Abstract
In recent years, the parameterized level set method (PLSM), which rests on radial basis functions in most early work, has gained growing attention in structural optimization. However, little work has been carried out to investigate the effect of the basis functions in the parameterized level set method. This paper examines the basis functions of the parameterized level set method, including radial basis functions, B-spline functions, and shape functions in the finite element method (FEM) for topology optimization of continuums. The effects of different basis functions in the PLSM are examined by analyzing and comparing the required storage, convergence speed, computational efficiency, and optimization results, with the benchmark minimum compliance problems subject to a volume constraint. The linear basis functions show relatively satisfactory overall performance. Besides, several schemes to boost computational efficiency are proposed. The study on examples with unstructured 2D and 3D meshes can also be considered as a tentative investigation of prospective possible commercial applications of this method.
Issue Section:
Design Automation
Keywords:
topology optimization,
parameterized level set method,
basis functions,
computer-aided design,
design methodology,
multidisciplinary design and optimization,
simulation-based design
Topics:
B-splines,
Cantilever beams,
Finite element analysis,
Optimization,
Topology,
Shapes,
Design
References
1.
Bendse
, M. P.
, 1989
, “Optimal Shape Design as a Material Distribution Problem
,” Struct. Optim.
, 1
(4
), pp. 193
–202
. 10.1007/BF016509492.
Rozvany
, G. I. N.
, 2000
, “The SIMP Method in Topology Optimization—Theoretical Background, Advantages and New Applications
,” The 8th AIAA/USAF/NASA/ISSMO Symposium on Multidisciplinary Analysis and Optimization
, Long Beach, CA
, Sept. 6–8
.3.
Sigmund
, O.
, 2001
, “A 99 Line Topology Optimization Code Written in Matlab
,” Struct. Multidiscip. Optim.
, 21
(2
), pp. 120
–127
. 10.1007/s0015800501764.
Guest
, J. K.
, Prévost
, J. H.
, and Belytschko
, T.
, 2004
, “Achieving Minimum Length Scale in Topology Optimization Using Nodal Design Variables and Projection Functions
,” Internat. J. Numer. Methods Eng.
, 61
(2
), pp. 238
–254
. 10.1002/nme.10645.
Xie
, Y. M.
, and Steven
, G. P.
, 1993
, “A Simple Evolutionary Procedure for Structural Optimization
,” Comput. Struct.
, 49
(5
), pp. 885
–896
. 10.1016/0045-7949(93)90035-C6.
Huang
, X.
, and Xie
, Y. M.
, 2007
, “Convergent and Mesh-Independent Solutions for the Bi-Directional Evolutionary Structural Optimization Method
,” Finite Elem. Anal. Des.
, 43
(14
), pp. 1039
–1049
. 10.1016/j.finel.2007.06.0067.
Sethian
, J. A.
, and Wiegmann
, A.
, 2000
, “Structural Boundary Design via Level Set and Immersed Interface Methods
,” J. Comput. Phys.
, 163
(2
), pp. 489
–528
. 10.1006/jcph.2000.65818.
Wang
, M. Y.
, Wang
, X.
, and Guo
, D.
, 2003
, “A Level Set Method for Structural Topology Optimization
,” Comput. Methods Appl. Mech. Eng.
, 192
(1–2
), pp. 227
–246
. 10.1016/S0045-7825(02)00559-59.
Allaire
, G.
, Jouve
, F.
, and Toader
, A. M.
, 2004
, “Structural Optimization Using Sensitivity Analysis and a Level-Set Method
,” J. Comput. Phys.
, 194
(1
), pp. 363
–393
. 10.1016/j.jcp.2003.09.03210.
Wei
, P.
, Ma
, H.
, and Wang
, M. Y.
, 2014
, “The Stiffness Spreading Method for Layout Optimization of Truss Structures
,” Struct. Multidiscip. Optim.
, 49
(4
), pp. 667
–682
. 10.1007/s00158-013-1005-711.
Guo
, X.
, Zhang
, W.
, and Zhong
, W.
, 2014
, “Doing Topology Optimization Explicitly and Geometrically—A New Moving Morphable Components Based Framework
,” ASME J. Appl. Mech.
, 81
(8
), p. 081009
. 10.1115/1.402760912.
Zhang
, W.
