A numerical approach to space-time finite elements for the wave equation

Anderson, Matthew; Kimn, Jung‐Han

doi:10.1016/j.jcp.2007.04.021

Cited by 18 publications

(18 citation statements)

References 26 publications

(36 reference statements)

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“…In this paragraph we present our technique of space-time mesh adaptations. In the literature, 169 many articles on mesh adaptations, [7], [9], [12,13], [23,24], [8], [10] and [22], can be found. Among interpolation from those of the old mesh, which we will call "mesh-to-mesh transfer".…”

Section: Remeshing Technique 168mentioning

confidence: 99%

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

Dumont¹,

Jourdan

Madani

2018

MCA

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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.

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Section: Remeshing Technique 168mentioning

confidence: 99%

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

Dumont¹,

Jourdan

Madani

2018

MCA

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“…We investigate a space-time Finite element method similar to using continuous approximation functions in both space and time to explore its use for numerical relativity simulations. The discretization for the KG equation in this paper is an extension of the discretization for the nonhomogeneous wave equation in [9].…”

Section: Klein-gordon Equationmentioning

confidence: 99%

“…First of all, we consider numerical stability, which is vital to simulate time dependent problems. Therefore, we implement a fully implicit method through space time finite element methods, which solved for a nonhomogeneous wave equation in the 3+1 dimension successfully in [9]. We discretize…”

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confidence: 99%

A Numerical Implementation of the Space-Time Finite Elements method for the 1+1 Klein-Gordon equation

Lim¹

2013

SIURO

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Abstract. We have implemented a fully implicit numerical approach based on space-time finite element methods for the Klein-Gordon equation in the 1(space)+1(time) dimension. The purpose of this paper is to present a stable and parallelizable numerical method. The proposed numerical method is applied to generate successful simulation results of spin-0 particle propagation in a charged scalar field. The time additive Schwarz method is vital to make successful simulations with KSP (Krylov Subspace Methods) solvers. The time parallelizable algorithm is implemented through PETSc(Portable, Extensible, Toolkit for Scientific Computation, developed by Argonne National Laboratory).1. Introduction. The Klein-Gordon(KG) equation is a partial differential equation(PDE) that governs the quantum evolution of wave functions for relativistic spinless particles. The KG equation describes a wide variety of physical phenomena. In paper [1], the KG equation includes classical wave systems such as the displacement of a string attached to an elastic bed and quantum systems based on scalar fields. The KG equation is also related to the Dirac equation. Modeling light matter interaction with the relativistic effect can be explained by the Dirac equation, and the KG equation can be applied to the relativistic effect.For another example, the electron-positron pair creation process in the supercritical breakdown of the fermionic vacuum is a striking prediction of the Dirac equation in [2]. It turns out that a quantum field theory based on the KG equation can predict similar phenomena from the Dirac equation in [3]. Therefore, the KG equation can provide new physical explanations about the universe, and can also provide clues to the Dirac equation because any solution to the Dirac equation is automatically a solution to the KG equation.In addition, the KG equation has many applications. The KG equation can be modified to a non-homogeneous model and a nonlinear model. A modified version of the KG equation can be used for many engineering models to explain solitary waves, and wave propagations. Paper [4] shows the Klein-Gordon system transmitting monochromatic waves. Paper [5] shows an Einstein-Klein-Gordon system (a massive scalar field coupled to gravity) responding to a numerical simulation of oscillation.Solutions of the KG equation both homogeneous and non-homogeneous are generally unavailable in [6]. Therefore, numerical simulation is required to estimate solutions. Papers [7] introduces numercial methods to the KG equation and KleinGordon-Schrödinger(KGS) equation without a forcing term. Paper [8] shows numerical approaches to the KGS equation without a forcing term.In this paper, we present to generate successful simulation results of spin-0 particle propagation in a charged scalar field. This problem is defined as a KG equation with forcing term. To efficiently solve the KG equation using numerical methods, we use various numerical procedures. First of all, we consider numerical stability, which is vital to simulate time dependent p...

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“…In this case of interpolation of the nodal velocity it is not possible to appoint all parts of (23). Moreover, it is impossible to reduce the order of the derivative as a result of integration by parts due to the distribution in (24). The Renaudot formula (23) can be written in the equivalent form d 2 wðvt,tÞ…”

Section: Numerical Description Of the Moving Mass Particlementioning

confidence: 99%

“…Fried [18] and Argyriss, Scharpf and Chan [19][20][21] began to treat the spatial and time variables equally in the formulation of physical problems. The space-time finite element method was applied successfully to wave problems [22][23][24], and also to acoustics [25] and fluid mechanics [26]. Unfortunately, vast majority of problems are solved with classical, stationary discretizations and the space-time element approach is less important in engineering practice.…”

Section: Introductionmentioning

confidence: 99%

Space–time finite element approach to general description of a moving inertial load

Dyniewicz¹

2012

Finite Elements in Analysis and Design

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A numerical approach to space-time finite elements for the wave equation

Cited by 18 publications

References 26 publications

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

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

A Numerical Implementation of the Space-Time Finite Elements method for the 1+1 Klein-Gordon equation

Space–time finite element approach to general description of a moving inertial load

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