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Topology optimization for microstructural design under stress constraints

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Abstract

This work aims at introducing stress responses within a topology optimization framework applied to the design of periodic microstructures. The emergence of novel additive manufacturing techniques fosters research towards new approaches to tailor materials properties. This paper derives a formulation to prevent the occurrence of high stress concentrations, often present in optimized microstructures. Applying macroscopic test strain fields to the material, microstructural layouts, reducing the stress level while exhibiting the best overall stiffness properties, are sought for. Equivalent stiffness properties of the designed material are predicted by numerical homogenization and considering a metallic base material for the microstructure, it is assumed that the classical Von Mises stress criterion remains valid to predict the material elastic allowable stress at the microscale. Stress constraints with arbitrary bounds are considered, assuming that a sizing optimization step could be applied to match the actual stress limits under realistic service loads. Density–based topology optimization, relying on the SIMP model, is used and the qp–approach is exploited to overcome the singularity phenomenon arising from the introduction of stress constraints with vanishing material. Optimization problems are solved using mathematical programming schemes, in particular MMA, so that a sensitivity analysis of stress responses at the microstructural level is required and performed considering the adjoint approach. Finally, the developed method is first validated with classical academic benchmarks and then illustrated with an original application: tailoring metamaterials for a museum anti–seismic stand.

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References

  • Aboudi J, Arnold SM, Bednarcyk BA (2012) Micromechanics of composite materials: A generalized multiscale analysis approach. Butterworth-Heinemann

  • Andreassen E, Andreasen CS (2014) How to determine composite material properties using numerical homogenization. Comput Mater Sci 83:488–495

    Article  Google Scholar 

  • Andreassen E, Jensen JS, Sigmund O, Thomsen JJ (2015) Optimal design of porous materials. DTU mechanical engineering. (DCAMM Special Report; No. S172)

  • Avellaneda S (1998) Calculating the performance of 1–3 piezoelectric composites for hydrophone applications: an effective medium approach. J Acoust Soc Am 103(3):1449–1467

    Article  Google Scholar 

  • Bendsøe M, Kikuchi N (1988) Generating optimal topologies in structural design using a homogeneization method. Comp Meth Appl Mech Eng 71:197–224

    Article  Google Scholar 

  • Bendsøe MP (1989) Optimal Shape as a material distribution problem. Structural Optimization 1:193–202

    Article  Google Scholar 

  • Bendsøe M, Sigmund O (1999) Material interpolation schemes in topology optimization. Arch Appl Mech 69(1999):635–654

    MATH  Google Scholar 

  • Bendsøe M, Sigmund O (2003) Topology optimization - Theory, methods and applications. Springer, EUA, New York

    MATH  Google Scholar 

  • Bensoussan A, Lions JL, Papanicolaou G (1978) Asymptotic analysis for periodic structures. North-Holland

  • Bourdin B (2001) Filters in topology optimization. Int J Numer Methods Eng 50:2143–2158

    Article  MathSciNet  Google Scholar 

  • Bruggi M (2008) On an alternative approach to stress constraints relaxation in topology optimization. Struct Multidisc Optim 36:125–141

    Article  MathSciNet  Google Scholar 

  • Bruggi M (2016) Topology optimization with mixed finite elements on regular grids. Comp Meth Appl Mech Eng 305:133–153

    Article  MathSciNet  Google Scholar 

  • Bruggi M, Dusyinx P (2012) Topology optimization for minimum weight with compliance and stress constraints. Struct Multidisc Optim 46(3):369–384

    Article  MathSciNet  Google Scholar 

  • Bruns TE, Tortorelli DA (2001) Topology optimization of non–linear elastic structures and compliant mechanisms. Comp Meth Appl Mech Eng 190:3443–3459

    Article  Google Scholar 

  • Cadman JE, Zhou S, Chen Y, Li Q (2013) On design of multi-functional microstructural materials. J Mater Sci 48:51–66

    Article  Google Scholar 

  • Cheng GD, Guo X (1997) ε–relaxed approach in topology optimization. Struct Optim 13:258–266

    Article  Google Scholar 

  • Chenm R (1985) Solution of minimax problems using equivalent differentiable functions. Comp & Maths with Appls 11(12):1165–1169

