A posteriori error estimates in finite element acoustic analysis

We present an a posteriori error estimator for the approximations of the acoustic vibration modes obtained by a finite element method which does not present spurious or circulation modes for non zero frequencies. We prove that the proposed estimator is equivalent to the error in the approximation of the eigenvectors up to higher order terms with constants independent of the eigenvalues. Numerical results for some test examples are presented which show the good behavior of the estimator when it is used as local error indicator for adaptive refinement.Facultad de Ciencias Exacta

An implicit-explicit time discretization scheme for second-order semilinear wave equations with application to dynamic boundary conditions

We construct and analyze a second-order implicit-explicit (IMEX) scheme for the time integration of semilinear second-order wave equations. The scheme treats the stiff linear part of the problem implicitly and the nonlinear part explicitly. This makes the scheme unconditionally stable and at the same time very efficient, since it only requires the solution of one linear system of equations per time step. For the combination of the IMEX scheme with a general, abstract, nonconforming space discretization we prove a full discretization error bound. We then apply the method to a nonconforming finite element discretization of an acoustic wave equation with a kinetic boundary condition. This yields a fully discrete scheme and a corresponding a-priori error estimate

A quadratic nonconforming vector finite element for H (curl ; Ω) ∩ H (div ; Ω)

We present a quadratic nonconforming vector finite element for problems posed on the space H (curl ; Ω) ∩ H (div ; Ω), where Ω ⊂ R . Generalizations to higher order and higher dimension are also discussed. © 2008 Elsevier Ltd. All rights reserved.

An efficient multi-time step FEM–SFEM iterative coupling procedure for elastic–acoustic interaction problems

An iterative coupling methodology between the Finite Element Method (FEM) and the Spectral Finite Element Method (SFEM) for the modeling of coupled elastic-acoustic problems in the time domain is presented here. Since the iterative coupling procedure allows the use of a nonconforming mesh at the interface between the subdomains, the difference in the element sizes concerning the FEM and SFEM is handled in a straightforward and efficient manner, thereby retaining all the advantages of the SFEM. By means of the HHT time integration method, controllable numerical damping can be introduced in one of the subdomains, increasing the robustness of the method and improving the accuracy of the results; besides, independent time-step sizes can be considered within each subdomain, resulting in a more efficient algorithm. In this work, a modification in the subcycling procedure is proposed, ensuring not only an efficient and accurate methodology but also avoiding the computation of a relaxation parameter. Numerical simulations are presented in order to illustrate the accuracy and potential of the proposed methodology.CAPES, UFJF, UFSJ, FAPEMIG and CNP

Kernel-free boundary integral method for two-phase Stokes equations with discontinuous viscosity on staggered grids

A discontinuous viscosity coefficient makes the jump conditions of the velocity and normal stress coupled together, which brings great challenges to some commonly used numerical methods to obtain accurate solutions. To overcome the difficulties, a kernel free boundary integral (KFBI) method combined with a modified marker-and-cell (MAC) scheme is developed to solve the two-phase Stokes problems with discontinuous viscosity. The main idea is to reformulate the two-phase Stokes problem into a single-fluid Stokes problem by using boundary integral equations and then evaluate the boundary integrals indirectly through a Cartesian grid-based method. Since the jump conditions of the single-fluid Stokes problems can be easily decoupled, the modified MAC scheme is adopted here and the existing fast solver can be applicable for the resulting linear saddle system. The computed numerical solutions are second order accurate in discrete $\ell^2$-norm for velocity and pressure as well as the gradient of velocity, and also second order accurate in maximum norm for both velocity and its gradient, even in the case of high contrast viscosity coefficient, which is demonstrated in numerical tests

The boundary element method applied to static and dynamic crack problems using hypersingular boundary integral equations

The need for hypersingular boundary integral equations in crack problems is motivated through acoustic and elastic wave scattering from a thin screen and crack. By integrating over a small (not necessarily symmetric) neighborhood about the singular point, and the rest of the boundary, and identifying terms from the integrals over the two surfaces which cancel each other, the finite-part of the hypersingular integral is defined for curved surfaces in both two and three dimensions. Stokes\u27 theorem is used to regularize the hypersingular integrals to a form conducive to simple numerical integration techniques. With no prior assumptions on the discretizasion or integration by parts, this method results in integrals which are at most weakly singular. The equivalence of this approach to the finite-part of the hypersingular integral is established;The necessary condition on the density function for the hypersingular integral equation to have meaning and the consequences on the solution of not satisfying the necessary conditions is discussed. This new formulation places restrictions on the choice of shape functions and the possible location of the collocation points within elements due to the smoothness requirement on the density function. Such restrictions for regular boundary integral equations with Cauchy principal value integrals are also discussed. The different kinds of integrals encountered in a hypersingular boundary integral equation such as weakly singular integrals, nearly singular integrals and regular area and line integrals are studied. Discretization considerations for precise and efficient numerical computation of these integrals in the context of the boundary element method is established and the influence of discretization on the solution is highlighted through numerical examples. Examples are chosen from problems of acoustic and elastic wave scattering from thin screens and cracks in three dimensions

Coupled structural, thermal, phase-change and electromagnetic analysis for superconductors, volume 1

This research program has dealt with the theoretical development and computer implementation of reliable and efficient methods for the analysis of coupled mechanical problems that involve the interaction of mechanical, thermal, phase-change and electromagnetic subproblems. The focus application has been the modeling of superconductivity and associated quantum-state phase-change phenomena. In support of this objective the work has addressed the following issues: (1) development of variational principles for finite elements; (2) finite element modeling of the electromagnetic problem; (3) coupling of thermal and mechanical effects; and (4) computer implementation and solution of the superconductivity transition problem. The research was carried out over the period September 1988 through March 1993. The main accomplishments have been: (1) the development of the theory of parametrized and gauged variational principles; (2) the application of those principled to the construction of electromagnetic, thermal and mechanical finite elements; and (3) the coupling of electromagnetic finite elements with thermal and superconducting effects; and (4) the first detailed finite element simulations of bulk superconductors, in particular the Meissner effect and the nature of the normal conducting boundary layer. The grant has fully supported the thesis work of one doctoral student (James Schuler, who started on January 1989 and completed on January 1993), and partly supported another thesis (Carmelo Militello, who started graduate work on January 1988 completing on August 1991). Twenty-three publications have acknowledged full or part support from this grant, with 16 having appeared in archival journals and 3 in edited books or proceedings

On Trefftz and weak Trefftz discontinuous Galerkin approaches for medium-frequency acoustics

International audienceIn this paper, the wave approach called the Variational Theory of Complex Rays (VTCR), which was developed for medium-frequency acoustics and vibrations, is revisited as a discontinuous Galerkin method. Extensions leading to a weak Trefftz constraint are introduced. This weak Trefftz discontinuous Galerkin approach enables hybrid FEM/VTCR strategies to be developed easily, and paves the way for new computational techniques for the resolution of engineering problems. This paper presents some of the fundamental properties of the approach, which is illustrated by several numerical examples