Isogeometric Analysis for Topology Optimization with a Phase Field Model

Archives of Computational Methods in Engineering

Volumn 19, Issue 3, 2012, Pages 427-465

  Dedè, Luca ^a,b   Borden, Micheal J ^a   Hughes, Thomas J R ^a  

^a UNIVERSITY OF TEXAS AT AUSTIN   (United States)

^b EPFL   (Switzerland)

Author keywords

[No Author keywords available]

Indexed keywords

CAHN-HILLIARD EQUATION; COUPLED SYSTEMS; DESIGN DOMAINS; DIMENSIONLESS PARAMETERS; ISOGEOMETRIC ANALYSIS; MINIMUM COMPLIANCE; NUMERICAL APPROXIMATIONS; OPTIMAL TOPOLOGIES; PHASE FIELD MODELS; PHASE FIELDS; STEADY STATE; VOLUME CONSTRAINT;

MATHEMATICAL MODELS; OPTIMIZATION; PHASE INTERFACES; TWO PHASE FLOW;

TOPOLOGY;

EID: 84865587878     PISSN: 11343060     EISSN: None     Source Type: Journal    
DOI: 10.1007/s11831-012-9075-z     Document Type: Article

References (120)

1. Aage N, Poulsen TH, Gersborg-Hansen A, Sigmund O (2008) Topology optimization of large scale Stokes flow problems. Struct Multidiscip Optim 35: 175-180.
2. Adams RA (1975) Sobolev spaces. Academic Press, New York.
3. Allaire G, Jouve F, Maillot H (2004) Topology optimization for minimum stress design with the homogenization method. Struct Multidiscip Optim 28: 87-98.
4. Allaire G, Jouve F, Toader A (2004) Structural optimization using sensitivity analysis and level set-method. J Comput Phys 194: 363-393.
5. Allaire G, de Gournay F, Jouve F, Toader AM (2005) Structural optimization using topological and shape sensitivity via a level set method. Control Cybern 34: 59-81.
6. Allen SM, Cahn JW (1979) A microscopic theory for antiphase boundary motion and its application to antiphase domain coarsening. Acta Metall 27: 1085-1095.
7. Ambrosio L, Buttazzo G (1993) An optimal design problem with perimeter penalization. Calc Var Partial Differ Equ 1: 55-69.
8. Bazilevs Y, Calo VM, Cottrell JA, Hughes TJR, Reali A, Scovazzi G (2007) Variational multiscale residual-based turbulence modeling for large eddy simulation of incompressible flows. Comput Methods Appl Mech Eng 197: 173-201.
9. Bendsøe MP (1983) On obtaining a solution to optimization problems for solid, elastic plates by restriction of the design space. J Struct Mech 11: 501-521.
10. Bendsøe MP, Haber RB (1993) The michell layout problem as a low volume fraction limit of the perforated plate topology optimization problem: an asymptotic study. Struct Optim 6: 263-267.
11. Bendsøe MP, Kikuchi N (1988) Generating optimal topologies in structural design using a homogenization method. Comput Methods Appl Mech Engrg 71: 197-224.
12. Bendsøe MP, Sigmund O (1999) Material interpolations schemes in topology optimization. Arch Appl Mech 69: 635-654.
13. Bendsøe MP, Sigmund O (2003) Topology optimization: theory, methods and applications. Springer, Berlin.
14. Bischoff M, Wall WA, Bletzinger KU, Ramm E (2004) Models and finite elements for thin-walled structures. In: Stein E, de Borst R, Hughes TJR (eds) Encyclopedia of computational mechanics. Solids struct coupled prob 3, vol 2. Wiley, New York, pp 59-137.
15. Bletzinger KU, Maute K (1997) Towards generalized shape and topology optimization. Eng Optim 29: 201-216.
16. Bletzinger KU, Ramm E (1993) Form finding of shells by structural optimization. Eng Comput 9: 27-35.
17. Bletzinger KU, Reitnger R, Kimmich S, Ramm E (1993) Shape optimization with program CARAT. In: Hörnlein H, Schittkowski K (eds) Software systems for structural optimization. Internat ser numer math, vol 110. Birkhäuser, Basel, pp 97-124.
18. Borden MJ, Scott MA, Evans JA, Hughes TJR (2011) Isogeometric finite element data structures based on Bézier extraction of NURBS. Int J Numer Methods Eng 87: 15-47.