    Article  MathSciNet  Google Scholar 

  • Collet M, Bruggi M, Duysinx P (2017) Topology optimization for minimum weight with compliance and simplified nominal stress constraints for fatigue resistance. Struct Multidiscip Optim 55:839–855

    Article  MathSciNet  Google Scholar 

  • Deaton JD, Grandhi R (2014) A survey of structural and multidisciplinary continuum topology optimization: post 2000. Struct Multidiscip Optim 49:1–38

    Article  MathSciNet  Google Scholar 

  • Deshpande V, Fleck N, Ashby M (2001) Effective properties of the octet-truss lattice material. J Mech Phys Solids 49:1747–1769

    Article  Google Scholar 

  • Duysinx P, Bendsøe MP (1998) Topology optimization of continuum structures with local stress constraints. Int J Numer Methods Eng 43:1453–1478

    Article  MathSciNet  Google Scholar 

  • Duysinx P, Sigmund O (1998) New developments in handling stress constraints in optimal material distribution. 7th Symposium on Multidisciplinary Analysis and Optimization AIAA–98–4906: 1501–1509

  • Evans KE, Nkansah MA, Hutchinson IJ, Rogers SC (1991) Molecular network design. Nature 353:124

    Article  Google Scholar 

  • Gibiansky LV, Sigmund O (2000) Multiphase composites with extremal bulk modulus. J Mech Phys Solids 48:461–498

    Article  MathSciNet  Google Scholar 

  • Guedes JM, Kikuchi N (1990) Preprocessing and postprocessing for materials based on the homogenization method with adaptive nite element methods. Comput Methods Appl Mech Eng 83:143–198

    Article  Google Scholar 

  • Guest J (2009) Imposing maximum length scale in topology optimization. Struct Multidiscip Optim 37:463–473

    Article  MathSciNet  Google Scholar 

  • Guest JK, Prévost JH (2006) Optimizing multifunctional materials: design of microstructures for maximized stiffness and fluid permeability. Int J Solids Struct 43:7028–7047

    Article  Google Scholar 

  • Gurson AL (1977) Continuum theory of ductile rupture by void nucleation and growth: Part I - Yield criterion and flow rules for porous ductile media. J Eng Mater Technol 99(1):2–15

    Article  Google Scholar 

  • Grabovsky Y, Kohn R (1995) Microstructures minimizing the energy of a two phase elastic composite in two space dimensions. ii: The Vigdergauz microstructure. J Mech Phys Solids 43(6):949–972

    Article  MathSciNet  Google Scholar 

  • Hashin Z, Shtrikman S (1963) A variational approach to the theory of the elastic behaviour of multiphase materials. J Mech Phys Solids 11(127):140

    MathSciNet  MATH  Google Scholar 

  • Hassani B, Hinton E (1997) A review of homogenization and topology optimization i:homogenization theory for media with periodic structure. Comput Struct 69(707):717

    Google Scholar 

  • Hassani B, Hinton E (1998) A review of homogenization and topology optimization ii:homogenization theory for media with periodic structure. Comput Struct 69(719):738

    Google Scholar 

  • Hassani B, Hinton E (1998) A review of homogenization and topology optimization iii:homogenization theory for media with periodic structure. Comput Struct 69(739):756

    MATH  Google Scholar 

  • Holmberg E, Torstenfelt B, Klarbring A (2013) Stress constrained topology optimization. Struc Multidisc Optim 48:33–47

    Article  MathSciNet  Google Scholar 

  • Jensen J, Sigmund O (2011) Topology optimization for nano-photonics. Laser Photonics Rev 5:308–321

    Article  Google Scholar 

  • Jia H, Misra A, Poorsolhjouy P, Liu C (2016) Optimal structural topology of materials with micro-scale tension-compression asymmetry simulated using granular micromechanics. Mater Des 115:422–432

    Article  Google Scholar 

  • Kirsch U (1990) On singular topologies in optimal structural design. Struct Optim 2:133–142

    Article  Google Scholar 

  • Kouznetsova V, Brekelmans WA, Baaijens FP (2001) An approach to micro-macro modeling of heterogeneous materials. Comput Mech 27(1):37–48

