Alfredo Bermúdez

Upwind methods for hyperbolic conservation laws with source terms

Abstract This paper deals with the extension of some upwind schemes to hyperbolic systems of conservation laws with source term. More precisely we give methods to get natural upwind discretizations of the source term when the flux is approximated by using flux-difference or flux-splitting techniques. In particular, the Q- schemes of Roe and van Leer and the flux-splitting techniques of Steger-Warming and Vijayasundaram are considered. Numerical results for a scalar advection equation with nonlinear source and for the one-dimensional shallow water equations are presented. In the last case we compare the different schemes proposed in terms of a conservation property. When this property does not hold, spurious numerical waves can appear which is the case for the centred discretization of the source term.

Upwind schemes for the two-dimensional shallow water equations with variable depth using unstructured meshes

In this paper, certain well-known upwind schemes for hyperbolic equations are extended to solve the two-dimensional Saint-Venant (or shallow water) equations. We consider unstructured meshes and a new type of finite volume to obtain a suitable treatment of the boundary conditions. The source term involving the gradient of the depth is upwinded in a similar way as the flux terms. The resulting schemes are compared in terms of a conservation property. For the time discretization we consider both explicit and implicit schemes. Finally, we present the numerical results for tidal flows in the Pontevedra ria, Galicia, Spain.

Finite element computation of the vibration modes of a fluid—solid system

Abstract In this paper we solve the interior elastoacoustic problem by a finite element method which does not present spurious or circulation modes for nonzero frequencies. It consists of classical triangular lagrangian elements for the solid and lowest order triangular Raviart-Thomas elements for the fluid. Transmission conditions at the fluid-solid interface are taken into account in a weak sense. Numerical results for some test examples are presented which show the good performance of the proposed methodology.

Finite element vibration analysis of fluid-solid systems without spurious modes

This paper deals with the finite element approximation of the vibration modes of a problem with fluid–structure interaction. Displacement variables are used for both the fluid and the solid. To avoid the typical spurious modes of this formulation we introduce a nonconforming discretization. Error estimates for the approximation of eigenvalues and eigenvectors are given.

A Finite Element Method with Lagrange Multipliers for Low-Frequency Harmonic Maxwell Equations

The aim of this paper is to analyze a finite element method to solve the low-frequency harmonic Maxwell equations in a bounded domain containing conductors and dielectrics. This system of partial differential equations is a model for the so-called eddy currents problem. After writing this problem in terms of the magnetic field, it is discretized by Nedelec edge finite elements on a tetrahedral mesh. Error estimates are easily obtained if the curl-free condition is imposed on the elements in the dielectric domain. Then, the curl-free condition is imposed, at a discrete level, by introducing a piecewise linear multivalued potential. The resulting problem is shown to be a discrete version of other continuous formulation in which the magnetic field in the dielectric part of the domain has been replaced by a magnetic potential. Moreover, this approach leads to an important saving in computational effort. Problems related to the topology are also considered in that the possibility of having a nonsimply connected dielectric domain is taken into account. Implementation issues are discussed, including an amenable procedure to impose the boundary conditions by means of a Lagrange multiplier. Finally, the method is applied to solve a three-dimensional model problem: a cylindrical electrode surrounded by dielectric.

A finite element solution of an added mass formulation for coupled fluid-solid vibrations

Summary. A finite element method to approximate the vibration modes of a structure in contact with an incompressible fluid is analyzed in this paper. The effect of the fluid is taken into account by means of an added mass formulation, which is one of the most usual procedures in engineering practice. Gravity waves on the free surface of the liquid are also considered in the model. Piecewise linear continuous elements are used to discretize the solid displacements, the variables to compute the added mass terms and the vertical displacement of the free surface, yielding a non conforming method for the spectral coupled problem. Error estimates are settled for approximate eigenfunctions and eigenfrequencies. Implementation issues are discussed and numerical experiments are reported. In particular the method is compared with other numerical scheme, based on a pure displacement formulation, which has been recently analyzed.

Finite element analysis of compressible and incompressible fluid-solid systems

This paper deals with a finite element method to solve interior fluid-structure vibration problems valid for compressible and incompressible fluids. It is based on a displacement formulation for both the fluid and the solid. The pressure of the fluid is also used as a variable for the theoretical analysis yielding a well posed mixed linear eigenvalue problem. Lowest order triangular Raviart-Thomas elements are used for the fluid and classical piece-wise linear elements for the solid. Transmission conditions at the fluid-solid interface are taken into account in a weak sense yielding a nonconforming discretization. The method does not present spurious or circulation modes for nonzero frequencies. Convergence is proved and error estimates independent of the acoustic speed are given. For incompressible fluids, a convenient equivalent stream function formulation and a post-process to compute the pressure are introduced.

Finite Element Analysis of a Quadratic Eigenvalue Problem Arising in Dissipative Acoustics

A quadratic eigenvalue problem arising in the determination of the vibration modes of an acoustic fluid contained in a cavity with absorbing walls is considered. The problem is shown to be equivalent to the spectral problem for a noncompact operator and a thorough spectral characterization is given. A numerical discretization based on Raviart--Thomas finite elements is analyzed. The method is proved to be free of spurious modes and to converge with optimal order. Implementation issues and numerical experiments confirming the theoretical results are reported.

FINITE ELEMENT SOLUTION OF INCOMPRESSIBLE FLUID–STRUCTURE VIBRATION PROBLEMS

In this paper we solve an eigenvalue problem arising from the computation of the vibrations of a coupled system, incompressible fluid – elastic structure, in absence of external forces. We use displacement variables for both the solid and the fluid but the fluid displacements are written as curls of a stream function. Classical linear triangular finite elements are used for the solid displacements and for the stream function in the fluid. The kinematic transmission conditions at the fluid–solid interface are taken into account in a weak sense by means of a Lagrange multiplier. The method does not present spurious or circulation modes for non-zero frequencies. Numerical results are given for some test cases.

Transient numerical simulation of a thermoelectrical problem in cylindrical induction heating furnaces

Abstract This paper concerns the mathematical modelling and numerical solution of thermoelectrical phenomena taking place in an axisymmetric induction heating furnace. We formulate the problem in a two-dimensional domain and propose a finite element method and an iterative algorithm for its numerical solution. We also provide a family of one-dimensional analytical solutions which are used to test the two-dimensional code and to predict the behaviour of the furnace under special conditions. Some numerical results for an industrial furnace used in silicon purification are shown.