Isogeometric topology optimization based on energy penalization for symmetric structure

Abstract

We present an energy penalization method for isogeometric topology optimization using moving morphable components (ITO–MMC), propose an ITO–MMC with an additional bilateral or periodic symmetric constraint for symmetric structures, and then extend the proposed energy penalization method to an ITO–MMC with a symmetric constraint. The energy penalization method can solve the problems of numerical instability and convergence for the ITO–MMC and the ITO–MMC subjected to the structural symmetric constraint with asymmetric loads. Topology optimization problems of asymmetric, bilateral symmetric, and periodic symmetric structures are discussed to validate the effectiveness of the proposed energy penalization approach. Compared with the conventional ITO–MMC, the energy penalization method for the ITO–MMC can improve the convergence rate from 18.6% to 44.5% for the optimization of the asymmetric structure. For the ITO–MMC under a bilateral symmetric constraint, the proposed method can reduce the objective value by 5.6% and obtain a final optimized topology that has a clear boundary with decreased iterations. For the ITO–MMC under a periodic symmetric constraint, the proposed energy penalization method can dramatically reduce the number of iterations and obtain a speedup of more than 2.

Keywords

topology optimization / moving morphable component / isogeometric analysis / energy penalization method / symmetric constraint

References

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

[2] Gholizadeh S, Ebadijalal M. Performance based discrete topology optimization of steel braced frames by a new metaheuristic. Advances in Engineering Software, 2018, 123: 77–92

[3] Zhang Y, Ge W, Zhang Y, et al. Topology optimization of hyperelastic structure based on a directly coupled finite element and element-free Galerkin method. Advances in Engineering Software, 2018, 123: 25–37

[4] Csébfalvi A, Lógó J. A critical analysis of expected-compliance model in volume-constrained robust topology optimization with normally distributed loading directions, using a minimax-compliance approach alternatively. Advances in Engineering Software, 2018, 120: 107–115

[5] Gao J, Li H, Gao L, et al. Topological shape optimization of 3D micro-structured materials using energy-based homogenization method. Advances in Engineering Software, 2018, 116: 89–102

[6] Sigmund O, Petersson J. Numerical instabilities in topology optimization: A survey on procedures dealing with checkerboards, mesh-dependencies and local minima. Structural Optimization, 1998, 16(1): 68–75

[7] Sigmund O. A 99 line topology optimization code written in Matlab. Structural and Multidisciplinary Optimization, 2001, 21(2): 120–127

[8] Andreassen E, Clausen A, Schevenels M, et al. Efficient topology optimization in MATLAB using 88 lines of code. Structural and Multidisciplinary Optimization, 2011, 43(1): 1–16

[9] Liao Z, Zhang Y, Wang Y, et al. A triple acceleration method for topology optimization. Structural and Multidisciplinary Optimization, 2019, 60(2): 727–724

[10] Zhou M, Liu Y, Lin Z. Topology optimization of thermal conductive support structures for laser additive manufacturing. Computer Methods in Applied Mechanics and Engineering, 2019, 353: 24–43

[11] Zhou M, Lian H, Sigmund O, et al. Shape morphing and topology optimization of fluid channels by explicit boundary tracking. International Journal for Numerical Methods in Fluids, 2018, 88(6): 296–313

[12] Liu T, Wang S, Li B, et al. A level-set-based topology and shape optimization method for continuum structure under geometric constraints. Structural and Multidisciplinary Optimization, 2014, 50(2): 253–273

[13] Liu T, Li B, Wang S, et al. Eigenvalue topology optimization of structures using a parameterized level set method. Structural and Multidisciplinary Optimization, 2014, 50(4): 573–591

[14] Mei Y, Wang X. A level set method for structural topology optimization and its applications. Computer Methods in Applied Mechanics and Engineering, 2004, 35(7): 415–441

[15] Wang Y, Benson D J. Geometrically constrained isogeometric parameterized level-set based topology optimization via trimmed elements. Frontiers of Mechanical Engineering, 2016, 11(4): 328–343

[16] Wang Y, Benson D J. Isogeometric analysis for parameterized LSM-based structural topology optimization. Computational Mechanics, 2016, 57(1): 19–35

[17] Xia Z, Wang Y, Wang Q, et al. GPU parallel strategy for parameterized LSM-based topology optimization using isogeometric analysis. Structural and Multidisciplinary Optimization, 2017, 56(2): 413–434

