Elsevier

Computers & Structures

Volume 237, September 2020, 106273
Computers & Structures

Transient wave propagation in inhomogeneous media with enriched overlapping triangular elements

https://doi.org/10.1016/j.compstruc.2020.106273Get rights and content

Highlights

  • We study and use overlapping finite elements in transient wave propagation dynamics.

  • The Bathe time integration is used for solutions in inhomogeneous media.

  • The total dispersion error is split into the spatial error and temporal error.

  • We analyze the errors and show monotonic convergence to exact solutions.

Abstract

We study and use overlapping triangular finite elements enriched by trigonometric functions and implicit time integration to solve transient wave propagations in inhomogeneous media. We show explicitly that the total dispersion error of the calculated solutions can be split into two parts, the spatial error and temporal error. The study of the spatial dispersion error shows the effectiveness of the enriched overlapping finite elements compared to the traditional finite elements and the overlapping finite elements without enrichment. The study of the temporal error of the Bathe time integration scheme shows monotonic convergence to zero with decreasing time step size. The result is that we see monotonic convergence to exact solutions as the mesh of enriched overlapping finite elements is refined and the time step is decreased. We demonstrate the effectiveness of using the proposed scheme in the solution of waves traveling in inhomogeneous media at different speeds, where reflected and transmitted waves are predicted accurately by “simply” using a fine enough mesh and small enough time step.

Introduction

The solution of transient wave propagation problems is of great importance in practical engineering analyses. Exact solutions can only be obtained for relatively simple problems, such as a single wave propagating in a one-dimensional space. Solutions of waves in complex geometric shapes can only be obtained by resorting to numerical methods.

The classical finite element method is a popular and widely used numerical approach for solving transient wave propagation problems. However, the solutions using the standard method suffer from significant dispersion errors induced by the spatial discretizations [1], [2], [3], [4], [5]. As a result, inaccurate numerical solutions are frequently obtained, especially in the relatively high wave number range. It is also found that the solutions show significant numerical anisotropy [5], [6], [7], that is, the accuracy of solutions depends on the directions of wave propagations even when the medium is isotropic and a seemingly uniform mesh is used.

Significant research efforts have focused on reducing the dispersion error in the solution of transient wave propagations in solids and various methods have been proposed, see e.g. [8], [9], [10], [11], [12], [13], [14], including the spectral element method, see e.g. [15], [16], [17], [18]. This scheme is a higher-order numerical technique combining spectral methods and the classical finite element discretization. The spectral element method can be used to solve problems with much less numerical dispersion error than the traditional finite element method; however, the procedure is difficult to use to solve general two- and three-dimensional problems in complex geometries. This limitation significantly impedes its wider application in practical engineering computations.

The method of finite spheres [19], [20], which is a typical meshfree numerical method, has also been proposed for wave propagation problems [12], [13]. If uniform spatial discretizations are employed, the method of finite sphere is quite effective in eliminating the numerical dispersion error and numerical anisotropy. However, a main shortcoming of the method is that a very expensive numerical integration is required in constructing the system matrices for non-uniform spatial discretizations.

Recently, a new paradigm of using “overlapping elements” was proposed for general static and dynamic analyses of solids and structures [21], [22], [23], [24], [25]. Numerical results show that the procedure can provide much more accurate solutions than the traditional finite element method without an expensive computational effort. Quite importantly, it was shown that the predictive behavior of the overlapping finite elements is almost insensitive to the geometric distortions of the mesh. The reason is that the local interpolations used in the method are not affected by (reasonable) geometric distortions of the mesh. This property is very valuable, significantly distinguishes the overlapping finite element method from the traditional finite element method, and makes the procedure an important ingredient in the AMORE scheme [24], [25].

Kim et al. successfully used overlapping finite elements enriched by trigonometric functions to solve transient wave propagations in homogeneous media [14]. The numerical results demonstrated that the proposed enriched overlapping finite element method with the Bathe time integration scheme shows several excellent but related solution properties for wave propagation problems. The key property is that the solutions using the scheme monotonically converge to the exact solution as the element size and time step decrease. Namely, when using a sufficiently fine mesh and small time step, the numerical dispersion error and numerical anisotropy are small, which means also that when multiple waves travel through the medium, these can all be solved for accurately. The property of monotonic convergence is very useful in practical engineering computations because accurate solutions can be reached by “simply” using a sufficiently fine mesh (as in static analyses) and a sufficiently small time step.

We focus in this paper on exploring the application of the ‘enriched overlapping finite elements’ to solve transient wave propagation problems in inhomogeneous media. Our study adds to the results published earlier by Kim et al. [14] who considered homogeneous media. We give now new insights regarding the dispersion errors, and also show that using the enriched overlapping finite element discretization provides much more accurate solutions than when using traditional finite elements or the overlapping finite elements without enrichment. Based on our observations we can expect that the overlapping finite element method using enrichments has much potential in solving complex wave propagation problems involving also anisotropic and composite media.

Section snippets

Governing equations of wave propagation in inhomogeneous media.

We consider a general problem domain Ω=Ω1Ω2 consisting of two sub-domains filled with two different media, as shown in Fig. 1. The governing wave equations are given as2u1-1c12u¨1=0,inΩ12u2-1c22u¨2=0,inΩ2in which uI (I = 1, 2) is the solution variable of wave propagation (such as pressure [26] or displacement [14]) in the sub-domains ΩI, I = 1, 2; 2 is the Laplace operator, cI (I = 1, 2) is the wave propagation velocity in the different media, and an overdot represents a derivative with

The interpolation scheme of enriched overlapping elements

We use the enriched triangular overlapping elements [21], [22], [23], [24]. For every overlap region with three nodes I, L, M the interpolation of the solution variable u is given byuhx=ρIuI+ρLuL+ρMuMin which, with J = I, L, M, the ρJ are the new interpolation functions and the uJ are the nodal unknowns which can be functions.

The three new interpolation functions are given byρJ=ϕJIhI+ϕJLhL+ϕJMhMwhere with J, K = I, L, MϕJK=i=16ĥiϕJiKin which hI, hL and hM are the usual shape functions of the

Dispersion analysis

The numerical solutions of wave propagation problems suffer from numerical dispersion errors. As a result, when using traditional finite element procedures, the solution accuracy in general becomes worse with an increase of the considered wave number, k=2πλ, where λ is the (exact) wave length. Therefore, it is important to examine the dispersion properties of a numerical technique. In this section, we conduct a dispersion analysis using the uniform mesh in a homogeneous medium shown in Fig. 2:

Numerical examples

In the previous section, we examined the dispersion and dissipation properties of the EOFEM in the solution of wave propagation problems. Although the investigation was based on using a uniform mesh pattern, the conclusions have significance for solving wave propagation problems for which non-uniform meshes need generally be used.

In this section, we solve several wave propagation problems to illustrate the performance of the EOFEM with the Bathe time integration scheme. We consider wave

Concluding remarks

We focused on employing the EOFEM with the Bathe time integration method to solve transient wave propagation problems in inhomogeneous media. We investigated the performance and properties of the approach in comparison to the use of the standard FEM and the OFEM.

A dispersion analysis shows that the total dispersion error contains contributions from the spatial discretization and the temporal discretization. As the time step size using the Bathe time integration scheme tends to zero (that is,

Declaration of Competing Interest

The authors declared that there is no conflict of interest.

Acknowledgements

We would like to thank Dr. Ki-Tae Kim, formerly in our MIT research group, for valuable comments regarding this work.

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