoidfl phases are distrib uted within this UC to identify the cellular material topology. The stiffness ratio between solid and fivoidfl phase s, E(1) /E(2) , is set to 10 12 and one assigns unitary Young Modulus to the solid phase, E(1) =1GPa, just for demonstration purposes in the results section. Both phases are assumed linear and isotropic with Poiss on ratio n = 0.3. The UC domain is Y, the volume is |Y| and i ts characteristic size is d. Size d is much smaller than the cellular material global size D. The behaviour of the periodic material is extract ed from the UC analysis through asymptotic homogenization that ass umes periodic boundary conditions on Y and infinite periodicity of the UC. The homogenized stiffness tensor for the periodic m aterial, Hijkm E, is given by, () () dY Y1YH ¶- ¶-=sij rsj ri qkm pqm pk pqrs ijkm yyEEdcdcrr (1)
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The base material stiffness tensor E pqrs depends on the topology design variable, r, according to the following power law (bi-material SIMP approach, see Bendsøe and Sigmund 1999), ()()()()NÎ-+=p,EEEpqrs ppqrs ppqrs 211rrr (2) Besides a direct dependence on the base material, t he homogenized stiffness tensor also depends on the UC geometry. This dependence is characterized by th e micro-displacements ckl (Y-periodic), solution of the following set of equilibrium equations defined in Y (six equations in 3-D), () () periodic YYdYdYY-“¶¶=¶¶¶wjiijkl jiqkl pijpq ,ywEywyErr (3) The methodology used here to obtain the homogenized elastic coefficients in (1) and micro-displacement s ckl is based on the FE software PREMAT (see Guedes and Kikuchi 1990 and Ferreira et al. 2014). The periodic boundary conditions are imposed by setting the same micro-displacements ckl values in opposite (corresponding) nodes on the UC boundaries. The micro-stresses sij (at the level of the material microstructure) are defined based on the micro- displacement fields rs k as, () rs mrs kms kr ijkm ij yEecddrs ¶¶-= (4) where rs e is the macroscopic (average) applied strain tensor , related to the macroscopic stress tensor pq s through pq rspq rs CseH= (5) where CH is the homogenized compliance tensor computed as t he inverse of the stiffness tensor EH, ()() ()jk il jl ik mnkl ijmn CE+=21HH (6) The micro-stress field (4) represents the stress te nsor field components in the UC domain Y (see also Yu and Zhang 2011). The methodology used to obtain the stress field is based on the software POSTMAT (Guedes and Kikuchi 1990). Note that the macroscopi c average stress (or average strain) is assumed con stant and known, corresponding to an applied uniform macr oscopic tensor field. 3. Stress control in topology optimization In this study the UC is made of a ductile base mate rial and a von-Mises criterion is used to character ize material failure. Moreover, the UC macro loading is defined by a given strain field and the UC stiffne ss is bounded from below to guarantee non-trivial designs . The problem will be set as a minimum weight (volu me fraction) design problem with constraints both on s tiffness and local stress (see Bruggi 2016 and disc ussion therein for an equivalent design formulation):
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() () () () () ()() () [] () eexz yz xy xx zz zz yy yy xx eejjqeeeeNNeeeeNNeSNNjSSggNNeggVggYdY 321where 10 and ,, 110Y~21~,1 ~2with ,1 ,, 1~~with ,1 :.t.s Y~~with min Y222222VM 3min min H**VM 100+++-+-+-= =´=££<=+´=-=£´==£==-´=sssssssssE (7) Uniform material distribution implies uniform strai n/stress distribution, and thus renders maximum str ess control in the UC, constraint eg, meaningless. The global stiffness constraint jg is thus necessary to ensure non-uniform designs. Also in terms of mechanical de sign it seems appropriate to enforce concurrently stiffness and strength criteria. In this formulation, ~ is the filtered density, a function of the neighbo uring design variables ir, as defined in Bruns and Tortorelli (2001). Assuming th e density uniform in each FE, the vectors and ~ contain the elements design variables and densities , respectively. The original design variables have no physical meaning. It is the density ~ that defines the UC layout. Stress distributions are computed through equations (2) and (4) for the filtered density ~. This helps to prevent black-white designs that exhibit jagged edg es, which contain unphysical stresses (singularitie s). In fact, the somewhat blurred smooth boundary produced by the density filter is preferable over a sharp j agged boundary in order to smooth the local stress field and suppress unphysical singularities (see discussi