Mass Lumping Technique Research Articles

Sparse and explicit FETD via approximate inverse Hodge (mass) matrix

Finite-element time-domain (FETD) simulations of Maxwell equations in irregular simplicial grids require the solution of a sparse linear system involving the Hodge (mass) matrix at each time step. This can be avoided by mass lumping techniques that approximate the mass matrix by a diagonal matrix, but not without shortcomings. In this communication, we propose an alternative approach to yield a conditionally stable, fully explicit, and sparse FETD by using a sparse approximate inverse of the mass matrix.

Discontinuous Galerkin methods for Maxwell's equations in the time domain

In this article, we describe a new high-order Discontinuous Galerkin approach to Maxwell's equations in the time domain. This approach is based on hexahedral meshes and uses a mass-lumping technique. Thanks to the orthogonality of the basis functions and a judicious choice of the approximation spaces, it provides an efficient solver for these equations in terms of storage and CPU time. To cite this article: G. Cohen et al., C. R. Physique 7 (2006).

A generalized mass lumping technique for vector finite-element solutions of the time-dependent Maxwell equations

Time-domain finite-element solutions of Maxwell's equations require the solution of a sparse linear system involving the mass matrix at every time step. This process represents the bulk of the computational effort in time-dependent simulations. As such, mass lumping techniques in which the mass matrix is reduced to a diagonal or block-diagonal matrix are very desirable. In this paper, we present a special set of high order 1-form (also known as curl-conforming) basis functions and reduced order integration rules that, together, allow for a dramatic reduction in the number of nonzero entries in a vector finite element mass matrix. The method is derived from the Nedelec curl-conforming polynomial spaces and is valid for arbitrary order hexahedral basis functions for finite-element solutions to the second-order wave equation for the electric (or magnetic) field intensity. We present a numerical eigenvalue convergence analysis of the method and quantify its accuracy and performance via a series of computational experiments.

The Andra Couplex 1 Test Case: Comparisons Between Finite-Element, Mixed Hybrid Finite Element and Finite Volume Element Discretizations

We present here results for the Andra Couplex 1 test case, obtained with the code Cast3m. This code is developped at the CEA (Commissariat р l'щnergie atomique) and is used mainly to solve problems of solid mechanics, fluid mechanics and heat transfers. Different types of discretization are available, among them finite element, finite volume and mixed hybrid finite element method. Cast3m is also a componant of the platteform Alliances (co-developped by Andra, CEA), which will be used by Andra for the safety calculation of an underground waste disposal in year 2004. We solve the Darcy equation for the water flow and a convection–diffusion transport equation for the Iodine 129 which escapes from a repository cave into the water. The water flow is calculated with a MHFE discretization. It is shown that this method provides sharp results even on relatively coarse grids. The convection–diffusion transport equation is discretized with FE (Finite Element), MHFE (Mixed Hybrid Finite Element) and FV (Finite Volume) methods. In our comparison, we point out the differences of these methods in term of accuracy, respect of the maximum principle and calculations cost. Neither the finite element nor the mixed hybrid finite element approach respects the maximum principle. This results in the presence of negative concentrations near the repository cave, whereas FV calculations respect the monotonicity. We show that mass lumping techniques suppress this problem but with strong restrictions on the grid. FE and MHFE approaches are more accurate than FV for the diffusion equation, but the overall results are equivalent since the advective terms are dominant in the far field and are discretized with centered schemes. We conclude by studying the influence of the grid: a very fine grid near the repository solves almost all the problems of monotonicity, without employing mass lumping techniques. We also observed a very important increase of the accuracy on a structured grid made up of rectangles.

A high‐order mass‐lumping procedure for B‐spline collocation method with application to incompressible flow simulations

AbstractThis paper presents new developments of the staggered spline collocation method for cost‐effective solution to the incompressible Navier–Stokes equations. Maximal decoupling of the velocity and the pressure is obtained by using the fractional step method of Gresho and Chan, allowing the solution to sparse elliptic problems only. In order to preserve the high‐accuracy of the B‐spline method, this fractional step scheme is used in association with a sparse approximation to the inverse of the consistent mass matrix. Such an approximation is constructed from local spline interpolation method, and represents a high‐order generalization of the mass‐lumping technique of the finite‐element method. A numerical investigation of the accuracy and the computational efficiency of the resulting semi‐consistent spline collocation schemes is presented. These schemes generate a stable and accurate unsteady Navier–Stokes solver, as assessed by benchmark computations. Copyright © 2003 John Wiley & Sons, Ltd.

An efficient finite-element time-domain method for the analysis of the coupling between wave and shielded enclosure

In this paper, we present an efficient model to study the coupling of an electromagnetic wave with two electronic equipment enclosures. The first one has two large apertures, and the second one presents a fine slot and two metallic plates. The three-dimensional Maxwell's equations are solved using a finite-element time-domain method with a mass lumping technique. The proposed model is shown to avoid numerical problems and decrease memory and CPU time. Numerical results obtained with or without lumping are presented and compared with those obtained by measurements.

On a collocation B-spline method for the solution of the Navier–Stokes equations

A novel B-spline collocation method for the solution of the incompressible Navier–Stokes equations is presented. The discretization employs B-splines of maximum continuity, yielding schemes with high-resolution power. The Navier–Stokes equations are solved by using a fractional step method, where the projection step is considered as a Div–Grad problem, so that no pressure boundary conditions need to be prescribed. Pressure oscillations are prevented by introducing compatible B-spline bases for the velocity and pressure, yielding efficient schemes of arbitrary order of accuracy. The method is applied to two-dimensional benchmark flows, and mass lumping techniques for cost-effective computation of unsteady problems are discussed.

