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In this work, the edge-based smoothed finite element method (ES-FEM) is incorporated with the Bathe time integration scheme to solve the transient wave propagation problems. The edge-based gradient smoothing technique (GST) can properly soften the “overly soft” system matrices from the standard finite element approach; then, the spatial numerical dispersion error of the calculated solutions for wave problems can be significantly reduced. To effectively solve the transient wave propagation problems, the Bathe time integration scheme is employed to perform the involved time integration. Due to the appropriate “numerical dissipation effects” from the Bathe time integration method, the spurious oscillations in the relatively large wave numbers (high frequencies) can be effectively suppressed; then, the temporal numerical dispersion error in the calculated solutions can also be notably controlled. A number of supporting numerical examples are considered to examine the capabilities of the present approach. The numerical results show that ES-FEM works very well with the Bathe time integration technique, and much more numerical solutions can be reached for solving transient wave propagation problems compared to the standard FEM.

The wave propagation problems are usually encountered in real engineering applications. For several simple wave problems (such as a single wave propagating in a one-dimensional space), the exact solutions can be obtained using the analytical approach. When it comes to the relatively complex wave propagation problems, we have to resort to the numerical methods.

Actually, a lot of different numerical techniques can be employed to deal with the wave problems, such as the finite element method (FEM) [

Among them, the classical finite element method is one of the most widely used and well-developed numerical approaches for analyzing wave problems. Based on the finite element method, several powerful and versatile commercial software packages have also been successfully developed for engineering computation. However, the finite element solutions for wave problems usually suffer from the numerical dispersion error issue for large wave numbers (namely, high frequencies) [

In order to address the numerical dispersion error issue. Much research effort has been made and a variety of advanced or modified finite element schemes have been proposed (see, e.g., [

The present work mainly focuses on tackling the transient wave propagation problems. Many direct time integration techniques can be employed to solve the transient wave propagation dynamics, such as the central difference method [

In this work, the Bathe implicit time integration technique is combined with the edge-based FEM (ES-FEM) to solve the transient wave propagation problems. The Bathe time integration scheme is a typical two-substep method. The Newmark trapezoidal rule is used in the first substep and the 3-point Euler backward method is employed in the second substep. In the Bathe method, the appropriate numerical dissipation effects are introduced to suppress the spurious wave modes in high frequency range; then spurious oscillations in the calculated numerical solutions can be effectively eliminated and the solution accuracy can be significantly improved. The numerical examples in this paper show that the edge-based smoothed FEM (ES-FEM) works very well with the Bathe implicit time integration technique in solving transient wave propagation problems. The numerical dispersion error in the calculated numerical results can be significantly suppressed and much more accurate numerical solutions can be obtained compared to the conventional FEM.

The rest of the present work is structured as follows: in the next section the formulation of the ES-FEM for wave problems is briefly retrospected. A comprehensive dispersion analysis of the present ES-FEM with Bathe time integration scheme for transient wave propagation problems is given in Section

For a typical scalar wave problem (such as the wave propagation in a prestressed membrane), the governing equation is given by [

Following the basic finite element discretizations [

In this work, the edge-based smoothed FEM [

The description of forming the edge-based smoothing domains.

In this work, the gradient smoothing operation is performed by smoothing the particle velocity

For the wave propagation in a prestressed membrane, the relationship between the particle velocity

Using (

Following the conventional finite element interpolation steps, we have

In this work, the Gauss integration scheme is used to perform the related numerical integration along the boundary of smoothing domain which consists of

Once the smoothed gradient matrix

The numerical solutions of the transient wave propagation problems usually suffer from the dispersion error issue. Both the spatial discretization and temporal discretization are able to lead to dispersion error. In this section, both the spatial discretization error and temporal discretization error will be investigated in detail by using the uniform node distributions with average nodal space

The employed uniform mesh pattern and the illustration of involved nodes for dispersion analysis. (a) FEM. (b) ES-FEM.

For the time-independent form of the wave equation, the following matrix equation can be obtained without considering the boundary condition [

For an interior node

Since no additional boundary conditions are considered here, so the numerical solutions corresponding to the involved nodes in Figure

By referring to the uniform mesh pattern shown in Figure

For the standard FEM and ES-FEM models, the matrices

To ensure that the nontrivial solutions to (

From (

Figure

Comparisons of the spatial discretization error as a function of the normalized numerical wave number in different wave propagation directions for the standard FEM and the present ES-FEM. (a) FEM. (b) ES-FEM.