, Yuan
, J.
, Zhang
, J.
, and Guo
, X.
, 2016
, “A New Topology Optimization Approach Based on Moving Morphable Components (MMC) and the Ersatz Material Model
,” Struct. Multidiscip. Optim.
, 53
(6
), pp. 1243
–1260
. 10.1007/s00158-015-1372-313.
Norato
, J. A.
, Bell
, B. K.
, and Tortorelli
, D. A.
, 2015
, “A Geometry Projection Method for Continuum-Based Topology Optimization With Discrete Elements
,” Comput. Methods Appl. Mech. Eng.
, 293
, pp. 306
–327
. 10.1016/j.cma.2015.05.00514.
de Ruiter
, M. J.
, and van Keulen
, F.
, 2000
, “Topology of Optimization: Approaching the Material Distribution Problem Using a Topological Function Description,” Computational Techniques for Materials, Composites and Composite Structures
, B. H. V.
Topping
, ed., Civil-Comp Press
, Edinburgh, UK
, pp. 111
–119
.15.
Lorensen
, W. E.
, and Cline
, H. E.
, 1987
, “Marching Cubes: A High Resolution 3D Surface Construction Algorithm
,” ACM SIGGRAPH Comput. Graph.
, 21
(4
), pp. 163
–169
. 10.1145/37402.3742216.
Osher
, S.
, and Fedkiw
, R.
, 2003
, Level Set Methods and Dynamic Implicit Surfaces
, Springer
, Berlin
.17.
Sethian
, J. A.
, 1999
, Level Set Methods and Fast Marching Methods: Evolving Interfaces in Computational Geometry, Fluid Mechanics, Computer Vision, and Materials Science
, Cambridge University Press
, Cambridge, UK
.18.
Zhu
, B.
, Zhang
, X.
, and Fatikow
, S.
, 2014
, “A Velocity Predictor–Corrector Scheme in Level Set-Based Topology Optimization to Improve Computational Efficiency
,” ASME J. Mech. Des.
, 136
(9
), p. 091001
. 10.1115/1.402772019.
Geiss
, M. J.
, Barrera
, J. L.
, Boddeti
, N.
, and Maute
, K.
, 2019
, “A Regularization Scheme for Explicit Level-Set XFEM Topology Optimization,” Front
,” Mech. Eng.
, 14
(2
), pp. 153
–170
. 10.1007/s11465-019-0533-220.
Maute
, K.
, Tkachuk
, A.
, Wu
, J.
, Qi
, H. J.
, Ding
, Z.
, and Dunn
, M. L.
, 2015
, “Level Set Topology Optimization of Printed Active Composites
,” ASME J. Mech. Des.
, 137
(11
), p. 111402
. 10.1115/1.403099421.
Liu
, J.
, and Ma
, Y.
, 2017
, “Sustainable Design-Oriented Level Set Topology Optimization
,” ASME J. Mech. Des.
, 139
(1
), p. 011403
. 10.1115/1.403505222.
Geiss
, M. J.
, Boddeti
, N.
, Weeger
, O.
, Maute
, K.
, and Dunn
, M. L.
, 2019
, “Combined Level-Set-XFEM-Density Topology Optimization of Four-Dimensional Printed Structures Undergoing Large Deformation
,” ASME J. Mech. Des.
, 141
(5
), p. 051405
. 10.1115/1.404194523.
Allaire
, G.
, Gournay
, F. D.
, Jouve
, F.
, and Toader
, A. M.
, 2005
, “Structural Optimization Using Topological and Shape Sensitivity via a Level Set Method
,” Control Cybern.
, 34
(1
), pp. 59
–80
.24.
Yamada
, T.
, Izui
, K.
, Nishiwaki
, S.
, and Takezawa
, A.
, 2010
, “A Topology Optimization Method Based on the Level Set Method Incorporating a Fictitious Interface Energy
,” Comput. Methods Appl. Mech. Engrg.
, 199
(45–48
), pp. 2876
–2891
. 10.1016/j.cma.2010.05.01325.
de Ruiter
, M. J.
, and van Keulen
, F.
, 2004
, “Topology Optimization Using a Topology Description Function
,” Struct. Multidiscip. Optim.
, 26
(6
), pp. 406
–416
. 10.1007/s00158-003-0375-726.
