Multiscale concurrent topology optimization for cellular structures with multiple microstructures based on ordered SIMP interpolation

Abstract This paper presents a novel multiscale concurrent topology optimization method for cellular structures with various types of microstructures to obtain a superior structural performance at an affordable computation cost. At macroscale, different microstructures are regarded as different materials in a multi-material topology optimization. An ordered solid isotropic material with penalization (SIMP) interpolation model, which is efficient for solving multi-material topology optimization problems without introducing any new variables, is employed to generate an overall structure layout with predetermined multiple materials. At microscale, each macroscale element is viewed as an individual microstructure. All macroscale elements with the same material are represented by a unique microstructure. A parametric level set method (PLSM) is applied to evolve the shape and topology of the representative microstructures, whose effective properties are evaluated by a numerical homogenization method. The connectivity of adjacent microstructures is guaranteed by a kinematical connective constraint. Using the proposed method, the macrostructural topology as well as the locations and configurations of microstructures can be simultaneously optimized. Numerical examples are provided to test the effectiveness of the proposed method.

[1]  Yi Min Xie,et al.  Topology optimization of functionally graded cellular materials , 2013, Journal of Materials Science.

[2]  Gengdong Cheng,et al.  Optimum structure with homogeneous optimum truss-like material , 2008 .

[3]  Yi Min Xie,et al.  Two-scale optimal design of structures with thermal insulation materials , 2015 .

[4]  Z. Kang,et al.  Topological shape optimization of microstructural metamaterials using a level set method , 2014 .

[5]  Y. Xie,et al.  A simple evolutionary procedure for structural optimization , 1993 .

[6]  G. Allaire,et al.  Shape optimization by the homogenization method , 1997 .

[7]  Shiwei Zhou,et al.  Topological design of structures and composite materials with multiobjectives , 2007 .

[8]  K. Saitou,et al.  Multi-material topology optimization using ordered SIMP interpolation , 2016, Structural and Multidisciplinary Optimization.

[9]  M. Bendsøe,et al.  Material interpolation schemes in topology optimization , 1999 .

[10]  Z. Kang,et al.  A level set method for structural shape and topology optimization using radial basis functions , 2009 .

[11]  P. Breitkopf,et al.  Multiscale structural topology optimization with an approximate constitutive model for local material microstructure , 2015 .

[12]  James K. Guest,et al.  Optimizing multifunctional materials: Design of microstructures for maximized stiffness and fluid permeability , 2006 .

[13]  Ole Sigmund,et al.  A 99 line topology optimization code written in Matlab , 2001 .

[14]  Liang Gao,et al.  Topology optimization for functionally graded cellular composites with metamaterials by level sets , 2018 .

[15]  Ole Sigmund,et al.  On the (non-)optimality of Michell structures , 2016, Structural and Multidisciplinary Optimization.

[16]  G. Allaire,et al.  Structural optimization using sensitivity analysis and a level-set method , 2004 .

[17]  Liang Gao,et al.  Integrated design of cellular composites using a level-set topology optimization method , 2016 .

[18]  K. Svanberg.  The method of moving asymptotes—a new method for structural optimization , 1987 .

[19]  Liang Gao,et al.  Topology optimization for concurrent design of structures with multi-patch microstructures by level sets , 2018 .

[20]  Shutian Liu,et al.  Concurrent topology design of structure and material using a two-scale topology optimization , 2017 .

[21]  Helder C. Rodrigues,et al.  Hierarchical topology optimization addressing material design constraints and application to sandwich-type structures , 2015 .

[22]  Jakob S. Jensen,et al.  Topology optimization of periodic microstructures for enhanced dynamic properties of viscoelastic composite materials , 2014 .

[23]  Peter D. Dunning,et al.  Simultaneous material and structural optimization by multiscale topology optimization , 2016 .

[24]  Xiaoming Wang,et al.  Color level sets: a multi-phase method for structural topology optimization with multiple materials , 2004 .