In addition, we also can find that the spatial discretization error results from the standard FEM are strongly affected by the wave propagation directions; namely, the standard FEM shows clear “numerical anisotropy” property in solving wave propagation problems even if the uniform mesh pattern is used. From the results shown in Figure

The temporal discretization is also an indispensable ingredient for solving the transient wave propagation problems. Many direct time integration schemes can be used to deal with the transient wave propagation dynamics. However, the temporal discretization always can result in additional temporal discretization error which can degrade the quality of the calculated numerical solutions. In this section, we mainly focus on investigating the additional dispersion effects from the temporal discretization. Since the Bathe time integration technique is a very effective approach in dynamic analysis and shows several evident advantages compared to other time integration schemes [

If the additional boundary conditions are not considered, the fundamental solution to the matrix equation shown in (

By using (

Using the standard Bathe time integration scheme [

Based on the standard Bathe time integration scheme [

For any given numerical wave number

From (

The total dispersion error of the numerical solutions for transient wave propagation problems can be split into two different parts: the first part, which is defined by

The spatial discretization error

For the given wave speed

For a fixed time step

Comparisons of the total dispersion error results as a function of the normalized wave number in various wave propagation directions for both FEM and ES-FEM. (a) FEM. (b) ES-FEM.

Comparisons of the additional dispersion error results induced by the temporal discretization as a function of the normalized wave number in various wave propagation directions for both FEM and ES-FEM. (a) FEM. (b) ES-FEM.

Comparisons of the percentage amplitude decay (AD) as a function of the normalized wave number in various wave propagation directions for both FEM and ES-FEM. (a) FEM. (b) ES-FEM.

In this section, a number of supporting numerical examples are given to examine the effectiveness of the present ES-FEM with Bathe time integration scheme in solving transient wave propagation problems. It should be pointed out that the nonreflecting boundary conditions [

The first considered numerical example is the two-dimensional scalar wave propagation in a square prestressed membrane. This problem is described in Figure

The problem description and used mesh patterns for the two-dimensional scalar wave propagation. (a) The geometry description of the square prestressed membrane. (b) The used uniform mesh.

The first concentrated force is a Ricker wavelet which is defined by [

Figure

The calculated displacement results from the standard FEM and the present ES-FEM along the wave propagation directions

Figures

Comparisons of the calculated displacement results from the standard FEM along various propagation directions for the first concentrated force case.

Comparisons of the calculated displacement results from the present ES-FEM along various propagation directions for the first concentrated force case.

Next, we consider another concentrated force which is defined by [

For this case, the displacements from the standard FEM and the present ES-FEM along the wave propagation direction

The calculated displacement results from the standard FEM and the present ES-FEM along the wave propagation directions

Comparisons of the calculated displacement results from the standard FEM along various propagation directions for the second concentrated force case.

Comparisons of the calculated displacement results from the present ES-FEM along various propagation directions for the second concentrated force case.

The second considered numerical example is still the two-dimensional scalar wave propagation in a square prestressed membrane. In this case, four same holes are located uniformly in the problem domain. This problem is described in Figure

The problem description and used mesh patterns for the two-dimensional scalar wave scattering problem. (a) The geometry description of the considered problem. (b) The used mesh pattern.

At the observation time

Comparisons of the calculated displacement results from the standard FEM and the ES-FEM along the wave propagation direction

Then, the corresponding displacement results along the other two different wave propagation directions (

Comparisons of the calculated displacement results from the standard FEM and the ES-FEM along the wave propagation direction

Comparisons of the calculated displacement results from the standard FEM and the ES-FEM along the wave propagation direction

Furthermore, several snapshots of the displacement distributions in the global computation domain from the standard FEM and ES-FEM at several observation times are shown in Figures

Snapshots of the displacement distributions from the FEM for the two-dimensional scalar wave scattering problem at different observation times: (a)

Snapshots of the displacement distributions from the ES-FEM for the two-dimensional scalar wave scattering problem at different observation times: (a)

This paper focuses on presenting an edge-based smoothed FEM (ES-FEM) with the Bathe time integration technique for solving two-dimensional transient wave propagation problems. The dispersion error properties of the present approach for solving transient wave problems are investigated detailedly. It is found that the present ES-FEM works quite well with the Bathe time integration technique, and the dispersion error can be significantly reduced compared to the standard FEM. Quite importantly, the troublesome “numerical anisotropy” property of the standard FEM for transient wave problems can be clearly suppressed by the present ES-FEM; hence, the obtained numerical results are almost the same in the different wave propagation directions. The numerical results show that the present ES-FEM indeed surpasses the standard FEM in solving transient wave problems and can provide much more accurate numerical results in predicting the behaviors of the waves; hence, it can be regarded as a very promising numerical approach for solving practical transient wave dynamic problems.

The data used to support the findings of this study are available from the corresponding author upon request.

The authors declare that they have no conflicts of interest regarding the publication of this paper.

The authors wish to express their gratitude to the National Natural Science Foundation of China (Grant no. 51809208), the Fundamental Research Funds for the Central Universities (Grant no. WUT: 2019IVB012), and the China Postdoctoral Science Foundation (Grant nos. 2018M632866 and 2018M642940).