Free access
Issue
ESAIM: COCV
Volume 9, March 2003
Page(s) 19 - 48
DOI http://dx.doi.org/10.1051/cocv:2002070
Published online 15 September 2003
  1. G. Alberti, Variational models for phase transitions, an approach via Formula -convergence, in Calculus of Variations and Partial Differential Equations, edited by G. Buttazzo et al. Springer-Verlag (2000) 95-114.
  2. G. Allaire, É. Bonnetier, G.A. Francfort and F. Jouve, Shape optimization by the homogenization method. Numer. Math. 76 (1997) 27-68. [CrossRef] [MathSciNet]
  3. G. Allaire, Shape optimization by the homogenization method. Springer-Verlag, New York (2002).
  4. L. Ambrosio and G. Buttazzo, An optimal design problem with perimeter penalization. Calc. Var. Partial Differential Equations 1 (1993) 55-69. [CrossRef] [MathSciNet]
  5. H. Attouch, Variational convergence for functions and operators. Applicable Mathematics Series. Pitman (Advanced Publishing Program), Boston, Mass.-London (1984).
  6. S. Baldo, Minimal interface criterion for phase transitions in mixtures of Cahn-Hilliard fluids. Ann. Inst. H. Poincaré Anal. Non Linéaire 7 (1990) 67-90.
  7. A.C. Barroso and I. Fonseca, Anisotropic singular perturbations - the vectorial case. Proc. Roy. Soc. Edinburgh Sect. A 124 (1994) 527-571. [MathSciNet]
  8. M.P. Bendsøe, Optimization of Structural Topology, Shape and Material. Springer Verlag, Berlin Heidelberg (1995).
  9. M.P. Bendsøe and O. Sigmund, Material interpolation schemes in topology optimization. Arch. Appl. Mech. 69 (1999) 635-654. [CrossRef]
  10. É. Bonnetier and A. Chambolle, Computing the equilibrium configuration of epitaxially strained crystalline films. SIAM J. Appl. Math. 62 (2002) 1093-1121. [CrossRef] [MathSciNet]
  11. B. Bourdin and A. Chambolle, Implementation of an adaptive finite-element approximation of the Mumford-Shah functional. Numer. Math. 85 (2000) 609-646. [CrossRef] [MathSciNet]
  12. B. Bourdin, G.A. Francfort and J.-J. Marigo, Numerical experiments in revisited brittle fracture. J. Mech. Phys. Solids 48 (2000) 797-826. [CrossRef] [MathSciNet]
  13. J.W. Cahn and J.E. Hilliard, Free energy of a nonuniform system I - interfacial free energy. J. Chem. Phys. 28 (1958) 258-267. [CrossRef]
  14. A. Chambolle, Finite-differences discretizations of the Mumford-Shah functional. ESAIM: M2AN 33 (1999) 261-288. [CrossRef] [EDP Sciences]
  15. B.-C. Chen and N. Kikuchi, Topology optimization with design-dependent loads. Finite Elem. Anal. Des. 37 (2001) 57-70. [CrossRef]
  16. L.Q. Chen and J. Shen, Application of semi implicit Fourier-spectral method to phase field equations. Comput. Phys. Comm. 108 (1998) 147-158. [CrossRef]
  17. A. Cherkaev, Variational methods for structural optimization. Springer-Verlag, New York (2000).
  18. A. Cherkaev and R.V. Kohn, Topics in the mathematical modelling of composite materials. Birkhäuser Boston Inc., Boston, MA (1997).
  19. P.G. Ciarlet, Mathematical elasticity. Vol. I. North-Holland Publishing Co., Amsterdam (1988). Three-dimensional elasticity.
  20. G. Dal Maso, An introduction to Formula -convergence. Birkhäuser, Boston (1993).
  21. I. Ekeland and R. Témam, Convex analysis and variational problems. Society for Industrial and Applied Mathematics (SIAM), Philadelphia, PA, English Edition (1999). Translated from the French.
  22. L.C. Evans and R.F. Gariepy, Measure theory and fine properties of functions. CRC Press, Boca Raton, FL (1992).
  23. D. Eyre, Systems of Cahn-Hilliard equations. SIAM J. Appl. Math. 53 (1993) 1686-1712. [CrossRef] [MathSciNet]
  24. K.J. Falconer, The geometry of fractal sets. Cambridge University Press, Cambridge (1986).
  25. H. Federer, Geometric measure theory. Springer-Verlag, New York (1969).
  26. E. Giusti, Minimal surfaces and functions of bounded variation. Birkhäuser, Boston (1984).
  27. R.B. Haber, C.S. Jog and M.P. Bendsøe, A new approach to variable-topology shape design using a constraint on the perimeter. Struct. Optim. 11 (1996) 1-12. [CrossRef]
  28. V.B. Hammer and N. Olhoff, Topology optimization of continuum structures subjected to pressure loading. Struct. Multidisc. Optim. 19 (2000) 85-92. [CrossRef]
  29. R.V. Kohn and G. Strang, Optimal design and relaxation of variational problems I-III. Comm. Pure Appl. Math. 39 (1986) 113-137, 139-182, 353-377. [CrossRef] [MathSciNet]
  30. R.V. Kohn and G. Strang, Optimal design in elasticity and plasticity. Internat. J. Numer. Methods Engrg. 22 (1986) 183-188. [CrossRef] [MathSciNet]
  31. P.H. Leo, J.S Lowengrub and H.J. Jou, A diffuse interface model for microstructural evolution in elastically stressed solids. Acta Mater. 46 (1998) 2113-2130. [CrossRef]
  32. L. Modica and S. Mortola. Il limite nella Formula -convergenza di una famiglia di funzionali ellittici. Boll. Un. Mat. Ital. A (5) 14 (1977) 526-529.
  33. L. Modica and S. Mortola, Un esempio di Formula -convergenza. Boll. Un. Mat. Ital. B (5) 14 (1977) 285-299. [MathSciNet]
  34. M. Negri, The anisotropy introduced by the mesh in the finite element approximation of the Mumford-Shah functional. Numer. Funct. Anal. Optim. 20 (1999) 957-982. [CrossRef] [MathSciNet]
  35. R.H. Nochetto, S. Rovida, M. Paolini and C. Verdi, Variational approximation of the geometric motion of fronts, in Motion by mean curvature and related topics (Trento, 1992) de Gruyter, Berlin (1994) 124-149.
  36. 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 (2001) 272-288. [CrossRef] [MathSciNet]
  37. M. Paolini and C. Verdi, Asympto. and numerical analyses of the mean curvature flow with a space-dependent relaxation parameter. Asymptot. Anal. 5 (1992) 553-574.
  38. J.A. Sethian and A. Wiegmann, Structural boundary design via level set and immersed interface methods. J. Comput. Phys. 163 (2000) 489-528. [CrossRef] [MathSciNet]
  39. R. Temam, Problèmes mathématiques en plasticité. Gauthier-Villars, Paris (1983).
  40. W.P. Ziemer, Weakly Differentiable Functions. Springer-Verlag, Berlin (1989).