Implicit Heaviside filter with high continuity based on suitably graded THB splines

Abstract

The variable density topology optimization (TO) method has been applied to various engineering fields because it can effectively and efficiently generate the conceptual design for engineering structures. However, it suffers from the problem of low continuity resulting from the discreteness of both design variables and explicit Heaviside filter. In this paper, an implicit Heaviside filter with high continuity is introduced to generate black and white designs for TO where the design space is parameterized by suitably graded truncated hierarchical B-splines (THB). In this approach, the fixed analysis mesh of isogeometric analysis is decoupled from the design mesh, whose adaptivity is implemented by truncated hierarchical B-spline subjected to an admissible requirement. Through the intrinsic local support and high continuity of THB basis, an implicit adaptively adjusted Heaviside filter is obtained to remove the checkboard patterns and generate black and white designs. Threefold advantages are attained in the proposed filter: a) The connection between analysis mesh and adaptive design mesh is easily established compared with the traditional adaptive TO method using nodal density; b) the efficiency in updating design variables is remarkably improved than the traditional implicit sensitivity filter based on B-splines under successive global refinement; and c) the generated black and white designs are preliminarily compatible with current commercial computer aided design system. Several numerical examples are used to verify the effectiveness of the proposed implicit Heaviside filter in compliance and compliant mechanism as well as heat conduction TO problems.

Graphical abstract

Keywords

topology optimization / truncated hierarchical B-spline / isogeometric analysis / black and white designs / Heaviside filter

References

[1] BendsøeM P, KikuchiN. Generating optimal topologies in structural design using a homogenization method. Computer Methods in Applied Mechanics and Engineering, 1988, 71( 2): 197– 224

[2] BendsøeM P. Optimal shape design as a material distribution problem. Structural Optimization, 1989, 1( 4): 193– 202

[3] SigmundO. A 99 line topology optimization code written in matlab. Structural and Multidisciplinary Optimization, 2001, 21( 2): 120– 127

[4] WangM Y, WangX M, GuoD M. A level set method for structural topology optimization. Computer Methods in Applied Mechanics and Engineering, 2003, 192( 1–2): 227– 246

[5] AllaireG, JouveF, ToaderA M. Structural optimization using sensitivity analysis and a level-set method. Journal of Computational Physics, 2004, 194( 1): 363– 393

[6] XieY M, StevenG P. A simple evolutionary procedure for structural optimization. Computers & Structures, 1993, 49( 5): 885– 896

[7] GuoX, ZhangW S, ZhongW L. Doing topology optimization explicitly and geometrically—a new moving morphable components based framework. Journal of Applied Mechanics, 2014, 81( 8): 081009

[8] GuoX, ZhangW S, ZhangJ, YuanJ. Explicit structural topology optimization based on moving morphable components (MMC) with curved skeletons. Computer Methods in Applied Mechanics and Engineering, 2016, 310 : 711– 748

[9] ZhangW S, YangW Y, ZhouJ H, LiD, GuoX. Structural topology optimization through explicit boundary evolution. Journal of Applied Mechanics, 2017, 84( 1): 011011

[10] CaiS Y, ZhangW H. An adaptive bubble method for structural shape and topology optimization. Computer Methods in Applied Mechanics and Engineering, 2020, 360 : 112778

[11] ZhouY, ZhangW H, ZhuJ H, XuZ. Feature-driven topology optimization method with signed distance function. Computer Methods in Applied Mechanics and Engineering, 2016, 310 : 1– 32

[12] JiuL P, ZhangW H, MengL, ZhouY, ChenL. A CAD-oriented structural topology optimization method. Computers & Structures, 2020, 239 : 106324

[13] BourdinB. Filters in topology optimization. International Journal for Numerical Methods in Engineering, 2001, 50( 9): 2143– 2158

[14] SigmundO. Morphology-based black and white filters for topology optimization. Structural and Multidisciplinary Optimization, 2007, 33( 4–5): 401– 424

[15] HughesT J R, CottrellJ A, BazilevsY. Isogeometric analysis: CAD, finite elements, NURBS, exact geometry and mesh refinement. Computer Methods in Applied Mechanics and Engineering, 2005, 194( 39–41): 4135– 4195

[16] WangY J, BensonD J. Geometrically constrained isogeometric parameterized level-set based topology optimization via trimmed elements. Frontiers of Mechanical Engineering, 2016, 11( 4): 328– 343

[17] HouW B, GaiY D, ZhuX F, WangX, ZhaoC, XuL K, JiangK, HuP. Explicit isogeometric topology optimization using moving morphable components. Computer Methods in Applied Mechanics and Engineering, 2017, 326 : 694– 712

[18] XieX D, WangS T, XuM M, WangY J. A new isogeometric topology optimization using moving morphable components based on R-functions and collocation schemes. Computer Methods in Applied Mechanics and Engineering, 2018, 339 : 61– 90

[19] XieX D, WangS T, YeM, XiaZ H, ZhaoW, JiangN, XuM M. Isogeometric topology optimization based on energy penalization for symmetric structure. Frontiers of Mechanical Engineering, 2020, 15( 1): 100– 122

