Linear convergence in the approximation of rank-one convex envelopes
Bartels, Sören
ESAIM: Mathematical Modelling and Numerical Analysis - Modélisation Mathématique et Analyse Numérique, Tome 38 (2004), p. 811-820 / Harvested from Numdam

A linearly convergent iterative algorithm that approximates the rank-1 convex envelope f rc of a given function f: n×m , i.e. the largest function below f which is convex along all rank-1 lines, is established. The proposed algorithm is a modified version of an approximation scheme due to Dolzmann and Walkington.

Publié le : 2004-01-01
DOI : https://doi.org/10.1051/m2an:2004040
Classification:  65K10,  74G15,  74G65,  74N99
@article{M2AN_2004__38_5_811_0,
     author = {Bartels, S\"oren},
     title = {Linear convergence in the approximation of rank-one convex envelopes},
     journal = {ESAIM: Mathematical Modelling and Numerical Analysis - Mod\'elisation Math\'ematique et Analyse Num\'erique},
     volume = {38},
     year = {2004},
     pages = {811-820},
     doi = {10.1051/m2an:2004040},
     mrnumber = {2104430},
     zbl = {1083.65058},
     language = {en},
     url = {http://dml.mathdoc.fr/item/M2AN_2004__38_5_811_0}
}
Bartels, Sören. Linear convergence in the approximation of rank-one convex envelopes. ESAIM: Mathematical Modelling and Numerical Analysis - Modélisation Mathématique et Analyse Numérique, Tome 38 (2004) pp. 811-820. doi : 10.1051/m2an:2004040. http://gdmltest.u-ga.fr/item/M2AN_2004__38_5_811_0/

[1] J.M. Ball, A version of the fundamental theorem for Young measures. Partial differential equations and continuum models of phase transitions. M Rascle, D. Serre, M. Slemrod Eds. Lect. Notes Phys. 344 (1989) 207-215. | Zbl 0991.49500

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

[3] S. Bartels, Reliable and efficient approximation of polyconvex envelopes. SIAM J. Numer. Anal. (accepted) [Preprints of the DFG Priority Program “Multiscale Problems”, No. 76 (2002) (http://www.mathematik.uni-stuttgart.de/~mehrskalen/)]. | Zbl 1089.65052

[4] S. Bartels, Error estimates for adaptive Young measure approximation in scalar nonconvex variational problems. SIAM J. Numer. Anal. 42 (2004) 505-529. | Zbl 1077.65069

[5] S. Bartels and A. Prohl, Multiscale resolution in the computation of crystalline microstructure. Numer. Math. 96 (2004) 641-660. | Zbl 1098.74044

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

[7] C. Carstensen and T. Roubíček, Numerical approximation of Young measures in non-convex variational problems. Numer. Math. 84 (2000) 395-414. | Zbl 0945.65070

[8] M. Chipot and S. Müller, Sharp energy estimates for finite element approximations of non-convex problems, in Variations of domain and free boundary problems, in solid mechanics, Solid Mech. Appl. 66 (1997) 317-327.

[9] B. Dacorogna, Direct methods in the calculus of variations. Appl. Math. Sci. 78 (1989). | MR 990890 | Zbl 0703.49001

[10] B. Dacorogna and J.-P. Haeberly, Some numerical methods for the study of the convexity notions arising in the calculus of variations. RAIRO Modél. Math. Anal. Numér. 32 (1998) 153-175. | Numdam | Zbl 0905.65075

[11] G. Dolzmann, Numerical computation of rank-one convex envelopes. SIAM J. Numer. Anal. 36 (1999) 1621-1635. | Zbl 0941.65062

[12] G. Dolzmann and N.J. Walkington, Estimates for numerical approximations of rank one convex envelopes. Numer. Math. 85 (2000) 647-663. | Zbl 0961.65063

[13] J.L. Ericksen, Constitutive theory for some constrained elastic crystals. Int. J. Solids Struct. 22 (1986) 951-964. | Zbl 0595.73001

[14] K. Hackl and U. Hoppe, On the calculation of microstructures for inelastic materials using relaxed energies. IUTAM symposium on computational mechanics of solid materials at large strains, C. Miehe Ed., Solid Mech. Appl. 108 (2003) 77-86. | Zbl 1040.74006

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

[16] R.V. Kohn and G. Strang, Optimal design and relaxation of variational problems. I.-III. Commun. Pure Appl. Math. 39 (1986) 353-377. | Zbl 0694.49004

[17] M. Kružik, Numerical approach to double well problems. SIAM J. Numer. Anal. 35 (1998) 1833-1849. | Zbl 0929.49016

[18] M. Luskin, On the computation of crystalline microstructure. Acta Numerica 5 (1996) 191-257. | Zbl 0867.65033

[19] C. Miehe and M. Lambrecht, Analysis of micro-structure development in shearbands by energy relaxation of incremental stress potentials: large-strain theory for standard dissipative materials. Internat. J. Numer. Methods Engrg. 58 (2003) 1-41. | Zbl 1032.74526

[20] S. Müller, Variational models for microstructure and phase transitions. Lect. Notes Math. 1713 (1999) 85-210. | Zbl 0968.74050

[21] R.A. Nicolaides, N. Walkington and H. Wang, Numerical methods for a nonconvex optimization problem modeling martensitic microstructure. SIAM J. Sci. Comput. 18 (1997) 1122-1141. | Zbl 0898.65035

[22] T. Roubíček, Relaxation in optimization theory and variational calculus. De Gruyter Series in Nonlinear Analysis Appl. 4 New York (1997). | MR 1458067 | Zbl 0880.49002