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Structural and Multidisciplinary Optimization

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12-06-2019 | Research Paper

A novel subdomain level set method for structural topology optimization and its application in graded cellular structure design

Authors: Hui Liu, Hongming Zong, Ye Tian, Qingping Ma, Michael Yu Wang

Published in: Structural and Multidisciplinary Optimization | Issue 6/2019

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Abstract

A novel subdomain structural topology optimization method is proposed for the minimum compliance problem based on the level sets with the parameterization of radial basis function (RBF). In this method, the level set function evolves on each subdomain separately and independently according to the requirements of objective functions and additional constraints. This makes the parameterization in the proposed subdomain method much faster and more cost-effective than that in the classical global method, as well as the evolution of the level set function since it can be achieved on each subdomain in parallel. In addition, the microstructures on arbitrary two adjacent subdomains can be connected perfectly, without any mismatch around the interfaces of the microstructures. Several typical examples are conducted to verify the correctness and effectiveness of the developed subdomain method. The effects of some factors on the optimized results are also investigated in detail, such as the RBF types, the connectivity types of microstructures, and the size of subdomain division. Without scale separation assumption, several layered graded cellular structures are successfully designed by employing the proposed method under the condition of corresponding repetition constraints. To improve the computational efficiency, a multi-node extended multiscale finite element method (EMsFEM) is used to solve the structural static equilibrium equation for the three-dimensional layered structure optimization problems. Furthermore, a MATLAB code is also provided in the Appendix for readers to reproduce the results of the two-dimensional problems in this work.

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Alexandersen J, Lazarov BS (2015a) Tailoring macroscale response of mechanical and heat transfer systems by topology optimization of microstructural details. In: Eng Appl Sci Optim Springer, pp 267–288. doi:https://doi.org/10.1007/978-3-319-18320-6_15

Alexandersen J, Lazarov BS (2015b) Topology optimisation of manufacturable microstructural details without length scale separation using a spectral coarse basis preconditioner. Comput Methods Appl Mech Eng 290:156–182. https://doi.org/10.1016/j.cma.2015.02.028 MathSciNetCrossRefMATH

Allaire G, Jouve F, Toader A-M (2004) Structural optimization using sensitivity analysis and a level-set method. J Comput Phys 194:363–393. https://doi.org/10.1016/j.jcp.2003.09.032 MathSciNetCrossRefMATH

Andreassen E, Clausen A, Schevenels M, Lazarov BS, Sigmund O (2010) Efficient topology optimization in MATLAB using 88 lines of code. Struct Multidiscip Optim 43:1–16. https://doi.org/10.1007/s00158-010-0594-7 CrossRefMATH

Bendsoe MP, Kikuchi N (1988) Generating optimal topologies in structural design using a homogenization method. Comput Methods Appl Mech Eng 71:197–224MathSciNetCrossRef

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

Bourdin B, Chambolle A (2003) Design-dependent loads in topology optimization. ESAIM: Control Optimisation and Calculus of Variations 9:19–48MathSciNetMATH

Challis VJ (2010) A discrete level-set topology optimization code written in Matlab. Struct Multidiscip Optim 41:453–464. https://doi.org/10.1007/s00158-009-0430-0 MathSciNetCrossRefMATH

Chen W, Tong L, Liu S (2017) Concurrent topology design of structure and material using a two-scale topology optimization. Comput Struct 178:119–128. https://doi.org/10.1016/j.compstruc.2016.10.013 CrossRef

Choi JS, Yamada T, Izui K, Nishiwaki S, Yoo J (2011) Topology optimization using a reaction–diffusion equation. Comput Methods Appl Mech Eng 200:2407–2420. https://doi.org/10.1016/j.cma.2011.04.013 MathSciNetCrossRefMATH

Clausen A, Wang F, Jensen JS, Sigmund O, Lewis JA (2015) Topology optimized architectures with programmable Poisson’s ratio over large deformations. Adv Mater 27:5523–5527. https://doi.org/10.1002/adma.201502485 CrossRef

Da DC, Cui XY, Long K, Li GY (2017) Concurrent topological design of composite structures and the underlying multi-phase materials. Comput Struct 179:1–14. https://doi.org/10.1016/j.compstruc.2016.10.006 CrossRef

Deaton JD, Grandhi RV (2013) A survey of structural and multidisciplinary continuum topology optimization: post 2000. Struct Multidiscip Optim 49:1–38. https://doi.org/10.1007/s00158-013-0956-z MathSciNetCrossRef

Efendiev Y, Hou TY (2009) Multiscale finite element methods: theory and applications. Springer

Groen JP, Sigmund O (2018) Homogenization-based topology optimization for high-resolution manufacturable microstructures. Int J Numer Methods Eng 113:1148–1163. https://doi.org/10.1002/nme.5575 MathSciNetCrossRef