    Article  Google Scholar 

  • Lazarov BS, Wang F, Sigmund O (2016) Length scale and manufacturability in density-based topology optimization. Arch Appl Mech 86:189–218

    Article  Google Scholar 

  • Le C, Norato J, Bruns TE, Ha C, Tortorelli DA (2010) Stress–based Topology Optimization for Continua. Struct Multidiscip Optim 41:605–620

    Article  Google Scholar 

  • Lipton R, Stuebner M (2006) Inverse homogenization and design of microstructure for pointwise stress control. Q J Mech Appl Math 59:131–169

    Article  MathSciNet  Google Scholar 

  • Lipton R, Stuebner M (2007) Optimal design of composite structures for strength and stiffness: an inverse homogenization approach. Struct Multidisc Optim 33(2007):351–362

    Article  MathSciNet  Google Scholar 

  • Liu Q (2006) Literature Review: Materials with Negative Poisson’s ratios and potential application to aerospace and defence. DSTO Research Library Thesaurus

  • Liu Q, Chan R, Huang X (2016) Concurrent topology optimization of macrostructures and material microstructures for natural frequency. Mater Des 106:380–390

    Article  Google Scholar 

  • Luo Y, Yu W, Yu M, Kang Z (2013) An enhanced aggregation method for topology optimization with local stress constraints. Comput Method Appl Mech Engrg 254:31–41

    Article  MathSciNet  Google Scholar 

  • Michel JC, Suquet PM (1993) On the strength of composite materials: variational bounds and computational aspects. In: Bendsoe MP, Mota Soares C (eds) Topology design of structures. Kluwer Academic Publishers, pp 355–374

  • Michel JC, Moulinec H, Suquet P (1998) Effective properties of composite materials with periodic microstructure: a computational approach. Comput Methods Appl Mech Engrg 172(1999):109–143

    MathSciNet  MATH  Google Scholar 

  • Miehe C, Koch A (2002) Computational micro-to-macro transitions of discretized microstructures undergoing small strains. Arch Appl Mech 72:300–317

    Article  Google Scholar 

  • Mlejnek H, Schirrmacher R (1993) An engineer’s approach to optimal material distribution and shape finding. Comput Methods Appl Mech Eng 106(1):1–26

    Article  Google Scholar 

  • Nguyen V-D, Béchet E, Gueuzaine C, Noels L (2012) Imposing periodic boundary conditions on arbitrary meshes by polynomial interpolation. Comput Mater Sci 55:390–406

    Article  Google Scholar 

  • Noël L, Duysinx P (2016) Shape optimization of microstructural designs subject to local stress constraints within an XFEM-level set framework. Struct Multidisc Optim. 55(6):2323–2338

    Article  MathSciNet  Google Scholar 

  • Oest J, Lund E (2017) Topology optimization with finite-life fatigue constraints. Struct Multidisc Optim. 56(5):1045–1059

    Article  MathSciNet  Google Scholar 

  • París J, Navarrina F, Colominas I, Casteleiro M (2010) Block aggregation of stress constraints in topology optimization of structures. Adv Eng Softw 41:433–441

    Article  Google Scholar 

  • Ponte-Castenada P, De Botton G (1992) On the homogenized yield strength two-phase composites. Proc. R. Soc. London. A. (438) 439–444

  • Rozvany GIN, Zhou M, Sigmund O (1994) Topology optimization in structural design. In: Adeli H (ed) Advances in design optimization, Chapman and Hall, London, pp 340–399

  • Sanchez-Palencia E (1983) Homogenization method for the study of composite media. In: Verhulst F (ed) asymptotic analysis II, lecture notes in mathematics, vol 985. Springer, Berlin, pp. 192:214

    Google Scholar 

  • Sanchez-Hubert J, Sanchez-Palencia E (1998) Introduction aux méthodes asymptotiques et à l’homogénéisation. Collection mathématiques appliquées pour la maîtrise. Masson. Partis. 1992

  • Sigmund O (1994) Materials with prescribed constitutive parameters: an inverse homogenization problem. Int J Solids Struct 31(17):2313–2329