Belytschko
, T.
, Xiao
, S. P.
, and Parimi
, C.
, 2003
, “Topology Optimization With Implicitly Function and Regularization
,” Internat. J. Numer. Methods Eng.
, 57
(8
), pp. 1177
–1196
. 10.1002/nme.82427.
Wei
, P.
, and Wang
, M. Y.
, 2009
, “Piecewise Constant Level Set Method for Structural Topology Optimization
,” Internat. J. Numer. Methods Eng.
, 78
(4
), pp. 379
–402
. 10.1002/nme.247828.
Wang
, S.
, and Wang
, M. Y.
, 2006
, “Radial Basis Functions and Level Set Method for Structural Topology Optimization
,” Internat. J. Numer. Methods Eng.
, 65
(12
), pp. 2060
–2090
. 10.1002/nme.153629.
Wang
, S. Y.
, Lim
, K. M.
, Khoo
, B. C.
, and Wang
, M. Y.
, 2007
, “An Extended Level Set Method for Shape and Topology Optimization
,” J. Comput. Phys.
, 221
(1
), pp. 395
–421
. 10.1016/j.jcp.2006.06.02930.
Wei
, P.
, Li
, Z.
, Li
, X.
, and Wang
, M. Y.
, 2018
, “An 88-Line MATLAB Code for the Parameterized Level Set Method Based Topology Optimization Using Radial Basis Functions
,” Struct. Multidiscip. Optim.
, 58
(2
), pp. 831
–849
. 10.1007/s00158-018-1904-831.
Liu
, Y.
, Li
, Z.
, Wei
, P.
, and Wang
, W.
, 2018
, “Parameterized Level-Set Based Topology Optimization Method Considering Symmetry and Pattern Repetition Constraints
,” Comput. Methods Appl. Mech. Eng.
, 340
, pp. 1079
–1101
. 10.1016/j.cma.2018.04.03432.
Wei
, P.
, and Paulino
, G. H.
, 2020
, “A Parameterized Level Set Method Combined With Polygonal Finite Elements in Topology Optimization
,” Struct. Multidiscip. Optim.
, 61
(5
), pp. 1913
–1928
. 10.1007/s00158-019-02444-y33.
Liu
, H.
, Tian
, Y.
, Zong
, H. M.
, Ma
, Q. P.
, Wang
, M. Y.
, and Zhang
, L.
, “Fully Parallel Level Set Method for Large-Scale Structural Topology Optimization
,” Comput. Struct.
, 221
, pp. 13
–27
. 10.1016/j.compstruc.2019.05.01034.
Wang
, M. Y.
, and Wang
, S.
, 2006
, “Parametric Shape and Topology Optimization With Radial Basis Functions,” IUTAM Symposium on Topological Design Optimization of Structures, Machines and Materials, Machines and Materials: Status and Perspectives, Solid Mechanics and its Applications
, Vol. 137
, M. P.
Bendsoe
, N.
Olhoff
, and O.
Sigmund
, eds., Springer
, Dordrecht, Netherlands
, pp. 13
–22
.35.
Wei
, P.
, and Wang
, M. Y.
, 2006
, “Parametric Structural Shape and Topology Optimization Method With Radial Basis Functions and Level-Set Method
,” Proceedings of ASME 2006 International Design Engineering Technical Conferences & Computers and Information in Engineering Conference
, Philadelphia, PA
, Sept. 10–13
.36.
Wei
, P.
, and Wang
, M. Y.
, 2006
, “The Augmented Lagrangian Method in Structural Shape and Topology Optimization With RBF Based Level Set Method
,” The Forth China-Japan-Korea Joint Symposium on Optimization of Structural and Mechanical Systems
, Kunming, China
, Nov. 6–9
.37.
Luo
, Z.
, Tong
, L.
, Wang
, M. Y.
, and Wang
, S.
, 2007
, “Shape and Topology Optimization of Compliant Mechanisms Using a Parameterization Level Set Method
,” J. Comput. Phys.
, 227
(1
), pp. 680
–705
. 10.1016/j.jcp.2007.08.01138.
Luo
, Z.
, Wang
, M. Y.
, Wang
, S. Y.
, and Wei
, P.
