Multiscale isogeometric topology optimization for lattice materials

Abstract This paper presents isogeometric topology optimization (ITO) for periodic lattice materials, where non-uniform rational B-spline (NURBS) basis functions of CAD models are directly used in the finite element analysis to improve computational accuracy and efficiency. Two TO schemes that use asymptotic homogenization (AH) for the calculation of the mechanical properties are proposed for lattice materials with uniform and graded relative density respectively. To accelerate ITO for graded lattice materials, the mechanical properties are expressed as a function of the relative density of the unit cell, a step that avoids their iterative calculations during ITO. Three benchmark examples are presented to validate the proposed scheme with results that show tangible advantages, such as reduced computational time and faster convergence, of ITO over conventional TO.

[1]  O. Sigmund,et al.  Checkerboard patterns in layout optimization , 1995 .

[2]  Zhenyu Liu,et al.  Topology Optimization Theory for Laminar Flow , 2018 .

[3]  O. Sigmund.  Morphology-based black and white filters for topology optimization , 2007 .

[4]  Daniel A. Tortorelli,et al.  Nonlinear structural design using multiscale topology optimization. Part I: Static formulation , 2013 .

[5]  V. Fascio,et al.  Numerical and experimental design of graded cellular sandwich cores for multi-functional aerospace applications , 2012 .

[6]  N. Kikuchi,et al.  NORTH-HOLLAND PREPROCESSING AND POSTPROCESSING FOR MATERIALS BASED ON THE HOMOGENIZATION METHOD WITH ADAPTIVE FINITE ELEMENT METHODS Jos , 2002 .

[7]  F. Windmeijer,et al.  An R-squared measure of goodness of fit for some common nonlinear regression models , 1997 .

[8]  Thomas J. R. Hughes,et al.  Isogeometric Analysis: Toward Integration of CAD and FEA , 2009 .

[9]  D. McDowell,et al.  In-Plane Stiffness and Yield Strength of Periodic Metal Honeycombs , 2004 .

[10]  M. Bruyneel,et al.  Topology optimization of membranes made of orthotropic material , 2006 .

[11]  Jun Yan,et al.  Optimum structure with homogeneous optimum cellular material for maximum fundamental frequency , 2009 .

[12]  A. Korobenko,et al.  Isogeometric Fatigue Damage Prediction in Large-Scale Composite Structures Driven by Dynamic Sensor Data , 2015 .

[13]  A. Kalamkarov,et al.  Asymptotic Homogenization of Composite Materials and Structures , 2009 .

[14]  Thomas J. R. Hughes,et al.  Isogeometric Analysis for Topology Optimization with a Phase Field Model , 2012 .

[15]  Sung-Kie Youn,et al.  Isogeometric topology optimization using trimmed spline surfaces , 2010 .

[16]  بهروز حسنی,et al.  Homogenization and Structural Topology Optimization, Theory, Practice and Software , 1999 .

[17]  M. Wang,et al.  Radial basis functions and level set method for structural topology optimization , 2006 .

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

[19]  P. Nakshatrala.  Nonlinear structural design using multiscale topology optimization , 2013 .

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

[21]  M. Meo,et al.  Multiscale damage modelling of 3D weave composite by asymptotic homogenisation , 2013 .

[22]  Thomas J. R. Hughes,et al.  Fluid–structure interaction analysis of bioprosthetic heart valves: significance of arterial wall deformation , 2014, Computational Mechanics.

[23]  D. Benson,et al.  Isogeometric analysis for parameterized LSM-based structural topology optimization , 2016 .

[24]  David J. Benson,et al.  A multi-patch nonsingular isogeometric boundary element method using trimmed elements , 2015, Computational Mechanics.

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

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

[27]  E. Hinton,et al.  Homogenization and Structural Topology Optimization: Theory, Practice and Software , 2011 .

[28]  Mohammad S. Al-Homoud,et al.  Performance characteristics and practical applications of common building thermal insulation materials , 2005 .

[29]  S. Shojaee,et al.  A Level Set Method for Structural Shape and Topology Optimization Using Radial Basis Functions , 2011 .

[30]  M. Ashby,et al.  Micro-architectured materials: past, present and future , 2010, Proceedings of the Royal Society A: Mathematical, Physical and Engineering Sciences.