Guo X, Cheng G-D (2010) Recent development in structural design and optimization. Acta Mech Sinica 26:807–823. https://doi.org/10.1007/s10409-010-0395-7 MathSciNetCrossRefMATH

Guo X, Zhang W, Zhong W (2014) Doing topology optimization explicitly and geometrically—a new moving morphable components based framework. J Appl Mech 81:081009. https://doi.org/10.1115/1.4027609 CrossRef

Ho HS, Lui BFY, Wang MY (2011) Parametric structural optimization with radial basis functions and partition of unity method. Optim Methods Software 26:533–553. https://doi.org/10.1080/10556788.2010.546399 MathSciNetCrossRefMATH

Ho HS, Wang MY, Zhou M (2012) Parametric structural optimization with dynamic knot RBFs and partition of unity method. Struct Multidiscip Optim 47:353–365. https://doi.org/10.1007/s00158-012-0848-7 MathSciNetCrossRefMATH

Kansa EJ, Power H, Fasshauer GE, Ling L (2004) A volumetric integral radial basis function method for time-dependent partial differential equations. I. Formulation. Eng Anal Bound Elem 28:1191–1206. https://doi.org/10.1016/j.enganabound.2004.01.004 CrossRefMATH

Kato J, Yachi D, Kyoya T, Terada K (2018) Micro-macro concurrent topology optimization for nonlinear solids with a decoupling multiscale analysis. Int J Numer Methods Eng 113:1189–1213. https://doi.org/10.1002/nme.5571 MathSciNetCrossRef

Lazarov BS (2013) Topology optimization using multiscale finite element method for high-contrast media. In: 9th International Conference on Large-Scale Scientific Computing. Springer, Sozopol, Bulgaria, pp 339–346. https://doi.org/10.1007/978-3-662-43880-0 CrossRef

Li H, Gao L, Xiao M, Gao J, Chen H, Zhang F, Meng W (2016a) Topological shape optimization design of continuum structures via an effective level set method. Cogent Eng 3 doi:https://doi.org/10.1080/23311916.2016.1250430

Li H, Li P, Gao L, Zhang L, Wu T (2015) A level set method for topological shape optimization of 3D structures with extrusion constraints. Comput Methods Appl Mech Eng 283:615–635. https://doi.org/10.1016/j.cma.2014.10.006 MathSciNetCrossRefMATH

Li H, Luo Z, Gao L, Qin Q (2018a) Topology optimization for concurrent design of structures with multi-patch microstructures by level sets. Comput Methods Appl Mech Eng 331:536–561. https://doi.org/10.1016/j.cma.2017.11.033 MathSciNetCrossRef

Li H, Luo Z, Gao L, Walker P (2018b) Topology optimization for functionally graded cellular composites with metamaterials by level sets. Comput Methods Appl Mech Eng 328:340–364. https://doi.org/10.1016/j.cma.2017.09.008 MathSciNetCrossRef

Li H, Luo Z, Zhang N, Gao L, Brown T (2016b) Integrated design of cellular composites using a level-set topology optimization method. Comput Methods Appl Mech Eng 309:453–475. https://doi.org/10.1016/j.cma.2016.06.012 MathSciNetCrossRef

Liu C, Du Z, Zhang W, Zhu Y, Guo X (2017) Additive manufacturing-oriented design of graded lattice structures through explicit topology. optimization. J Appl Mech 84:081008–081001–081012. https://doi.org/10.1115/1.4036941 CrossRef

Liu H, Wang Y, Zong H, Wang MY (2018) Efficient structure topology optimization by using the multiscale finite element method. Struct Multidiscip Optim 58:1411–1430. https://doi.org/10.1007/s00158-018-1972-9 MathSciNetCrossRef

Liu H, Zhang HW (2013) A uniform multiscale method for 3D static and dynamic analyses of heterogeneous materials. Comput Mater Sci 79:159–173. https://doi.org/10.1016/j.commatsci.2013.06.006 CrossRef

Luo Z, Tong L, Luo J, Wei P, Wang MY (2009) Design of piezoelectric actuators using a multiphase level set method of piecewise constants. J Comput Phys 228:2643–2659. https://doi.org/10.1016/j.jcp.2008.12.019 MathSciNetCrossRefMATH

Luo Z, Tong L, Wang MY, Wang S (2007) Shape and topology optimization of compliant mechanisms using a parameterization level set method. J Comput Phys 227:680–705. https://doi.org/10.1016/j.jcp.2007.08.011 MathSciNetCrossRefMATH

Luo Z, Wang MY, Wang S, Wei P (2008) A level set-based parameterization method for structural shape and topology optimization. Int J Numer Methods Eng 76:1–26. https://doi.org/10.1002/nme.2092 MathSciNetCrossRefMATH