on in Le et al 2010). For a detailed study on the filtering effect in the stress distribution see appendix A. A s an alternative solution for jagged boundary induced pr oblems see also Svärd (2015). The design problem constraints (7) are set as g £ 1, the standard form required by the MMA (Method o f Moving Asymptotes, see Svanberg 1987) optimizer. The N´N local constraints ge £ 1 contain the equivalent von-Mises stress values ( the average value within each element volume) within the limit value s*. According to Duysinx and Bendsøe (1998), the same SIMP penalization should be assumed for stiffness, p, and local stress interpolation, q, to achieve full physical consistence. However, if p = q then the optimum becomes singular (Bruggi 2008). T o overcome singularity one applies a suitable mathematical relaxation to t he stress constraint equations, adopting an exponen t q < p, known as the qp -approach (see e.g. Bruggi 2008, 2016; Bruggi and D uysinx 2012). Additionally, the constraint gj £ 1, imposes a lower bound S* on the overall stiffness S (measured here in terms of strain energy). This is required when the loading condition is set imposing a given macro str ain field . In the case of a given macro stress field there is an upper bound imposed on compliance, i.e . gj = C/C* where ()2Y~HC=C(see Bruggi 2016). So, the optimal design formulation (7) aims at weig ht reduction, satisfying both stiffness and strengt h criteria. In particular, this formulation can be us ed to reduce stress concentrations in a standard mi nimum compliance optimal design. In this case, one assume s the lower bound S* equal to the strain energy of the
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optimal compliance design, while the s* bound is set to reduce the peak stress values. A s trength-oriented optimal design can then be obtained through (7) at the expense of the UC volume fraction increase but keeping the same structural stiffness. This design approach, of reducing stress concentration in the o ptimal stiffness-based design, is the one pursued in this work and results are shown in the next section. The optimizer used requires the gradients of object ive and constraint functions w.r.t. the design vari able r. When the density filter is used, the sensitivitie s of the objective function g0, constraints ge and gj are given by (Sigmund 2007), () () {} ijjNjjieNiieiNiiexxRxxxxRgggiee--= --¶¶=¶¶¶¶=¶¶ÎÎÎmin min ,0 max ;~~~r (8) where Ne defines the set of elements neighbouring element e and R min is the filter radius. Assuming a uniform density in each element and a sq uare-grid mesh, the required derivatives w.r.t. ~are: eigY~0=¶¶ (9) ()() ¶¶+-=¶¶+ieqeei qeeieqg~~~1~~~VM **)1 (VM d (10) Y~21~H*km inlkm nl ijESg¶¶-=¶¶ (11) where eY is the element volume and dei is the Kronecker symbol. In (11), the derivative of the constitutive tensor w.r.t. density can be obtained by the adjoint metho d (see Bendsøe and Sigmund 2003), which for the power law (2) becomes: ()() () isnl rsl rn qkm pqm pk pqrs pqrs piinlkm iyyEEpEdY ~Y1~Y211H ¶- ¶--=¶¶-dcdc (12) In (10), the derivative of the element von-Mises st ress w.r.t. ~ is obtained using a semi-analytical method calculated by a first-order forward finite d ifference, ()()[]()aaiieieie~~1~~~VM VM VM -+»¶¶ (13) with a as the perturbation parameter. After performing te sts to assess numerical precision, a = 10 -3 is chosen. When 1~=ir, (13) is replaced by the respective backward finit e difference version. The finite difference approach is easy to implement, although computationally more costly when compared with adjoint or direct differentiation methods. The algorithm developed to solve problem (7) is sum marized in the flowchart in Fig. 2. The initializat ion assigns initial values to r followed by density filtering to obtain r~ Homogenization (3) and local stress calculation (4) follows. Volume, stiffness and stre ss-constraint functions can then be evaluated. A convergence criterion is tested (e.g. iteration num ber, design change). If it is not satisfied one pro ceeds to the sensitivity analysis. The numerical gradient (8) of the von-Mises stress in each FE, with e=1,, N´N, is evaluated in parallel. The required N2 density perturbations are distributed through mult iple core processors. Homogenization runs for each perturbation, and the corresponding stress output is used to compute the respective finite difference derivative of each str ess-constraint (13). The stress, volume (9) and sti ffness (11) derivatives w.r.t. design variables r are computed using the chain rule (8). Then, MMA u pdates r and the