FICTITIOUS DOMAINS, MIXED FINITE ELEMENTS AND PERFECTLY MATCHED LAYERS FOR 2-D ELASTIC WAVE PROPAGATION

We design a new and efficient numerical method for the modelization of elastic wave propagation in domains with complex topographies. The main characteristic is the use of the fictitious domain method for taking into account the boundary condition on the topography: the elastodynamic problem is extended in a domain with simple geometry, which permits us to use a regular mesh. The free boundary condition is enforced introducing a Lagrange multiplier, defined on the boundary and discretized with a nonuniform boundary mesh. This leads us to consider the first-order velocity-stress formulation of the equations and particular mixed finite elements. These elements have three main nonstandard properties: they take into account the symmetry of the stress tensor, they are compatible with mass lumping techniques and lead to explicit time discretization schemes, and they can be coupled with the Perfectly Matched Layer technique for the modeling of unbounded domains. Our method permits us to model wave propagation in complex media such as anisotropic, heterogeneous media with complex topographies, as it will be illustrated by several numerical experiments.

Fictitious Domains, Mixed Finite Elements and Perfectly Matched Layers for 2-D Elastic Wave Propagation

We design a new and efficient numerical method for the modelization of elastic wave propagation in domains with complex topographies. The main characteristic is the use of the fictitious domain method for taking into account the boundary condition on the topography: the elastodynamic problem is extended in a domain with simple geometry, which permits us to use a regular mesh. The free boundary condition is enforced introducing a Lagrange multiplier, defined on the boundary and discretized with a nonuniform boundary mesh. This leads us to consider the first-order velocity-stress formulation of the equations and particular mixed finite elements. These elements have three main nonstandard properties: they take into account the symmetry of the stress tensor, they are compatible with mass lumping techniques and lead to explicit time discretization schemes, and they can be coupled with the Perfectly Matched Layer technique for the modeling of unbounded domains. Our method permits us to model wave propagation in complex media such as anisotropic, heterogeneous media with complex topographies, as it will be illustrated by several numerical experiments.

Comparison of mass lumping techniques for solving the 3D Maxwell's equations in the time domain

This paper presents a comparison of mass lumping techniques for the time-domain solution of the 3D Maxwell's equations through edge elements. These techniques allow obtaining a fully explicit time integration procedure. We present two different techniques: the so-called classical method and the Lacoste method as well as a modified scheme to avoid numerical problems due to the shape of mesh elements. Numerical results are presented and compared with those obtained with a consistent method (without lumping).

An explicit 2D finite element time domain scheme for electromagnetic wave propagation

An explicit 2D FETD formulation for electromagnetic wave propagation is presented. The space discretization is obtained using finite element triangles. The domain is truncated using a Higdon absorbing boundary condition. The time discretization is performed using a four level scheme. Mass lumping technique makes it explicit. Several numerical parameters are studied, such as the mesh density, the order of the ABC and the numerical stability.

Massively parallel computational methods for finite element analysis of transient structural responses

With the emphasis on the finitely damped system (e.g. control structure interaction) case, two fully implicit and two semi-implicit sets of finite element method-based numerical algorithms are formulated for transient response analysis of space frame and truss structures in a massively parallel processing (MPP) environment. All algorithm sets use an implicit force calculation/vector equation of motion assembly procedure. The semi-implicit algorithms are based on the explicit central difference (CD) and the fourth-order Runge-Kutta (RK4) schemes, respectively, in conjunction with the use of mass lumping techniques so that solution of the recurrence equations for unknown displacements is reduced to a trivial diagonal matrix inversion operation. The fully implicit algorithm sets are based on the Newmark Beta (NB) and CD schemes, respectively, in conjunction with use of the (iterative) preconditioned conjugate gradient (PCG) method for solving the linear algebraic recurrence equations. The semi-implicit algorithm sets are fully implemented and assessed on an MPP CM-2 computer. A preliminary assessment of the fully implicit sets of algorithms is made on a Sun Workstation. These numerical study results show that the newly formulated MPP algorithms are, to a varying degree, superefficient (or potentially superefficient) on the CM-2 compared with, and even highly competitive against, the conventional sequential algorithms on an advanced serial computer.

Numerical modeling of axisymmetric electron beam devices using a coupled particle-finite element method

A coupled particle finite-element method is developed for the numerical resolution of the time-dependent relativistic Vlasov-Maxwell equations in axisymmetric geometries. The use of a conforming finite-element method for the Maxwell solver supplemented with a mass-lumping technique leads to an explicit system for the computation of the electromagnetic fields components. For the Vlasov solver, a fully vectorized algorithm is used for the location of the particles in an unstructured mesh made up with triangles. The numerical values of the electromagnetic fields radiated by a dipole are compared with the analytical solution. The whole Vlasov-Maxwell solver is tested on a simplified free-electron laser electron injector, designed for an RF-based free electron laser and the results are compared with those obtained with a finite-difference code. >

Developments in ultrasonic modeling with finite element analysis

An overview is given of finite element analysis and its application to the modeling of ultrasonic nondestructive evaluation phenomena. Following a discussion of the underlying weighted residual methodology, a mass-lumping technique is described which results in an efficient computer implementation for 2D geometries. Code predictions are compared with both analytical and experimental results, and data from studies of attenuation, anisotropy, defect interactions, and surface waves are given. Initial results from a full 3D formulation are also shown.