Frontiers of Mechanical Engineering

, Volume 11, Issue 4, pp 328–343 | Cite as

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

Research Article

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 

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References

  1. 1.
    Zuo K, Chen L, Zhang Y, et al. Manufacturing-and machiningbased topology optimization. International Journal of Advanced Manufacturing Technology, 2006, 27(5–6): 531–536CrossRefGoogle Scholar
  2. 2.
    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): 735–747CrossRefGoogle Scholar
  3. 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: 615–635MathSciNetCrossRefGoogle Scholar
  4. 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): 2060–2090MathSciNetCrossRefMATHGoogle Scholar
  5. 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): 373–396MATHGoogle Scholar
  6. 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): 227–246MathSciNetCrossRefMATHGoogle Scholar
  7. 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): 197–224MathSciNetCrossRefMATHGoogle Scholar
  8. 8.
    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: 31–41CrossRefGoogle Scholar
  9. 9.
    Gao X, Ma H. Topology optimization of continuum structures under buckling constraints. Computers & Structures, 2015, 157: 142–152CrossRefGoogle Scholar
  10. 10.
    Borrvall T, Petersson J. Topology optimization of fluids in stokes flow. International Journal for Numerical Methods in Fluids, 2003, 41(1): 77–107MathSciNetCrossRefMATHGoogle Scholar
  11. 11.
    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): 251–259MathSciNetCrossRefMATHGoogle Scholar
  12. 12.
    Zhou S, Li W, Li Q. Level-set based topology optimization for electromagnetic dipole antenna design. Journal of Computational Physics, 2010, 229(19): 6915–6930MathSciNetCrossRefMATHGoogle Scholar
  13. 13.
    Suzuki K, Kikuchi N. A homogenization method for shape and topology optimization. Computer Methods in Applied Mechanics and Engineering, 1991, 93(3): 291–318CrossRefMATHGoogle Scholar
  14. 14.
    Allaire G, Bonnetier E, Francfort G, et al. Shape optimization by the homogenization method. Numerische Mathematik, 1997, 76(1): 27–68MathSciNetCrossRefMATHGoogle Scholar
  15. 15.
    Bendse M P. Optimal shape design as a material distribution problem. Structural Optimization, 1989, 1(4): 193–202CrossRefGoogle Scholar
  16. 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): 309–336CrossRefGoogle Scholar
  17. 17.
    Xie Y M, Steven G P. A simple evolutionary procedure for structural optimization. Computers & Structures, 1993, 49(5): 885–896CrossRefGoogle Scholar
  18. 18.
    Tanskanen P. The evolutionary structural optimization method: Theoretical aspects. Computer Methods in Applied Mechanics and Engineering, 2002, 191(47–48): 5485–5498CrossRefMATHGoogle Scholar
  19. 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): 363–393MathSciNetCrossRefMATHGoogle Scholar
  20. 20.
    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: 55–64CrossRefGoogle Scholar
  21. 21.
    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): 313–346MathSciNetCrossRefMATHGoogle Scholar
  22. 22.
    Chen J, Freytag M, Shapiro V. Shape sensitivity of constructively represented geometric models. Computer Aided Geometric Design, 2008, 25(7): 470–488MathSciNetCrossRefMATHGoogle Scholar
  23. 23.
    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): 318–331CrossRefMATHGoogle Scholar
  24. 24.
    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): 253–273MathSciNetCrossRefGoogle Scholar
  25. 25.
    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): 573–591MathSciNetCrossRefGoogle Scholar
  26. 26.
    Liu J, Ma Y S. 3D level-set topology optimization: A machining feature-based approach. Structural and Multidisciplinary Optimization, 2015, 52(3): 563–582MathSciNetCrossRefGoogle Scholar
  27. 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: 525–542MathSciNetCrossRefGoogle Scholar
  28. 28.
    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: 711–748MathSciNetCrossRefGoogle Scholar
  29. 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): 4135–4195MathSciNetCrossRefMATHGoogle Scholar
  30. 30.
    Cottrell J A, Hughes T J R, Bazilevs Y. Isogeometric Analysis: Toward Integration of CAD and FEA. Chichester Wiley, 2009CrossRefMATHGoogle Scholar
  31. 31.
    Hughes T J R. The Finite Element Method: Linear Static and Dynamic Finite Element Analysis. Mineola: Courier Dover Publications, 2000MATHGoogle Scholar
  32. 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): 3270–3296MathSciNetCrossRefMATHGoogle Scholar
  33. 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): 2982–2995CrossRefMATHGoogle Scholar
  34. 34.
    Kumar A, Parthasarathy A. Topology optimization using B-spline finite element. Structural and Multidisciplinary Optimization, 2011, 44(4): 471–481MathSciNetCrossRefMATHGoogle Scholar
