Isaac Harari

The Discontinuous Enrichment Method

We propose a finite element based discretization method in which the standard polynomial field is enriched within each element by a nonconforming field that is added to it. The enrichment contains free-space solutions of the homogeneous differential equation that are not represented by the underlying polynomial field. Continuity of the enrichment across element interfaces is enforced weakly by Lagrange multipliers. In this manner, we expect to attain high coarse-mesh accuracy without significant degradation of conditioning, assuring good performance of the computation at any mesh resolution. Examples of application to acoustics and advection-diffusion are presented.

Galerkin/least-squares finite element methods for the reduced wave equation with non-reflecting boundary conditions in unbounded domains

Abstract Finite element methods are constructed for the reduced wave equation in unbounded domains. Exterior boundary conditions for a computational problem are derived from an exact relation between the solution and its derivatives on an artificial boundary by the DtN method, precluding singular behavior in finite element models. Galerkin and Galerkin/least-squares finite element methods are presented. Model problems of radiation with inhomogeneous Neumann boundary conditions in plane and spherical configurations are employed to design and evaluate the numerical methods in the entire range of propagation and decay. The Galerkin/least-squares method with DtN boundary conditions is designed to exhibit superior behavior for problems of acoustics, providing accurate solutions with relatively low mesh resolution and allowing numerical damping of unresolved waves. General convergence results guarantee the good performance of Galerkin/least-squares methods on all configurations of practical interest. Numerical tests validate these conclusions.

A discontinuous Galerkin method with Lagrange multipliers for the solution of Helmholtz problems in the mid-frequency regime

We present a discontinuous Galerkin method (DGM) for the solution of the Helmholtz equation in the mid-frequency regime. Our approach is based on the discontinuous enrichment method in which the standard polynomial field is enriched within each finite element by a non-conforming field that contains free-space solutions of the homogeneous partial differential equation to be solved. Hence, for the Helmholtz equation, the enrichment field is chosen here as the superposition of plane waves. We enforce a weak continuity of these plane waves across the element interfaces by suitable Lagrange multipliers. Preliminary results obtained for two-dimensional model problems discretized by uniform meshes reveal that the proposed DGM enables the development of elements that are far more competitive than both the standard linear and the standard quadratic Galerkin elements for the discretization of Helmholtz problems.

Finite element methods for the Helmholtz equation in an exterior domain: model problems

Abstract Finite element methods are presented for the reduced wave equation in unbounded domains. Model problems of radiation with inhomogeneous Neumann boundary conditions, including the effects of a moving acoustic medium, are examined for the entire range of propagation and decay. Exterior boundary conditions for the computational problem over a finite domain are derived from an exact relation between the solution and its derivatives on that boundary. Galerkin, Galerkin/least-squares and Galerkin/gradient least-squares finite element methods are evaluated by comparing errors pointwise and in integral norms. The Galerkin/least-squares method is shown to exhibit superior behavior for this class of problems.

What are C and h ?: inequalities for the analysis and design of finite element methods

Abstract Increasing mathematical analysis of finite element methods is motivating the inclusion of mesh-dependent terms in new classes of methods for a variety of applications. Several inquualities of functional analysis are often employed in convergence proofs. In the following, Poincare-Friedrichs inequalities, inverse estimates and least-squares bounds are characterized as tools for the error analysis and practical design of finite element methods with terms that depend on the mesh parameter. Sharp estimates of the constants of these inequalities are provided, and precise definitions of mesh size that arise naturally in the context of different problems in terms of element geometry are derived.

Analysis of continuous formulations underlying the computation of time-harmonic acoustics in exterior domains

Abstract Potential non-uniqueness of boundary representations of the Helmholtz equation underscores the importance of investigating continuous boundary-based and domain-based formulations, which is the main purpose of this work. Uniqueness properties of the solutions of boundary integral equations are reviewed. We analyze formulations for domain-based computation that are derived by the DtN method, which imposes a relation between the function and its normal derivative on an artificial boundary. The DtN formulation is shown to possess non-reflective boundary conditions and to give rise to exact (and thereby unique) solutions. In practical implementation the DtN map is often truncated. The truncated DtN operator fails to completely inhibit reflection of higher modes, resulting in loss of uniqueness at characteristic wave numbers of higher harmonics. However, simple expressions that determine a sufficient number of terms in the operator for unique solutions at any given wave number are derived. We prove that a local approximation of the boundary conditions restores uniqueness for all wave numbers, and derive its three-dimensional version.