Meza LR, Das S, Greer JR (2014) Strong, lightweight, and recoverable three-dimensional ceramic nanolattices. Science 345:1322–1326CrossRef

Morse BS, Yoo TS, Chen DT, Rheingans P, Subramanian KR (2001) Interpolating implicit surfaces from scattered surface data using compactly supported radial basis functions. International conference on shape modeling and applications 15:89–98

Nakshatrala PB, Tortorelli DA, Nakshatrala KB (2013) Nonlinear structural design using multiscale topology optimization. Part I: static formulation. Comput Methods Appl Mech Eng 261-262:167–176. https://doi.org/10.1016/j.cma.2012.12.018 MathSciNetCrossRefMATH

Nocedal J, Wright SJ (1999) Numerical optimization. Springer, New YorkCrossRef

Osher S, Fedkiw R (2002) Level set methods and dynamic implicit surfaces. Springer,

Otomori M, Yamada T, Izui K, Nishiwaki S (2014) Matlab code for a level set-based topology optimization method using a reaction diffusion equation. Struct Multidiscip Optim 51:1159–1172. https://doi.org/10.1007/s00158-014-1190-z MathSciNetCrossRef

Radman A, Huang X, Xie YM (2012) Topology optimization of functionally graded cellular materials. J Mater Sci 48:1503–1510. https://doi.org/10.1007/s10853-012-6905-1 CrossRef

Rojas-Labanda S, Stolpe M (2015) Automatic penalty continuation in structural topology optimization. Struct Multidiscip Optim 52:1205–1221. https://doi.org/10.1007/s00158-015-1277-1 MathSciNetCrossRef

Rozvany GIN, Zhou M, Birker T (1992) Generalized shape optimization without homogenization. Structural Optimization 4:250–252CrossRef

Sethian JA (1999) Level set methods and fast marching methods: evolving interfaces in computational geometry, fluid mechanics, computer vision, and materials science. Cambridge University Press

Shojaee S, Mohammadian M (2011) Piecewise constant level set method for structural topology optimization with MBO type of projection. Struct Multidiscip Optim 44:455–469. https://doi.org/10.1007/s00158-011-0646-7 MathSciNetCrossRefMATH

Sigmund O, Aage N, Andreassen E (2016) On the (non-)optimality of Michell structures. Struct Multidiscip Optim 54:361–373. https://doi.org/10.1007/s00158-016-1420-7 MathSciNetCrossRef

Sigmund O, Maute K (2013) Topology optimization approaches. Struct Multidiscip Optim 48:1031–1055. https://doi.org/10.1007/s00158-013-0978-6 MathSciNetCrossRef

Silva ECN, Walters MC, Paulino GH (2006) Modeling bamboo as a functionally graded material: lessons for the analysis of affordable materials. J Mater Sci 41:6991–7004. https://doi.org/10.1007/s10853-006-0232-3 CrossRef

Sivapuram R, Dunning PD, Kim HA (2016) Simultaneous material and structural optimization by multiscale topology optimization. Struct Multidiscip Optim 54:1267–1281. https://doi.org/10.1007/s00158-016-1519-x MathSciNetCrossRef

Stolpe M, Svanberg K (2001) On the trajectories of penalization methods for topology optimization. Struct Multidiscip Optim 21:128–139CrossRef

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

van Dijk NP, Maute K, Langelaar M, van Keulen F (2013) Level-set methods for structural topology optimization: a review. Struct Multidiscip Optim 48:437–472. https://doi.org/10.1007/s00158-013-0912-y MathSciNetCrossRef

Vicente WM, Zuo ZH, Pavanello R, Calixto TKL, Picelli R, Xie YM (2016) Concurrent topology optimization for minimizing frequency responses of two-level hierarchical structures. Comput Methods Appl Mech Eng 301:116–136. https://doi.org/10.1016/j.cma.2015.12.012 MathSciNetCrossRefMATH

Wang MY, Wang X, Guo D (2003) A level set method for structural topology optimization. Comput Methods Appl Mech Eng 192:227–246MathSciNetCrossRef

Wang MY, Zhou S (2004) Phase field: a Variational method for structural topology optimization. Comput Model Eng Sci 6:547–566MathSciNetMATH

Wang S, Wang MY (2006) Radial basis functions and level set method for structural topology optimization. Int J Numer Methods Eng 65:2060–2090. https://doi.org/10.1002/nme.1536 MathSciNetCrossRefMATH

Wang SY, Lim KM, Khoo BC, Wang MY (2007) An extended level set method for shape and topology optimization. J Comput Phys 221:395–421. https://doi.org/10.1016/j.jcp.2006.06.029 MathSciNetCrossRefMATH

