A MATLAB code for the material-field series-expansion topology optimization method

doi:10.1007/s11465-021-0637-3

Front. Mech. Eng.

2021, Vol. 16

Issue (3) : 607-622 https://doi.org/10.1007/s11465-021-0637-3

RESEARCH ARTICLE

A MATLAB code for the material-field series-expansion topology optimization method

Pai LIU¹, Yi YAN², Xiaopeng ZHANG¹, Yangjun LUO^1,²(

)

¹. State Key Laboratory of Structural Analysis for Industrial Equipment, Dalian University of Technology, Dalian 116024, China
². School of Aeronautics and Astronautics, Dalian University of Technology, Dalian 116024, China

Download: PDF(884 KB) HTML
Export: BibTeX | EndNote | Reference Manager | ProCite | RefWorks

Abstract

This paper presents a MATLAB implementation of the material-field series-expansion (MFSE) topo-logy optimization method. The MFSE method uses a bounded material field with specified spatial correlation to represent the structural topology. With the series-expansion method for bounded fields, this material field is described with the characteristic base functions and the corresponding coefficients. Compared with the conventional density-based method, the MFSE method decouples the topological description and the finite element discretization, and greatly reduces the number of design variables after dimensionality reduction. Other features of this method include inherent control on structural topological complexity, crisp structural boundary description, mesh independence, and being free from the checkerboard pattern. With the focus on the implementation of the MFSE method, the present MATLAB code uses the maximum stiffness optimization problems solved with a gradient-based optimizer as examples. The MATLAB code consists of three parts, namely, the main program and two subroutines (one for aggregating the optimization constraints and the other about the method of moving asymptotes optimizer). The implementation of the code and its extensions to topology optimization problems with multiple load cases and passive elements are discussed in detail. The code is intended for researchers who are interested in this method and want to get started with it quickly. It can also be used as a basis for handling complex engineering optimization problems by combining the MFSE topology optimization method with non-gradient optimization algorithms without sensitivity information because only a few design variables are required to describe relatively complex structural topology and smooth structural boundaries using the MFSE method.

Keywords MATLAB implementation topology optimization material-field series-expansion method bounded material field dimensionality reduction

Corresponding Author(s): Yangjun LUO

Just Accepted Date: 11 June 2021 Online First Date: 04 August 2021 Issue Date: 24 September 2021

Cite this article:

Pai LIU,Yi YAN,Xiaopeng ZHANG, et al. A MATLAB code for the material-field series-expansion topology optimization method[J]. Front. Mech. Eng., 2021, 16(3): 607-622.

URL:

https://academic.hep.com.cn/fme/EN/10.1007/s11465-021-0637-3
https://academic.hep.com.cn/fme/EN/Y2021/V16/I3/607

Fig.1 Realizations of two material fields with (a) a small and (b) a large correlation length and the same material-field point settings.

Fig.2 Relationship between the material field values and interpolated Young’s modulus with different projection parameters.

Fig.3 Numbering of (a) the finite elements and nodes and (b) the material-field points within the design domain for the case “refine=2”.

Fig.4 MBB beam design problem considered in the main program.

Fig.5 Elemental density distribution (left) and material-field contour (right) of the optimized designs corresponding to truncation errors of (a)

10 − 5

, (b)

10 − 4

, and (c)

10 − 3

in the material field series expansion with “corlencoe=0.30”.

Fig.6 Elemental density distribution (left) and material-field contour (right) of the optimized designs corresponding to truncation errors of (a)

10 − 5

, (b)

10 − 4

, and (c)

10 − 3

in the material field series expansion with “corlencoe=0.10”.

Correlation length coefficient	Truncation error	Compliance	Number of design variables	Average number of active constraints
0.1	$10 − 5$	9.32	1230	255
0.1	$10 − 4$	9.40	978	265
0.1	$10 − 3$	9.36	728	268
0.3	$10 − 5$	9.46	176	248
0.3	$10 − 4$	9.51	138	235
0.3	$10 − 3$	9.64	100	227

Tab.1 Effects of the correlation length and the truncation error in the material-field series expansion

Fig.7 Elemental density distribution (left) and material-field contour (right) of the optimized design obtained with “corlencoe=0.05”.

Fig.8 Elemental density distribution (left) and material-field contour (right) of the optimized designs corresponding to “refine” values of (a) 2, (b) 3, and (c) 4 with “corlencoe=0.3”.

Tab.2 Effects of the parameter “refine”

Fig.9 Cantilever beam design problem with two load cases: (a) Design domain and boundary conditions; (b) elemental density distribution and material-field contour of the optimized design.

Fig.10 Cantilever beam design problem with passive elements: (a) Design domain and the boundary conditions; and (b) elemental density distribution and material-field contour of the optimized design.

