Geometrically constrained isogeometric parameterized level-set based topology optimization via trimmed elements

doi:10.1007/s11465-016-0403-0

Front. Mech. Eng.

2016, Vol. 11

Issue (4) : 328-343 https://doi.org/10.1007/s11465-016-0403-0

RESEARCH ARTICLE

Geometrically constrained isogeometric parameterized level-set based topology optimization via trimmed elements

Yingjun WANG¹(

),David J. BENSON²

¹. School of Mechanical and Automotive Engineering, South China University of Technology, Guangzhou 510641, China; Department of Mechanical Engineering, McGill University, Montreal H3A0C3, Canada
². Department of Structural Engineering, University of California, San Diego 92093, USA

Download: PDF(635 KB) HTML
Export: BibTeX | EndNote | Reference Manager | ProCite | RefWorks

Abstract

In this paper, an approach based on the fast point-in-polygon (PIP) algorithm and trimmed elements is proposed for isogeometric topology optimization (TO) with arbitrary geometric constraints. The isogeometric parameterized level-set-based TO method, which directly uses the non-uniform rational basis splines (NURBS) for both level set function (LSF) parameterization and objective function calculation, provides higher accuracy and efficiency than previous methods. The integration of trimmed elements is completed by the efficient quadrature rule that can design the quadrature points and weights for arbitrary geometric shape. Numerical examples demonstrate the efficiency and flexibility of the method.

Keywords isogeometric analysis topology optimization level set method arbitrary geometric constraint trimmed element

Corresponding Author(s): Yingjun WANG

Online First Date: 19 October 2016 Issue Date: 29 November 2016

Cite this article:

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.

URL:

https://academic.hep.com.cn/fme/EN/10.1007/s11465-016-0403-0
https://academic.hep.com.cn/fme/EN/Y2016/V11/I4/328

Fig.1 An example of geometric constraint in TO problems

Fig.2 Two mesh schemes for the example in Fig. 1. (a) Meshes by conventional finite element methods; (b) regular mesh

Fig.3 A 2D design domain and LSM

Fig.4 An example of the Greville abscissae collocation for the surface formed from knot vectors

Fig.5 Ray crossing test: One crossing denotes P₁ is inside and two crossings denote P₂ is outside

Fig.6 An example of element classification

Fig.7 An illustration for the approximation representation of the trimmed domain: Domain with (a) initial and (b) refined trimming control polygon

Fig.8 The element filling density computation for a boundary-crossing element: (a) Polyline approximation of the boundary and (b) the integration scheme

Fig.9 The Michell type structure with a geometric constraint: (a) Design domain and (b) distribution of initial holes

Fig.10 Optimization stages of the Michell type structure with geometric constraint: (a) Initial design, (b) Stage 2, (c) Stage 4, (d) Stage 10, (e) Stage 20 and (f) Stage 25 (final result)

Fig.11 Convergence histories of the Michell type structure with the geometric constraint

Fig.12 Optimization result of the Michell type structure without the geometric constraint [36]

Fig.13 The spanner structure with profile geometric constraint: (a) Design domain and (b) distribution of initial holes

Fig.14 Optimization stages of the spanner structure with geometric constraint: (a) Initial design, (b) Stage 2, (c) Stage 5, (d) Stage 10, (e) Stage 20, (f) Stage 30, (g) Stage 40 and (h) Stage 55 (final result)

Fig.15 Convergent histories of the spanner type structure with geometric constrain

Fig.16 Construction of geometrically constrained domain of the spanner model by R-functions. The operator

∧

represents the R-conjunction operation [66] which can be also regarded as an intersection operation

