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A robust preconditioning technique for the extended finite element method

Authors

  • Alexander Menk,

    1. Robert Bosch GmbH, P. O. Box 300240, 70442 Stuttgart, Germany
    2. Department of Civil Engineering, Glasgow University, Glasgow G12 8LT, U.K.
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  • Stéphane P. A. Bordas

    Corresponding author
    1. Institute of Theoretical, Applied and Computational Mechanics, Cardiff University, Cardiff CF24 3AA, U.K.
    • School of Engineering, Institute of Theoretical, Applied and Computational Mechanics, Cardiff University, Queen's Building, Cardiff CF24 3AA, U.K.
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    • Professor of Engineering, Royal Academy of Engineering/Leverhulme Trust Senior Research Fellow.


Abstract

The extended finite element method enhances the approximation properties of the finite element space by using additional enrichment functions. But the resulting stiffness matrices can become ill-conditioned. In that case iterative solvers need a large number of iterations to obtain an acceptable solution. In this paper a procedure is described to obtain stiffness matrices whose condition number is close to the one of the finite element matrices without any enrichments. A domain decomposition is employed and the algorithm is very well suited for parallel computations. The method was tested in numerical experiments to show its effectiveness. The experiments have been conducted for structures containing cracks and material interfaces. We show that the corresponding enrichments can result in arbitrarily ill-conditioned matrices. The method proposed here, however, provides well-conditioned matrices and can be applied to any sort of enrichment. The complexity of this approach and its relation to the domain decomposition is discussed. Computation times have been measured for a structure containing multiple cracks. For this structure the computation times could be decreased by a factor of 2. Copyright © 2010 John Wiley & Sons, Ltd.

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