Hostname: page-component-586b7cd67f-tf8b9 Total loading time: 0 Render date: 2024-11-26T09:55:57.412Z Has data issue: false hasContentIssue false
  • English
  • Français

A domain splitting method for heat conduction problemsin composite materials

Published online by Cambridge University Press:  15 April 2002

Friedrich Karl Hebeker
Friedrich Karl Hebeker*
Affiliation:
Fachbereich Mathematik, Justus-Liebig-Universit t Gießen, Arndtstr. 2, 35392 Gießen, Germany. ([email protected])
Get access

Abstract

We consider a domain decomposition method for some unsteadyheat conduction problem in composite structures.This linear model problem is obtained by homogenization of thin layersof fibres embedded into some standard material.For ease of presentation we consider the case of two space dimensions only.The set of finite element equations obtained by the backward Euler schemeis parallelized in a problem-oriented fashion by some noniterative overlappingdomain splitting method,eventually enhanced by inexpensive local iterationsto reduce the overlap.We present a detailed convergence analysis of this algorithmwhich is particularly well appropriate to handle fibre layersof nonlinear material.Special emphasis is to take into account the specific regularity propertiesof the present mathematical model.Numerical experiments show the reliability of the theoretical predictions.

Keywords

Fibre layers of adaptive materialhomogenizationheat conductionfinite element methodnoniterative overlapping domain decomposition.
Type
Research Article
Information
ESAIM: Mathematical Modelling and Numerical Analysis , Volume 34 , Issue 1 , January 2000 , pp. 47 - 62
Copyright
© EDP Sciences, SMAI, 2000

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

Blum, H., Lisky, S. and Rannacher, R., A domain splitting algorithm for parabolic problems. Computing 49 (1992) 11-23. CrossRef
Braess, D., Dahmen, W. and Chr. Wieners, A multigrid algorithm for the mortar finite element method. SIAM J. Numer. Anal. 37 (1999) 48-69. CrossRef
Chen, H. and Lazarov, R.D., Domain splitting algorithm for mixed finite element approximations to parabolic problems. East-West J. Numer. Math. 4 (1996) 121-135.
Chen, Z. and Zou, J., Finite element methods and their convergence analysis for elliptic and parabolic interface problems. Numer. Math. 79 (1998) 175-202. CrossRef
W. Hackbusch, Theorie und Numerik elliptischer Differentialgleichungen. Teubner, Stuttgart (1986).
Hackbusch, W. and Sauter, S., Composite finite elements for the approximation of PDEs on domains with complicated microstructures. Numer. Math. 75 (1997) 447-472. CrossRef
H. Haller, Composite materials of shape-memory alloys: micromechanical modelling and homogenization (in German). Ph.D. thesis, Technische Universitt Mnchen (1997).
F.H. Hebeker, An a posteriori error estimator for elliptic boundary and interface problems. Preprint 97-46 (SFB 359), Universitt Heidelberg (1997); submitted.
F.K. Hebeker, Multigrid convergence analysis for elliptic problems arising in composite materials (in preparation).
F.K. Hebeker and Yu.A. Kuznetsov, Unsteady convection and convection-diffusion problems via direct overlapping domain decomposition methods. Preprint 93-54 (SFB 359), Universitt Heidelberg, 1993; Numer. Methods Partial Differential Equations 14 (1998) 387-406.
Hoffmann, K.H. and Zou, J., Finite element analysis on the Lawrence-Doniach model for layered superconductors. Numer. Funct. Anal. Optim. 18 (1997) 567-589. CrossRef
J. Jäger, An overlapping domain decomposition method to parallelize the solution of parabolic differential equations (in German). Ph.D. thesis, Universitt Heidelberg (1994).
C. Kober, Composite materials of shape-memory alloys: modelling as layers and numerical simulation (in German). Ph.D. thesis, Technische Universitt Mnchen (1997).
Kuznetsov, Yu.A., New algorithms for approximate realization of implicit difference schemes. Sov. J. Numer. Anal. Modell. 3 (1988) 99-114.
Yu.A. Kuznetsov, Domain decomposition methods for unsteady convection diffusion problems. Comput. Methods Appl. Sci. Engin. (Proceedings of the Ninth International Conference, Paris 1990) SIAM, Philadelphia (1990) 211-227.
Yu.A. Kuznetsov, Overlapping domain decomposition methods for finite element problems with singular perturbed operators. in Domain decomposition Methods for Partial differential equations, R. Glowinski et al. Eds., SIAM, Philadelphia. Proc. of the 4th Intl. Symp. (1991) 223-241
A. Quarteroni and A. Valli, Numerical Approximation of Partial Differential Equations. Springer, Berlin etc. (1994).
Rannacher, R. and Zhou, J., Analysis of a domain splitting method for nonstationary convection-diffusion problems. East-West J. Numer. Math. 2 (1994) 151-172.
J. Wloka, Partielle Differentialgleichungen. Teubner, Stuttgart (1982).