[25]  Michael Yu Wang,et al.  Concurrent design with connectable graded microstructures , 2017 .

[26]  S. Torquato,et al.  Composites with extremal thermal expansion coefficients , 1996 .

[27]  Z. Kang,et al.  A multi-material level set-based topology and shape optimization method , 2015 .

[28]  G. Rozvany.  Aims, scope, methods, history and unified terminology of computer-aided topology optimization in structural mechanics , 2001 .

[29]  Y. Xie,et al.  Topological design of microstructures of cellular materials for maximum bulk or shear modulus , 2011 .

[30]  S. Torquato,et al.  Design of materials with extreme thermal expansion using a three-phase topology optimization method , 1997 .

[31]  Shiwei Zhou,et al.  Topology optimization of microstructures of cellular materials and composites for macrostructures , 2013 .

[32]  Xiaoming Wang,et al.  A level set method for structural topology optimization , 2003 .

[33]  Shiwei Zhou,et al.  Design of graded two-phase microstructures for tailored elasticity gradients , 2008, Journal of Materials Science.

[34]  B. C. Chen,et al.  Composite material design of two‐dimensional structures using the homogenization design method , 2001 .

[35]  M. Zhou,et al.  The COC algorithm, Part II: Topological, geometrical and generalized shape optimization , 1991 .

[36]  Shiwei Zhou,et al.  Topology optimization for 3D microstructures of viscoelastic composite materials , 2021 .

[37]  Boyan S. Lazarov,et al.  Robust topology optimisation of microstructural details without length scale separation - using a spectral coarse basis preconditioner , 2014, ArXiv.

[38]  Richard M. Christensen,et al.  Mechanics of cellular and other low-density materials , 2000 .

[39]  Michael Yu Wang,et al.  Shape and topology optimization of compliant mechanisms using a parameterization level set method , 2007, J. Comput. Phys..

[40]  P. Breitkopf,et al.  Concurrent topology optimization design of material and structure within FE2 nonlinear multiscale analysis framework , 2014 .

[41]  Seung Chul Han,et al.  A New Type of Low Density Material: Shellular , 2015, Advanced materials.

[42]  Gengdong Cheng,et al.  Two-scale concurrent topology optimization with multiple micro materials based on principal stress orientation , 2018 .

[43]  M. Wang,et al.  Structure-material integrated design by level sets , 2016 .

[44]  M. Bendsøe,et al.  Topology Optimization: "Theory, Methods, And Applications" , 2011 .

[45]  Y. Xie,et al.  Bi-directional evolutionary topology optimization of continuum structures with one or multiple materials , 2009 .

[46]  Liang Gao,et al.  Topological shape optimization of 3D micro-structured materials using energy-based homogenization method , 2018, Adv. Eng. Softw..

[47]  Helder C. Rodrigues,et al.  A hierarchical model for concurrent material and topology optimisation of three-dimensional structures , 2008 .

[48]  Weihong Zhang,et al.  Scale‐related topology optimization of cellular materials and structures , 2006 .

[49]  Shutian Liu,et al.  Size-dependent optimal microstructure design based on couple-stress theory , 2010 .

[50]  Zhan Kang,et al.  Layout design of reinforced concrete structures using two-material topology optimization with Drucker–Prager yield constraints , 2013 .

[51]  O. Sigmund,et al.  Topology optimization approaches , 2013, Structural and Multidisciplinary Optimization.

[52]  Shiwei Zhou,et al.  On design of multi-functional microstructural materials , 2012, Journal of Materials Science.

[53]  Z. Kang,et al.  Structural topology optimization based on non-local Shepard interpolation of density field , 2011 .

[54]  Casper Schousboe Andreasen,et al.  How to determine composite material properties using numerical homogenization , 2014 .

[55]  Xianguang Gu,et al.  Concurrent topology optimization of composite macrostructure and microstructure constructed by constituent phases of distinct Poisson's ratios for maximum frequency , 2017 .