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Machine Computation Using the Exponentially Convergent Multiscale Spectral Generalized Finite Element Method∗

Published online by Cambridge University Press:  11 March 2014

Ivo Babuška ,
Xu Huang  and
Robert Lipton
Ivo Babuška
Affiliation:
Institute for Computational Engineering and Science and Department of Aerospace Engineering, University of Texas, Austin, TX 78712, USA. [email protected]
Xu Huang
Affiliation:
Department of Mathematics, Louisiana State University, Baton Rouge, LA 70803, USA; [email protected]
Robert Lipton
Affiliation:
Department of Mathematics and Center for Computation and Technology, Louisiana State University, Baton Rouge, LA 70803, USA; [email protected]
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Abstract

A multiscale spectral generalized finite element method (MS-GFEM) is presented for the solution of large two and three dimensional stress analysis problems inside heterogeneous media. It can be employed to solve problems too large to be solved directly with FE techniques and is designed for implementation on massively parallel machines. The method is multiscale in nature and uses an optimal family of spectrally defined local basis functions over a coarse grid. It is proved that the method has an exponential rate of convergence. To fix ideas we describe its implementation for a two dimensional plane strain problem inside a fiber reinforced composite. Here fibers are separated by a minimum distance however no special assumption on the fiber configuration such as periodicity or ergodicity is made. The implementation of MS-GFEM delivers the discrete solution operator using the same order of operations as the number of fibers inside the computational domain. This implementation is optimal in that the number of operations for solution is of the same order as the input data for the problem. The size of the MS-GFEM matrix used to represent the discrete inverse operator is controlled by the scale of the coarse grid and the convergence rate of the spectral basis and can be of order far less than the number of fibers. This strategy is general and can be applied to the solution of very large FE systems associated with the discrete solution of elliptic PDE.

Keywords

Generalized finite elementsmultiscale methodspectral methodheterogeneous mediafiber reinforced composites
Type
Research Article
Information
ESAIM: Mathematical Modelling and Numerical Analysis , Volume 48 , Issue 2: Multiscale problems and techniques , March 2014 , pp. 493 - 515
Copyright
© EDP Sciences, SMAI, 2014

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