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observed). The difference of energy values between optimal topology and shape results are due to the presence of "grey" elements. A similar comparative analysis holds for another ex ample selected from Appendix B. An applied triaxial macro-strain ( xe,ye,ze) = (1.7,0.4,-0.9) ´10 -3 , equivalent to the previous plane stress in a homo geneous plate, produces the optimal topologies in Fig. 5a a nd 5b using the same FE mesh for stiffness and stif fness + strength designs, respectively. Note that in the la st case a lower bound on stiffness S should be used instead of an upper bound on compliance C. In the topology problem the stresses are bounded by the optimal stress value obtained from shape optimization solving the min-max stress problem (Fig. 5c), as detailed in Appendix B. Comparing the compliance and the strength-oriented topology optimization problems (see Fig 3a and 3b or Fig 5a and 5b), the volume is basically the same and energy values are equal. However, the local st ress distributions around the hole and the respective pe ak values do not compare so well. The topology optimization problem with stress constraints reprod uces the ESP and its related optimal shape with goo d accuracy. This means, as expected, that stress conc entration is very sensitive to the geometrical desi gn details while the total strain energy measure is ra ther insensitive (Pedersen 2000). This may indicate the advantage of formulating the topology optimization problem oriented also for strength. Case b): average principal stresses/strains are of opposite sign (fishearfl type) For the shear load cases the ESP is no longer valid , however a M-equi-stress principle is satisfied (Vindergauz 2001, Noël and Duysinx 2017) for subopt imal shapes. In the following examples the density and stress results are obtained for a square-grid m esh 64 ´64 ´1. This mesh is chosen based on a mesh convergence analysis for the shear loads, similar t o the one described in Appendix A. The same optimiz ation formulation (7) is used and the following applied m acro-strains are considered (plane strain): Shear 1: in-plane distortion ( 01.0 =xy e) Shear 2: shear rotated 45º ( 01.0 =xe, 01.0 -=ye) The shear load case 1 applies a distortion field su ch that stiffness S is maximized when a cross-shape UC is attained. As can be observed from Fig. 6, consid ering a volume fraction of 60%, the first design (n o stress control) exhibits a square hole with sharp corners and slightly rounded sides as expected (Vigdergauz- microstructure). The change in geometry due to the imposed stress limit reveals a smaller central hole and material removed from members where the stress leve l was lower. The redistribution of stresses taking place approximates a fully-stressed design solution as ma y be expected in optimality. A significant stress r eduction is achieved with a small volume fraction increase. Equivalent von-Mises stresses are maximum at the cr oss arms intersections. The stress-constrained desi gn exhibits slender holes along the cross arms and hal fway from the intersections. These generated fine h oles are well defined for the discretization level used and their boundaries are smooth enough so that no singularities arise. The shear load case 2 is equivalent to the previous one but rotated by 45º. The stiffest design is als o a square hole with sharp corners and slightly rounded sides but also rotated 45º, see Fig. 7a. Note that there is a scale factor ( 2) between these two designs. This is a consequence of the implicit constraint imposed by the fixed UC design domain. As the stress-limit is tigh tened the design changes are similar to the previou s example for the same range of stress limit s*, see Fig. 7. In reality, these topologies recall the results obt ained by Sigmund (2000) for extremal microstructure s, where a square cell with square inclusions and a fi ner scale laminates are optimal for the bulk modulu s, and
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provide shear modulus for an extreme square symmetr ic rank2 laminate. This optimality hinges in the fa ct that the strain and stress fields are constant with in the microstructure constituents. The obtained re sults show that when the stress constraint is tightened it tri ggers the appearance of a finer scale laminate like region, thus providing a more uniform stress field distribu tion. As general comments, all these cases studied imply symmetry both in density and equivalent stress distributions. So only a quarter of the UC is model led, reducing significantly the number of degrees o f freedom, design variables and stress constraints. N ote