A level set method for shape and topology optimization of large-displacement compliant mechanisms

International Journal for Numerical Methods in Engineering

Volumn 76, Issue 6, 2008, Pages 862-892

  Luo, Zhen ^a   Tong, Liyong ^a  

^a UNIVERSITY OF SYDNEY   (Australia)

Author keywords

Compliant mechanisms; Geometrical non linearity; Level set methods; Radial basis functions; Shape and topology optimization

Indexed keywords

ARCHITECTURAL DESIGN; CONVEX OPTIMIZATION; DROP BREAKUP; EQUATIONS OF MOTION; EVOLUTIONARY ALGORITHMS; IMAGE SEGMENTATION; LINEARIZATION; MACHINE DESIGN; MECHANICAL ENGINEERING; MECHANISMS; MEDICAL IMAGING; RADIAL BASIS FUNCTION NETWORKS; RECONSTRUCTION (STRUCTURAL); SENSITIVITY ANALYSIS; SET THEORY; SHAPE OPTIMIZATION; STRUCTURAL DESIGN; STRUCTURAL OPTIMIZATION; TOPOLOGY; WIRELESS LOCAL AREA NETWORKS (WLAN);

COMPLIANT MECHANISMS; GEOMETRICAL NON-LINEARITY; LEVEL SET METHODS; RADIAL BASIS FUNCTIONS; SHAPE AND TOPOLOGY OPTIMIZATION;

LEVEL MEASUREMENT;

EID: 55049140084     PISSN: 00295981     EISSN: 10970207     Source Type: Journal    
DOI: 10.1002/nme.2352     Document Type: Article

References (80)

