Space-time FEM Research Articles

Space-Time FEM for the Vectorial Wave Equation under Consideration of Ohm’s Law

Abstract The ability to deal with complex geometries and to go to higher orders is the main advantage of space-time finite element methods. Therefore, we want to develop a solid background from which we can construct appropriate space-time methods. In this paper, we will treat time as another space direction, which is the main idea of space-time methods. First, we will briefly discuss how exactly the vectorial wave equation is derived from Maxwell’s equations in a space-time structure, taking into account Ohm’s law. Then we will derive a space-time variational formulation for the vectorial wave equation using different trial and test spaces. This paper has two main goals. First, we prove unique solvability for the resulting Galerkin–Petrov variational formulation. Second, we analyze the discrete equivalent of the equation in a tensor product and show conditional stability, i.e., under a CFL condition. Understanding the vectorial wave equation and the corresponding space-time finite element methods is crucial for improving the existing theory of Maxwell’s equations and paves the way to computations of more complicated electromagnetic problems.

Space–time FEM with block-iterative algorithm for nonlinear dynamic fracture analysis of concrete gravity dam

In this paper, space–time finite element method is developed for the dynamic fracture analysis of dam–reservoir (DR) system which is supported on a perfectly rigid foundation. In this method, an auxiliary variable q representing the first-order time derivative of hydrodynamic pressure is treated as the primary unknown for the reservoir domain. Similarly, velocity v is the primary unknown for the solid domain. A three point Gauss–Lobatto quadrature rule is employed for computing time integral of those terms which contain stress term. Further, a partitioned method based on a block-iterative scheme is employed to incorporate the material nonlinearity of the concrete and to enforce the coupling between the two domains in a single iteration loop. A co-axially rotating crack model with exponential strain softening rule is employed to model the fracture of the concrete. Afterwards, numerical performance of the proposed scheme is demonstrated by analyzing two situations of dynamic fracture of Koyna concrete gravity dam.

Space\u2013time computations in practical engineering applications: a summary of the 25-year history

In an article published online in July 2018 it was stated that the algorithm proposed in the article is “enabling practical implementation of the space–time FEM for engineering applications.” In fact, space–time computations in practical engineering applications were already enabled in 1993. We summarize the computations that have taken place since then. These computations started with finite element discretization and are now also with isogeometric discretization. They were all in 3D space and were all carried out on parallel computers. For quarter of a century, these computations brought solution to many classes of complex problems ranging from Orion spacecraft parachutes to wind turbines, from patient-specific cerebral aneurysms to heart valves, from thermo-fluid analysis of ground vehicles and tires to turbocharger turbines and exhaust manifolds.

An efficient solution algorithm for space–time finite element method

An efficient solution algorithm has been developed for space–time finite element method that is derived from time discontinuous Galerkin (TDG) formulation. The proposed algorithm features an iterative solver accelerated by a novel and efficient preconditioner. This preconditioner is constructed based on the block structure of coupled space–time system matrix, which is expressed as addition of Kronecker products of temporal and spatial submatrices. With this unique decomposition, the most computationally intensive operations in the iterative solver, i.e. matrix operations, are subsequently optimized and accelerated employing the inverse property of Kronecker product. Theoretical analysis and numerical examples both demonstrate that the proposed algorithm provides significantly better performance than the already developed implementations for TDG-based space–time FEM. It reduces the computational cost of solving space–time equations to the same order of solving stiffness equations associated with regular FEM, thereby enabling practical implementation of the space–time FEM for engineering applications.

On stabilized space‐time FEM for anisotropic meshes: Incompressible Navier–Stokes equations and applications to blood flow in medical devices

SummaryIn complex applications, such as the analysis of hydraulic performance of blood pumps (ventricular assist devices), the Navier–Stokes equations have to be discretized on very anisotropic meshes. If stabilized finite element formulations are applied, standard definitions of the stabilization parameter are usually not appropriate to handle elements with a high aspect ratio. If, in addition, rotating objects, moving meshes, or turbulence has to be considered in the simulation, further modifications of the stabilization procedure have to be applied.In this paper, we present stabilized space‐time finite element formulations of the incompressible Navier–Stokes equations that show very good convergence properties on complex anisotropic meshes and lead to reasonable numerical accuracy in complex flows when compared with experimental data. Copyright © 2017 John Wiley & Sons, Ltd.

Enriched space–time finite element method: a new paradigm for multiscaling from elastodynamics to molecular dynamics

SUMMARYThe main objective of this paper is to present an enriched version of the space–time FEM method to incorporate multiple temporal scale features with a focus on dynamics problems. The method is established by integrating the basic framework of the space–time discontinuous Galerkin method with the extended finite element method. Two versions of the method have been developed: one at the continuum scale (elastodynamics) and the other focuses on the dynamics at the molecular level. After an initial outline of the formulation, we explore the incorporation of different types of enrichments based on the length scale of interest. The effects of both continuous and discontinuous enrichments are demonstrated through numerical examples involving wave propagations and dynamic fracture in harmonic lattice. The robustness of the method is evaluated in terms of convergence and the ability to capture the fine scale features. It is shown that the enriched space–time FEM leads to an improvement in the convergence properties over the traditional space–time FEM for problems with multiple temporal features. It is also highly effective in integrating atomistic with continuum representations with a coupled framework. Copyright © 2012 John Wiley & Sons, Ltd.

PARALLEL GMRES SOLUTION OF CONVECTIVE TRANSPORT PROBLEMS ON DISTRIBUTED MEMORY COMPUTERS

Parallel GMRES algorithms for solution of convective transport problems on distributed memory com-puters are investigated in this paper. The space-time FEM stabilised by the GLS formulation is applied as the most accurate and stable. The parallel algorithms based on the domain decomposition and iterative GMRES solver are implemented in order to reduce long computing time associated with space-time GLS method. The developed data structures, the static load balancing and the inter-processor communication algorithms are particularly suited for homo-geneous distributed memory PC clusters. The benchmark problem of the rotating cone is solved in order to validate efficiency of the developed software. The performance of parallel computations, the speed-up and the efficiency is measured on several PC clusters as well as on the IBM SP2 supercomputer.

Parallel FEM Software for CFD Problems

The present paper describes the development and the performance of parallel FEM software for solving various CFD problems. Domain decomposition strategy and parallel iterative GMRES solver have been adapted to the universal space-time FEM code FEMTOOL, which allows implementation of any partial differential equation with minor expenses. The developed data structures, the static load balancing and the inter-processor communication algorithms have been particularly suited for homogeneous distributed memory PC clusters. The universality of the considered parallel algorithms has been validated solving applications described by the Poisson equation, by the convective transport equation and by the Navier-Stokes equations. Three typical benchmark problems have been solved in order to perform the efficiency study. The performance of parallel computations, the speed-up and the efficiency have been measured on three BEOWULF PC clusters as well as on the cluster of IBM RISC workstations and on the IBM SP2 supercomputer.