Approximation of a nonlinear thermoelastic problem with a moving boundary via a fixed-domain method

Jindřich Nečas; Tomáš Roubíček

Aplikace matematiky (1990)

  • Volume: 35, Issue: 5, page 361-372
  • ISSN: 0862-7940

Abstract

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The thermoelastic stresses created in a solid phase domain in the course of solidification of a molten ingot are investigated. A nonlinear behaviour of the solid phase is admitted, too. This problem, obtained from a real situation by many simplifications, contains a moving boundary between the solid and the liquid phase domains. To make the usage of standard numerical packages possible, we propose here a fixed-domain approximation by means of including the liquid phase domain into the problem (in this way we get the fixed domain involving the whole ingot) and by replacing the liquid phase with a solid phase having, however, a small shear modulus. The weak L 2 -convergence of thus approximated stresses in the solid phase domain is demonstrated. Besides, this convergence is shown to be strong on subsets whose closure belongs to the solid phase domain.

How to cite

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Nečas, Jindřich, and Roubíček, Tomáš. "Approximation of a nonlinear thermoelastic problem with a moving boundary via a fixed-domain method." Aplikace matematiky 35.5 (1990): 361-372. <http://eudml.org/doc/15637>.

@article{Nečas1990,
abstract = {The thermoelastic stresses created in a solid phase domain in the course of solidification of a molten ingot are investigated. A nonlinear behaviour of the solid phase is admitted, too. This problem, obtained from a real situation by many simplifications, contains a moving boundary between the solid and the liquid phase domains. To make the usage of standard numerical packages possible, we propose here a fixed-domain approximation by means of including the liquid phase domain into the problem (in this way we get the fixed domain involving the whole ingot) and by replacing the liquid phase with a solid phase having, however, a small shear modulus. The weak $L^2$-convergence of thus approximated stresses in the solid phase domain is demonstrated. Besides, this convergence is shown to be strong on subsets whose closure belongs to the solid phase domain.},
author = {Nečas, Jindřich, Roubíček, Tomáš},
journal = {Aplikace matematiky},
keywords = {nonlinear thermoelasticity; solidification; moving boundary; weak L(sup 2) convergence; strong convergence; solid phase domain; solidification; molten ingot; nonlinear behaviour of the solid phase; fixed-domain approximation},
language = {eng},
number = {5},
pages = {361-372},
publisher = {Institute of Mathematics, Academy of Sciences of the Czech Republic},
title = {Approximation of a nonlinear thermoelastic problem with a moving boundary via a fixed-domain method},
url = {http://eudml.org/doc/15637},
volume = {35},
year = {1990},
}

TY - JOUR
AU - Nečas, Jindřich
AU - Roubíček, Tomáš
TI - Approximation of a nonlinear thermoelastic problem with a moving boundary via a fixed-domain method
JO - Aplikace matematiky
PY - 1990
PB - Institute of Mathematics, Academy of Sciences of the Czech Republic
VL - 35
IS - 5
SP - 361
EP - 372
AB - The thermoelastic stresses created in a solid phase domain in the course of solidification of a molten ingot are investigated. A nonlinear behaviour of the solid phase is admitted, too. This problem, obtained from a real situation by many simplifications, contains a moving boundary between the solid and the liquid phase domains. To make the usage of standard numerical packages possible, we propose here a fixed-domain approximation by means of including the liquid phase domain into the problem (in this way we get the fixed domain involving the whole ingot) and by replacing the liquid phase with a solid phase having, however, a small shear modulus. The weak $L^2$-convergence of thus approximated stresses in the solid phase domain is demonstrated. Besides, this convergence is shown to be strong on subsets whose closure belongs to the solid phase domain.
LA - eng
KW - nonlinear thermoelasticity; solidification; moving boundary; weak L(sup 2) convergence; strong convergence; solid phase domain; solidification; molten ingot; nonlinear behaviour of the solid phase; fixed-domain approximation
UR - http://eudml.org/doc/15637
ER -

References

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  1. J. Nečas I. Hlaváček, Mathematical Theory of Elastic and Elasto-Plastic Bodies: An introduction, Elsevier, Amsterdam, 1981. (1981) Zbl0448.73009MR0600655
  2. J. Nečas, Introduction to the Theory of Nonlinear Elliptic Equations, Teubner, Leipzig, 1983. (1983) Zbl0526.35003MR0731261

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