[18] Liu Y, Li Z, Wei P, et al. Parameterized level-set based topology optimization method considering symmetry and pattern repetition constraints. Computer Methods in Applied Mechanics and Engineering, 2018, 340: 1079–1101

[19] Li Z, Shi T, Xia Q. Eliminate localized eigenmodes in level set based topology optimization for the maximization of the first eigenfrequency of vibration. Advances in Engineering Software, 2017, 107: 59–70

[20] Liu J, Li L, Ma Y. Uniform thickness control without pre-specifying the length scale target under the level set topology optimization framework. Advances in Engineering Software, 2017, 115: 204–216

[21] Xia Q, Shi T, Xia L. Stable hole nucleation in level set based topology optimization by using the material removal scheme of BESO. Computer Methods in Applied Mechanics and Engineering, 2019, 343: 438–452

[22] Li Z, Shi T, Xia L, et al. Maximizing the first eigenfrequency of structures subjected to uniform boundary erosion through the level set method. Engineering with Computers, 2019, 35(1): 21–33

[23] Xie Y M, Steven G P. A simple evolutionary procedure for structural optimization. Computers & Structures, 1993, 49(5): 885–896

[24] Huang X, Xie Y M. Convergent and mesh-independent solutions for the bi-directional evolutionary structural optimization method. Finite Elements in Analysis and Design, 2007, 43(14): 1039–1049

[25] Stanford B, Ifju P. Aeroelastic topology optimization of membrane structures for micro air vehicles. Structural and Multidisciplinary Optimization, 2009, 38(3): 301–316

[26] Zhu J H, Zhang W H, Xia L. Topology optimization in aircraft and aerospace structures design. Archives of Computational Methods in Engineering, 2016, 23(4): 595–622

[27] Zegard T, Paulino G H. Bridging topology optimization and additive manufacturing. Structural and Multidisciplinary Optimization, 2016, 53(1): 175–192

[28] Guo X, Zhou J, Zhang W, et al. Self-supporting structure design in additive manufacturing through explicit topology optimization. Computer Methods in Applied Mechanics and Engineering, 2017, 323: 27–63

[29] Li Y F, Huang X, Meng F, et al. Evolutionary topological design for phononic band gap crystals. Structural and Multidisciplinary Optimization, 2016, 54(3): 595–617

[30] Li Y F, Huang X, Zhou S. Topological design of cellular phononic band gap crystals. Materials, 2016, 9(3): 186–208

[31] Wang Y, Arabnejad S, Tanzer M, et al. Hip implant design with three-dimensional porous architecture of optimized graded density. Journal of Mechanical Design, 2018, 140(11): 111406

[32] Aage N, Andreassen E, Lazarov B S, et al. Giga-voxel computational morphogenesis for structural design. Nature, 2017, 550(7674): 84–86

[33] Sigmund O. Materials with prescribed constitutive parameters: An inverse homogenization problem. International Journal of Solids and Structures, 1994, 31(17): 2313–2329

[34] Zhang W, Sun S. Scale-related topology optimization of cellular materials and structures. International Journal for Numerical Methods in Engineering, 2006, 68(9): 993–1011

[35] Vicente W M, Picelli R, Pavanello R, Xie Y M. Topology optimization of periodic structures for coupled acoustic-structure systems. In: Proceedings of the VII European Congress on Computational Methods in Applied Sciences and Engineering. Crete Island, 2016

[36] He G, Huang X, Wang H, et al. Topology optimization of periodic structures using BESO based on unstructured design points. Structural and Multidisciplinary Optimization, 2016, 53(2): 271–275

[37] Guo X, Zhang W, Zhong W. Doing topology optimization explicitly and geometrically—A new moving morphable components based framework. Frontiers in Applied Mechanics, 2014, 81(8): 081009

[38] Hou W, Gai Y, Zhu X, et al. Explicit isogeometric topology optimization using moving morphable components. Computer Methods in Applied Mechanics and Engineering, 2017, 326: 694–712

[39] Hughes T J R, Cottrell J A, Bazilevs Y. Isogeometric analysis: CAD, finite elements, NURBS, exact geometry and mesh refinement. Computer Methods in Applied Mechanics and Engineering, 2005, 194(39–41): 4135–4195

[40] Cottrell J A, Hughes T J R, Bazilevs Y. Isogeometric Analysis: Toward Integration of CAD and FEA. Hoboken: John Wiley & Sons, 2009