that these stress constraints should be seen as a measure of the local stress concentration factor, a nd do not correspond to a real stress value, since the applied average stress/strains and the Young™s modu lus of the UC solid part are fiunitaryfl. One also po ints out that an adaptive version of the stress relaxati on along the iterations was implemented. This is th e so- called continuum approach where q starts from a relative low number and gradually in creases with the iteration number, i.e. as one moves toward a "0-1" design. In this work one tries not only this appro ach ( q = 2 ® 3) but also q fixed to 3, with SIMP penalization p = 4. Both approaches proved to be equally effectiv e when solving the examples chosen in this work. Also , the behaviour of the iteration-history curves lea ding to the results shown here (Fig. 4 is given as an examp le) offers evidence that the finite difference deri vatives are accurate. Neither convergence problems nor osci llating behaviour toward the optima are detected. Finally, a speed-up of 22 is achieved here with 28 cores when solving the topology problem with stress constraints. This indicates very-good problem scala bility. Note that, in problems with similar structu re, parallel processing would be useful independently o f the methodology used to obtain sensitivities. 5. Conclusions Topology optimization has been extensively applied to strength design of (macro) structures, but only recently extended to material microstructure design . The primary challenges of stress control in struc tural optimization are non-linearity, design singularity and high computational cost due to the stress const raints local nature (Deaton and Grandhi 2014). Application s in UC material design also have to overcome these challenges. Further difficulties will appear in mul ti-scale design (macro/micro) due to the separation of scales in modelling heterogeneous media by asymptot ic homogenization models, where micro and macro- structural descriptions are required. This work considers optimal topology design of peri odic materials UCs taking into consideration stiffn ess and strength requirements. The stress field is highly non-linear and strongly dependent on the structural design. Stress levels a re drastically affected by the boundary local geometri c characteristics. As shown, a density filtering me thod has a smoothing (beneficial) effect on stresses althoug h leading to less sharp contour between void and so lid material. A crucial issue is the choice of the FE d iscretization assuring a robust and accurate stress field evaluation while preserving a reasonable computatio nal cost. The mesh convergence analysis performed indicates that a UC mesh between 64×64 and 128×128 is a balanced choice. A suitable qp -relaxation of the equivalent stress is used here t o overcome the design singularity phenomenon. The damage approach (Verbart et al. 201 6) can be explored in future developments since it relaxes the problem and simultaneously allows local stress control with just one design constraint. The optimization problem is solved using a first-or der mathematical programming method (MMA), so sensitivity analysis is required. Most of the compu ting effort and data storage involved are spent in the stress constraints first-order derivatives computation. Th e problem is worsened here since a local stress app roach was taken. As a preliminary approach, finite differ ence approximations of the stress derivatives are u sed here. The obtained results show that they are relia ble. Since stress-constraint derivatives can be com puted independently, parallel computing techniques are us ed to reduce the CPU-time. Further efficiency gains can
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be obtained using analytical derivatives and an app ropriate treatment of the strength constraints (see Collet et al 2016a,b). For load-bearing periodic cellular material microst ructures, topology optimization can bring significa nt weight reduction while guaranteeing structural beha viour compliant with strength design criteria. This formulation also allows a better description of the microstructure topology, due the high sensitivity of the stress distribution to the geometrical design, expe diting the satisfaction of the ESP principle for bu lk macro loadings. Remarkably, in the shear macro-loading si tuations, peak stresses reductions of 20% to 25% ar e achieved with only 1 to 2% increase in UC volume fr action, this being accomplished by triggering the appearance of what seems to be an additional length scale, similar to an extremal microstructure (Sigm und 2000).
