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Math. Comput. Appl. 2018, 23(2), 29;

4D Remeshing Using a Space-Time Finite Element Method for Elastodynamics Problems

Institut Montpelliérain Alexander Grothendieck, University Montpellier, CNRS, 34090 Montpellier, France
Laboratory of Mechanics and Civil Engineering, University Montpellier, CNRS, 34090 Montpellier, France
Author to whom correspondence should be addressed.
Received: 10 March 2018 / Revised: 15 May 2018 / Accepted: 22 May 2018 / Published: 25 May 2018
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In this article, a Space-Time Finite Element Method (STFEM) is proposed for the resolution of mechanical problems involving three dimensions in space and one in time. Special attention will be paid to the non-separation of the space and time variables because this kind of interpolation is well suited to mesh adaptation. For that purpose, we have developed a technique of 4D mesh generation adapted to space-time remeshing. A difficulty arose in the representation of 4D finite elements and meshes. This original technique does not require coarse-to-fine and fine-to-coarse mesh-to-mesh transfer operators and does not increase the size of the linear systems to be solved, compared to traditional finite element methods. Space-time meshes are composed of simplex finite elements. Computations are carried out in the context of the continuous Galerkin method. We have tested the method on a linearized elastodynamics problem. Our technique of mesh adaptation was validated on elementary examples and applied to a problem of mobile loading. The convergence and stability of the method are studied and compared with existing methods. This work is a first implementation of 4D space-time remeshing. A stability criterion for the method is established, as well as a convergence rate of about two. Using simplex elements, it is possible to develop a technique of mesh adaptation able to follow a mobile loading zone. View Full-Text
Keywords: finite elements; space-time; elastodynamics; mesh adaptation finite elements; space-time; elastodynamics; mesh adaptation

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Dumont, S.; Jourdan, F.; Madani, T. 4D Remeshing Using a Space-Time Finite Element Method for Elastodynamics Problems. Math. Comput. Appl. 2018, 23, 29.

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