Keywords: model reduction, component mode synthesis, adaptivity, a posteriori, error estimation.

Many important problems in industry are the so called multiphysics problems which involve several different types of physics. One such problem is thermoelastic stress analysis where the objective is to predict the elastic strain of a material caused by heat flow in order to prevent structural failure. A common technique for simulating thermoelasticity is to connect two finite element solvers, one for heat transfer and one for elastic deformation, into a network where each physics is solved for and data exchanged between the solvers.

In this paper we present a method to automatically control the reduction error in both the thermal and elastic solver for a one-way coupled thermoelastic problem where each of the physics is approximated using the component mode synthesis (CMS) model reduction method. The method combines the methodology for a posteriori error estimation for CMS developed in [1,2], with that for a posteriori error estimation for multiphysics problems developed in [3,4,5].

The error estimate measures the difference between the reduced and the full finite element solution. A feature of the estimate is that it automatically gives a quantitative measure of the propagation of the error between the thermal and elastic solvers with respect to a certain computational goal. The results presented extend the results in [6] by allowing temperature dependendent elastic parameters, leading to a linearized thermal dual problem. The analytical results are accompanied with a numerical example.

1

H. Jakobsson, F. Bengzon, M.G. Larson, "Adaptive Component Mode Synthesis in Linear Elasticity", Int. J. Numer. Meth. Engng, 86, 829-844, 2011. doi:10.1002/nme.3078

2

H. Jakobsson, M.G. Larson, "A posteriori error analysis of component mode synthesis for the elliptic eigenvalue problem", Computer Methods in Applied Mechanics and Engineering, 200(41-44), 2840-2847, 2011. doi:10.1016/j.cma.2011.05.002

3

M. Larson, F. Bengzon, "Adaptive finite element approximation of multiphysics problems", Comm. Numer. Methods Engrg., 24(6), 505-521, 2007. doi:10.1002/cnm.1087

4

M. Larson, R. Söderlund, F. Bengzon, "Adaptive finite element approximation of coupled flow and transport problems with applications in heat transfer", Internat. J. Numer. Methods Fluids, 57(9), 1397-1420, 2008. doi:10.1002/fld.1818

5

F. Bengzon, M.G. Larson, "Adaptive Finite Element Approximation of Multiphysics Problems: A Fluid Structure Interaction Model Problem", Technical report, Department of Mathematics and Mathematical Statistics, Umeå University, 2009.

6

H. Jakobsson, F. Bengzon, M.G. Larson, "Duality Based Adaptive Model Reduction for One-way Coupled Thermoelastic Problems", Int. J. Numer. Meth. Engng, to appear. doi:10.1002/nme.4273

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