1	M P Bendsøe, N Kikuchi. Generating optimal topologies in structural design using a homogenization method. Computer Methods in Applied Mechanics and Engineering, 1988, 71(2): 197–224 https://doi.org/10.1016/0045-7825(88)90086-2
2	M P Bendsøe. Optimal shape design as a material distribution problem. Structural Optimization, 1989, 1(4): 193–202 https://doi.org/10.1007/BF01650949
3	G I N Rozvany, M Zhou, T Birker. Generalized shape optimization without homogenization. Structural Optimization, 1992, 4(3–4): 250–252 https://doi.org/10.1007/BF01742754
4	M Y Wang, X Wang, D Guo. A level set method for structural topology optimization. Computer Methods in Applied Mechanics and Engineering, 2003, 192(1–2): 227–246 https://doi.org/10.1016/S0045-7825(02)00559-5
5	G Allaire, F Jouve, A Toader. Structural optimization using sensitivity analysis and a level-set method. Journal of Computational Physics, 2004, 194(1): 363–393 https://doi.org/10.1016/j.jcp.2003.09.032
6	Y M Xie, G P Steven. A simple evolutionary procedure for structural optimization. Computers & Structures, 1993, 49(5): 885–896 https://doi.org/10.1016/0045-7949(93)90035-C
7	X Huang, Y M Xie. Bi-directional evolutionary topology optimization of continuum structures with one or multiple materials. Computational Mechanics, 2009, 43(3): 393–401 https://doi.org/10.1007/s00466-008-0312-0
8	O Sigmund, K Maute. Topology optimization approaches. Structural and Multidisciplinary Optimization, 2013, 48(6): 1031–1055 https://doi.org/10.1007/s00158-013-0978-6
9	N P van Dijk, K Maute, M Langelaar, et al.. Level-set methods for structural topology optimization: A review. Structural and Multidisciplinary Optimization, 2013, 48(3): 437–472 https://doi.org/10.1007/s00158-013-0912-y
10	X Huang, Y M Xie. A further review of ESO type methods for topology optimization. Structural and Multidisciplinary Optimization, 2010, 41(5): 671–683 https://doi.org/10.1007/s00158-010-0487-9
11	O A Sigmund. 99 line topology optimization code written in MATLAB. Structural and Multidisciplinary Optimization, 2001, 21(2): 120–127 https://doi.org/10.1007/s001580050176
12	E Andreassen, A Clausen, M Schevenels, et al.. Efficient topology optimization in MATLAB using 88 lines of code. Structural and Multidisciplinary Optimization, 2011, 43(1): 1–16 https://doi.org/10.1007/s00158-010-0594-7
13	K Suresh. A 199-line MATLAB code for Pareto-optimal tracing in topology optimization. Structural and Multidisciplinary Optimization, 2010, 42(5): 665–679 https://doi.org/10.1007/s00158-010-0534-6
14	M Otomori, T Yamada, K Izui, et al.. MATLAB code for a level set-based topology optimization method using a reaction diffusion equation. Structural and Multidisciplinary Optimization, 2015, 51(5): 1159–1172 https://doi.org/10.1007/s00158-014-1190-z
15	L Xia, P Breitkopf. Design of materials using topology optimization and energy-based homogenization approach in MATLAB. Structural and Multidisciplinary Optimization, 2015, 52(6): 1229–1241 https://doi.org/10.1007/s00158-015-1294-0
16	V J Challis. A discrete level-set topology optimization code written in MATLAB. Structural and Multidisciplinary Optimization, 2010, 41(3): 453–464 https://doi.org/10.1007/s00158-009-0430-0
17	K Liu, A Tovar. An efficient 3D topology optimization code written in MATLAB. Structural and Multidisciplinary Optimization, 2014, 50(6): 1175–1196 https://doi.org/10.1007/s00158-014-1107-x
18	C Talischi, G H Paulino, A Pereira, et al.. PolyTop: A MATLAB implementation of a general topology optimization framework using unstructured polygonal finite element meshes. Structural and Multidisciplinary Optimization, 2012, 45(3): 329–357 https://doi.org/10.1007/s00158-011-0696-x
19	P Wei, Z Y Li, X P Li, et al.. An 88-line MATLAB code for the parameterized level set method based topology optimization using radial basis functions. Structural and Multidisciplinary Optimization, 2018, 58(2): 831–849 https://doi.org/10.1007/s00158-018-1904-8
20	Y Luo, J Bao. A material-field series-expansion method for topology optimization of continuum structures. Computers & Structures, 2019, 225: 106122 https://doi.org/10.1016/j.compstruc.2019.106122
21	Y Luo, J Xing, Z Kang. Topology optimization using material-field series expansion and Kriging-based algorithm: An effective non-gradient method. Computer Methods in Applied Mechanics and Engineering, 2020, 364: 112966 https://doi.org/10.1016/j.cma.2020.112966
22	K Svanberg. The method of moving asymptotes—A new method for structural optimization. International Journal for Numerical Methods in Engineering, 1987, 24(2): 359–373 https://doi.org/10.1002/nme.1620240207
23	Y J Luo, J J Zhan, J Xing, et al.. Non-probabilistic uncertainty quantification and response analysis of structures with a bounded field model. Computer Methods in Applied Mechanics and Engineering, 2019, 347: 663–678 https://doi.org/10.1016/j.cma.2018.12.043
24	J K Guest, J Prevost, T Belytschko. 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 https://doi.org/10.1002/nme.1064
25	A V Knyazev, M E Argentati, I Lashuk, et al.. Block locally optimal preconditioned eigenvalue xolvers (BLOPEX) in HYPRE and PETSc. SIAM Journal on Scientific Computing, 2007, 29(5): 2224–2239 https://doi.org/10.1137/060661624

