Keywords: finite elements, space-time, four-dimensional mesh generation, elastodynamics, mesh adaptation.

A space-time finite element method (STFEM) is proposed, in this paper, for the resolution of mechanical problems including three dimensions in space and one in time. For that purpose, we have developed a technique of four-dimesnional mesh generation adapted to space-time remeshing. The method has been tested on a linearized elastodynamics problem. This original technique does not require coarse-to-fine and fine-to-coarse mesh transfer operators and does not increase the size of the linear systems to be solved, compared to the traditional finite element methods. Space-time meshes are made of simplex finite elements. Computations are realized in the context of the continuous Galerkin method. The technique of mesh adaptation, propsed by the authors, was applied to a problem of mobile loading. The evolutionary mesh is able to follow the mobile loading zone.

The STFEM can be regarded as an extension of the classical finite element method, applied to a boundary problem resulting from a non-stationary problem. Currently, several approaches exist. For example the large time increment method (LATIN), the Discontinuous Galerkin method, and the method proposed by the authors which is a continuous Galerkin method. In most publications on the discontinuous Galerkin method, the functions of interpolation are assumed to be the product of functions of space variables and functions of time variables. In this paper special attention is paid to the non-separation of the space and time variables. The reason of this choice is not motivated by the accuracy of numerical results, but rather by what constitutes the aim of this study: the remeshing. It can be seen that this kind of interpolation is well-suited to mesh adaptation. The space-time mesh adaptation developed is based on a method of mesh generation not structured in space and time. The construction of four-dimesnional meshes collides with the limits of the representation. To overcome this drawback, an automatic method of construction, is propsed that is inspired by the two- and three-dimensional technique. The authors' technique of mesh adaptation was applied to a problem of mobile load such as contact forces. The approach makes it possible to build an evolutionary mesh able to follow the clamping zone. However, this technique does not require a mesh-to-mesh transfer operator and makes it possible to preserve the same size of linear system on each time slab.

It should be noted that one of the drawbacks of the STFEM is the size of the systems to be solved. The use of a laminated mesh does not require the assembly of the total matrix of the problem, but only the submatrices. This reduces the size of the systems to be solved. The size of these linear systems is exactly the same as that obtained in the case of approaches coupling an incremental method of the finite difference type to solve time integration, with the "classic" finite element method to solve the space problem.

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