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Appendix A: Mesh convergence analysis In this study a volume fraction of 0.85 and a macro -strain ( xe,ye,ze) = (-1,-1,1) ´10 -2 (in-plane hydrostatic) are considered. Under these conditions the optimal topology design is a hole of circular shape (Vigdergauz 2001, 2002). The stress distributions o btained from topology optimization in square meshes (N´N´1), progressively refined, are compared to the anal yses of stresses in non-square meshes specially tailored to the circular shape. Performing optimization with a square-grid mesh wit hout filtering technique, a jagged hole boundary is obtained (see Fig. A1a) and stress singularities ap pear (see Fig. A1d). If stress control is of concer n this can lead to solutions that are optimal only in terms of reducing the amplitude of unphysical peak stress. This phenomenon tends to obscure more important features , in the correct solution, as the existence of a st ress concentration or equi-stress state around the hole (see Fig. A1g). So, one must ensure that the stress contours in the highly stressed areas are smooth, thus overcoming the FE jagged boundary effects. One way to promote this , while still using a square-grid mesh, is by densi ty- based filtering techniques and mesh refinement. The results shown in Fig. A1b-c,e-f correspond to opti mal compliance designs with a density filter as defined by Bruns and Tortorelli (2001). Fig. A1b refers to a Filter = 1 meaning that the fi lter radius Rmin collects for averaging only the immediate adjacent element neighbors of element i (8 neighbors in 2D, Rmin = N/2). Filter = 2 in Fig. A1c includes also the elements immediate adjacent to the previou s neighborhood (total of 24 neighbors, Rmin = N/22). The total number of neighbors is preserved with mes h refinement thus limiting the presence of "grey" t o Rmin thickness. The qualitative evaluation of results that can be e xtracted from Fig. A1 is complemented with a more quantitative one based on stress and energy values, as depicted in Fig. A2. As the mesh is refined the re is strong convergence of the Strain Energy for both me shes and filters. In the maximum von-Mises stress c ase one sees that it deviates strongly in the case with out filter, overestimating the correct solution. Wi th the density filter one achieves a much more expectable stress distribution and accurate stress prediction (compare the colored contours in Fig. A1e-f with th e tailored mesh result in Fig. A1g). For example, t he stress distributions and peak values, of mesh = 64 ´64 with Filter = 1 (Fig. A1e) and mesh = 256 ´256 with Filter = 2 (Fig. A1f), mirror the reference distrib utions shown in Fig. A1g for the same level of discretization. The benefit of the filter in stress distribution is thus obvious. While diffusing the jagged boundary it smoothes the local stress field, suppre ssing unphysical singularities. This mesh convergence analysis aims at accurate app roximations of peak stresses, as they play an important role in stress-constrained optimization p roblems. However, the higher the accuracy the highe r the computational cost. So, one also wants here to iden tify a mesh size that satisfactorily balances accur acy and computational resources. On one hand, since peak st resses predicted by the 64 ´64 and 256 ´256 meshes differ only by 5% one considers that convergence is attained at 64 ´64 discretization level. On the other hand, the computational cost becomes prohibitively high f or meshes finer than 128 ´128. Stress-based topology optimization problems, which are typically time con suming, may become interesting to the engineering practice if one finds a reasonable trade-off betwee n accuracy and runtime. Although the example analys ed in this appendix is relatively simple, it points to a square-grid mesh with a level of discretization bet ween 64 ´64 and 128 ´128 with the filter radius kept to a minimum, Filte r = 1, as the best option. This is taken into consideration in section 4.
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