[1]

Download

[1]	Kai LONG, Xiaoyu YANG, Nouman SAEED, Ruohan TIAN, Pin WEN, Xuan WANG. Topology optimization of transient problem with maximum dynamic response constraint using SOAR scheme[J]. Front. Mech. Eng., 2021, 16(3): 593-606.
[2]	Liang XUE, Jie LIU, Guilin WEN, Hongxin WANG. Efficient, high-resolution topology optimization method based on convolutional neural networks[J]. Front. Mech. Eng., 2021, 16(1): 80-96.
[3]	Peng WEI, Wenwen WANG, Yang YANG, Michael Yu WANG. Level set band method: A combination of density-based and level set methods for the topology optimization of continuums[J]. Front. Mech. Eng., 2020, 15(3): 390-405.
[4]	Xianda XIE, Shuting WANG, Ming YE, Zhaohui XIA, Wei ZHAO, Ning JIANG, Manman XU. Isogeometric topology optimization based on energy penalization for symmetric structure[J]. Front. Mech. Eng., 2020, 15(1): 100-122.
[5]	Emmanuel TROMME, Atsushi KAWAMOTO, James K. GUEST. Topology optimization based on reduction methods with applications to multiscale design and additive manufacturing[J]. Front. Mech. Eng., 2020, 15(1): 151-165.
[6]	Mariana MORETTI, Emílio C. N. SILVA. Topology optimization of piezoelectric bi-material actuators with velocity feedback control[J]. Front. Mech. Eng., 2019, 14(2): 190-200.
[7]	Manman XU, Shuting WANG, Xianda XIE. Level set-based isogeometric topology optimization for maximizing fundamental eigenfrequency[J]. Front. Mech. Eng., 2019, 14(2): 222-234.
[8]	Markus J. GEISS, Jorge L. BARRERA, Narasimha BODDETI, Kurt MAUTE. A regularization scheme for explicit level-set XFEM topology optimization[J]. Front. Mech. Eng., 2019, 14(2): 153-170.
[9]	Yu-Chin CHAN, Kohei SHINTANI, Wei CHEN. Robust topology optimization of multi-material lattice structures under material and load uncertainties[J]. Front. Mech. Eng., 2019, 14(2): 141-152.
[10]	Long JIANG, Yang GUO, Shikui CHEN, Peng WEI, Na LEI, Xianfeng David GU. Concurrent optimization of structural topology and infill properties with a CBF-based level set method[J]. Front. Mech. Eng., 2019, 14(2): 171-189.
[11]	Junjie ZHAN, Yangjun LUO. Robust topology optimization of hinge-free compliant mechanisms with material uncertainties based on a non-probabilistic field model[J]. Front. Mech. Eng., 2019, 14(2): 201-212.
[12]	Jiadong DENG, Claus B. W. PEDERSEN, Wei CHEN. Connected morphable components-based multiscale topology optimization[J]. Front. Mech. Eng., 2019, 14(2): 129-140.
[13]	Jikai LIU, Qian CHEN, Xuan LIANG, Albert C. TO. Manufacturing cost constrained topology optimization for additive manufacturing[J]. Front. Mech. Eng., 2019, 14(2): 213-221.
[14]	Yingjun WANG,David J. BENSON. Geometrically constrained isogeometric parameterized level-set based topology optimization via trimmed elements[J]. Front. Mech. Eng., 2016, 11(4): 328-343.
[15]	Shutian LIU,Quhao LI,Wenjiong CHEN,Liyong TONG,Gengdong CHENG. An identification method for enclosed voids restriction in manufacturability design for additive manufacturing structures[J]. Front. Mech. Eng., 2015, 10(2): 126-137.

Viewed

Full text

Abstract

Cited

Shared

Discussed