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

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

References

[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

[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

[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

[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

[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

[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

[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

[9] Gao X, Ma H. Topology optimization of continuum structures under buckling constraints. Computers & Structures, 2015, 157: 142– 152

[10] Borrvall T, Petersson J. Topology optimization of fluids in stokes flow. International Journal for Numerical Methods in Fluids, 2003, 41(1): 77–107

[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

[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

[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

[14] Allaire G, Bonnetier E, Francfort G, Shape optimization by the homogenization method. Numerische Mathematik, 1997, 76(1): 27–68

[15] Bends�e M P. Optimal shape design as a material distribution problem. Structural Optimization, 1989, 1(4): 193–202

[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

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

[18] Tanskanen P. The evolutionary structural optimization method: Theoretical aspects. Computer Methods in Applied Mechanics and Engineering, 2002, 191(47–48): 5485–5498

[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

[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

[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

[22] Chen J, Freytag M, Shapiro V. Shape sensitivity of constructively represented geometric models. Computer Aided Geometric Design, 2008, 25(7): 470–488

[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

[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

[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

[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

[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

[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

[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

[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

[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

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

[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

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

[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

[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

[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

[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

[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

[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

[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

[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

[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

[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

[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

[48] Mei Y, Wang X, Cheng G. A feature-based topological optimization for structure design. Advances in Engineering Software, 2008, 39(2): 71–87

[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

[51] Wendland H. Piecewise polynomial, positive definite and compactly supported radial functions of minimal degree. Advances in Computational Mathematics, 1995, 4(1): 389–396

[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

[54] Benson D J, Hartmann S, Bazilevs Y, Blended isogeometric shells. Computer Methods in Applied Mechanics and Engineering, 2013, 255: 133–146

[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

[56] Li K, Qian X. Isogeometric analysis and shape optimization via boundary integral. Computer Aided Design, 2011, 43(11): 1427–1437

[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

[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

[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

[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

[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

Acknowledgements

The support for this research and Yingjun Wang by National Science Foundation (Grant No. CMMI-1068106) is gratefully acknowledged.

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2016 Higher Education Press and Springer-Verlag Berlin Heidelberg