Fig.17 The R-function LSF of the spanner domain

Ω15

in Fig. 16

1	Zuo K, Chen L, Zhang Y, Manufacturing- and machining-based topology optimization. International Journal of Advanced Manufacturing Technology, 2006, 27(5–6): 531–536 https://doi.org/10.1007/s00170-004-2210-8
2	Xia Q, Shi T, Wang M Y, A level set based method for the optimization of cast part. Structural and Multidisciplinary Optimization, 2010, 41(5): 735–747 https://doi.org/10.1007/s00158-009-0444-7
3	Li H, Li P, Gao L, et al. A level set method for topological shape optimization of 3D structures with extrusion constraints. Computer Methods in Applied Mechanics and Engineering, 2015, 283: 615–635 https://doi.org/10.1016/j.cma.2014.10.006
4	Wang S, Wang M Y. Radial basis functions and level set method for structural topology optimization. International Journal for Numerical Methods in Engineering, 2006, 65(12): 2060–2090 https://doi.org/10.1002/nme.1536
5	Wang M Y, Wang X. PDE-driven level sets, shape sensitivity and curvature flow for structural topology optimization. Computer Modeling in Engineering & Sciences, 2004, 6 (4): 373–396
6	Wang M Y, Wang X, Guo D. 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
7	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 https://doi.org/10.1016/0045-7825(88)90086-2
8	Luo Y, Wang M Y, Zhou M, Topology optimization of reinforced concrete structures considering control of shrinkage and strength failure. Computers & Structures, 2015, 157: 31–41 https://doi.org/10.1016/j.compstruc.2015.05.009
9	Gao X, Ma H. Topology optimization of continuum structures under buckling constraints. Computers & Structures, 2015, 157: 142– 152 https://doi.org/10.1016/j.compstruc.2015.05.020
10	Borrvall T, Petersson J. Topology optimization of fluids in stokes flow. International Journal for Numerical Methods in Fluids, 2003, 41(1): 77–107 https://doi.org/10.1002/fld.426
11	Gersborg-Hansen A, Bends�e M P, Sigmund O. Topology optimization of heat conduction problems using the finite volume method. Structural and Multidisciplinary Optimization, 2006, 31(4): 251–259 https://doi.org/10.1007/s00158-005-0584-3
12	Zhou S, Li W, Li Q. Level-set based topology optimization for electromagnetic dipole antenna design. Journal of Computational Physics, 2010, 229(19): 6915–6930 https://doi.org/10.1016/j.jcp.2010.05.030
13	Suzuki K, Kikuchi N. A homogenization method for shape and topology optimization. Computer Methods in Applied Mechanics and Engineering, 1991, 93(3): 291–318 https://doi.org/10.1016/0045-7825(91)90245-2
14	Allaire G, Bonnetier E, Francfort G, Shape optimization by the homogenization method. Numerische Mathematik, 1997, 76(1): 27–68 https://doi.org/10.1007/s002110050253
15	Bends�e M P. Optimal shape design as a material distribution problem. Structural Optimization, 1989, 1(4): 193–202 https://doi.org/10.1007/BF01650949
16	Zhou M, Rozvany G I N. The COC algorithm, Part II: Topological, geometrical and generalized shape optimization. Computer Methods in Applied Mechanics and Engineering, 1991, 89(1–3): 309–336 https://doi.org/10.1016/0045-7825(91)90046-9
17	Xie Y M, Steven G P. A simple evolutionary procedure for structural optimization. Computers & Structures, 1993, 49(5): 885–896 https://doi.org/10.1016/0045-7949(93)90035-C
18	Tanskanen P. The evolutionary structural optimization method: Theoretical aspects. Computer Methods in Applied Mechanics and Engineering, 2002, 191(47–48): 5485–5498 https://doi.org/10.1016/S0045-7825(02)00464-4
19	Allaire G, Jouve F, Toader A M. 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
20	Xia Q, Shi T, Liu S, A level set solution to the stress-based structural shape and topology optimization. Computers & Structures, 2012, 90–91: 55–64 https://doi.org/10.1016/j.compstruc.2011.10.009
21	Chen J, Shapiro V, Suresh K, Shape optimization with topological changes and parametric control. International Journal for Numerical Methods in Engineering, 2007, 71(3): 313–346 https://doi.org/10.1002/nme.1943
22	Chen J, Freytag M, Shapiro V. Shape sensitivity of constructively represented geometric models. Computer Aided Geometric Design, 2008, 25(7): 470–488 https://doi.org/10.1016/j.cagd.2008.01.005
23	Luo J, Luo Z, Chen S, A new level set method for systematic design of hinge-free compliant mechanisms. Computer Methods in Applied Mechanics and Engineering, 2008, 198(2): 318–331 https://doi.org/10.1016/j.cma.2008.08.003
24	Liu T, Wang S, Li B, A level-set-based topology and shape optimization method for continuum structure under geometric constraints. Structural and Multidisciplinary Optimization, 2014, 50(2): 253–273 https://doi.org/10.1007/s00158-014-1045-7