, 2008
, “A Level Set-Based Parameterization Method for Structural Shape and Topology Optimization
,” Internat. J. Numer. Methods Eng.
, 76
(1
), pp. 1
–26
. 10.1002/nme.209239.
Luo
, Z.
, Tong
, L.
, and Kang
, Z.
, 2009
, “A Level Set Method for Structural Shape and Topology Optimization Using Radial Basis Functions
,” Comput. Struct.
, 87
(7–8
), pp. 425
–434
. 10.1016/j.compstruc.2009.01.00840.
Li
, H.
, Luo
, Z.
, Gao
, L.
, and Wu
, J.
, 2018
, “An Improved Parametric Level Set Method for Structural Frequency Response Optimization Problems
,” Adv. Eng. Softw.
, 126
, pp. 75
–89
. 10.1016/j.advengsoft.2018.10.00141.
Jiang
, L.
, and Chen
, S.
, 2017
, “Parametric Structural Shape & Topology Optimization With a Variational Distance-Regularized Level Set Method
,” Comput. Methods Appl. Mech. Eng.
, 321
, pp. 316
–336
. 10.1016/j.cma.2017.03.04442.
Jiang
, L.
, Chen
, S.
, and Jiao
, X.
, 2017
, “Parametric Shape and Topology Optimization: A New Level Set Approach Based on Cardinal Basis Functions
,” Internat. J. Numer. Methods Eng.
, 114
(1
), pp. 66
–87
. 10.1002/nme.573343.
Wang
, Y.
, Luo
, Z.
, Kang
, Z.
, and Zhang
, N.
, 2015
, “A Multi-Material Level Set-Based Topology and Shape Optimization Method
,” Comput. Methods Appl. Mech. Engrg.
, 283
, pp. 1570
–1586
. 10.1016/j.cma.2014.11.00244.
Li
, H.
, Luo
, Z.
, Xiao
, M.
, Gao
, L.
, and Gao
, J.
, 2019
, “A New Multiscale Topology Optimization Method for Multiphase Composite Structures of Frequency Response With Level Sets
,” Comput. Methods Appl. Mech. Eng.
, 356
, pp. 116
–144
. 10.1016/j.cma.2019.07.02045.
Deng
, X.
, Wang
, Y.
, Yan
, J.
, Liu
, T.
, and Wang
, S.
, 2016
, “Topology Optimization of Total Femur Structure: Application of Parameterized Level Set Method Under Geometric Constraints
,” ASME J. Mech. Des.
, 138
(1
), p. 011402
. 10.1115/1.403180346.
Wang
, Y.
, and Kang
, Z.
, 2018
, “A Velocity Field Level Set Method for Shape and Topology Optimization
,” Int. J. Numer. Methods Eng.
, 115
(11
), pp. 1315
–1336
. 10.1002/nme.584547.
Wang
, S.
, and Wang
, M. Y.
, 2006
, “A Moving Superimposed Finite Element Method for Structural Topology Optimization
,” Int. J. Numer. Methods Eng.
, 65
(11
), pp. 1892
–1922
. 10.1002/nme.152748.
Liu
, Z.
, and Korvink
, J. G.
, 2008
, “Adaptive Moving Mesh Level Set Method for Structure Topology Optimization
,” Eng. Optim.
, 40
(6
), pp. 529
–558
. 10.1080/0305215080198554449.
Ha
, S. H.
, and Cho
, S.
, 2008
, “Level Set Based Topological Shape Optimization of Geometrically Nonlinear Structures Using Unstructured Mesh
,” Comput. Struct.
, 86
(13
), pp. 1447
–1455
. 10.1016/j.compstruc.2007.05.02550.
Yamasaki
, S.
, Kawamoto
, A.
, Nomura
, T.
, and Fujita
, K.
, 2014
, “A Consistent Grayscale-Free Topology Optimization Method Using the Level-Set Method and Zero-Level Boundary Tracking Mesh
,” Int. J. Numer. Methods Eng.
, 101
(10
), pp. 744
–773
. 10.1002/nme.482651.
Paulino
, G. H.
, and Le
, C. H.
, 2009
, “A Modified Q4/Q4 Element for Topology Optimization
,” Struct. Multidiscip. Optim.
, 37
(3
), pp. 255
–264
. 10.1007/s00158-008-0228-552.
van Dijk
, N. P.
