Numerical analysis of a relaxed variational model of hysteresis in two-phase solids

Mathematical Modelling and Numerical Analysis

Volumn 35, Issue 5, 2001, Pages 865-878

  Carstensen, Carsten ^a   Plecháč, Petr ^b  

^a VIENNA UNIVERSITY OF TECHNOLOGY   (Austria)

^b UNIVERSITY OF WARWICK   (United Kingdom)

Author keywords

A posteriori error estimates; A priori error estimates; Adaptive algorithms; Elasticity; Hysteresis; Microstructure; Non convex minimization; Phase transitions; Variational problems

Indexed keywords

EID: 0035617838     PISSN: 0764583X     EISSN: None     Source Type: Journal    
DOI: 10.1051/m2an:2001139     Document Type: Article

References (25)

1. J.M. Ball and R.D. James, Fine phase mixtures as minimisers of energy. Arch. Rational Mech. Anal. 100 (1987) 13-52.
2. J.M. Ball and R.D. James, Proposed experimental tests of the theory of fine microstructure and the two-well problem. Philos. Trans. Roy. Soc. London Ser. A 338 (1992) 389-450.
3. M. Bildhauer, M. Fuchs and G. Seregin, Local regularity of solutions of variational problems for the equilibrium configuration of an incompressible, multiphase elastic body. Nonlin. Diff. Equations Appl. 8 (2001) 53-81.
4. S.C. Brenner and L.R. Scott, The mathematical theory of finite element methods, in Texts in Applied Mathematics 15. Springer-Verlag, New York (1994).
5. C. Carstensen and S. A. Funken, Constants in Clément-interpolation error and residual based a posteriori estimates in finite element methods. East-West J. Numer. Math. 8 (2000) 153-175.
6. C. Carstensen and S. A. Funken, Fully reliable localised error control in the FEM. SIAM J. Sci. Comput. 21 (2000) 1465-1484.
7. C. Carstensen and S. Müller, Local stress regularity in scalar non-convex variational problems. In preparation.
8. C. Carstensen and P. Plecháč, Numerical solution of the scalar double-well problem allowing microstructure. Math. Comp. 66 (1997) 997-1026.
9. C. Carstensen and Petr Plecháč, Numerical analysis of compatible phase transitions in elastic solids. SIAM J. Numer. Anal. 37 (2000) 2061-2081.
10. P.G. Ciarlet, The Finite Element Method for Elliptic Problems. North-Holland Publishing Company, Amsterdam, New York, Oxford (1978).
11. H. Le Dret and A. Raoult, Variational convergence for nonlinear shell models with directors and related semicontinuity and relaxation results. Arch. Rational Mech. Anal. 154 (1999) 101-134.
12. K. Eriksson, D. Estep, P. Hansbo and C. Johnson, Introduction to adaptive methods for differential equations, in Acta Numerica, A. Iserles, Ed., Cambridge University Press, Cambridge (1995) 105-158.
13. I. Fonseca and S. Müller, A-quasiconvexity, lower semicontinuity and Young measures. SIAM J. Math. Anal. 30 (1999) 1355-1390.
14. J. Goodman, R.V. Kohn and L. Reyna, Numerical study of a relaxed variational problem from optimal design. Comput. Methods Appl. Mech. Engrg. 57 (1986) 107-127.
15. A. G. Khachaturyan, Theory of Structural Transformations in Solids. John Wiley & Sons, New York (1983).
16. M.S. Kuczma, A. Mielke and E. Stein, Modelling of hysteresis in two-phase systems. Solid Mechanics Conference (1999); Arch. Mech. 51 (1999) 693-715.
17. M.S. Kuczma, A. Mielke and E. Stein, Modelling of hysteresis in two-phase systems. Solid Mechanics Conference (1999); Arch. Mech. 51 (1999) 693-715.
18. R.V. Kohn, The relaxation of a double-well energy. Contin. Mech. Thermodyn. 3 (1991) 193-236.
19. M. Luskin, On the computation of crystalline microstructure, in Acta Numerica, A. Iserles, Ed., Cambridge University Press, Cambridge (1996) 191-257.
20. A. Mielke and F. Theil, A mathematical model for rate-independent phase transformations with hysteresis, in Workshop of Continuum Mechanics in Analysis and Engineering, H.-D. Alber, D. Bateau and R. Farwig, Eds. , Shaker-Verlag, Aachen (1999) 117-129.
21. A. Mielke, F. Theil and V. I. Levitas, A variational formulation of rate-independent phase transformations using an extremum principle. Submitted to Arch. Rational Mech. Anal.
22. A. L. Roitburd, Martensitic transformation as a typical phase transformation in solids, in Solid State Physics 33, Academic Press, New York (1978) 317-390.
23. G. A. Seregin, The regularity properties of solutions of variational problems in the theory of phase transitions in an elastic body. St. Petersbg. Math. J. 7 (1996) 979-1003, English translation from Algebra Anal. 7 (1995) 153-187.
24. G. A. Seregin, The regularity properties of solutions of variational problems in the theory of phase transitions in an elastic body. St. Petersbg. Math. J. 7 (1996) 979-1003, English translation from Algebra Anal. 7 (1995) 153-187.
25. R. Verfürth, A review of a posteriori error estimation and adaptive mesh-refinement techniques, in Wiley-Teubner Series Advances in Numerical Mathematics, John Wiley & Sons, Chichester; Teubner, Stuttgart (1996).