[20] XuJ, GaoL, XiaoM, GaoJ, LiH. Isogeometric topology optimization for rational design of ultra-lightweight architected materials. International Journal of Mechanical Sciences, 2020, 166 : 105103

[21] GaoJ, XiaoM, GaoL, YanJ H, YanW T. Isogeometric topology optimization for computational design of re-entrant and chiral auxetic composites. Computer Methods in Applied Mechanics and Engineering, 2020, 362 : 112876

[22] GaoJ, WangL, XiaoM, GaoL, LiP G. An isogeometric approach to topological optimization design of auxetic composites with tri-material micro-architectures. Composite Structures, 2021, 271 : 114163

[23] HassaniB, KhanzadiM, TavakkoliS M. An isogeometrical approach to structural topology optimization by optimality criteria. Structural and Multidisciplinary Optimization, 2012, 45( 2): 223– 233

[24] WangY J, XuH, PasiniD. Multiscale isogeometric topology optimization for lattice materials. Computer Methods in Applied Mechanics and Engineering, 2017, 316 : 568– 585

[25] BurmanE, ElfversonD, HansboP, LarsonM G, LarssonK. Cut topology optimization for linear elasticity with coupling to parametric nondesign domain regions. Computer Methods in Applied Mechanics and Engineering, 2019, 350 : 462– 479

[26] XuM M, XiaL, WangS Y, LiuL H, XieX D. An isogeometric approach to topology optimization of spatially graded hierarchical structures. Composite Structures, 2019, 225 : 111171

[27] BazilevsY, CaloV M, CottrellJ A, EvansJ A, HughesT J R, LiptonS, ScottM A, SederbergT W. Isogeometric analysis using T-splines. Computer Methods in Applied Mechanics and Engineering, 2010, 199( 5–8): 229– 263

[28] ScottM A, BordenM J, VerhooselC V, SederbergT W, HughesT J R. Isogeometric finite element data structures based on Bézier extraction of T-splines. International Journal for Numerical Methods in Engineering, 2011, 88( 2): 126– 156

[29] ScottM A LiX Sederberg T W HughesT J R. Local refinement of analysis-suitable T-splines. Computer Methods in Applied Mechanics and Engineering, 2012, 213–216: 206– 222

[30] KraftR. Adaptive and linearly independent multilevel B-splines. In: Proceedings of the 3rd International Conference on Curves and Surfaces. 1997, 2: 209– 218

[31] VuongA V, GiannelliC, JüttlerB, SimeonB. A hierarchical approach to adaptive local refinement in isogeometric analysis. Computer Methods in Applied Mechanics and Engineering, 2011, 200( 49–52): 3554– 3567

[32] XieX D, WangS T, XuM M, JiangN, WangY J. A hierarchical spline based isogeometric topology optimization using moving morphable components. Computer Methods in Applied Mechanics and Engineering, 2020, 360 : 112696

[33] JohannessenK A, KvamsdalT, DokkenT. Isogeometric analysis using LR B-splines. Computer Methods in Applied Mechanics and Engineering, 2014, 269 : 471– 514

[34] JohannessenK A, RemonatoF, KvamsdalT. On the similarities and differences between classical hierarchical, truncated hierarchical and LR B-splines. Computer Methods in Applied Mechanics and Engineering, 2015, 291 : 64– 101

[35] KandučT, GiannelliC, PelosiF, SpeleersH. Adaptive isogeometric analysis with hierarchical box splines. Computer Methods in Applied Mechanics and Engineering, 2017, 316 : 817– 838

[36] XiaS T, QianX P. Isogeometric analysis with Bézier tetrahedra. Computer Methods in Applied Mechanics and Engineering, 2017, 316 : 782– 816

[37] GiannelliC, JüttlerB, KleissS K, MantzaflarisA, SimeonB, ŠpehJ. THB-splines: an effective mathematical technology for adaptive refinement in geometric design and isogeometric analysis. Computer Methods in Applied Mechanics and Engineering, 2016, 299 : 337– 365

[38] GiannelliC, JüttlerB, SpeleersH. THB-splines: the truncated basis for hierarchical splines. Computer Aided Geometric Design, 2012, 29( 7): 485– 498

[39] XieX D, WangS T, WangY J, JiangN, ZhaoW, XuM M. Truncated hierarchical B-spline-based topology optimization. Structural and Multidisciplinary Optimization, 2020, 62( 1): 83– 105

[40] LiY, XieY M. Evolutionary topology optimization for structures made of multiple materials with different properties in tension and compression. Composite Structures, 2021, 259 : 113497

[41] QianX P. Topology optimization in B-spline space. Computer Methods in Applied Mechanics and Engineering, 2013, 265 : 15– 35

[42] GaoJ, GaoL, LuoZ, LiP G. Isogeometric topology optimization for continuum structures using density distribution function. International Journal for Numerical Methods in Engineering, 2019, 119( 10): 991– 1017

[43] GaoY L, GuoY J, ZhengS J. A NURBS-based finite cell method for structural topology optimization under geometric constraints. Computer Aided Geometric Design, 2019, 72 : 1– 18