19. Borden MJ, Verhoosel CV, Scott MA, Hughes TJR, Landis CM (2012) A phase-field description of dynamic brittle fracture. Comput Methods Appl Mech Eng 217-220: 77-95.
20. Bourdin B (2011) Filters in topology optimization. Int J Numer Methods Eng 50: 2143-2158.
21. Bourdin B, Chambolle S (2003) Design-dependent loads in topology optimization. Control Optim Calc Var 9: 19-48.
22. Bourdin B, Chambolle S (2006) The phase field method in optimal design. In: Bendsøe MP, Olhoff N, Sigmund O (eds) IUTAM symposium on topological design optimization of structures. Solid mech appl, vol 137. Springer, Dordrecht, pp 207-215.
23. Brezzi F, Fortin M (1991) Mixed and hybrid finite element methods. Springer, New York.
24. Burger M, Stainko R (2006) Phase-field relaxation of topology optimization with local stress constraints. SIAM J Control Optim 45: 1447-1466.
25. ∞ bound for solutions of the Cahn-Hilliard equation. Arch Ration Mech Anal 133: 129-144.
26. Cahn JW (1961) On spinodal decomposition. Acta Metall 9: 795-801.
27. Cahn JW, Hilliard JE (1958) Free energy of a non-uniform system. I. Interfacial free energy. J Chem Phys 28: 258-267.
28. Cahn JW, Hilliard JE (1959) Free energy of a non-uniform system. III. Nucleation in a two-component incompressible fluid. J Chem Phys 31: 688-699.
29. Carter W, Taylor J, Cahn J (1997) Variational methods for microstructural-evolution theories. J Min Metals Mater Soc 49: 30-36.
30. Cho S, Ha SH (2008) Isogeometric shape design optimization: exact geometry and enhanced sensitivity. Struct Multidiscip Optim 38: 53-70.
31. Choi JS, Yamada T, Izui K, Nishiwaki S, Yoo J (2011) Topology optimization using a reaction-diffusion equation. Comput Methods Appl Mech Eng 200: 2407-2420.
32. Chung J, Hulbert GM (1993) A time integration algorithm for structural dynamics with improved numerical dissipation: the generalized α-method. J Appl Mech 60: 371-375.
33. Ciarlet P (1978) The finite element method for elliptic problems. North-Holland, Amsterdam.
34. Cottrell JA, Hughes TJR, Reali A (2007) Studies of refinement and continuity in isogeometric structural analysis. Comput Methods Appl Mech Eng 196: 4160-4183.
35. Cottrell JA, Hughes TJR, Bazilevs Y (2009) Isogeometric analysis; towards integration of CAD and FEA. Wiley, New York.
36. Dal Maso G (1993) An introduction to Γ-convergence. Birkhäuser, Boston.
37. de Gournay F, Allaire G, Jouve F (2008) Shape and topology optimization of the robust compliance via the level set method. Control Optim Calc Var 14: 43-70.
38. Duysinx P, Bendsøe MP (1998) Topology optimization of continuum structures with local stress constraints. Int J Numer Methods Eng 43: 1453-1478.
39. Elliott CM (1989) The Cahn-Hilliard model for the kinetics of phase separation. In: Rodrigues JF (ed) Mathematical models for phase change problems. Internat ser numer math, vol 88. Birkhäuser, Basel, pp 35-73.
40. Elliott CM, Garcke H (1996) On the Cahn-Hilliard equation with degenerate mobility. SIAM J Math Anal 27: 404-423.
41. Evans JA, Bazilevs Y, Babuška I, Hughes TJR (2009) n-widths, supinfs, and optimality ratios for the k-version of the isogeometric finite element method. Comput Methods Appl Mech Eng 198: 1726-1741.
42. Fife PC (2000) Models for phase separation and their mathematics. Electron J Differ Equ 48: 1-26.
43. Frieboes HB, Lowengrub JS, Wise S, Zheng X, Macklin P, Bearer EL, Cristini V (2007) Computer simulation of glioma growth and morphology. NeuroImage 37: 59-70.
44. Gain AL, Paulino GH (2012) Phase-field based topology optimization with polygonal elements: a finite volume approach for the evolution equation. J Struct Multidiscip Optim. doi: 10. 1007/s00158-012-0781-9.