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

[32]  O. Sigmund.  Materials with prescribed constitutive parameters: An inverse homogenization problem , 1994 .

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

[34]  K. Matsui,et al.  Continuous approximation of material distribution for topology optimization , 2004 .

[35]  M. Zako,et al.  Microstructure-based deep-drawing simulation of knitted fabric reinforced thermoplastics by homogenization theory , 2001 .

[36]  A. H. Wilkins,et al.  Design of three dimensional isotropic microstructures for maximized stiffness and conductivity , 2007, 0712.3101.

[37]  P. Breitkopf,et al.  Design of materials using topology optimization and energy-based homogenization approach in Matlab , 2015 .

[38]  A. Korobenko,et al.  Isogeometric analysis of continuum damage in rotation-free composite shells , 2015 .

[39]  Damiano Pasini,et al.  Multiscale design and multiobjective optimization of orthopedic hip implants with functionally graded cellular material. , 2012, Journal of biomechanical engineering.

[40]  A. Korobenko,et al.  Computational free-surface fluid–structure interaction with application to floating offshore wind turbines , 2016 .

[41]  L. Piegl,et al.  The NURBS Book , 1995, Monographs in Visual Communications.

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

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

[44]  Anders Clausen,et al.  Efficient topology optimization in MATLAB using 88 lines of code , 2011 .

[45]  M. Wang,et al.  A level set‐based parameterization method for structural shape and topology optimization , 2008 .

[46]  E. Hinton,et al.  A review of homogenization and topology optimization I- homogenization theory for media with periodic structure , 1998 .

[47]  David J. Benson,et al.  Multi-patch nonsingular isogeometric boundary element analysis in 3D , 2015 .

[48]  D. Pasini,et al.  Linear multiscale analysis and finite element validation of stretching and bending dominated lattice materials , 2012 .

[49]  Mostafa Khanzadi,et al.  An isogeometrical approach to structural topology optimization by optimality criteria , 2012 .

[50]  Liang Gao,et al.  A level-set-based topology and shape optimization method for continuum structure under geometric constraints , 2014 .

[51]  O. Sigmund,et al.  Minimum length scale in topology optimization by geometric constraints , 2015 .

[52]  Sung-Kie Youn,et al.  Shape optimization and its extension to topological design based on isogeometric analysis , 2010 .

[53]  Fengwen Wang,et al.  Using strain energy-based prediction of effective elastic properties in topology optimization of material microstructures , 2007 .

[54]  Jian Cao,et al.  A dual homogenization and finite element approach for material characterization of textile composites , 2002 .

[55]  John A. Evans,et al.  An Isogeometric design-through-analysis methodology based on adaptive hierarchical refinement of NURBS, immersed boundary methods, and T-spline CAD surfaces , 2012 .

[56]  Gengdong Cheng,et al.  Novel implementation of homogenization method to predict effective properties of periodic materials , 2013 .

[57]  N. Kikuchi,et al.  A comparison of homogenization and standard mechanics analyses for periodic porous composites , 1992 .

[58]  Yi Min Xie,et al.  Evolutionary structural optimisation (ESO) using a bidirectional algorithm , 1998 .

[59]  Damiano Pasini,et al.  Mechanical properties of lattice materials via asymptotic homogenization and comparison with alternative homogenization methods , 2013 .

[60]  D. McDowell,et al.  Generalized continuum modeling of 2-D periodic cellular solids , 2004 .

[61]  C. D. Boor,et al.  On Calculating B-splines , 1972 .

[62]  Robert R. Mayer,et al.  Characterization of Aluminum Honeycomb Material Failure in Large Deformation Compression, Shear, and Tearing , 2002 .

[63]  T. Hughes,et al.  Isogeometric analysis : CAD, finite elements, NURBS, exact geometry and mesh refinement , 2005 .

[64]  Damiano Pasini,et al.  Fatigue design of a mechanically biocompatible lattice for a proof-of-concept femoral stem. , 2013, Journal of the mechanical behavior of biomedical materials.

[65]  Q. Qin,et al.  Topological shape optimization of multifunctional tissue engineering scaffolds with level set method , 2016, Structural and Multidisciplinary Optimization.

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

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