1. Howell LL. Compliant Mechanisms. Wiley: New York, 2001.
2. Howell LL, Midha A. A loop-closure theory for the analysis and synthesis of compliant mechanisms. ASME Journal of Mechanical Design 1996; 118(1):121-125.
3. Nathan D, Howell LL. A self-retracting fully compliant bistable micromechanism. Journal of Microelectromechanical Systems 2003; 12(3):273-280.
4. Cheng KT, Olhoff N. An investigation concerning optimal design of solid elastic plate. International Journal of Solids and Structures 1981; 17:305-323.
5. Bendsøe MP, Kikuchi N. Generating optimal topology in structural design using a homogenization method. Computer Methods in Applied Mechanics and Engineering 1988; 71(1):197-224.
6. Ananthasuresh GK, Kota S, Gianchandani Y. A methodical approach to the design of compliant micromechanisms. Solid-State Sensor and Actuator Workshop, 1994; 189-192.
7. Sigmund O. Some inverse problems in topology design of materials and mechanisms. Proceedings of the IUTAM Symposium on Optimization of Mechanical Systems. Kluwer: Dordrecht, 1996; 277-284.
8. Yin L, Ananthasuresh GK. Design of distributed compliant mechanisms. Mechanics Based Design of Structures and Machines 2001; 31(2):151-179.
9. Larsen VD, Sigmund O, Bouwstra S. Design and fabrication of compliant micromechanisms and structures with negative Poisson's ratio. Journal of Microelectromechanical Systems 1996; 6(2):99-106.
10. Kota S, Lu KJ, Kreiner Z, Trease B, Arenas J, Geiger J. Design and application of compliant mechanisms for surgical tools. Journal of Biomechanical Engineering 2005; 127(6):981-989.
11. Wang MY, Chen SK, Wang XM, Mei YL. Design of multi-material compliant mechanisms using level set methods. Journal of Mechanical Design 2005; 127(5):941-956.
12. Luo Z, Chen LP, Yang JZ, Zhang YQ, Abdel-Malek K. Compliant mechanism design using multi-objective topology optimization scheme of continuum structures. Structural and Multidisciplinary Optimization 2005; 30(2):142-154.
13. Sigmund O. Design of multiphysics actuator using topology optimization - Part I: one material structure. Computer Methods in Applied Mechanics and Engineering 2001; 190(49-50):6577-6604.
14. Ananthasuresh GK, Howell LL. Mechanical design of compliant microsystems-a perspective and prospects. ASME Journal of Mechanical Design 2005; 127(4):736-738.
15. Bendsøe MP, Sigmund O. Topology Optimization: Theory, Methods, and Applications. Springer: Berlin, Heidelberg, 2003.
16. Bendsøe MP, Sigmund O. Material interpolation schemes in topology optimization. Archive of Applied Mechanics 1999; 69(9-10):635-654.
17. Zhou M, Rozvany GIN. The COC algorithm. Part II: topological, geometry and generalized shape optimization. Computer Methods in Applied Mechanics and Engineering 1991; 89(1-3):197-224.
18. Mlejnek HP. Some aspects of the genesis of structures. Structural and Multidisciplinary Optimization 1992; 5(1-2):64-69.
19. Sigmund O, Petersson J. Numerical instabilities in topology optimization: a survey on procedures dealing with checkerboards, mesh-dependencies and local minima. Structural and Multidisciplinary Optimization 1998; 16:68-75.
20. Xie YM, Steven GP. A simple evolution procedure for structural optimization. Computers and Structures 1993; 49(5):885-896.
21. Tang WY, Tong LY, Gu YX. Improved genetic algorithm for design optimization of truss structures with size, shape and topology variables. International Journal for Numerical Methods in Engineering 2005; 62(13):1737-1762.
22. Osher S, Sethian JA. Front propagating with curvature dependent speed: algorithms based on Hamilton-Jacobi formulations. Journal of Computational Physics 1988; 78:12-49.
23. Sethian JA. Level set methods and fast marching methods: evolving interfaces in computational geometry, fluid mechanics, computer version and material science. Cambridge Monograph on Applied and Computational Mathematics. Cambridge University Press: Cambridge, U.K., 1999.
24. Sethian JA, Wiegmann A. Structural boundary design via level set and immersed interface methods. Journal of Computational Physics 2000; 163(2):489-528.
25. Osher S, Santosa F. Level-set methods for optimization problem involving geometry and constraints: I. Frequencies of a two-density inhomogeneous drum. Journal of Computational Physics 2001; 171:272-288.
26. Wang MY, Wang XM, Guo DM. A level set method for structural topology optimization. Computer Methods in Applied Mechanics and Engineering 2003; 192:227-246.
27. Allaire G, Jouve F, Toader AM. Structural optimization using sensitivity analysis and a level-set method. Journal of Computational Physics 2004; 194(1):363-393.
