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Math. Comput. Appl. 2018, 23(2), 29; https://doi.org/10.3390/mca23020029

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

1
Institut Montpelliérain Alexander Grothendieck, University Montpellier, CNRS, 34090 Montpellier, France
2
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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Abstract

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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