45. Garcke H (2005) On a Cahn-Hilliard model for phase separation with elastic misfit. Ann Inst Henri Poincaré 22: 165-185.
46. Garcke H, Weikard U (2005) Numerical approximation of the Cahn-Larché equation. Numer Math 100: 639-662.
47. Gee MW, Siefert CM, Hu JJ, Tuminaro RS, Sala MG (2006) ML 5. 0 smoothed aggregation user's guide. Sandia National Laboratories. SAND2006-2649.
48. Gersborg-Hansen A, Bendsøe MP, Sigmund O (2006) Topology optimization of heat conduction problems using the finite volume method. Struct Multidisc Optim 31: 251-259.
49. Gomez H, Hughes TJR (2011) Provably unconditionally stable, second-order time accurate, mixed variational methods for phase-field models. J Comput Phys 230: 5310-5327.
50. Gomez H, Calo VM, Bazilevs Y, Hughes TJR (2008) Isogeometric analysis of the Cahn-Hilliard phase-field model. Comput Methods Appl Mech Eng 197: 4333-4352.
51. Gomez H, Hughes TJR, Nogueira X, Calo VM (2010) Isogeometric analysis of the Navier-Stokes-Korteweg equations. Comput Methods Appl Mech Eng 199: 1828-1840.
52. Guest JK (2009) Topology optimization with multiple phase projection. Comput Methods Appl Mech Eng 199: 123-135.
53. Ha SH, Choi KK, Cho S (2010) Numerical method for shape optimization using T-spline based isogeometric method. Struct Multidiscip Optim 42: 417-428.
54. Haber RB, Jog CS, Bendsøe MP (1996) A new approach to variable-topology shape design using a constraint on the perimeter. Struct Optim 11: 1-12.
55. Haftka RT, Grandhi RV (1986) Structural shape optimization-a survey. Comput Methods Appl Mech Eng 57: 91-106.
56. Heroux M, Bartlett R, Hoekstra VHR, Hu J, Kolda T, Lehoucq R, Long K, Pawlowski R, Phipps E, Salinger A, Thornquist H, Tuminaro R, Willenbring J, Williams A (2003) An overview of trilinos. Sandia National Laboratories. SAND2003-2927.
57. Hughes TJR (2000) The finite element method: linear static and dynamic finite element analysis. Dover Publications, Mineola.
58. Hughes TJR, Cottrell JA, Bazilevs Y (2005) Isogeometric analysis: CAD, finite elements, NURBS, exact geometry and mesh refinement. Comput Methods Appl Mech Eng 194: 4135-4195.
59. Hughes TJR, Reali A, Sangalli G (2010) Efficient quadrature for NURBS-based isogeometric analysis. Comput Methods Appl Mech Eng 199: 301-313.
60. Jansen KE, Whiting CH, Hulbert GM (2000) A generalized-α method for integrating the filtered Navier-Stokes equations with a stabilized finite element method. Comput Methods Appl Mech Eng 190: 305-319.
61. Jensen JS, Sigmund O (2005) Topology optimization of photonic crystal structures: a high-bandwidth low-loss T-junction waveguide. J Opt Soc Am B 22: 1191-1198.
62. Jog CS, Haber RB (1996) Stability of finite element models for distributed-parameter optimization and topology design. Comput Methods Appl Mech Eng 130: 203-226.
63. Kohn RV, Strang G (1986) Optimal design and relaxation of variational problems. II. Commun Pure Appl Math 39: 139-182.
64. Kumar AV, Parthasarathy A (2011) Topology optimization using B-spline finite elements. Struct Multidiscip Optim 44: 471-481.
65. Larché FC, Cahn JW (1978) Thermochemical equilibrium of multiphase solids under stress. Acta Metall 26: 1579-1589.
66. Lee S, Kwak BM, Kim IY (2007) Smooth boundary topology optimization using B-spline and hole generation. Int J CAD/CAM 7.
67. Lipton R (1994) A saddle-point theorem with application to structural optimization. J Optim Theory Appl 81: 549-568.
68. Maute K (1998) Topologie- und formoptimierung von dünnwandigen tragwerken. PhD Thesis, University of Stuttgart.
