Relating phase field and sharp interface approaches to structural topology optimization

 Access to full text

1. [1] H. Abels, H. Garcke and G. Grün, Thermodynamically consistent, frame indifferent diffuse interface models for incompressible two-phase flows with different densities. Math. Models Methods Appl. Sci. 22 (2012) 1150013. Zbl1242.76342MR2890451
2. [2] G. Allaire, Optimization by the Homogenization Method. Springer, Berlin (2002). Zbl0990.35001MR1859696
3. [3] G. Allaire, F. Jouve and A.-M. Toader, Structural optimization using sensitivity analysis and a level set method. J. Comput. Phys.194 (2004) 363–393. Zbl1136.74368MR2033390
4. [4] S. Baldo, Minimal interface criterion for phase transitions in mixtures of Cahn−Hillard fluids. Ann. Inst. Henri Poincaré 7 (1990) 67–90. Zbl0702.49009MR1051228
5. [5] J.W. Barrett, H. Garcke and R. Nürnberg, On sharp interface limits of Allen−Cahn/Cahn−Hilliard variational inequalities. Discrete Contin. Dyn. Syst. Ser. S1 (2008) 1–14. Zbl1165.35406MR2375577
6. [6] J.W. Barrett, R. Nürnberg and V. Styles, Finite Element approximation of a phase field model for void electromigration. SIAM J. Numer. Anal.46 (2004) 738–772. Zbl1076.78012MR2084234
7. [7] M.P. Bendsoe and O. Sigmund, Topology Optimization. Springer, Berlin (2003). Zbl1059.74001MR2008524
8. [8] L. Blank, H. Garcke, L. Sarbu and V. Styles, Non-local Allen-Cahn systems: analysis and a primal dual active set method. IMA J. Numer. Anal.33 (2013) 1126–1155. Zbl1279.65087MR3119711
9. [9] L. Blank, H. Garcke, L. Sarbu, T. Srisupattarawanit, V. Styles and A. Voigt, Phase-field approaches to structural topology optimization. Constrained Optim. Opt. Control for Partial Differ. Eqs., edited by G. Leugering, S. Engell, A. Griewank, M. Hinze, R. Rannacher, V. Schulz, M. Ulbrich, S. Ulbrich. In vol. 160, Int. Ser. Numer. Math. (2012) 245–255. MR3060477
10. [10] J.F. Blowey and C.M. Elliott, Curvature dependent phase boundary motion and parabolic double obstacle problems. IMA J. Math. Appl.47 (1993) 19–60. Zbl0794.35092MR1246337
11. [11] B. Bourdin and A. Chambolle, Design-dependent loads in topology optimization. ESAIM: COCV 9 (2003) 19–48. Zbl1066.49029MR1957089
12. [12] B. Bourdin and A. Chambolle, The phase-field method in optimal design, in vol. 137 of IUTAM Symposium on Topological Design Optimization of Structures, Machines and Materials (2006) 207–215. 
13. [13] L. Bronsard, H. Garcke and B. Stoth, A multi-phase Mullins-Sekerka system: matched asymptotic expansions and an implicit time discretization for the geometric evolution problem. SIAM J. Appl. Math.60 (1999) 295–315. Zbl0942.35095MR1632810
14. [14] L. Bronsard, C. Gui and M. Schatzman, A three layered minimizer in R2 for a variational problem with a symmetric three-well potential. Commun. Pure Appl. Math.47 (1996) 677–715. Zbl0855.35035MR1387190
15. [15] L. Bronsard and R. Reitich, On singular three-phase boundary motion and the singular limit of a vector-valued Ginzburg-Landau equation. Arch. Rat. Mech. Anal.124 (1993) 355–379. Zbl0785.76085MR1240580
16. [16] M. Burger and R. Stainko, Phase-field relaxation of topology optimization with local stress constraints. SIAM J. Control Optim.45 (2006) 1447–1466. Zbl1116.74053MR2257229
17. [17] M. Burger, A framework for the construction of level set methods for shape optimization and reconstruction. Interfaces Free Bound.5 (2003) 301–332. Zbl1081.35134MR1998617
18. [18] M. Burger, B. Hackl and W. Ring, Incorporating topological derivatives into level set methods. J. Comput. Phys.194 (2004) 344–362. Zbl1044.65053MR2033389
19. [19] J.W. Cahn and J.E. Hilliard, Free energy of a nonuniform system. I. Interfacial free energy. J. Chem. Phys. 28 (1958) 258–267. 
20. [20] L.Q. Chen, Phase-field models for microstructure evolution. Ann. Rev. Mater. Research32 (2002) 113–140. 
21. [21] P.G. Ciarlet, Mathematical Elasticity, Three Dimensional Elasticity, vol. 1. Elsevier (1988). Zbl0648.73014MR936420
22. [22] T.A. Davis, UMFPACK Version 5.2.0 User Guide. University of Florida (2007). 