Wang Y, Chen F, Wang MY (2017a) Concurrent design with connectable graded microstructures. Comput Methods Appl Mech Eng 317:84–101. https://doi.org/10.1016/j.cma.2016.12.007 MathSciNetCrossRef

Wang Y, Wang MY, Chen F (2016) Structure-material integrated design by level sets. Struct Multidiscip Optim 54:1145–1156. https://doi.org/10.1007/s00158-016-1430-5 MathSciNetCrossRef

Wang Y, Xu H, Pasini D (2017b) Multiscale isogeometric topology optimization for lattice materials. Comput Methods Appl Mech Eng 316:568–585. https://doi.org/10.1016/j.cma.2016.08.015 MathSciNetCrossRef

Wei P, Li Z, Li X, Wang MY (2018) An 88-line MATLAB code for the parameterized level set method based topology optimization using radial basis functions. Struct Multidiscip Optim 58:831–849. https://doi.org/10.1007/s00158-018-1904-8 MathSciNetCrossRef

Wei P, Wang MY (2009) Piecewise constant level set method for structural topology optimization. Int J Numer Methods Eng 78:379–402. https://doi.org/10.1002/nme.2478 MathSciNetCrossRefMATH

Xia L, Breitkopf P (2014a) Concurrent topology optimization design of material and structure within FE² nonlinear multiscale analysis framework. Comput Methods Appl Mech Eng 278:524–542. https://doi.org/10.1016/j.cma.2014.05.022 CrossRefMATH

Xia L, Breitkopf P (2014b) A reduced multiscale model for nonlinear structural topology optimization. Comput Methods Appl Mech Eng 280:117–134. https://doi.org/10.1016/j.cma.2014.07.024 MathSciNetCrossRefMATH

Xia L, Breitkopf P (2015) Multiscale structural topology optimization with an approximate constitutive model for local material microstructure. Comput Methods Appl Mech Eng 286:147–167. https://doi.org/10.1016/j.cma.2014.12.018 MathSciNetCrossRefMATH

Xia L, Breitkopf P (2016) Recent advances on topology optimization of multiscale nonlinear structures. Arch Comput Methods Eng 24:227–249. https://doi.org/10.1007/s11831-016-9170-7 MathSciNetCrossRefMATH

Xie YM, Steven GP (1993) A simple evolutionary procedure for structural optimization. Comput Struct 49:885–896CrossRef

Yang XY, Xei YM, Steven GP, Querin OM (1999) Bidirectional evolutionary method for stiffness. Optimization. AIAA J 37:1483–1488. https://doi.org/10.2514/2.626 CrossRef

Zhang HW, Liu H, Wu JK (2013) A uniform multiscale method for 2D static and dynamic analyses of heterogeneous materials. Int J Numer Methods Eng 93:714–746. https://doi.org/10.1002/nme.4404 MathSciNetCrossRefMATH

Zhang W, Li D, Zhou J, Du Z, Li B, Guo X (2018) A moving morphable void (MMV)-based explicit approach for topology optimization considering stress constraints. Comput Methods Appl Mech Eng 334:381–413. https://doi.org/10.1016/j.cma.2018.01.050 MathSciNetCrossRef

Zhang W, Sun S (2006) Scale-related topology optimization of cellular materials and structures. Int J Numer Methods Eng 68:993–1011. https://doi.org/10.1002/nme.1743 CrossRefMATH

Zheng X et al (2014) Ultralight, ultrastiff mechanical metamaterials. Science 344:1373–1377CrossRef

Zhou M, Rozvany GIN (1991) The COC algorithm, part II- topological, geometrical and generalized shape optimization. Comput Methods Appl Mech Eng 89:309–336CrossRef

Zhu S, Wu Q, Liu C (2011) Shape and topology optimization for elliptic boundary value problems using a piecewise constant level set method. Appl Numer Math 61:752–767. https://doi.org/10.1016/j.apnum.2011.01.005 MathSciNetCrossRefMATH

Zhu Y, Li S, Du Z, Liu C, Guo X, Zhang W (2019) A novel asymptotic-analysis-based homogenisation approach towards fast design of infill graded microstructures. J Mech Physics Solids 124:612–633. https://doi.org/10.1016/j.jmps.2018.11.008 MathSciNetCrossRef

Title: A novel subdomain level set method for structural topology optimization and its application in graded cellular structure design
Authors: Hui Liu
Hongming Zong
Ye Tian
Qingping Ma
Michael Yu Wang
Publication date: 12-06-2019
Publisher: Springer Berlin Heidelberg
Published in: Structural and Multidisciplinary Optimization / Issue 6/2019
Print ISSN: 1615-147X
Electronic ISSN: 1615-1488
DOI: https://doi.org/10.1007/s00158-019-02318-3

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