Numerical analysis of a relaxed variational model of hysteresis in two-phase solids
ESAIM: Mathematical Modelling and Numerical Analysis - Modélisation Mathématique et Analyse Numérique, Volume 35 (2001) no. 5, p. 865-878

$\phantom{\rule{-0.166667em}{0ex}}$This paper presents the numerical analysis for a variational formulation of rate-independent phase transformations in elastic solids due to Mielke et al. The new model itself suggests an implicit time-discretization which is combined with the finite element method in space. A priori error estimates are established for the quasioptimal spatial approximation of the stress field within one time-step. A posteriori error estimates motivate an adaptive mesh-refining algorithm for efficient discretization. The proposed scheme enables numerical simulations which show that the model allows for hysteresis.

Classification:  65N30,  73C05
Keywords: variational problems, phase transitions, elasticity, hysteresis, a priori error estimates, a posteriori error estimates, adaptive algorithms, non-convex minimization, microstructure
@article{M2AN_2001__35_5_865_0,
author = {Carstensen, Carsten and Plech\'a\v c, Petr},
title = {Numerical analysis of a relaxed variational model of hysteresis in two-phase solids},
journal = {ESAIM: Mathematical Modelling and Numerical Analysis - Mod\'elisation Math\'ematique et Analyse Num\'erique},
publisher = {EDP-Sciences},
volume = {35},
number = {5},
year = {2001},
pages = {865-878},
zbl = {1007.74062},
mrnumber = {1866271},
language = {en},
url = {http://www.numdam.org/item/M2AN_2001__35_5_865_0}
}

Carstensen, Carsten; Plecháč, Petr. Numerical analysis of a relaxed variational model of hysteresis in two-phase solids. ESAIM: Mathematical Modelling and Numerical Analysis - Modélisation Mathématique et Analyse Numérique, Volume 35 (2001) no. 5, pp. 865-878. http://www.numdam.org/item/M2AN_2001__35_5_865_0/

[1] J.M. Ball and R.D. James, Fine phase mixtures as minimisers of energy. Arch. Rational Mech. Anal. 100 (1987) 13-52. | Zbl 0629.49020

[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. | Zbl 0758.73009

[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. | Zbl 0984.49021

[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). | MR 1278258 | Zbl 0804.65101

[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. | Zbl 0973.65091

[6] C. Carstensen and S. A. Funken, Fully reliable localised error control in the FEM. SIAM J. Sci. Comput. 21 (2000) 1465-1484. | Zbl 0956.65099

[7] C. Carstensen and S. Müller, Local stress regularity in scalar non-convex variational problems. In preparation. | Zbl 1012.49027

[8] C. Carstensen and P. Plecháč, Numerical solution of the scalar double-well problem allowing microstructure. Math. Comp. 66 (1997) 997-1026. | Zbl 0870.65055

[9] C. Carstensen and Petr Plecháč, Numerical analysis of compatible phase transitions in elastic solids. SIAM J. Numer. Anal. 37 (2000) 2061-2081. | Zbl 1049.74062

[10] P.G. Ciarlet, The Finite Element Method for Elliptic Problems. North-Holland Publishing Company, Amsterdam, New York, Oxford (1978). | MR 520174 | Zbl 0383.65058

[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. | Zbl 0969.74040

[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. | Zbl 0829.65122

[13] I. Fonseca and S. Müller, A-quasiconvexity, lower semicontinuity and Young measures. SIAM J. Math. Anal. 30 (1999) 1355-1390. | Zbl 0940.49014

[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. | Zbl 0591.73119

[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. | Zbl 0990.74048

[17] R.V. Kohn, The relaxation of a double-well energy. Contin. Mech. Thermodyn. 3 (1991) 193-236. | Zbl 0825.73029

[18] M. Luskin, On the computation of crystalline microstructure, in Acta Numerica, A. Iserles, Ed., Cambridge University Press, Cambridge (1996) 191-257. | Zbl 0867.65033

[19] 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.

[20] 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. | Zbl 1012.74054

[21] A. L. Roitburd, Martensitic transformation as a typical phase transformation in solids, in Solid State Physics 33, Academic Press, New York (1978) 317-390.

[22] 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. | Zbl 0947.74019

[23] 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). | Zbl 0853.65108