23. [23] K. Deckelnick, G. Dziuk and C.M. Elliott, Computation of geometric pdes and mean curvature flow. Acta Numerica (2005) 139–232. Zbl1113.65097MR2168343
24. [24] L. Dedè, M.J. Borden, T.J.R. Hughes, Isogeometric analysis for topology optimization with a phase field model, Arch. Comput. Methods Eng.19 (2012) 427–465. MR2969359
25. [25] C.M. Elliott and S. Luckhaus, A generalised diffusion equation for phase separation of a multi-component mixture with interfacial free energy. SFB256, Preprint 195, University Bonn (1999). 
26. [26] P.C. Fife, Dynamics of internal layers and diffusive interfaces. Vol. 53 of CBMS-NSF Regional Conf. Ser. Appl. Math. SIAM, Philadelphia (1988). Zbl0684.35001MR981594
27. [27] P.C. Fife and O. Penrose, Interfacial dynamics for thermodynamically consistent phase-field models with nonconserved order parameter. EJDE (1995) 1–49. Zbl0851.35059MR1361512
28. [28] P. Fratzl, O. Penrose and J.L. Lebowitz, Modeling of phase separation in alloys with coherent elastic misfit. J. Statist. Phys. 95 (1999). Zbl0952.74052MR1712452
29. [29] H. Garcke, The Γ-limit of the Ginzburg-Landau energy in an elastic medium. AMSA18 (2008) 345–379. Zbl1193.49054MR2489134
30. [30] H. Garcke, On Cahn−Hilliard systems with elasticity. Proc. Roy. Soc. Edinburgh Sect. A 133 (2003) 307–331. Zbl1130.74037MR1969816
31. [31] H. Garcke, B. Nestler and B. Stoth, On anisotropic order parameter models for multi-phase systems and their sharp interface limits. Phys. D115 (1998) 87–108. Zbl0936.82010MR1616772
32. [32] H. Garcke, B. Nestler and B. Stoth, A multi phase field concept: numerical simulations for moving phase boundaries and multiple junctions. SIAM J. Appl. Math.60 (1999) 295–315. Zbl0942.35095MR1740846
33. [33] H. Garcke and A. Novick-Cohen, A singular limit for a system of degenerate Cahn−Hilliard equations. Adv. Differ. Eqs. 5 (2000) 401–434. Zbl0988.35019MR1750107
34. [34] H. Garcke, R. Nürnberg, V. Styles, Stress and diffusion induced interface motion: Modelling and numerical simulations. Eur. J. Appl. Math.18 (2007) 631–657. Zbl1128.74004MR2381562
35. [35] H. Garcke and B. Stinner, Second order phase field asymptotics for multicomponent systems. Interfaces Free Boundaries8 (2006) 131–157. Zbl1106.35116MR2256839
36. [36] M. Giaquinta and L. Martinazzi, An introduction to the regularity theory for elliptic systems, harmonic maps and minimal graphs. Edizioni della normale, Scuola Normale Superiore Pisa (2005). Zbl1262.35001MR2192611
37. [37] M.E. Gurtin. An introduction to continuum mechanics. Math. Sci. Engrg. 158 (2003). Zbl0559.73001
38. [38] I. Hlavacek and J. Necas, On inequalities of Korn’s type, I. Boundary value problems for elliptic systems of partial differential equations. Arch. Rat. Mech. Anal. 36 (1970) 312–334. Zbl0193.39001MR252844
39. [39] F.C. Larché and J.W. Cahn, The effect of self-stress on diffusion in solids. Acta Metall.30 (1982) 1835–1845. 