, Maute
, K.
, Langelaar
, M.
, and van Keulen
, F.
, 2013
, “Level-Set Methods for Structural Topology Optimization: A Review
,” Struct. Multidiscip. Optim.
, 48
(3
), pp. 437
–472
. 10.1007/s00158-013-0912-y53.
van Dijk
, N. P.
, Langelaar
, M.
, and van Keulen
, F.
, 2012
, “Explicit Level-Set-Based Topology Optimization Using an Exact Heaviside Function and Consistent Sensitivity Analysis
,” Inte. J. Numer. Methods Eng.
, 91
(1
), pp. 67
–97
. 10.1002/nme.425854.
Liu
, Z.
, Korvink
, J. G.
, and Huang
, R.
, 2005
, “Structure Topology Optimization: Fully Coupled Level Set Method via FEMLAB
,” Struct. Multidiscip. Optim.
, 29
(6
), pp. 407
–417
. 10.1007/s00158-004-0503-z55.
Xing
, X.
, Wei
, P.
, and Wang
, M. Y.
, 2010
, “A Finite Element-Based Level Set Method for Structural Optimization
,” Int. J. Numer. Methods Eng.
, 82
(7
), pp. 805
–842
. 10.1002/nme.278556.
Wang
, M. Y.
, Zong
, H. M.
, Ma
, Q. P.
, Tian
, Y.
, and Zhou
, M. D.
, 2019
, “Cellular Level Set in B-Splines (CLIBS): A Method for Modeling and Topology Optimization of Cellular Structures
,” Comput. Methods Appl. Mech. Eng.
, 349
, pp. 378
–404
. 10.1016/j.cma.2019.02.02657.
Wei
, P.
, 2007
, “Level Set Methods for Shape and Topology Optimization of Structures
,” Ph.D. thesis
, The Chinese University of Hong Kong
, Hong Kong
.58.
Wang
, M. Y.
, and Wei
, P.
, 2005
, “Topology Optimization with Level Set Method Incorporating Topological Derivative
,” Proceedings of 6th World Congress of Structural and Multidisciplinary Optimization (WCSMO6)
, Rio de Janeiro, Brazil
, May 30–June 3
.59.
Gordon
, W. J.
, and Riesenfeld
, R. F.
, 1974
, “B-spline Curves and Surfaces
,” Comput. Aided Geom. Des.
, 4
, pp. 95
–126
. 10.1016/B978-0-12-079050-0.50011-460.
Lopes
, C. G.
, Santos
, R. B.
, and Novotny
, A. A.
, 2015
, “Topological Derivative-Based Topology Optimization of Structures Subject to Multiple Load-Cases
,” Lat. Am. J. Solids Struct.
, 12
(5
), pp. 834
–860
. 10.1590/1679-7825125261.
Talischi
, C.
, Paulino
, G. H.
, Pereira
, A.
, and Menezes
, I. F. M.
, 2012a
, “PolyMesher: A General-Purpose Mesh Generator for Polygonal Elements Written in Matlab
,” Struct. Multidiscip. Optim.
, 45
(3
), pp. 309
–328
. 10.1007/s00158-011-0706-z62.
Talischi
, C.
, Paulino
, G. H.
, Pereira
, A.
, and Menezes
, I. F. M.
, 2012b
, “Polytop: A Matlab Implementation of a General Topology Optimization Framework Using Unstructured Polygonal Finite Element Meshes
,” Struct. Multidiscip. Optim.
, 45
(3
), pp. 329
–357
. 10.1007/s00158-011-0696-x63.
Wei
, P.
, Liu
, Y.
, and Li
, Z.
, 2020
, “A Multi-Discretization Scheme for Topology Optimization Based on Parameterized Level Set Method
,” Int. J. Simul. Multidisci. Des. Optim.
, 11
(3
), pp. 1
–10
. 10.1051/smdo/201901964.
Da
, D.
, Xia
, L.
, Li
, G.
, and Huang
, X.
, 2018
, “Evolutionary Topology Optimization of Continuum Structures With Smooth Boundary Representation
,” Struct. Multidiscip. Optim.
, 57
(6
), pp. 2143
–2159
. 10.1007/s00158-017-1846-6Copyright © 2020 by ASME
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