28. Osher S, Fedkiw RP. Level Set Methods and Dynamic Implicit Surface. Springer: New York, 2002.
29. Mitchell IM. A toolbox of level set methods. Technical Report, TR-2004-09, Department of Computer Science, University of British Columbia, Canada, 2004.
30. Burger M, Hacker B, Ring W. Incorporating topological derivatives into level set methods. Journal of Computational Physics 2004; 194(1):344-362.
31. Allaire G, Jouve F. Coupling the level set method and the topological gradient in structural optimization. IUTAM Symposium on Topological Design Optimization of Structures, Machines and Materials, vol. 137, Bendsoe MP, Olhoff N, Sigmund O (eds), 2006; 3-12.
32. Sokolowski J, Zochowski A. On the topological derivative in shape optimization. SIAM Journal on Control Optimization 1999; 37(4):1251-1272.
33. Norato J, Bendsøe MP, Haber JR, Tortorelli D. A topological derivative method for topology optimization. Structural and Multidisciplinary Optimization 2007; 33(4-5):375-386.
34. Nocedal J, Wright SJ. Numerical Optimization. Springer: New York, 1999.
35. Belytschko T, Xiao SP, Parimi C. Topology optimization with implicitly function and regularization. International Journal for Numerical Method in Engineering 2003; 57(1):1177-1196.
36. Norato J, Haber JR, Tortorelli D, Bendsøe MP. A geometry projection method for shape optimization. International Journal for Numerical Methods in Engineering 2004; 60(14):2289-2312.
37. Haber H. A multilever, level-set method for optimizing eigenvalues in shape design problems. Journal of Computing 2004; 198(2):518-534.
38. Chen JQ, Shapiro V, Suresh K, Tsukanov I. Shape optimization with topological changes and parametric control. International Journal for Numerical Methods in Engineering 2007; 71(3):313-346.
39. Wang SY, Wang MY. Radial basis functions and level set method for structural topology optimization. International Journal for Numerical Methods in Engineering 2006; 65(12):2060-2090.
40. Wei P, Wang MY. A piecewise constant level set method for structural shape and topology optimization. Seventh World Congress of Structural and Multidisciplinary Optimization (WCSM07), Seoul, Korea, 2007.
41. Luo Z, Wang MY, Wang SY, Wei P. A level set-based parameterization method for structural shape and topology optimization. International Journal for Numerical Methods in Engineering 2007. DOI: 10.1002/ nme2092.
42. Luo JZ, Luo Z, Chen LP, Tong LY, Wang MY. A semi-implicit level set method for structural shape and topology optimization. Journal of Computational Physics 2008; DOI: 10.1016/j.jcp.2008.02.003.
43. Belytschko T, Parimi C, Moës N, Sukumar N, Usui S. Structured extended finite element methods for solids defined by implicit surfaces. International Journal for Numerical Methods in Engineering 2002; 56(4):609-635.
44. Wang SY, Lim KM, Khoo BC, Wang MY. An extended level set method for shape and topology optimization. Journal of Computational Physics 2007; 221(1):395-421.
45. Nishiwaki S, Frecker MI, Min S, Kikuchi N. Topology optimization of compliant mechanisms using the homogenization method. International Journal for Numerical Methods in Engineering 1998; 42(3):535-559.
46. Buhl T, Pedersen CBW, Sigmund O. Stiffness design of geometrically nonlinear structures using topology optimization. Structural and Multidisciplinary Optimization 2000; 19(2):93-104.
47. Jung D, Gea HC. Topology optimization of nonlinear structures. Finite Element in Analysis and Design 2004; 40:1417-1427.
48. Bruns TE, Tortorelli DA. Topology optimization of nonlinear elastic structures and compliant mechanisms. Computer Methods in Applied Mechanics and Engineering 2001; 190:3443-3459.
49. Pedersen CBW, Buhl T, Sigmund O. Topology synthesis of large-displacement compliant mechanisms. International Journal for Numerical Methods in Engineering 2001; 50(12):2683-2705.
50. Yoon GH, Kim YY. Element connectivity parameterization for topology optimization of geometrically nonlinear structures. International Journal of Solids and Structures 2005; 42(7):1983-2009.
51. Cho S, Kwak J. Topological design optimization of geometrically non-linear structures using meshfree method. Computer Methods in Applied Mechanics and Engineering 2006; 195:5909-5925.
52. Kwak J, Cho S. Topological shape optimization of geometrically nonlinear structures using level set method. Computers and Structures 2005; 83(27):2257-2268.
53. Haug E, Rousselet B. Design sensitivity analysis in structural mechanics. II. Eigenvalue variations. Journal of Structural Mechanics 1980; 8:161-186.
