An adaptive finite element method for solving a double well problem describing crystalline microstructure
- Volume: 33, Issue: 4, page 781-796
- ISSN: 0764-583X
Access Full Article
topHow to cite
topProhl, Andreas. "An adaptive finite element method for solving a double well problem describing crystalline microstructure." ESAIM: Mathematical Modelling and Numerical Analysis - Modélisation Mathématique et Analyse Numérique 33.4 (1999): 781-796. <http://eudml.org/doc/193946>.
@article{Prohl1999,
author = {Prohl, Andreas},
journal = {ESAIM: Mathematical Modelling and Numerical Analysis - Modélisation Mathématique et Analyse Numérique},
keywords = {nonconvex energy functional; laminated microstructure; improved convergence; discontinuous ansatz functions; adaptivity criterion; height of interelement jumps},
language = {eng},
number = {4},
pages = {781-796},
publisher = {Dunod},
title = {An adaptive finite element method for solving a double well problem describing crystalline microstructure},
url = {http://eudml.org/doc/193946},
volume = {33},
year = {1999},
}
TY - JOUR
AU - Prohl, Andreas
TI - An adaptive finite element method for solving a double well problem describing crystalline microstructure
JO - ESAIM: Mathematical Modelling and Numerical Analysis - Modélisation Mathématique et Analyse Numérique
PY - 1999
PB - Dunod
VL - 33
IS - 4
SP - 781
EP - 796
LA - eng
KW - nonconvex energy functional; laminated microstructure; improved convergence; discontinuous ansatz functions; adaptivity criterion; height of interelement jumps
UR - http://eudml.org/doc/193946
ER -
References
top- [1] R.E. Bank, PLTMG: A Software Package for Solving Elliptic Partial Differential Equations. User's Guide 6.0. SIAM Philadelphia (1990). Zbl0717.68001MR1052151
- [2] J. Ball and R. James, Fine phase mixtures as minimizers of energy. Arch. Rational Mech. Anal. 100 (1987) 13-52. Zbl0629.49020MR906132
- [3] J. Ball and R. James, Proposed experimental tests of a theory of fine microstructure and the two-well problem. Philos. Trans. Roy. Soc. London Ser. A 338 (1992) 389-450. Zbl0758.73009
- [4] C. Carstensen and P. Plechac, Numerical solution of the scalar double-well problem allowing microstructure. Math. Comp. 66 (1997) 997-1026. Zbl0870.65055MR1415798
- [5] M. Chipot, Numerical analysis of oscillations in nonconvex problems. Numer. Math. 56 (1991) 747-767. Zbl0712.65063MR1128031
- [6] M. Chipot and C. Collins, Numerical approximations in variational problems with potential wells. SIAM J. Numer. Anal. 29 (1992) 1002-1019. Zbl0763.65049MR1173182
- [7] M. Chipot, C. Collins and D. Kinderlehrer, Numerical analysis of oscillations in multiple well problems. Numer. Math. 70 (1995) 259-282. Zbl0824.65045MR1330864
- [8] C. Collins, Computation and Analysis of Twinning in Crystalline Solids, Ph.D. thesis, University of Minnesota, USA (1990).
- [9] P. Ciarlet, The finite element method for elliptic problems. North-Holland, Amsterdam (1978). Zbl0383.65058MR520174
- [10] C. Collins, D. Kinderlehrer and M. Luskin, Numerical approximation of the solution of a variational problem with a double well potential. SIAM J. Numer. Anal. 28 (1991) 321-332. Zbl0725.65067MR1087507
- [11] C. Collins and M. Luskin, Optimal order estimates for the finite element approximation of the solution of a nonconvex variational problem. Math. Comp. 57 (1991) 621-637. Zbl0735.65042MR1094944
- [12] J. Ericksen, Constitutive theory for some constrained elastic crystals. J. Solids and Structures 22 (1986) 951-964. Zbl0595.73001
- [13] J. Ericksen, Some constrained elastic crystals, in Material Instabilites in Continuum Mechanics and Related Problems, J. Ball Ed., Oxford University Press, Oxford (1987) 119-137. Zbl0655.73022MR970522
- [14] J. Ericksen, Twinning of crystals I, in Metastability and Incompletely Posed Problems, S. Antman, J. Ericksen, D. Kinderlehrer and I. Muller Eds., Springer-Verlag, New York (1987) 77-96; IMA Volumes in Mathematics and Its Applications, Vol 3. Zbl0638.73006MR870011
- [15] M. Gobbert and A. Prohl, A discontinuous finite element method for solving a multi-well problem. Technical Report 1539, IMA (1998) and SIAM J. Numer. Anal. (to be published). Zbl0957.49019
- [16] M. Gobbert and A. Prohl, A survey of classical and new finite element methods for the computation of crystalline microstructure. Technical Report 1576, IMA (1998). Zbl1010.74063
- [17] P. Gremaud, Numerical analysis of a nonconvex variational problem related to solid-solid phase transitions. SIAM J. Numer. Anal. 31 (1994) 111-127. Zbl0797.65052MR1259968
- [18] P. Kloucek, B. Li and M. Luskin, Analysis of a class of nonconforming finite elements for crystalline microstructure. Math. Comp. 67 (1996) 1111-1125. Zbl0903.65081MR1344616
- [19] P. Kloucek and M. Luskin, The computation of the dynamics of martensitic microstructure. Contin. Mech. Thermodyn. 6 (1994) 209-240. Zbl0825.73047MR1285922
- [20] M. Kruzik, Numerical approach to double well problems. SIAM J. Numer. Anal. 35 (1998) 1833-1849. Zbl0929.49016MR1639950
- [21] M. Kruzik, Oscillations, Concentrations and Microstructure Modeling, Ph.D. thesis, Charles University, Prague, Czech Republic (1996).
- [22] B. Li, Analysis and Computation of Martensitic Microstructure, Ph.D. thesis, University of Minnesota, USA (1996).
- [23] B. Li and M. Luskin, Finite element analysis of microstructure for the cubic to tetragonal transformation. SIAM J. Numer. Anal. 35 (1998) 376-392. Zbl0919.49020MR1618484
- [24] B. Li and M. Luskin, Nonconforming Finite element approximation of crystalline microstructure. Math. Comp. 67 (1998) 917-946. Zbl0901.73076MR1459391
- [25] B. Li and M. Luskin, Approximation of a martensitic laminate with varying volume fractions. RAIRO ModéL Math. Anal. Numér. 33 (1999) 67-87. Zbl0928.74012MR1685744
- [26] M. Luskin, Approximation of a laminated microstructure for a rotationally invariant, double well energy density. Numer. Math. 75 (1997) 205-221. Zbl0874.73060MR1421987
- [27] M. Luskin, On the computation of crystalline microstructure. Acta Numer. 5 (1996) 191-257. Zbl0867.65033MR1624603
NotesEmbed ?
topTo embed these notes on your page include the following JavaScript code on your page where you want the notes to appear.