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.
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References
Zuo K, Chen L, Zhang Y, et al. Manufacturing-and machiningbased topology optimization. International Journal of Advanced Manufacturing Technology, 2006, 27(5–6): 531–536
Xia Q, Shi T, Wang M Y, et al. A level set based method for the optimization of cast part. Structural and Multidisciplinary Optimization, 2010, 41(5): 735–747
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
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
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
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
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
Luo Y, Wang M Y, Zhou M, et al. Topology optimization of reinforced concrete structures considering control of shrinkage and strength failure. Computers & Structures, 2015, 157: 31–41
Gao X, Ma H. Topology optimization of continuum structures under buckling constraints. Computers & Structures, 2015, 157: 142–152
Borrvall T, Petersson J. Topology optimization of fluids in stokes flow. International Journal for Numerical Methods in Fluids, 2003, 41(1): 77–107
Gersborg-Hansen A, Bendse M P, Sigmund O. Topology optimization of heat conduction problems using the finite volume method. Structural and Multidisciplinary Optimization, 2006, 31(4): 251–259
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
Suzuki K, Kikuchi N. A homogenization method for shape and topology optimization. Computer Methods in Applied Mechanics and Engineering, 1991, 93(3): 291–318
Allaire G, Bonnetier E, Francfort G, et al. Shape optimization by the homogenization method. Numerische Mathematik, 1997, 76(1): 27–68
Bendse M P. Optimal shape design as a material distribution problem. Structural Optimization, 1989, 1(4): 193–202
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
Xie Y M, Steven G P. A simple evolutionary procedure for structural optimization. Computers & Structures, 1993, 49(5): 885–896
Tanskanen P. The evolutionary structural optimization method: Theoretical aspects. Computer Methods in Applied Mechanics and Engineering, 2002, 191(47–48): 5485–5498
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
Xia Q, Shi T, Liu S, et al. A level set solution to the stress-based structural shape and topology optimization. Computers & Structures, 2012, 90–91: 55–64
Chen J, Shapiro V, Suresh K, et al. Shape optimization with topological changes and parametric control. International Journal for Numerical Methods in Engineering, 2007, 71(3): 313–346
Chen J, Freytag M, Shapiro V. Shape sensitivity of constructively represented geometric models. Computer Aided Geometric Design, 2008, 25(7): 470–488
Luo J, Luo Z, Chen S, et al. 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
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
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
Liu J, Ma Y S. 3D level-set topology optimization: A machining feature-based approach. Structural and Multidisciplinary Optimization, 2015, 52(3): 563–582
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
Guo X, Zhang W, Zhang J, et al. Explicit structural topology optimization based on moving morphable components (MMC) with curved skeletons. Computer Methods in Applied Mechanics and Engineering, 2016, 310: 711–748
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
Cottrell J A, Hughes T J R, Bazilevs Y. Isogeometric Analysis: Toward Integration of CAD and FEA. Chichester Wiley, 2009
Hughes T J R. The Finite Element Method: Linear Static and Dynamic Finite Element Analysis. Mineola: Courier Dover Publications, 2000
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
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
Kumar A, Parthasarathy A. Topology optimization using B-spline finite element. Structural and Multidisciplinary Optimization, 2011, 44(4): 471–481
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
Wang Y, Benson D J. Isogeometric analysis for parameterized LSMbased structural topology optimization. Computational Mechanics, 2016, 57(1): 19–35
Scott M A, Borden M J, Verhoosel C V, et al. Isogeometric finite element data structures based on Bzier extraction of T-splines. International Journal for Numerical Methods in Engineering, 2011, 88(2): 126–156
Nguyen-Thanh N, Kiendl J, Nguyen-Xuan H, et al. Rotation free isogeometric thin shell analysis using PHT-splines. Computer Methods in Applied Mechanics and Engineering, 2011, 200(47–48): 3410–3424
Speleers H, Manni C, Pelosi F, et al. Isogeometric analysis with Powell-Sabin splines for advection-diffusion-reaction problems. Computer Methods in Applied Mechanics and Engineering, 2012, 221–222: 132–148
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
Wang Y W, Huang Z D, Zheng Y, et al. Isogeometric analysis for compound B-spline surfaces. Computer Methods in Applied Mechanics and Engineering, 2013, 261–262: 1–15
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
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
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
Luo Z, Wang MY, Wang S, et al. A level-set-based parameterization method for structural shape and topology optimization. International Journal for Numerical Methods in Engineering, 2008, 76(1): 1–26
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
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
Mei Y, Wang X, Cheng G. A feature-based topological optimization for structure design. Advances in Engineering Software, 2008, 39 (2): 71–87
Osher S, Fedkiw R. Level Set Methods and Dynamic Implicit Surfaces. New York: Springer, 2003
Luo Z, Tong L, Wang M Y, et al. Shape and topology optimization of compliant mechanisms using a parameterization level set method. Journal of Computational Physics, 2007, 227(1): 680–705
Wendland H. Piecewise polynomial, positive definite and compactly supported radial functions of minimal degree. Advances in Computational Mathematics, 1995, 4(1): 389–396
Piegl L, Tiller W. The NURBS Book (Monographs in Visual Communication). Berlin: Springer, 1997
de Boor C. On calculating with B-splines. Journal of Approximation Theory, 1972, 6(1): 50–62
Benson D J, Hartmann S, Bazilevs Y, et al. Blended isogeometric shells. Computer Methods in Applied Mechanics and Engineering, 2013, 255: 133–146
Benson D J, Bazilevs Y, Hsu M C, et al. A large deformation, rotation-free, isogeometric shell. Computer Methods in Applied Mechanics and Engineering, 2011, 200(13–16): 1367–1378
Li K, Qian X. Isogeometric analysis and shape optimization via boundary integral. Computer Aided Design, 2011, 43(11): 1427–1437
Cai S, Zhang W. Stress constrained topology optimization with freeform design domains. Computer Methods in Applied Mechanics and Engineering, 2015, 289: 267–290
Hales T C. The Jordan curve theorem, formally and informally. American Mathematical Monthly, 2007, 114(10): 882–894
Shimrat M, Algorithm M. Algorithm 112: Position of point relative to polygon. Communications of the ACM, 1962, 5(8): 434–451
Nassar A, Walden P, Haines E, et al. Fastest point in polygon test. Ray Tracing News, 1992, 5(3)
Haines E. Point in Polygon Strategies. In: Heckbert S, ed. Graphics Gems IV. Elsevier, 1994, 24–26
Lasserre J. Integration on a convex polytope. Proceedings of the American Mathematical Society, 1998, 126(08): 2433–2441
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
Bendse M P, Sigmund O. Topology Optimization: Theory, Methods and Applications. Springer, 2003
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
Shapiro V. Theory of R-functions and Applications: A Primer. Technical Report CPA88-3. 1991
Gerstle T L, Ibrahim A M S, Kim P S, et al. A plastic surgery application in evolution: Three-dimensional printing. Plastic and Reconstructive Surgery, 2014, 133(2): 446–451
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Wang, Y., Benson, D.J. Geometrically constrained isogeometric parameterized level-set based topology optimization via trimmed elements. Front. Mech. Eng. 11, 328–343 (2016). https://doi.org/10.1007/s11465-016-0403-0
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DOI: https://doi.org/10.1007/s11465-016-0403-0