Linear convergence in the approximation of rank-one convex envelopes

Sören Bartels

ESAIM: Mathematical Modelling and Numerical Analysis (2010)

  • Volume: 38, Issue: 5, page 811-820
  • ISSN: 0764-583X

Abstract

top
A linearly convergent iterative algorithm that approximates the rank-1 convex envelope  f r c 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.

How to cite

top

Bartels, Sören. "Linear convergence in the approximation of rank-one convex envelopes." ESAIM: Mathematical Modelling and Numerical Analysis 38.5 (2010): 811-820. <http://eudml.org/doc/194241>.

@article{Bartels2010,
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. },
author = {Bartels, Sören},
journal = {ESAIM: Mathematical Modelling and Numerical Analysis},
keywords = {Nonconvex variational problem; calculus of variations; relaxed variational problems; rank-1 convex envelope; microstructure; iterative algorithm.; nonconvex variational problem; rank-one convex envelope; phase transitions; crystalline solids; algorithm; convergence; numerical experiments},
language = {eng},
month = {3},
number = {5},
pages = {811-820},
publisher = {EDP Sciences},
title = {Linear convergence in the approximation of rank-one convex envelopes},
url = {http://eudml.org/doc/194241},
volume = {38},
year = {2010},
}

TY - JOUR
AU - Bartels, Sören
TI - Linear convergence in the approximation of rank-one convex envelopes
JO - ESAIM: Mathematical Modelling and Numerical Analysis
DA - 2010/3//
PB - EDP Sciences
VL - 38
IS - 5
SP - 811
EP - 820
AB - 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.
LA - eng
KW - Nonconvex variational problem; calculus of variations; relaxed variational problems; rank-1 convex envelope; microstructure; iterative algorithm.; nonconvex variational problem; rank-one convex envelope; phase transitions; crystalline solids; algorithm; convergence; numerical experiments
UR - http://eudml.org/doc/194241
ER -

References

top
  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.  Zbl0991.49500
  2. J.M. Ball and R.D. James, Fine phase mixtures as minimizers of energy. Arch. Rational Mech. Anal.100 (1987) 13–52.  Zbl0629.49020
  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) (www.mathematik.uni-stuttgart.de/~mehrskalen/)].  
  4. S. Bartels, Error estimates for adaptive Young measure approximation in scalar nonconvex variational problems. SIAM J. Numer. Anal.42 (2004) 505–529.  Zbl1077.65069
  5. S. Bartels and A. Prohl, Multiscale resolution in the computation of crystalline microstructure. Numer. Math.96 (2004) 641–660.  Zbl1098.74044
  6. C. Carstensen and P. Plecháč, Numerical solution of the scalar double-well problem allowing microstructure. Math. Comp.66 (1997) 997–1026.  Zbl0870.65055
  7. C. Carstensen and T. Roubíček, Numerical approximation of Young measures in non-convex variational problems. Numer. Math.84 (2000) 395–414.  Zbl0945.65070
  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).  Zbl0703.49001
  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.  Zbl0905.65075
  11. G. Dolzmann, Numerical computation of rank-one convex envelopes. SIAM J. Numer. Anal.36 (1999) 1621–1635.  Zbl0941.65062
  12. G. Dolzmann and N.J. Walkington, Estimates for numerical approximations of rank one convex envelopes. Numer. Math.85 (2000) 647–663.  Zbl0961.65063
  13. J.L. Ericksen, Constitutive theory for some constrained elastic crystals. Int. J. Solids Struct.22 (1986) 951–964.  Zbl0595.73001
  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.  Zbl1040.74006
  15. R.V. Kohn, The relaxation of a double-well energy. Contin. Mech. Thermodyn.3 (1991) 193–236.  Zbl0825.73029
  16. R.V. Kohn and G. Strang, Optimal design and relaxation of variational problems. I.-III. Commun. Pure Appl. Math.39 (1986) 353–377.  Zbl0694.49004
  17. M. Kružik, Numerical approach to double well problems. SIAM J. Numer. Anal.35 (1998) 1833–1849.  Zbl0929.49016
  18. M. Luskin, On the computation of crystalline microstructure. Acta Numerica5 (1996) 191–257.  Zbl0867.65033
  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.  Zbl1032.74526
  20. S. Müller, Variational models for microstructure and phase transitions. Lect. Notes Math.1713 (1999) 85–210.  Zbl0968.74050
  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.  Zbl0898.65035
  22. T. Roubíček, Relaxation in optimization theory and variational calculus. De Gruyter Series in Nonlinear Analysis Appl.4 New York (1997).  

NotesEmbed ?

top

You must be logged in to post comments.

To embed these notes on your page include the following JavaScript code on your page where you want the notes to appear.

Only the controls for the widget will be shown in your chosen language. Notes will be shown in their authored language.

Tells the widget how many notes to show per page. You can cycle through additional notes using the next and previous controls.

    
                

Note: Best practice suggests putting the JavaScript code just before the closing </body> tag.