[41] Wang X, Zhu X, Hu P. Isogeometric finite element method for buckling analysis of generally laminated composite beams with different boundary conditions. International Journal of Mechanical Sciences, 2015, 104: 190–199

[42] Marussig B, Hughes T J R. A review of trimming in isogeometric analysis: Challenges, data exchange and simulation aspects. Archives of Computational Methods in Engineering, 2018, 25(4): 1059–1127

[43] Pan Q, Rabczuk T, Chen C, et al. Isogeometric analysis of minimal surfaces on the basis of extended Catmull–Clark subdivision. Computer Methods in Applied Mechanics and Engineering, 2018, 337: 128–149

[44] An Z, Yu T, Bui T Q, et al. Implementation of isogeometric boundary element method for 2-D steady heat transfer analysis. Advances in Engineering Software, 2018, 116: 36–49

[45] Rypl D, Patzák B. Construction of weighted dual graphs of NURBS-based isogeometric meshes. Advances in Engineering Software, 2013, 60–61: 31–41

[46] Rypl D, Patzák B. Object oriented implementation of the T-spline based isogeometric analysis. Advances in Engineering Software, 2012, 50(1): 137–149

[47] Lai W, Yu T, Bui T Q, et al. 3-D elasto-plastic large deformations: IGA simulation by Bézier extraction of NURBS. Advances in Engineering Software, 2017, 108: 68–82

[48] Rypl D, Patzák B. From the finite element analysis to the isogeometric analysis in an object oriented computing environment. Advances in Engineering Software, 2012, 44(1): 116–125

[49] Dedè L, Borden M J, Hughes T J R. Isogeometric analysis for topology optimization with a phase field model. Archives of Computational Methods in Engineering, 2012, 19(3): 427–465

[50] Kumar A V, Parthasarathy A. Topology optimization using B-spline finite elements. Structural and Multidisciplinary Optimization, 2011, 44(4): 471–481

[51] Wang Y, Xu H, Pasini D. Multiscale isogeometric topology optimization for lattice materials. Computer Methods in Applied Mechanics and Engineering, 2017, 316: 568–585

[52] Xu M, Xia L, Wang S, et al. An isogeometric approach to topology optimization of spatially graded hierarchical structures. Composite Structures, 2019, 225: 111171

[53] Xie X, Wang S, Xu M, et al. 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

[54] Zhang W, Zhou J, Zhu Y, et al. Structural complexity control in topology optimization via moving morphable component (MMC) approach. Structural and Multidisciplinary Optimization, 2017, 56(3): 535–552

[55] Piegl L, Tiller W. The NURBS Book. Berlin: Springer, 1997

[56] de Falco C, Reali A, Vázquez R. GeoPDEs: A research tool for isogeometric analysis of PDEs. Advances in Engineering Software, 2011, 42(12): 1020–1034

[57] de Boor C D. On calculating with B-splines. Journal of Approximation Theory, 1972, 6(1): 50–62

[58] Fougerolle Y D, Gribok A, Foufou S, et al. Boolean operations with implicit and parametric representation of primitives using R-functions. IEEE Transactions on Visualization and Computer Graphics, 2005, 11(5): 529–539

[59] Svanberg K. The method of moving asymptotes—A new method for structural optimization. International Journal for Numerical Methods in Engineering, 1987, 24(2): 359–373

[60] Huang X, Xie Y M. Optimal design of periodic structures using evolutionary topology optimization. Structural and Multidisciplinary Optimization, 2008, 36(6): 597–606

[61] Zhang W, Yuan J, Zhang J, et al. A new topology optimization approach based on moving morphable components (MMC) and the ersatz material model. Structural and Multidisciplinary Optimization, 2016, 53(6): 1243–1260

[62] Xia Z, Wang Q, Liu Q, et al. A novel approach for automatic reconstruction of boundary condition in structure analysis. Advances in Engineering Software, 2016, 96: 38–57

[63] Xia Z, Wang Q, Wang Y, et al. A CAD/CAE incorporate software framework using a unified representation architecture. Advances in Engineering Software, 2015, 87(C): 68–85

[64] Wang Y, Wang Q, Deng X, et al. Graphics processing unit (GPU) accelerated fast multipole BEM with level-skip M2L for 3D elasticity problems. Advances in Engineering Software, 2015, 82(2): 105–118

RIGHTS & PERMISSIONS

The Author(s) 2019. This article is published with open access at link.springer.com and journal.hep.com.cn