Modified integration rules for reducing dispersion error in finite element methods

doi:10.1016/j.cma.2003.09.010

Computer Methods in Applied Mechanics and Engineering

Volume 193, Issues 3–5, 23 January 2004, Pages 275-287

https://doi.org/10.1016/j.cma.2003.09.010 Get rights and content

Abstract

This paper describes a simple but effective technique for reducing dispersion errors in finite element solutions of time-harmonic wave propagation problems. The method involves a simple shift of the integration points to locations away from conventional Gauss or Gauss–Lobatto integration points. For bilinear rectangular elements, such a shift results in fourth-order accuracy with respect to dispersion error (error in wavelength), as opposed to the second-order accuracy resulting from conventional integration. Numerical experiments involving distorted meshes indicate that the method has superior performance in comparison with other dispersion reducing finite elements.

Introduction

Finite element methods, when applied to time-harmonic wave propagation problems, incur solution errors that can be classified into error in the amplitude and error in the wavelength. In large-scale problems where the domain size is much larger than the wavelength, the error in the wavelength, known as the dispersion error, tends to accumulate and produce large phase errors, resulting in completely erroneous results [1]. On the other hand, amplitude error is relatively benign in that it does not increase with growing domain size. This paper focuses on the reduction of dispersion error.

The dispersion error could be reduced either by using a finer mesh, or by using higher order finite elements. Both of these options come with large increase in computational cost, making some of the large-scale simulations prohibitively expensive. It is thus desirable to reduce the dispersion for lower order finite elements. It can be quickly realized that dispersion error in one-dimensional problems can be completely eliminated by resorting to analytical solutions [1]. For higher dimensions, the dispersion error cannot be completely eliminated [2], but can be reduced. Over the past two decades, several researchers have proposed various methods for reducing the dispersion in higher dimensions, especially for low-order finite elements. The following is a brief survey.

The peculiar aspect of dispersion error in higher dimensions is its anisotropy, i.e., the dispersion error depends on the angle between the wave propagation direction and mesh orientation. Mullen and Belytchko [3] have analyzed this property, and compared the performance of various finite element as well as standard finite difference techniques. Marfurt [4] reduced the dispersion error by taking generalized average of consistent and lumped mass matrices, but the anisotropy still remains. Later, Krenk [5] proposed selective scaling of stiffness and mass matrices to reduce the dispersion error. The implication of his method for distorted meshes is not clear. More recently, Harari and Hughes [6] have used Galerkin/least squares (GLS) method [7] to reduce dispersion on square meshes. The resulting dispersion error, although reduced, is still significant. Thompson and Pinsky [8] modified this method to further reduce the dispersion, but their method still suffers from anisotropy. Babuska and Ihlenburg [9] used the concepts of generalized finite element method (GFEM) and finite difference stencils to derive a method that reduces the dispersion by several orders of magnitude, but only for square meshes. Oberai and Pinsky [10] used the concept of edge residuals to derive the same scheme in the context of finite elements. However, when extended to rectangular or unstructured meshes, their scheme becomes heuristic in nature and loses its accuracy properties. Babuska and Melenk [11] used a partition of unity method to reduce the dispersion, but for the method to be practical, one needs to know the behavior of the solution a priori, which is often not the case. The method of residual free bubbles, developed by Franca et al. [12], is a robust method that can improve the accuracy by effective enrichment of shape functions, but such enrichment demands significant increase in computational cost, especially for non-uniform meshes. Similar comment applies to the sub-grid modeling approach proposed by Cipolla [13]. Weighted-averaging FEM proposed by Min et al. [14] significantly reduces dispersion, but only on square meshes and at significant increase in computational cost.

A fundamental limitation of most of the above methods is that they are applicable only for propagating waves in homogeneous media discretized by regular square meshes. Many important and practical problems of wave propagation involve complex geometries and material heterogeneities, and cannot be discretized using square meshes. Although dispersion reducing methods for triangular meshes may offer added flexibility [15], the current versions are limited to uniform structured meshes.

With the aim of reducing dispersion on unstructured meshes, we recently proposed a dispersion-reducing framework named the Local Mesh-Dependent Augmented Galerkin (L-MAG) methods [16]. L-MAG methods provide a systematic procedure to reduce the dispersion error on rectangular meshes. The methods are more flexible than the other dispersion reducing methods in that they are accurate for meshes containing rectangular elements of varying size. However, although they perform better than other dispersion reducing methods, L-MAG methods lose their accuracy properties when used for distorted meshes.

In this paper, we develop an effective dispersion reducing technique that is significantly simpler than all the existing dispersion reducing methods. The method is based on the simple observation that by shifting the Gauss quadrature to some unconventional locations would reduce the anisotropic dispersion significantly. In fact, for rectangular elements, the proposed method makes the numerical wavelength fourth-order accurate, as opposed to the second-order accuracy obtained from conventional integration rules. Needless to say, the proposed method can be directly applied to distorted meshes. The remainder of the paper focuses on the derivation of the method, its application to structured and unstructured meshes and the comparison of its performance with some existing dispersion reducing finite element methods.

