Hostname: page-component-586b7cd67f-rdxmf Total loading time: 0 Render date: 2024-11-23T00:48:30.180Z Has data issue: false hasContentIssue false
  • English
  • Français

Linear convergence in the approximation of rank-one convex envelopes

Published online by Cambridge University Press:  15 October 2004

Sören Bartels
Sören Bartels*
Affiliation:
Department of Mathematics, University of Maryland, College Park, MD 20742-4015, USA. [email protected].
Get access

Abstract

A linearly convergent iterative algorithm that approximates the rank-1 convex envelope $f^{rc}$ of a given function $f:\mathbb{R}^{n\times m} \to \mathbb{R}$, 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.

Keywords

Nonconvex variational problemcalculus of variationsrelaxed variational problemsrank-1 convex envelopemicrostructureiterative algorithm.
Type
Research Article
Information
ESAIM: Mathematical Modelling and Numerical Analysis , Volume 38 , Issue 5 , September 2004 , pp. 811 - 820
Copyright
© EDP Sciences, SMAI, 2004

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

References

Ball, J.M., 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. CrossRef
Ball, J.M. and James, R.D., Fine phase mixtures as minimizers of energy. Arch. Rational Mech. Anal. 100 (1987) 13–52. CrossRef
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) (www.mathematik.uni-stuttgart.de/~mehrskalen/)].
Bartels, S., Error estimates for adaptive Young measure approximation in scalar nonconvex variational problems. SIAM J. Numer. Anal. 42 (2004) 505–529. CrossRef
Bartels, S. and Prohl, A., Multiscale resolution in the computation of crystalline microstructure. Numer. Math. 96 (2004) 641–660. CrossRef
Carstensen, C. and Plecháč, P., Numerical solution of the scalar double-well problem allowing microstructure. Math. Comp. 66 (1997) 997–1026. CrossRef
Carstensen, C. and Roubíček, T., Numerical approximation of Young measures in non-convex variational problems. Numer. Math. 84 (2000) 395–414. CrossRef
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.
B. Dacorogna, Direct methods in the calculus of variations. Appl. Math. Sci. 78 (1989).
Dacorogna, B. and Haeberly, J.-P., 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. CrossRef
Dolzmann, G., Numerical computation of rank-one convex envelopes. SIAM J. Numer. Anal. 36 (1999) 1621–1635. CrossRef
Dolzmann, G. and Walkington, N.J., Estimates for numerical approximations of rank one convex envelopes. Numer. Math. 85 (2000) 647–663. CrossRef
Ericksen, J.L., Constitutive theory for some constrained elastic crystals. Int. J. Solids Struct. 22 (1986) 951–964. CrossRef
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.
Kohn, R.V., The relaxation of a double-well energy. Contin. Mech. Thermodyn. 3 (1991) 193–236. CrossRef
Kohn, R.V. and Strang, G., Optimal design and relaxation of variational problems. I.-III. Commun. Pure Appl. Math. 39 (1986) 353–377. CrossRef
Kružik, M., Numerical approach to double well problems. SIAM J. Numer. Anal. 35 (1998) 1833–1849. CrossRef
Luskin, M., On the computation of crystalline microstructure. Acta Numerica 5 (1996) 191–257. CrossRef
Miehe, C. and Lambrecht, M., 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. CrossRef
Müller, S., Variational models for microstructure and phase transitions. Lect. Notes Math. 1713 (1999) 85–210. CrossRef
Nicolaides, R.A., Walkington, N. and Wang, H., Numerical methods for a nonconvex optimization problem modeling martensitic microstructure. SIAM J. Sci. Comput. 18 (1997) 1122–1141. CrossRef
T. Roubíček, Relaxation in optimization theory and variational calculus. De Gruyter Series in Nonlinear Analysis Appl. 4 New York (1997).