25	Liu T, Li B, Wang S, Eigenvalue topology optimization of structures using a parameterized level set method. Structural and Multidisciplinary Optimization, 2014, 50(4): 573–591 https://doi.org/10.1007/s00158-014-1069-z
26	Liu J, Ma Y S. 3D level-set topology optimization: A machining feature-based approach. Structural and Multidisciplinary Optimization, 2015, 52(3): 563–582 https://doi.org/10.1007/s00158-015-1263-7
27	Xia Q, Shi T. Constraints of distance from boundary to skeleton: For the control of length scale in level set based structural topology optimization. Computer Methods in Applied Mechanics and Engineering, 2015, 295: 525–542 https://doi.org/10.1016/j.cma.2015.07.015
28	Guo X, Zhang W, Zhang J, Explicit structural topology optimization based on moving morphable components (MMC) with curved skeletons. Computer Methods in Applied Mechanics and Engineering, 2016, 310: 711–748 https://doi.org/10.1016/j.cma.2016.07.018
29	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 https://doi.org/10.1016/j.cma.2004.10.008
30	Cottrell J A, Hughes T J R, Bazilevs Y. Isogeometric Analysis: Toward Integration of CAD and FEA. Chichester Wiley, 2009
31	Hughes T J R. The Finite Element Method: Linear Static and Dynamic Finite Element Analysis. Mineola: Courier Dover Publications, 2000
32	Seo Y D, Kim H J, Youn S K. Isogeometric topology optimization using trimmed spline surfaces. Computer Methods in Applied Mechanics and Engineering, 2010, 199(49–52): 3270–3296 https://doi.org/10.1016/j.cma.2010.06.033
33	Kim H J, Seo Y D, Youn S K. Isogeometric analysis for trimmed CAD surfaces. Computer Methods in Applied Mechanics and Engineering, 2009, 198(37–40): 2982–2995 https://doi.org/10.1016/j.cma.2009.05.004
34	Kumar A, Parthasarathy A. Topology optimization using B-spline finite element. Structural and Multidisciplinary Optimization, 2011, 44(4): 471–481 https://doi.org/10.1007/s00158-011-0650-y
35	Ded� 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 https://doi.org/10.1007/s11831-012-9075-z
36	Wang Y, Benson D J. Isogeometric analysis for parameterized LSM-based structural topology optimization. Computational Mechanics, 2016, 57(1): 19–35 https://doi.org/10.1007/s00466-015-1219-1
37	Scott M A, Borden M J, Verhoosel C V, Isogeometric finite element data structures based on B�zier extraction of T-splines. International Journal for Numerical Methods in Engineering, 2011, 88(2): 126–156 https://doi.org/10.1002/nme.3167
38	Nguyen-Thanh N, Kiendl J, Nguyen-Xuan H, Rotation free isogeometric thin shell analysis using PHT-splines. Computer Methods in Applied Mechanics and Engineering, 2011, 200(47–48): 3410–3424 https://doi.org/10.1016/j.cma.2011.08.014
39	Speleers H, Manni C, Pelosi F, Isogeometric analysis with Powell-Sabin splines for advection-diffusion-reaction problems. Computer Methods in Applied Mechanics and Engineering, 2012, 221–222: 132–148 https://doi.org/10.1016/j.cma.2012.02.009
40	Kim H J, Seo Y D, Youn S K. Isogeometric analysis with trimming technique for problems of arbitrary complex topology. Computer Methods in Applied Mechanics and Engineering, 2010, 199(45–48): 2796–2812 https://doi.org/10.1016/j.cma.2010.04.015
41	Wang Y W, Huang Z D, Zheng Y, Isogeometric analysis for compound B-spline surfaces. Computer Methods in Applied Mechanics and Engineering, 2013, 261–262: 1–15 https://doi.org/10.1016/j.cma.2013.04.001
42	Beer G, Marussig B, Zechner J. A simple approach to the numerical simulation with trimmed CAD surfaces. Computer Methods in Applied Mechanics and Engineering, 2015, 285: 776–790 https://doi.org/10.1016/j.cma.2014.12.010
43	Nagy A P, Benson D J. On the numerical integration of trimmed isogeometric elements. Computer Methods in Applied Mechanics and Engineering, 2015, 284: 165–185 https://doi.org/10.1016/j.cma.2014.08.002
44	Wang Y, Benson D J, Nagy A P. A multi-patch nonsingular isogeometric boundary element method using trimmed elements. Computational Mechanics, 2015, 56(1): 173–191 https://doi.org/10.1007/s00466-015-1165-y
45	Luo Z, Wang M Y, Wang S, A level-set-based parameterization method for structural shape and topology optimization. International Journal for Numerical Methods in Engineering, 2008, 76(1): 1–26 https://doi.org/10.1002/nme.2092
46	Luo Z, Tong L, Kang Z. A level set method for structural shape and topology optimization using radial basis functions. Computers & Structures, 2009, 87(7–8): 425–434 https://doi.org/10.1016/j.compstruc.2009.01.008
47	Osher S, Sethian J A. Fronts propagating with curvature-dependent speed: Algorithms based on Hamilton-Jacobi formulations. Journal of Computational Physics, 1988, 79(1): 12–49 https://doi.org/10.1016/0021-9991(88)90002-2