    Article  MathSciNet  Google Scholar 

  • Sigmund O (1995) Tailoring materials with prescribed elastic properties. Mech Mater 20:351–368

    Article  Google Scholar 

  • Sigmund O (1999) A new class of extremal composites. J Mech Phys Solids 48(2000):397–428

    MathSciNet  MATH  Google Scholar 

  • Sigmund O, Maut K (2013) Topology optimization approaches: a comparative review. Struc Multidisc Optim 48:1031–1055

    Article  Google Scholar 

  • Sigmund O, Petersson J (1998) Numerical instabilities in topology optimization: a survey on procedures dealing with checkerboards, mesh-dependencies and local minima. Struct Optim 16:68–75

    Article  Google Scholar 

  • Sigmund O, Torquato S (1999) Design of smart composite materials using topology optimization. Smart Mater Struct 8:365–379

    Article  Google Scholar 

  • Suquet P (1982) Une méthode duale en homogénéisation : application aux milieux élastiques. Journal de Mé,canique Théorique et Appliquée 98:79

    MATH  Google Scholar 

  • Svanberg K (1987) Method of moving asymptotes - a new method for structural optimization. Int J Numer Methods Eng 24:359–373

    Article  MathSciNet  Google Scholar 

  • Tyrus J, Gosz M, DeStantiago E (2007) A local finite element implementation for imposing periodic boundary conditions on composite micromechanical models. Int J Solids Struct 44(9):2972–2989

    Article  Google Scholar 

  • Svärd H (2015) Interior value extrapolation: a new method for stress, evaluation during topology optimization. Struct Multidisc Optim 51:613–629

    Article  Google Scholar 

  • Vigdergauz S (2001) The effective properties of a perforated elastic plate numerical optimization by genetic algorithm. Int J Solids Struct 38:8593–8616

    Article  Google Scholar 

  • Xia L (2015) Towards optimal design of multiscale nonlinear structures dissertation submitted for the degree of doctor of philosophy of advanced mechanics. Université de Technologie de Compiègne, Sorbonne Universités

    Google Scholar 

  • Xia L, Breitkopf P (2015) Design of materials using topology optimization and energy-based homogenization approach in Matlab. Struct Multidisc Optim 52:1229–1241

    Article  MathSciNet  Google Scholar 

  • Yang RJ, Chen CJ (1996) Stress-based topology optimization. Struct Multidisc Optim 12(2):98–105

    Article  Google Scholar 

  • Zhang S, Gain AL, Norato J (2017) Stress-based topology optimization with discrete geometric components. Comput Methods Appl Mech Engrg 325:1–21

    Article  MathSciNet  Google Scholar 

Download references

Acknowledgements

Part of the work was carried out by the first author, M. Collet, during a research stay at Politecnico di Milano. M. Collet is supported by a grant from the Belgian National Fund for Scientific Research (FRIA), which is gratefully acknowledged. The third author, M. Bruggi, would like to acknowledge the contribution of MIUR through the project PRIN15-2015LYYXA8 Multi-scale mechanical models for the design and optimization of microstructured smart materials and metamaterials.

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Appendices

Appendix A

To validate the choice of the reference mesh (64 × 64 elements) used in this contribution, an additional simulation with a finer mesh (128 × 128 elements) is performed. The finer mesh is used to solve the bulk modulus maximization problem described in (15a) with the objective function given in (24). The resulting designs along with their scaled stress maps for both the unconstrained and the stress constrained problems are provided in Fig. 16.

Fig. 16
figure 16

Maximization of the bulk modulus under hydrostatic loading for a fine mesh , i.e. 128 × 128 elements: a optimized layout without stress constraints; b optimized layout with stress constraints; c scaled Von Mises stress map for the optimized layout without stress constraints; d scaled Von Mises stress map for the optimized layout with stress constraints

The cavity layouts as well as the corresponding stress maps in Fig. 16 are in good agreement with the results obtained with the reference mesh, see Fig. 4. As observed previously, the stress–constrained design leads to a decrease in the bulk modulus values with respect to the unconstrained design, i.e. \(\protect \mathcal {K}^{H} = 0.2400~N/m^{2}\) and \(\protect \mathcal {K}^{H} = 0.2438~N/m^{2}\) respectively. One should also notice that using a finer mesh yields bulk modulus values closer to the Hashin–Shtrikman theoretical bounds detailed in Table 1.