ANALYTICAL AND NUMERICAL STUDIES OF A FINITE ELEMENT PML FOR THE HELMHOLTZ EQUATION

A symmetric PML formulation that is suitable for finite element computation of time-harmonic acoustic waves in exterior domains is analyzed. Dispersion analysis displays the dependence of the discrete representation of the PML parameters on mesh refinement. Stabilization by modification of the coefficients is employed to improve PML performance, in conjunction with standard stabilized finite elements in the Helmholtz region. Numerical results validate the good performance of this finite element PML approach.

A cost comparison of boundary element and finite element methods for problems of time-harmonic acoustics

Abstract The cost of obtaining solutions to problems governed by the Helmholtz equation in both interior and exterior domains by means of boundary element and finite element methods is studied and compared. The main emphasis is on the computational effort required to solve the systems of equations emanating from the two methods. Boundary element methods require fewer equations to be solved by virtue of the fact that only boundaries are discretized. These equations, however, are less structured than those of finite element methods and hence cost-effectiveness is not as clear cut as might be expected. Both direct and iterative solution techniques are examined. For interior problems finite element methods are more economical on most practical configurations. Finite elements also appear to possess a certain computational advantage on the exterior problems examined, and, in general, are definitely competitive with boundary element methods. The cost-effectiveness of the two solution strategies is examined. Some issues of equation formation are also addressed.

Recent developments in finite element methods for structural acoustics

SummaryThe study of structural acoustics involves modeling acoustic radiation and scattering, primarily in exterior regions, coupled with elastic and structural wave propagation. This paper reviews recent progress in finite element analysis that renders computation a practical tool for solving problems of structural acoustics. The cost-effectiveness of finite element methods is composed of several ingredients. Boundary-value problems in unbounded domains are inappropriate for direct discretization. Employing DtN methodology yields an equivalent problem that is suitable for finite element analysis by posing impedance relations at an artificial exterior boundary. Vell-posedness of the resulting continuous formulations is discussed, leading to simple guidelines for practical implementation and verifying that DtN boundary conditions provide a suitable basis for computation.Approximation by Galerkin finite element methods results in spurious dispersion, degrading with reduced wave resolution. Accuracy is improved by Galerkin/least-squares and related technologies on the basis of detailed examinations of discrete errors in simplified settings, relaxing wave-resolution requirements. This methodology is applied to time-harmonic problems of acoustics and coupled problems of structural acoustics. Space-time finite methods based on time-discontinuous Galerkin/least-squares are derived for transient problems of structural acoustics. Numerical results validate the superior performance of Galerkin/least-squares finite elements for problems of structural acoustics.A comparative study of the cost of computation demonstrates that Galerkon/least-squares finite element methods are economically competitive with boundary element methods, the prevailing numerical approach to exterior problems of acoustics. Efficient iterative methods are derived for solving the large-scale matrix problems that arise in structural acoustics computation of realistics configuration at high wavenumbers. An a posteriori error estimator and adaptive strategy are developed for time-harmonic acoustic problems and the role of adaptivity in reducing the cost of computation is addressed.

Stabilization of electrostatically actuated microstructures using parametric excitation

Electrostatically actuated microstructures are inherently nonlinear and can become unstable. Pull-in instability is encountered as a basic instability mechanism. We demonstrate that the parametric excitation of a microstructure by periodic (ac) voltages may have a stabilizing effect and permits an increase of the steady (dc) component of the actuation voltage beyond the pull-in value. An elastic string as well as a cantilever beam are considered in order to illustrate the influence of fast-scale excitation on the slow-scale behavior. The main conclusions about the stability are drawn using the simplest model of a parametrically excited system described by Mathieu and Hills equations. Theoretical results are verified by numerical analysis of microstructure subject to nonlinear electrostatic forces and performed by using Galerkin decomposition with undamped linear modes as base functions. The parametric stabilization of a cantilever beam is demonstrated experimentally.