48	Mei Y, Wang X, Cheng G. A feature-based topological optimization for structure design. Advances in Engineering Software, 2008, 39(2): 71–87 https://doi.org/10.1016/j.advengsoft.2007.01.023
49	Osher S, Fedkiw R. Level Set Methods and Dynamic Implicit Surfaces. New York: Springer, 2003
50	Luo Z, Tong L, Wang M Y, Shape and topology optimization of compliant mechanisms using a parameterization level set method. Journal of Computational Physics, 2007, 227(1): 680–705 https://doi.org/10.1016/j.jcp.2007.08.011
51	Wendland H. Piecewise polynomial, positive definite and compactly supported radial functions of minimal degree. Advances in Computational Mathematics, 1995, 4(1): 389–396 https://doi.org/10.1007/BF02123482
52	Piegl L, Tiller W. The NURBS Book (Monographs in Visual Communication). Berlin: Springer, 1997
53	de Boor C. On calculating with B-splines. Journal of Approximation Theory, 1972, 6(1): 50–62 https://doi.org/10.1016/0021-9045(72)90080-9
54	Benson D J, Hartmann S, Bazilevs Y, Blended isogeometric shells. Computer Methods in Applied Mechanics and Engineering, 2013, 255: 133–146 https://doi.org/10.1016/j.cma.2012.11.020
55	Benson D J, Bazilevs Y, Hsu M C, A large deformation, rotation-free, isogeometric shell. Computer Methods in Applied Mechanics and Engineering, 2011, 200(13–16): 1367–1378 https://doi.org/10.1016/j.cma.2010.12.003
56	Li K, Qian X. Isogeometric analysis and shape optimization via boundary integral. Computer Aided Design, 2011, 43(11): 1427–1437 https://doi.org/10.1016/j.cad.2011.08.031
57	Cai S, Zhang W. Stress constrained topology optimization with free-form design domains. Computer Methods in Applied Mechanics and Engineering, 2015, 289: 267–290 https://doi.org/10.1016/j.cma.2015.02.012
58	Hales T C. The Jordan curve theorem, formally and informally. American Mathematical Monthly, 2007, 114(10): 882–894
59	Shimrat M, Algorithm M. Algorithm 112: Position of point relative to polygon. Communications of the ACM, 1962, 5(8): 434–451 https://doi.org/10.1145/368637.368653
60	Nassar A, Walden P, Haines E, Fastest point in polygon test. Ray Tracing News, 1992, 5(3)
61	Haines E. Point in Polygon Strategies. In: Heckbert S, ed. Graphics Gems IV. Elsevier, 1994, 24–26
62	Lasserre J. Integration on a convex polytope. Proceedings of the American Mathematical Society, 1998, 126(08): 2433–2441 https://doi.org/10.1090/S0002-9939-98-04454-2
63	Dunavant D A. High degree efficient symmetrical Gaussian quadrature rules for the triangle. International Journal for Numerical Methods in Engineering, 1985, 21(6): 1129–1148 https://doi.org/10.1002/nme.1620210612
64	Bends�e M P, Sigmund O. Topology Optimization: Theory, Methods and Applications. Springer, 2003
65	Wang S, Wang M Y. Structural shape and topology optimization using an implicit free boundary parametrization method. Computer Modeling in Engineering & Sciences, 2006, 13(2): 119–147
66	Shapiro V. Theory of R-functions and Applications: A Primer. Technical Report CPA88-3. 1991
67	Gerstle T L, Ibrahim A M S, Kim P S, A plastic surgery application in evolution: Three-dimensional printing. Plastic and Reconstructive Surgery, 2014, 133(2): 446–451 https://doi.org/10.1097/01.prs.0000436844.92623.d3

[1]	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.
[2]	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.
[3]	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.
[4]	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.
[5]	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.
[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]	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.
[8]	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.
[9]	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.
[10]	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.
[11]	Jiadong DENG, Claus B. W. PEDERSEN, Wei CHEN. Connected morphable components-based multiscale topology optimization[J]. Front. Mech. Eng., 2019, 14(2): 129-140.
[12]	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.
[13]	Li LI, Michael Yu WANG, Peng WEI. XFEM schemes for level set based structural optimization[J]. Front Mech Eng, 2012, 7(4): 335-356.
[14]	Xinjun LIU, Xiang CHEN, Zhidong LI. Modular design of typical rigid links in parallel kinematic machines: Classification and topology optimization[J]. Front Mech Eng, 2012, 7(2): 199-209.
[15]	Xinjun LIU, Zhidong LI, Xiang CHEN. Structural optimization of typical rigid links in a parallel kinematic machine[J]. Front Mech Eng, 2011, 6(3): 344-353.

Viewed

Full text

Abstract

Cited

Shared

Discussed