69. Maute K, Ramm E (1997) Adaptive topology optimization of shell structures. AIAA J 35: 1767-1773.
70. Michell AGM (1904) The limit of economy of material in frame structures. Philos Mag 8: 589-597.
71. Miehe C, Welschinger F, Hofacker M (2010) Thermodynamically consistent phase-field models of fracture: variational principles and multi-field FE implementations. Int J Numer Methods Eng 83: 1273-1311.
72. Mlejnek H, Schirrmacher R (1993) An engineering approach to optimal material distribution and shape finding. Comput Methods Appl Mech Eng 106: 1-26.
73. --convergenza. Boll Unione Mat Ital, B 5(14): 285-299.
74. Mohammadi B, Pironneau O (2010) Applied shape optimization for fluids. Oxford University Press, Oxford.
75. Nagy AP, Abdalla NM, Gürdal Z (2010) Isogeometric sizing and shape optimisation of beam structures. Comput Methods Appl Mech Eng 199: 1216-1230.
76. Norato JA, Bendsøe MP, Haber RB, Tortorelli DA (2007) A topological derivative method for topology optimization. Struct Multidisc Optim 33: 375-386.
77. Oden JT, Hawkins A, Prudhomme S (2010) General diffuse-interface theories and an approach to predictive tumor growth modeling. Math Models Methods Appl Sci 20: 477-517.
78. Onuki A (1989) Ginzburg-Landau approach to elastic effects in the phase separation of solids. J Phys Soc Jpn 58: 965-985.
79. Osher S, Sethian JA (1988) Front propagating with curvature dependent speed: algorithms based on Hamilton-Jacobi formulations. J Comput Phys 78: 12-49.
80. París J, Navarrina F, Colominas I, Casteleiro M (2009) Topology optimization of structures with local and global stress constraints. Struct Multidiscip Optim 39: 419-437.
81. Park KS, Youn SK (2008) Topology optimization of shell structures using adaptive inner-front (AIF) level set method. Struct Multidiscip Optim 36: 43-58.
82. Pedersen NL (2002) Topology optimization of laminated plates with prestress. Comput Struct 80: 559-570.
83. Petersson J (1999) Some convergence results in perimeter-controlled topology optimization. Comput Methods Appl Mech Eng 171: 123-140.
84. Petersson J, Sigmund O (1998) Slope constrained topology optimization. Int J Numer Methods Eng 41: 1417-1434.
85. Piegl L, Tiller W (1997) The NURBSbBook. Springer, New York.
86. Poulsen TA (2003) A new scheme for imposing a minimum length scale on topology optimization. Int J Numer Methods Eng 57: 741-760.
87. Quarteroni A, Valli A (1994) Numerical approximation of partial differential equations. Springer, Berlin.
88. Quarteroni A, Sacco R, Saleri F (2007) Numerical mathematics. Springer, Berlin.
89. Ramm E, Maute K, Schwarz S (1998) Adaptive topology and shape optimization. In: Idelsohn S, Oñate E, Dvorkin E (eds) Computational mechanics, new trends and applications. CIMNE, Barcelona.
90. Rozvany GIN (2001) Aims, scope, methods, history and unified terminology of computer aided topology optimization in structural mechanics. Struct Multidiscip Optim 21: 90-108.
91. Rozvany GIN, Birker T (1994) On singular topologies in exact layout optimization. Struct Optim 8: 228-235.
92. Rybka P, Hoffmann KH (1999) Convergence of solutions to Cahn-Hilliard equation. Commun Partial Differ Equ 24: 1055-1077.
93. Schwarz S, Maute K, Ramm E (2001) Topology and shape optimization for elastoplastic structural response. Comput Methods Appl Mech Eng 190: 2135-2155.
94. Sederberg TW, Zhengs JM, Bakenov JM, Nasri A (2003) T-splines and T-NURCCSs. ACM Trans Graph 22: 477-484.
95. Seo YD, Kim HJ, Youn SK (2010) Isogeometric topology optimization using trimmed spline surfaces. Comput Methods Appl Mech Eng 199: 3270-3296.
96. Seo YD, Kim HJ, Youn SK (2010) Shape optimization and its extension to topological design based on isogeometric analysis. Int J Solids Struct 47: 1618-1640.