However, although the quality of the structural responses evaluated is improved, the CPU time drastically increases when using such a fine mesh. Moreover, the number of selected constraints handled by the optimizer at the end of the optimization process is also significantly increased, \(\protect N_{s}^{end} = 2649\) over 16384 potential elements. Therefore, the gain in accuracy is obtained at the price of an important loss of efficiency.

For these reasons, the reference mesh is used in this paper as it constitutes a good balance between accuracy, i.e. the stress fields are correctly captured, and CPU time. Finally, one should note that the adopted mesh size is similar to the one used in multiple works addressing stress–based optimization, see e.g. Bruggi and Dusyinx (2012), Collet et al. (2017), Oest and Lund (2017), and Zhang et al. (2017).

Appendix B

For illustration purposes, let us consider two problems: a simple problem (P1), where we minimize the volume subject to stress constraints, and a second problem (P2), where we minimize the maximum local stress subject to a volume constraint. The maximum local stress can be approximated by replacing the max function by a smooth continuous function, see e.g. Chenm (1985). By writing down the stationary conditions on the Lagrangians for both problems ((34) and (35)), one can see that problems (P1) and (P2) are equivalent, provided that there exists a specific relation between the Lagrange multipliers λ1 and λ2. In fact, (P1) and (P2) are equivalent if: \(\protect \lambda _{1} = \frac {1}{\lambda _{2}}\), i.e. if the Lagrange multipliers are the inverse from one another. Swapping the objective and the constraint is therefore consistent with respect to the mathematical programming approaches, as considered in our paper.

$$(P_{1})\left\{ \begin{array}{lll} \displaystyle \min_{x} & \displaystyle \mathcal{V}\\ \text{s.t.} &\displaystyle \sigma^{\max} \leq \overline{\sigma},& \end{array} \right. (P_{2})\left\{ \begin{array}{lll} \displaystyle \min_{x} & \displaystyle \sigma^{\max}\\ \text{s.t.} &\displaystyle \mathcal{V}\leq\overline{V},& \end{array} \right. $$
$$\begin{array}{@{}rcl@{}} \mathcal{L}_{1}(x,\lambda_{1}) &=& \mathcal{V}+\lambda_{1}(\sigma^{\max}-\overline{\sigma}) \\ \frac{\partial \mathcal{L}_{1}}{\partial x} &=& \frac{\partial \mathcal{V}}{\partial x}+\lambda_{1}\frac{\partial \sigma^{\max}}{\partial x}\qquad\ \ \ = 0\\ \frac{\partial \mathcal{L}_{1}}{\partial \lambda_{1}} &=& \sigma^{\max}-\overline{\sigma} \qquad\qquad\quad \ = 0 \end{array} $$
(34)
$$\begin{array}{@{}rcl@{}} \mathcal{L}_{2}(x,\lambda_{2}) &=& \mathcal{\sigma^{\max}}+\lambda_{2}(\mathcal{V}-\overline{\mathcal{V}})\\ \frac{\partial \mathcal{L}_{2}}{\partial x} &=& \frac{\partial \sigma^{\max}}{\partial x}+\lambda_{2}\frac{\partial \mathcal{V}}{\partial x}\qquad\ \ \ = 0\\ \frac{\partial \mathcal{L}_{2}}{\partial \lambda_{2}} &=& \mathcal{V}-\overline{\mathcal{V}}\qquad \qquad \qquad \ \ = 0 \end{array} $$
(35)

(P1) and (P2) are are equivalent if one takes : \(\protect \lambda _{1} = \frac {1}{\lambda _{2}}\). Said otherwise, the two optimization problems are equivalent if the Lagrange multipliers are the inverse from one another which make sense since one has swapped the constraint and the objective from one problem to the other.

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Collet, M., Noël, L., Bruggi, M. et al. Topology optimization for microstructural design under stress constraints. Struct Multidisc Optim 58, 2677–2695 (2018). https://doi.org/10.1007/s00158-018-2045-9

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