[44] CostaG, MontemurroM, PailhèsJ. A 2D topology optimisation algorithm in NURBS framework with geometric constraints. International Journal of Mechanics and Materials in Design, 2018, 14( 4): 669– 696

[45] CostaG, MontemurroM. Eigen-frequencies and harmonic responses in topology optimisation: a CAD-compatible algorithm. Engineering Structures, 2020, 214 : 110602

[46] CostaG, MontemurroM, PailhèsJ. NURBS hyper-surfaces for 3D topology optimization problems. Mechanics of Advanced Materials and Structures, 2021, 28( 7): 665– 684

[47] CostaG, MontemurroM, PailhèsJ. Minimum length scale control in a NURBS-based SIMP method. Computer Methods in Applied Mechanics and Engineering, 2019, 354 : 963– 989

[48] CostaG, MontemurroM, PailhèsJ, PerryN. Maximum length scale requirement in a topology optimisation method based on NURBS hyper-surfaces. CIRP Annals-Manufacturing Technology, 2019, 68( 1): 153– 156

[49] GuestJ K, PrévostJ H, BelytschkoT. Achieving minimum length scale in topology optimization using nodal design variables and projection functions. International Journal for Numerical Methods in Engineering, 2004, 61( 2): 238– 254

[50] XuS L, CaiY W, ChengG D. Volume preserving nonlinear density filter based on heaviside functions. Structural and Multidisciplinary Optimization, 2010, 41( 4): 495– 505

[51] WangF W, LazarovB S, SigmundO. On projection methods, convergence and robust formulations in topology optimization. Structural and Multidisciplinary Optimization, 2011, 43( 6): 767– 784

[52] PolliniN, AmirO. Mixed projection- and density-based topology optimization with applications to structural assemblies. Structural and Multidisciplinary Optimization, 2020, 61( 2): 687– 710

[53] HuangX D. Smooth topological design of structures using the floating projection. Engineering Structures, 2020, 208 : 110330

[54] HuangX D. On smooth or 0/1 designs of the fixed-mesh element-based topology optimization. Advances in Engineering Software, 2021, 151 : 102942

[55] WangY Q, KangZ, HeQ Z. An adaptive refinement approach for topology optimization based on separated density field description. Computers & Structures, 2013, 117 : 10– 22

[56] WangY Q, HeJ J, LuoZ, KangZ. An adaptive method for high-resolution topology design. Acta Mechanica Sinica, 2013, 29( 6): 840– 850

[57] WangY Q, KangZ, HeQ Z. Adaptive topology optimization with independent error control for separated displacement and density fields. Computers & Structures, 2014, 135 : 50– 61

[58] BuffaA, GiannelliC. Adaptive isogeometric methods with hierarchical splines: error estimator and convergence. Mathematical Models and Methods in Applied Sciences, 2016, 26( 1): 1– 25

[59] BraccoC, GiannelliC, VázquezR. Refinement algorithms for adaptive isogeometric methods with hierarchical splines. Axioms, 2018, 7( 3): 43

[60] CarraturoM, GiannelliC, RealiA, VázquezR. Suitably graded THB-spline refinement and coarsening: towards an adaptive isogeometric analysis of additive manufacturing processes. Computer Methods in Applied Mechanics and Engineering, 2019, 348 : 660– 679

[61] De BoorC. On calculating with B-splines. Journal of Approximation Theory, 1972, 6( 1): 50– 62

[62] GarauE M, VázquezR. Algorithms for the implementation of adaptive isogeometric methods using hierarchical B-splines. Applied Numerical Mathematics, 2018, 123 : 58– 87

[63] XuJ C, ZhangZ M. Analysis of recovery type a posteriori error estimators for mildly structured grids. Mathematics of Computation, 2004, 73( 247): 1139– 1152

[64] JariR H, MuL. Application for superconvergence of finite element approximations for the elliptic problem by global and local L²-projection methods. American Journal of Computational Mathematics, 2012, 2( 4): 249– 257

[65] CostantiniP, ManniC, PelosiF, SampoliM L. Quasi-interpolation in isogeometric analysis based on generalized B-splines. Computer Aided Geometric Design, 2010, 27( 8): 656– 668

[66] LiM Z, ChenL J, MaQ. A meshfree quasi-interpolation method for solving burgers’ equation. Mathematical Problems in Engineering, 2014, 2014 : 492072

[67] MaZ D, KikuchiN, ChengH C. Topological design for vibrating structures. Computer Methods in Applied Mechanics and Engineering, 1995, 121( 1–4): 259– 280

[68] FerrariF, SigmundO. A new generation 99 line matlab code for compliance topology optimization and its extension to 3D. Structural and Multidisciplinary Optimization, 2020, 62( 4): 2211– 2228

[69] XieX D, YangA D, JiangN, ZhaoW, LiangZ S, WangS T. Adaptive topology optimization under suitably graded THB-spline refinement and coarsening. International Journal for Numerical Methods in Engineering, 2021, 122( 20): 5971– 5998

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