Effective relaxation for microstructure simulations: Algorithms and applications

Computer Methods in Applied Mechanics and Engineering

Volumn 193, Issue 48-51, 2004, Pages 5143-5175

  Bartels, S ^a   Carstensen, C ^b   Hackl, K ^c   Hoppe, U ^c  

^a University of Maryland   (United States)

^b HUMBOLDT UNIVERSITY   (Germany)

^c RUHR UNIVERSITY BOCHUM   (Germany)

Author keywords

Adaptive finite element methods; Computational microstructures; Multi scale problems; Phase transitions; Quasiconvexification; Relaxation; Stabilization

Indexed keywords

ALGORITHMS; BENCHMARKING; CONTINUUM MECHANICS; ELASTOPLASTICITY; NUMERICAL METHODS; PHASE TRANSITIONS;

CONVEXIFICATION; MUICROSTRUCTURE SIMULATIONS; RELAXATION THEORY;

MICROSTRUCTURE;

MICROSTRUCTURE;

EID: 8644252598     PISSN: 00457825     EISSN: None     Source Type: Journal    
DOI: 10.1016/j.cma.2003.12.065     Document Type: Article

References (31)

1. S. Aubry M. Fago M. Ortiz A constrained sequential-lamination algorithm for the simulation of sub-grid microstructure in martensitic materials Comput. Methods Appl. Mech. Engrg. 192 26-27 2003 2823-2843
2. J.M. Ball Convexity conditions and existence theorems in nonlinear elasticity Arch. Rational Mech. Anal. 63 4 1977 337-403
3. J.M. Ball Strict convexity, strong ellipticity, and regularity in the calculus of variations Math. Proc. Cambridge Philos. Soc. 87 3 1980 501-513
4. J.M. Ball, A version of the fundamental theorem for Young measures. in: Partial Differential Equations and Continuum Models of Phase Transitions, M. Rascle, D. Serre, M. Slemrod (Eds.), Lecture Notes in Physics, vol. 344, 1989, pp. 207-215
5. J.M. Ball R.D. James Fine phase mixtures as minimizers of energy Arch. Rational Mech. Anal. 100 1987 13-52
6. J.M. Ball B. Kirchheim J. Kristensen Regularity of quasiconvex envelopes Calc. Var. Partial Differ. Equat. 11 2000 333-359
7. S. Bartels C. Carstensen P. Plecháč A. Prohl Convergence for stabilisation of degenerately convex minimisation problems Interfaces and Free Boundaries 6 2 2004 253-269
8. S. Bartels, Reliable and efficient approximation of polyconvex envelopes, Preprints of the DFG Priority Program "Multiscale Problems" 76 (2003). Available from: www.mathematik.uni-stuttgart.de/~mehrskalen/
9. S. Bartels A. Prohl Multiscale resolution in the computation of crystalline microstructure Numer. Math. 96 4 2004 641-660
10. S.C. Brenner L.R. Scott The mathematical theory of finite element methods Texts in Applied Mathematics vol. 15 2002 Springer-Verlag New York p. xvi+361
11. x1 f(x, y, y′ )dx Trans. Am. Math. Soc. 7 2 1906 314-324
12. C. Carstensen All first-order averaging techniques for a posteriori finite element error control on unstructured grids are efficient and reliable Math. Comp. 73 247 2004 1153-1165
13. C. Carstensen G. Dolzmann An a priori error estimate for finite element discretizations in nonlinear elasticity for polyconvex materials under small loads Numer. Math. 97 1 2004 67-80
14. C. Carstensen K. Hackl On microstructures occurring in a model of finite-strain elastoplasticity involving a single slip-system ZAMM 80 2 2000 S421-S422
15. C. Carstensen K. Hackl A. Mielke Nonconvex potentials and microstructures in finite-strain plasticity Proc. R. Soc. Lond. A 458 2018 2002 299-317
16. C. Carstensen K. Jochimsen Adaptive finite element methods for microstructures? Numerical experiments for a 2-well benchmark Computing 71 2003 175-204
17. C. Carstensen R. Klose A posteriori finite element error control for the p-Laplace problem SIAM J. Sci. Comput. 25 3 2003 792-814
18. C. Carstensen M.O. Rieger Young-Measure approximations for elastodynamics with non-monotone stress-strain relations M2AN Math. Model. Numer. Anal. 38 3 2004 397-418
19. C. Carstensen S. Müller Local stress regularity in scalar non-convex variational problem SIAM J. Math. Anal. 34 2 2002 495-509
20. C. Carstensen P. Plecháč Numerical solution of the scalar double-well problem allowing microstructure Math. Comput. 66 219 1997 997-1026
21. C. Carstensen P. Plecháč Numerical analysis of compatible phase transitions in elastic solids SIAM J. Numer. Anal. 37 6 2000 2061-2081
22. C. Carstensen P. Plecháč Numerical analysis of a relaxed variational model of hysteresis in two-phase solids M2AN Math. Model. Numer. Anal. 35 5 2001 865-878
23. M. Chipot, S. Müller, Sharp energy estimates for finite element approximations of non-convex problems. Variations of domain and free-boundary problems in solid mechanics (Paris, 1997), Solid Mech. Appl., 66, Kluwer Acad. Publ., Dordrecht, 1991
24. B. Dacorogna Direct methods in the calculus of variations 1989 Springer-Verlag Berlin
25. G. Dolzmann Numerical computation of rank-one convex envelopes SIAM J. Numer. Anal. 36 5 1999 1621-1635
26. J. Goodman R.V. Kohn L. Reyna Numerical study of a relaxed variational problem from optimal design Comput. Methods Appl. Mech. Engrg. 57 1 1986 107-127
27. K. Hackl Generalized standard media and variational principles in classical and finite strain elastoplasticity J. Mech. Phys. Solids 45 5 1997 667-688
28. K. Hackl U. Hoppe On the calculation of microstructures for inelastic materials using relaxed energies In: C. Miehe (Ed.) IUTAM Symposium on Computational Mechanics of Solid Materials at Large Strains Solid Mechanics and its Applications 108 2003 Kluwer 77-86
29. B. Halphen Q.S. Nguyen Sur les matériaux standards géneralises J. Mech. 14 1975 39-63
30. D. Kinderlehrer P. Pedregal Weak convergence of integrands and the Young measure representation SIAM J. Math. Anal. 23 1991 1-19
31. R.V. Kohn The relaxation of a double-well energy Contin. Mech. Thermodyn. 3 3 1991 193-236