  35. 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): 427–465MathSciNetCrossRefMATHGoogle Scholar
  36. 36.
    Wang Y, Benson D J. Isogeometric analysis for parameterized LSMbased structural topology optimization. Computational Mechanics, 2016, 57(1): 19–35MathSciNetCrossRefMATHGoogle Scholar
  37. 37.
    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): 126–156MathSciNetCrossRefMATHGoogle Scholar
  38. 38.
    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): 3410–3424MathSciNetCrossRefMATHGoogle Scholar
  39. 39.
    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: 132–148MathSciNetCrossRefMATHGoogle Scholar
  40. 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): 2796–2812MathSciNetCrossRefMATHGoogle Scholar
  41. 41.
    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: 1–15MathSciNetCrossRefMATHGoogle Scholar
  42. 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: 776–790MathSciNetCrossRefGoogle Scholar
  43. 43.
    Nagy A P, Benson D J. On the numerical integration of trimmed isogeometric elements. Computer Methods in Applied Mechanics and Engineering, 2015, 284: 165–185MathSciNetCrossRefGoogle Scholar
  44. 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): 173–191MathSciNetCrossRefMATHGoogle Scholar
  45. 45.
    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): 1–26MathSciNetCrossRefMATHGoogle Scholar
  46. 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): 425–434CrossRefGoogle Scholar
  47. 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): 12–49MathSciNetCrossRefMATHGoogle Scholar
  48. 48.
    Mei Y, Wang X, Cheng G. A feature-based topological optimization for structure design. Advances in Engineering Software, 2008, 39 (2): 71–87CrossRefGoogle Scholar
  49. 49.
    Osher S, Fedkiw R. Level Set Methods and Dynamic Implicit Surfaces. New York: Springer, 2003CrossRefMATHGoogle Scholar
  50. 50.
    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): 680–705MathSciNetCrossRefMATHGoogle Scholar
  51. 51.
    Wendland H. Piecewise polynomial, positive definite and compactly supported radial functions of minimal degree. Advances in Computational Mathematics, 1995, 4(1): 389–396MathSciNetCrossRefMATHGoogle Scholar
  52. 52.
    Piegl L, Tiller W. The NURBS Book (Monographs in Visual Communication). Berlin: Springer, 1997CrossRefMATHGoogle Scholar
  53. 53.
    de Boor C. On calculating with B-splines. Journal of Approximation Theory, 1972, 6(1): 50–62MathSciNetCrossRefMATHGoogle Scholar
  54. 54.
    Benson D J, Hartmann S, Bazilevs Y, et al. Blended isogeometric shells. Computer Methods in Applied Mechanics and Engineering, 2013, 255: 133–146MathSciNetCrossRefMATHGoogle Scholar
  55. 55.
    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): 1367–1378MathSciNetCrossRefMATHGoogle Scholar
  56. 56.
    Li K, Qian X. Isogeometric analysis and shape optimization via boundary integral. Computer Aided Design, 2011, 43(11): 1427–1437CrossRefGoogle Scholar
  57. 57.
    Cai S, Zhang W. Stress constrained topology optimization with freeform design domains. Computer Methods in Applied Mechanics and Engineering, 2015, 289: 267–290MathSciNetCrossRefGoogle Scholar
  58. 58.
    Hales T C. The Jordan curve theorem, formally and informally. American Mathematical Monthly, 2007, 114(10): 882–894MathSciNetCrossRefMATHGoogle Scholar
  59. 59.
    Shimrat M, Algorithm M. Algorithm 112: Position of point relative to polygon. Communications of the ACM, 1962, 5(8): 434–451CrossRefGoogle Scholar
  60. 60.
    Nassar A, Walden P, Haines E, et al. Fastest point in polygon test. Ray Tracing News, 1992, 5(3)Google Scholar
  61. 61.
    Haines E. Point in Polygon Strategies. In: Heckbert S, ed. Graphics Gems IV. Elsevier, 1994, 24–26Google Scholar
  62. 62.
    Lasserre J. Integration on a convex polytope. Proceedings of the American Mathematical Society, 1998, 126(08): 2433–2441MathSciNetCrossRefMATHGoogle Scholar
  63. 63.
    Dunavant D A. High degree efficient symmetrical Gaussian quadrature rules for the triangle. International Journal for Numerical Methods in Engineering, 1985, 21(6): 1129–1148MathSciNetCrossRefMATHGoogle Scholar
  64. 64.
    Bendse M P, Sigmund O. Topology Optimization: Theory, Methods and Applications. Springer, 2003Google Scholar
  65. 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): 119–147MathSciNetMATHGoogle Scholar
  66. 66.
    Shapiro V. Theory of R-functions and Applications: A Primer. Technical Report CPA88-3. 1991Google Scholar
  67. 67.
    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): 446–451CrossRefGoogle Scholar

Copyright information

© Higher Education Press and Springer-Verlag Berlin Heidelberg 2016

Authors and Affiliations

  1. 1.School of Mechanical and Automotive EngineeringSouth China University of TechnologyGuangzhouChina
  2. 2.Department of Mechanical EngineeringMcGill UniversityMontrealCanada
  3. 3.Department of Structural EngineeringUniversity of CaliforniaSan DiegoUSA

Personalised recommendations