Numerical analysis of compatible phase transitions in elastic solids

SIAM Journal on Numerical Analysis

Volumn 37, Issue 6, 2000, Pages 2061-2081

  Carstensen, Carsten ^a   Plecháč, Petr ^b  

^a UNIVERSITY OF KIEL   (Germany)

^b UNIVERSITY OF DELAWARE   (United States)

Author keywords

A posteriori error estimates; A priori error estimates; Adaptive algorithms; Elasticity; Microstructure; Nonconvex minimization; Phase transitions; Variational problems

Indexed keywords

EID: 0000088421     PISSN: 00361429     EISSN: None     Source Type: Journal    
DOI: 10.1137/S0036142998337697     Document Type: Article

References (29)

1. G. ALLAIRE AND R. V. KOHN, Optimal design for minimum weight and compliance in plane stress using extremal microstructures, European J. Mech. A Solids, 12 (1993), pp. 839-878.
2. 6+x studied by powder diffraction and gas volumetry, Phys. C, 172 (1990), pp. 31-42.
3. J. BALL AND R. JAMES, Fine phase mixtures as minimisers of energy, Arch. Rational Mech. Anal, 100 (1987), pp. 13-52.
4. J. BALL AND R. JAMES, Proposed experimental tests of the theory of fine microstructure and the two-well problem, Phil. Trans. Roy. Soc. London Ser. A., 338 (1992), pp. 389-450.
5. K. BHATTACHARYA, personal communication.
6. S. BRENNER AND L. SCOTT, The Mathematical Theory of Finite Element Methods, Texts Appl. Math. 15, Springer-Verlag, New York, 1994.
7. C. CARSTENSEN, Weighted Clement-Type Interpolation and a Posteriori Analysis for FEM, Tech. Report, Christian-Albrechts-Universität zu Kiel, Berichtsreihe des Mathematischen Seminars Kiel 97-19, 1997.
8. C. CARSTENSEN AND S. A. FUNKEN, Constants in Clément-Interpolation Error and Residual Based a Posteriori Estimates in Finite Element Methods, Tech. Report, Christian-Albrechts-Universität zu Kiel. Berichtsreihe des Mathematischen Seminars Kiel 97-11, 1997.
9. C. CARSTENSEN AND S. A. FUNKEN, Fully reliable localized error control in the FEM, SIAM J. Sci. Comput., 21 (2000), pp. 1465-1484.
10. C. CARSTENSEN AND P. PLECHÁČ, Adaptive algorithms for scalar non-convex variational problems, Appl. Numer. Math., 26 (1998), pp. 203-216.
11. C. CARSTENSEN AND P. PLECHÁČ, Numerical solution of the scalar double-well problem allowing microstructure, Math. Comp., 66 (1997), pp. 997-1026.
12. C. CARSTENSEN AND R. VERFÜRTH, Edge residuals dominate a posteriori error estimates for low order finite element methods, SIAM J. Numer. Anal., 36 (1999), pp. 1571-1587.
13. B. DACAROGNA, Direct Methods in the Calculus of Variations, Appl. Math. Sci. 78, Springer-Verlag, Berlin, 1989.
14. K. ERIKSSON, D. ESTEP, P. HANSBO, AND C. JOHNSON, Introduction to adaptive methods for differential equations, Acta Numerica, (1995), pp. 105-158.
15. G. FRIESECKE, A necessary and sufficient condition for non-attainment and formation of microstructure almost everywhere in scalar variational problems, Proc. Roy. Soc. Edinburgh Sect. A, 124 (1994), pp. 437-471.
16. 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), pp. 107-127.
17. A. G. KHACHATURYAN, Theory of Structural Transformations in Solids, John Wiley & Sons, New York, 1983.
18. J. BALL, B. KIRCHHEIM, AND J. KRISTENSEN, Regularity of quasiconvex envelopes, Calc. Var. and Partial Differential Equations, to appear.
19. R. KOHN, The relaxation of a double-well energy, Contin. Mech. Thermodyn., 3 (1991), pp. 193-236.
20. B. LI AND M. LUSKIN, Finite element analysis of microstructure for the cubic to tetragonal transformation, SIAM J. Numer. Anal., 35 (1998), pp. 376-392.
21. B. LI AND M. LUSKIN, Nonconforming finite element approximation of crystalline microstructure, Math. Comp., 67 (1998), pp. 917-946.
22. M. LUSKIN, On the computation of crystalline microstructure, Acta Numerica, 5 (1996), pp. 191-257.
23. I. FONSECA AND S. MÜLLER, A-quasiconvexity, lower semicontinuity and Young measures, SIAM J. Math. Anal., 30 (1999), pp. 1355-1390.
24. S. NASH, Newton-type minimization via the Lánczos method, SIAM J. Numer. Anal., 21 (1984), pp. 770-788.
25. R. NICOLAIDES AND N. WALKINGTON, Strong convergence of numerical solutions to degenerate variational problems, Math. Comp., 64 (1995), pp. 117-127.
26. P. PEDREGAL, Parametrized Measures and Variational Principles, Progr. Nonlinear Differential Equations Appl. 30, Birkhäuser, Boston, 1997.
27. A. L. ROITBURD, Martensitic Transformation as a Typical Phase Transformation in Solids, Solid State Physics 33, Academic Press, New York, 1978, pp. 317-390.
28. G. SEREGIN, The regularity properties of solutions of variational problems in the theory of phase transitions in elastic, St. Petersburg Math. J., 4 (1996), pp. 979-1003.
29. R. VERFÜRTH, A Review of a Posteriori Error Estimation and Adaptive Mesh-Refinement Techniques, Wiley-Teubner, New York, 1996.