54. Wang MY, Wang XM. PDE-driven level sets, shape sensitivity and curvature flow for structural topology optimization. Computer Modeling in Engineering and Science 2004; 6(4):373-396.
55. Wendland H. Piecewise polynomial, positive definite and compactly supported radial functions of minimal degree. Advances in Computational Mathematics 1995; 4:389-396.
56. Buhmann MD. Radial basis functions: theory and implementations. Cambridge Monographs on Applied and Computational Mathematics, vol. 12. Cambridge University Press: Cambridge, NY, 2004.
57. Schaback R, Wendland H. Using compactly supported radial basis functions to solve partial differential equations. In Boundary Element Technology, vol. XIII, Chen CS, Brebbia CA, Pepper DW (eds). WitPress: Southampton, Boston, 1999; 311-324.
58. Wendland H. Computational aspects of radial basis function approximation. In Topics in Multivariate Approximation and Interpolation, Jetter K et al. (eds). Elsevier: Amsterdam, 2005; 231-256.
59. Cecil T, Qian JL, Osher S. Numerical methods for high dimensional Hamilton-Jacobi equations using radial basis functions. Journal of Computational Physics 2004; 196(1):327-347.
60. Dolbow J, Belytschko T. An introduction to programming the meshless element free Galerkin method. Archives of Computational Methods in Engineering 1998; 5(3):207-241.
61. Bathe KJ. Finite Element Procedures. Prentice-Hall: Englewood Cliffs, NJ, 1996.
62. Sigmund O. On the design of compliant mechanisms using topology optimization. Mechanics of Structures and Machines 1997; 25(4):493-524.
63. Frecker MI, Ananthasuresh GK, Nishiwaki S, Kikuchi N, Kota S. Topological synthesis of compliant mechanisms using multi-criteria optimization. Journal of Mechanical Design 1997; 119(2):238-245.
64. Lau GK, Du H, Lim MK. Use of functional specifications as objective functions in topological optimization of compliant mechanism. Computer Methods in Applied Mechanics and Engineering 2001; 190(34):4421-4433.
65. Bruns TE, Tortorelli DA. An element removal and reintroduction strategy for the topology optimization of structures and compliant mechanisms. International Journal for Numerical Methods in Engineering 2003; 57:1413-1430.
66. Sokolowski J, Zolesio JP. Introduction to Shape Optimization: Shape Sensitivity Analysis. Springer: Berlin, 1992.
67. Choi KK, Kim NH. Structural Sensitivity Analysis and Optimization I - Linear Systems. Springer: New York, 2005.
68. Lorensen WE, Cline HE. Marching cubes: a high resolution 3D surface construction algorithm. Computer Graphics 1987; 21(4):163-169.
69. Yang RJ, Choi KK. Accuracy of finite element based shape design sensitivity analysis. Journal of Structural Mechanics 1985; 13(2):223-239.
70. Wang MY, Wei P. Topology optimization with level set method incorporating topological derivatives. Proceedings of 6th World Congress of Structural and Multidisciplinary Optimization (WCSM06), Rio de Janeiro, Brazil, May 2005.
71. Belytschko T, Moës N, Usui S, Parimi C. Arbitrary discontinuities in finite elements. International Journal for Numerical Methods in Engineering 2001; 50(4):993-1013.
72. Wang SY, Wang MY. A moving superimposed finite element method for structural topology optimization. International Journal for Numerical Methods in Engineering 2006; 65(11):1892-1922.
73. Rozvany GIN, Kirsch U, Bendsøe MP, Sigmund O. Layout optimization of structures. Applied Mechanics Reviews 1995; 48(2):41-119.
74. Bruns TE, Tortorelli DA. Numerical methods for the topology optimization of nonlinear elastic structures that exhibit snap-through. International Journal for Numerical Methods in Engineering 2002; 55:1215-1237.
75. Svanberg K. The method of moving asymptotes: a new method for structural optimization. International Journal for Numerical Method in Engineering 1987; 24:359-373.
76. Svanberg K. A class of globally convergent optimization methods based on conservative convex separable approximations. SIAM Journal on Optimization 2002; 12(2):555-573.
77. Rahmatalla S, Swan CC. Sparse monolithic compliant mechanisms using continuum structural topology optimization. International Journal for Numerical Methods in Engineering 2005; 62(1):1579-1605.
78. Poulsen TA. A new scheme for imposing a minimum length scale in topology optimization. International Journal for Numerical Methods in Engineering 2003; 57(6):741-760.
79. Guest JK, Prévost JH, Belytschko T. 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.
80. Chen SK, Wang YW. Geometric width control in topology optimization using level set method and a quadratic energy functional. ASME Proceedings of IDETC/CIE, Philadelphia, PA, U.S.A., 10-13 September 2006.