Finite element solution of the Helmholtz equation with high wave number Part I: The h-version of the FEM

AbstractThe paper addresses the properties of finite element solutions for the Helmholtz equation. The h-version of the finite element method with piecewise linear approximation is applied to a one-dimensional model problem. New results are shown on stability and error estimation of the discrete model. In all propositions, assumptions are made on the magnitude of hk only, where k is the wavelength and h is the stepwidth of the FE-mesh. Previous analytical results had been shown with the assumption that k2h is small. For medium and high wavenumber, these results do not cover the meshsizes that are applied in practical applications. The main estimate reveals that the error in H1-norm of discrete solutions for the Helmholtz equation is polluted when k2h is not small. The error is then not quasioptimal; i.e., the relation of the FE-error to the error of best approximation generally depends on the wavenumber k. It is noted that the pollution term in the relative error is of the same order as the phase lead of the numerical solution. In the result of this analysis, thorough and rigorous understanding of error behavior throughout the range of convergence is gained. Numerical results are presented that show sharpness of the error estimates and highlight some phenomena of the discrete solution behavior. The h-p-version of the FEM is studied in Part II

Efficient implementation of high-order finite elements for Helmholtz problems

Computational modeling remains key to the acoustic design of various applications, but it is constrained by the cost of solving large Helmholtz problems at high frequencies. This paper presents an efficient implementation of the high-order Finite Element Method for tackling large-scale engineering problems arising in acoustics. A key feature of the proposed method is the ability to select automatically the order of interpolation in each element so as to obtain a target accuracy while minimising the cost. This is achieved using a simple local a priori error indicator. For simulations involving several frequencies, the use of hierarchic shape functions leads to an efficient strategy to accelerate the assembly of the finite element model. The intrinsic performance of the high-order FEM for 3D Helmholtz problem is assessed and an error indicator is devised to select the polynomial order in each element. A realistic 3D application is presented in detail to demonstrate the reduction in computational costs and the robustness of the a priori error indicator. For this test case the proposed method accelerates the simulation by an order of magnitude and requires less than a quarter of the memory needed by the standard FEM

On stability of discretizations of the Helmholtz equation (extended version)

We review the stability properties of several discretizations of the Helmholtz equation at large wavenumbers. For a model problem in a polygon, a complete $k$-explicit stability (including $k$-explicit stability of the continuous problem) and convergence theory for high order finite element methods is developed. In particular, quasi-optimality is shown for a fixed number of degrees of freedom per wavelength if the mesh size $h$ and the approximation order $p$ are selected such that $kh/p$ is sufficiently small and $p = O(\log k)$, and, additionally, appropriate mesh refinement is used near the vertices. We also review the stability properties of two classes of numerical schemes that use piecewise solutions of the homogeneous Helmholtz equation, namely, Least Squares methods and Discontinuous Galerkin (DG) methods. The latter includes the Ultra Weak Variational Formulation

A new fast multi-domain BEM to model seismic wave propagation and amplification in 3D geological structures

International audienceThe analysis of seismic wave propagation and amplification in complex geological structures raises the need for efficient and accurate numerical methods. The solution of the elastodynamic equations using traditional boundary element methods (BEMs) is greatly hindered by the fully-populated nature of the matrix equations arising from the discretization. In a previous study limited to homogeneous media, the present authors have established that the Fast Multipole (FM) method reduces the complexity of a 3-D elastodynamic BEM to $N \log N$ per GMRES iteration and demonstrated its effectiveness on 3-D canyon configurations. In this article, the frequency-domain FM-BEM methodology is extented to 3-D elastic wave propagation in piecewise-homogeneous domains in the form of a FM-accelerated multi-region BE-BE coupling approach. This new method considerably enhances the capability of the BEM for studying the propagation of seismic waves in 3-D alluvial basins of arbitrary geometry embedded in semi-infinite media. Several fully 3-D examples (oblique SV-waves) representative of such configurations validate and demonstrate the capabilities of the multi-domain fast multipole approach. They include comparisons with available (low-frequency) results for various types of incident wavefields, and time-domain results obtained by means of Fourier synthesis

Wavenumber-explicit continuity and coercivity estimates in acoustic scattering by planar screens

We study the classical first-kind boundary integral equation reformulations of time-harmonic acoustic scattering by planar sound-soft (Dirichlet) and sound-hard (Neumann) screens. We prove continuity and coercivity of the relevant boundary integral operators (the acoustic single-layer and hypersingular operators respectively) in appropriate fractional Sobolev spaces, with wavenumber-explicit bounds on the continuity and coercivity constants. Our analysis is based on spectral representations for the boundary integral operators, and builds on results of Ha-Duong (Jpn J Ind Appl Math 7:489--513 (1990) and Integr Equat Oper Th 15:427--453 (1992)).Comment: v2 has minor corrections compared to v1. arXiv admin note: substantial text overlap with arXiv:1401.280

Life Cycle Design. Teilvorhaben A5.1: Verbesserung der Daempfungsansaetze fuer die Berechnung von Schiffsschwingungen Abschlussbericht

In this project, a general model of damping is used including local effects of internal and external damping. The application of this approach to finite-element models leads to non-proportional damping matrices. Efficient solution methods for the corresponding discrete dynamic equations have been developed and implemented. Measurements of vibrations have been carried out on three container vessels. The main focus of the measurements was on the medium frequency range around the major forcing frequencies. The methods of experimental modal analysis are in general not applicable in this frequency range due to high mode density. A different approach of identifying damping parameters has therefore been taken here. The damping parameters in the physical models of damping are considered unknown. Their optimal values are sought from the comparison of the measurements with the results of finite element computations. In the computations, the damping coefficients depend on the forcing frequency as well as on the element location (spatial dependence). Conclusions on the optimal selection of the parameters are drawn. The influence of modifications in the hydrodynamic masses and in changes of the finite element model is also considered. (orig.)SIGLEAvailable from TIB Hannover: RR 7836(99-139) / FIZ - Fachinformationszzentrum Karlsruhe / TIB - Technische InformationsbibliothekBundesministerium fuer Bildung und Forschung (BMBF), Bonn (Germany)DEGerman