Geodesic
Efficient application of e-invariants in finite element method for an elastodynamic equation
Kybernetika, Tome 49 (2013) no. 5, pp. 765-779 Cet article a éte moissonné depuis la source Czech Digital Mathematics Library

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We introduce a new efficient way of computation of partial differential equations using a hybrid method composed from FEM in space and FDM in time domain. The overall computational scheme is explicit in time. The key idea of the suggested way is based on a transformation of standard basis functions into new basis functions. The results of this matrix transformation are e-invariants (effective invariants) with such suitable properties which save the number of arithmetical operations needed for a problem solution. The application of this procedure and its effectiveness for 2D problem was the first time published in [2]. Now we describe the generalization of this procedure for 3D problem. In order to present the main principle of our process and its advantage, we first explain the main idea of our approach on a simple 1D example and then the application of the e-invariants on an elastodynamics equation using hexahedral elements in 3D is described. Finally, the efficiency of the suggested method in both cases from the point of the required number of arithmetical operations is analyzed. The result of this analysis confirms computational efficiency the suggested method and the usefulness of e-invariants which save only the essential information needed for the computation. Moreover, the method can be used for various types of elements and equations.
We introduce a new efficient way of computation of partial differential equations using a hybrid method composed from FEM in space and FDM in time domain. The overall computational scheme is explicit in time. The key idea of the suggested way is based on a transformation of standard basis functions into new basis functions. The results of this matrix transformation are e-invariants (effective invariants) with such suitable properties which save the number of arithmetical operations needed for a problem solution. The application of this procedure and its effectiveness for 2D problem was the first time published in [2]. Now we describe the generalization of this procedure for 3D problem. In order to present the main principle of our process and its advantage, we first explain the main idea of our approach on a simple 1D example and then the application of the e-invariants on an elastodynamics equation using hexahedral elements in 3D is described. Finally, the efficiency of the suggested method in both cases from the point of the required number of arithmetical operations is analyzed. The result of this analysis confirms computational efficiency the suggested method and the usefulness of e-invariants which save only the essential information needed for the computation. Moreover, the method can be used for various types of elements and equations.
Classification : 65M60, 74B05, 74H15, 74S05
Keywords: e-invariants; finite element method; elastodynamics equation; efficient computation 
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Balazovjech, Martin; Halada, Ladislav. Efficient application of e-invariants in finite element method for an elastodynamic equation. Kybernetika, Tome 49 (2013) no. 5, pp. 765-779. http://geodesic.mathdoc.fr/item/KYB_2013_49_5_a6/

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