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

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.

Spacetime finite element methods for control problems subject to the wave equation

“…Suppose that (A) holds. Let κ < 2, p, q ∈ N + and let (u h , φ h ) in V p h × V q h be the solution of (16).…”

“…In this section we will study the finite element method (16) in the case that the continuum solution (u, φ) to the control problem ( 6) is in the natural energy class H 1 (M ) × L 2 (M ). We make the standing assumption that (A) holds, so that ( 16) has a unique solution.…”

“…The present paper falls into this category and aims, in the spirit of [8] devoted to the dual data assimilation problem, to provide some convergent results, including rate of convergence, with respect to the discrete parameter. We mention a growing interest for space-time (finite element) methods of approximation for the wave equation, initially advocated in [30,23,32] and more recently in [33], [1], [16], [17], [48].…”

A simplified thermomechanical approach to visualize hexagonal honeycomb construction

“…In order to capture the flow, one needs to perform dynamic simulations that would be a computationally difficult task as compared to the present steady-state case. In order to simulate the spatiotemporal behavior of the transformation, the current finite element formulation needs to be changed to space-time finite element formulation [19,20]. Thus, the future prospects of the present formulation remain in simulating the dynamics in the vicinity of the triple junction through which one might gain better understanding of the transformation process.…”

Special Issue “Mesh-Free and Finite Element-Based Methods for Structural Mechanics Applications”

“…A Space-Time Finite Element Method (STFEM) is proposed by Dumont et al [9] for the resolution of mechanical problems involving three dimensions in space and one in time. Special attention was paid to the nonseparation of the space and time variables because this kind of interpolation is well suited to mesh adaptation.…”