97. Sigmund O (1997) On the design of compliant mechanisms using topology optimization. Mech Struct Mach 25: 493-524.
98. Sigmund O (2001) Design of multiphysics actuators using topology optimization-part I: one-material structures. Comput Methods Appl Mech Eng 190: 6577-6604.
99. Sigmund O (2004) Morphology-based black and white filters for topology optimization. Struct Multidiscip Optim 61: 238-254.
100. Sigmund O, Petersson J (1998) Numerical instabilities in topology optimization: a survey on procedures dealing with checkerboards, mesh-dependencies and local minima. Struct Optim 16: 68-75.
101. Sokolowski J, Zolésio JP (2003) Introduction to shape optimization: shape sensitivity analysis. Springer, Berlin.
102. Spatschek R, Müller-Gugenberger C, Brener E, Nestler B (2007) Phase field modeling of fracture and stress-induced phase transitions. Phys Rev E 75: 066111.
103. Stolpe M, Svanberg K (2001) On the trajectories of penalization methods for topology optimization. Struct Multidiscip Optim 21: 128-139.
104. Stolpe M, Svanberg K (2003) Modeling topology optimization problems as linear mixed 0-1 programs. Int J Numer Methods Eng 57: 723-739.
105. Svanberg K (1987) The method of moving asymptotes-a new method for structural optimization. Int J Numer Methods Eng 24: 359-373.
106. Svanberg K (2002) A class of globally convergent optimization methods based on conservative convex separable approximations. SIAM J Optim 12: 555-573.
107. Takezawa A, Nishiwaki S, Kitamura M (2010) Shape and topology optimization based on the phase field method and sensitivity analysis. J Comput Phys 229: 2697-2718.
108. van der Zee KG, Oden JT, Prudhomme S, Hawkins AJ (2011) Goal-oriented error estimation for Cahn-Hilliard models of binary phase transition. Numer Methods Partial Differ Equ 27: 160-196.
109. Vese LA, Chan TF (2002) A multiphase level set framework for image segmentation using the Mumford and Shah model. Int J Comput Vis 50: 271-293.
110. Wall WA, Frenzel MA, Cyron C (2008) Isogeometric structural shape optimization. Comput Methods Appl Mech Eng 197: 2976-2988.
111. Wang MY, Zhou S (2004) Phase field: a variational method for structural topology optimization. Comput Model Eng Sci 6: 227-246.
112. Wang D, Zhang WH, Jiang JS (2002) Combined shape and sizing optimization of truss structures. Comput Mech 29: 307-312.
113. Wang MY, Wang X, Guo D (2003) A level set method for structural topology optimization. Comput Methods Appl Mech Eng 192: 227-246.
114. Weiss D (2010) Feature-based spline optimization in CAD; a step towards geometry-based structure creation. Struct Multidiscip Optim 42: 619-631.
115. Wu W, Yang DZ, Huang YY, Qi M, Wang WQ (2008) Topology optimization of a novel stent platform with drug reservoirs. Med Eng Phys 30: 1177-1185.
116. Xie YM, Steven GP (1997) Evolutionary structural optimization. Springer, London.
117. Yamada T, Izui K, Nishiwaki S, Takezawa A (2010) A topology optimization method based on the level set method incorporating a fictitious interface energy. Comput Methods Appl Mech Eng 199: 2876-2891.
118. Young V, Querin OM, Steven GP, Xie YM (1999) 3D and multiple load case bi-directional evolutionary optimization (BESO). Struct Optim 18: 183-192.
119. Zhou S, Wang MY (2006) 3D Multi-material structural topology optimization with the generalized Cahn-Hilliard equations. Comput Model Eng Sci 16: 69-82.
120. Zhou S, Wang MY (2007) Multimaterial structural topology optimization with a generalized Cahn-Hilliard model of multiphase transition. Struct Multidiscip Optim 33: 89-111.