The following is the outline of the paper. Section 2 contains discussion of the variational boundary value problem and the Galerkin finite element discretization of the problem. The idea of shifting the integration points is discussed in Section 3. The locations of the integration points that reduce the dispersion error are obtained in Section 4. Section 5 focuses on evaluating the performance of proposed method with the help of numerical examples. The paper is concluded in Section 6 with some closing remarks.

Section snippets

Finite element formulation of time-harmonic wave propagation problems

Helmholtz equation, also called the reduced wave equation, is used to analyze scalar wave propagation encountered in linear acoustic and anti-plane shear problems. The corresponding boundary value problem takes the form: find $u:Ω→ C$ , such that $− ∇^{2} u−k^{2} u=f in Ω$ with boundary conditions $u=g on Γ_{g},$ $∇ u· n +b(u)=F on Γ_{h} .$

In the above, u is the complex-valued field variable (acoustic pressure, or anti-plane displacement), k is the wave number associated with the excitation frequency (k=ω/c), g is the specified

Generalized integration rules for evaluation of stiffness and mass matrices

Since finite elements can be distorted, it is customary to map the finite elements from a square parent element ({−1<ξ<+1}×{−1<η<+1}), and transform the above integrals into the natural coordinate system as follows: $K_{e} =∫_{−1}^{+1} ∫_{−1}^{+1} (∇ N)^{T} ∇ N det (J) d ξ d η, M_{e} =∫_{−1}^{+1} ∫_{−1}^{+1} N^{T} N det (J) d ξ d η.$ The above integrals are traditionally evaluated using Gauss quadrature $∫_{−1}^{+1} ∫_{−1}^{+1} φ d ξ d η=∑_{i=1}^{n_{g}} W_{i} φ(ξ_{i},η_{i}),$ where n_g is the number of Gauss points, W_i are the weights and (ξ_i,η_i) are the local coordinates of Gauss points. For the

Dispersion reducing integration rule

In this section, by taking advantage of the flexibility offered by the generalized integration rule, we try to fix the integration points so that the dispersion error is minimized for rectangular meshes. The coefficient matrix ( $S = K −k^{2} M$ ) of a four-node rectangular finite element of size Δx×Δy, when computed using Eqs. , , is given by $S_{e} = S_{0} S_{x} S_{xy} S_{y} S_{x} S_{0} S_{y} S_{xy} S_{xy} S_{y} S_{0} S_{x} S_{y} S_{xy} S_{x} S_{0},$ where $S_{0} = (Δ x^{2} + Δ y^{2})(α^{2} +1) 4 Δ x Δ y − Δ x Δ y 16 (1+β^{2})^{2} k^{2},S_{x} = (Δ x^{2} − Δ y^{2})−α^{2} (Δ x^{2} + Δ y^{2}) 4 Δ x Δ y − Δ x Δ y 16 (1−β^{4})k^{2},S_{y} = (Δ y^{2} − Δ x^{2})−α^{2} (Δ x^{2} + Δ y^{2}) 4 Δ x Δ y − Δ x Δ y 16$

Numerical examples

The performance of the proposed modified integration rule is tested using three examples: simulation of plane wave, simulation of radiation from a point source, and simulation of asymmetric radiation from a circular cavity. All the problems are solved using the proposed method, as well as some existing methods, namely Galerkin method with consistent integration (referred to as Galerkin in the figures), modified Galerkin least squares method [8] (GLSm), generalized finite element [9] or

Concluding remarks

In this paper, we developed a simple but effective method for reducing numerical dispersion in finite element solutions of time-harmonic wave propagation problems. By simply shifting the integration points to the unconventional location of $(± 2/3,± 2/3)$ , we were able to obtain fourth-order accuracy in the wave number, as opposed to the second-order accuracy resulting from conventional methods. In spite of its simplicity, the proposed method appears to outperform more involved dispersion reducing

Acknowledgements

This material is based upon work supported by the National Science Foundation under Grant No. 0100188. Any opinions, findings, and conclusions or recommendations expressed in this material are those of the authors and do not necessarily reflect the views of the National Science Foundation.

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Modified integration rules for reducing dispersion error in finite element methods

Abstract

Introduction

Section snippets

Finite element formulation of time-harmonic wave propagation problems

Generalized integration rules for evaluation of stiffness and mass matrices

Dispersion reducing integration rule

Numerical examples

Concluding remarks

Acknowledgements

Comput. Methods Appl. Mech. Engrg.

Comput. Methods Appl. Mech. Engrg.

Comput. Methods Appl. Mech. Engrg.

Comput. Methods Appl. Mech. Engrg.

J. Computat. Phys.

Is the pollution effect of the FEM avoidable for the Helmholtz equation considering high wave numbers?

S.I.A.M. J. Numer. Anal.

Dispersion analysis of finite element semidiscretizations of the two-dimensional wave equation

Int. J. Numer. Methods Engrg.

Accuracy of finite difference and finite element modeling of the scalar and elastic wave equation

Geophysics