40. [40] L. Modica, The gradient theory of phase transitions and minimal interface criterion. Arch. Rat. Mech. Anal.98 (1987) 123–142. Zbl0616.76004MR866718
41. [41] F. Murat and S. Simon, Etudes des problèmes d’optimal design. Lect. Notes Comput. Sci. Springer Verlag, Berlin 41 (1976) 54–62. Zbl0334.49013
42. [42] A. Novick-Cohen and L. Peres Hari, Geometric motion for a degenerate Allen−Cahn/Cahn−Hillard system: The partial wetting case. Physica D 209 (2005) 205–235. Zbl1094.35065MR2167453
43. [43] O.A. Oleinik, A.S. Shamaev and G.A. Yosifian, Mathematical problems in elasticity and homogenization. In vol. 26 of Studies Math. Appl. (1992) 1–398. Zbl0768.73003MR1195131
44. [44] S.J. Osher and F. Santosa, Level set methods for optimization problems involving geometry and constraints. I. Frequencies of a two-density inhomogeneous drum. J. Comput. Phys. 171 (2011) 272–288. Zbl1056.74061MR1843648
45. [45] S.J. Osher and J.A. Sethian, Fronts propagating with curvature-dependent speed: algorithms based on Hamilton-Jacobi formulations. J. Comput. Phys.79 (1988) 12–49. Zbl0659.65132MR965860
46. [46] N. Owen, J. Rubinstein and P. Sternberg, Minimizers and gradient flows for singularly perturbed bi-stable potentials with a Dirichlet condition. Roc. Roy. Soc. London A429 (1990) 505–532. Zbl0722.49021MR1057968
47. [47] J. Petersson, Some convergence results in perimeter-controlled topology optimization. Comput. Meth. Appl. Mech. Eng.171 (1999) 123–140. Zbl0947.74050MR1684859
48. [48] O. Pironneau, Optimal Shape Design for Elliptic Systems. Springer-Verlag, New York (1984). Zbl0496.93029MR725856
49. [49] J. Rubinstein and P. Sternberg, Nonlocal reaction-diffusion equations and nucleation. IMA J. Appl. Math.48 (1992) 249–264. Zbl0763.35051MR1167735
50. [50] P. Penzler, M. Rumpf and B. Wirth, A phase-field model for compliance shape optimization in nonlinear elasticity. ESAIM: COCV 18 (2012) 229–258. Zbl1251.49054MR2887934
51. [51] A. Schmidt and K.G. Siebert, Design and adaptive finite element software. The finite element toolbox ALBERTA. In vol. 42 of Lect. Notes Comput. Sci. Eng. Springer-Verlag, Berlin (2005). Zbl1068.65138MR2127659
52. [52] O. Sigmund, J. Petersson, Numerical instabilities in topology optimization: A survey on procedures dealing with checkerboards, mesh-dependencies and local minima. Struct. Multidisc Optim.16 (1998) 68–75. 
53. [53] J. Simon, Differentiation with respect to domain boundary value problems. Numer. Funct. Anal. Optim.2 (1980) 649–687. Zbl0471.35077MR619172
54. [54] J. Sokolowski and J.P. Zolesio, Introduction to shape optimization: shape sensitivity analysis, vol. 10. Springer Ser. Comput. Math. Springer, Berlin (1992). Zbl0761.73003MR1215733
55. [55] A. Takezawa, S. Nishiwaki and M. Kitamura, Shape and topology optimization based on the phase field method and sensitivity analysis. J. Comput. Phys.229 (2010) 2697–2718. Zbl1185.65109MR2586210
56. [56] F. Tröltzsch, Optimal control of partial differential equations: theory, methods and applications, vol. 112. Graduate Studies Math. (2010). Zbl1195.49001MR2583281
57. [57] J.D. van der Waals, The thermodynamic theory of capillarity under the hypothesis of a continuous variation of density (in Dutch), Vol. 1. Verhaendel. Kronik. Akad. Weten. Amsterdam (1983); Engl. translation by J.S. Rowlinson. J. Stat. Phys. 20 (1979) 197–244. Zbl1245.82006MR523642
58. [58] M. Wallin and M. Ristinmaa, Howard’s algorithm in a phase-field topology optimization approach. Int. J. Numer. Meth. Eng.94 (2013) 43–59. MR3040512
59. [59] M.Y. Wang and S.W. Zhou, Phase field: A variational method for structural topology optimization. Comput. Model. Eng. Sci.6 (2004) 547–566. Zbl1152.74382MR2108636
60. [60] M.Y. Wang and S.W. Zhou, Multimaterial structural topology optimization with a generalized Cahn−Hilliard model of multiphase transition. Struct. Multidisc. Optim. 33 (2007) 89-111. Zbl1245.74077MR2291576
61. [61] M.Y. Wang and S.W. Zhou, 3D multi-material structural topology optimization with the generalized Cahn−Hilliard equations. Comput. Model. Eng. Sci. 16 (2006) 83–102. 
62. [62] E. Zeidler, Nonlinear Functional Analysis and its Applications, I: Fixed-point theorems. Springer-Verlag (1986). Zbl0583.47050MR816732
63. [63] E. Zeidler, Nonlinear Functional Analysis and its Applications, IV. Applications Math. Phys. Springer Verlag (1988). Zbl0648.47036MR932255
64. [64] E. Zeidler, Nonlinear Functional Analysis and its Applications, II/B. Nonlinear Monotone Operators. Springer Verlag (1990). Zbl0684.47029MR1033498
65. [65] J. Zowe and S. Kurcyusz, Regularity and stability for the mathematical programming problem in Banach spaces. Appl. Math. Optim.5 (1979) 49–62. Zbl0401.90104MR526427

1. Andrzej Myśliński, Konrad Koniarski, Hybrid level set phase field method for topology optimization of contact problems