1. Introduction
As is well known, the phenomenon of wave propagation [
1,
2,
3,
4,
5,
6,
7] widely exists in various areas of science and engineering, such as acoustics, elastodynamics, electromagnetics, and fluid dynamics. Numerical simulation plays an important role in understanding and mastering the fundamental laws of such wave propagation. Traditional numerical methods [
8,
9,
10], such as the finite difference method and finite element method (FEM), have been widely used in wave propagation analysis. However, they usually have the problems of low computational efficiency and poor computational accuracy due to the use of universal polynomial functions. To overcome these drawbacks, several basis functions, including wave characteristics, have been introduced, and then a series of semi-analytical numerical methods have been constructed, such as the wave-based method [
11], scaled boundary finite element method [
12] and boundary element method [
13,
14,
15], and so on. The aforementioned numerical methods belong to the mesh-based methods, which are sensitive to the mesh quality. To eliminate the effect of mesh generation, a group of meshless methods is developed. Similarly, by introducing the basis functions including the wave characteristics, several boundary-type meshless methods are proposed. They can be divided into two categories: weak-form boundary meshless methods and strong-form boundary meshless methods. The weak-form boundary meshless methods mainly include the local boundary integral equation method [
16], boundary node method [
17], hybrid boundary node method [
18], boundary face method [
19], null-field boundary integral equation method [
20], and so on. The strong-form boundary meshless methods mainly include the wave superposition method [
21,
22], method of fundamental solutions (MFS) [
23,
24], regularized meshless method [
25], boundary distributed source method [
26], singular boundary method (SBM) [
27,
28,
29,
30,
31], collocation Trefftz method (CTM) [
32,
33], and so on. Due to their simpler form, integral-free and easy-to-use merits, this study focused on the strong-form boundary meshless methods based on the semi-analytical basis functions.
A broader and more challenging problem in wave propagation analysis is the simulation of wave propagation in the time domain. The following three popular approaches have been widely used to treat transient wave propagation problems: (1) Time-stepping methods (TSM) [
23,
34] transform the transient wave propagation problem into a series of time-independent problems, and the accuracy and stability of this method highly depend on the time-step size. (2) Frequency domain techniques (FDT) [
35,
36] use the transformation technique to eliminate the time derivative leading to a time-independent equation in the frequency domain, and then employ a numerical inversion scheme to invert the frequency domain solutions back into the time-dependent solutions. The FDT does not require time stepping, and thus avoids the effect of the time step on numerical accuracy. However, for systems with no intrinsic damping and mismatched initial and ending responses, the numerical inversion transformation fails to produce accurate results. This is why in practical calculations often small artificial damping is added to the model. (3) Spatial–temporal semi-analytical basis function methods [
37,
38,
39] employ the spatial–temporal semi-analytical basis function a priori to satisfy the transient wave equation and then solve it directly. Among these three time-discretization schemes, the first two have been widely used for transient wave propagation analysis; the last one has not been widely used because the time-dependent semi-analytical basis functions are not easy to construct, in particular, the transient wave equation, including the source excitations.
Fortunately, the composite multiple reciprocity method (CMRM) [
40] has been proposed and applied to deal with some specified source terms in time-independent nonhomogeneous PDEs. In this study, the CMRM is extended from the spatial domain to the spatial–temporal domain, which transforms the original transient wave propagation problem with a source term into the high-order time-dependent problem without a source term. Then a group of spatial–temporal semi-analytical basis functions and spatial–temporal radial Trefftz functions are derived to satisfy the governing equation of such high-order time-dependent problem in advance. Correspondingly, the so-called spatial–temporal radial Trefftz collocation method (STRTCM) is constructed to solve 3D transient wave equations with specified sound source excitations, which only require the node discretization on the spatial–temporal domain boundaries.
In this paper, a novel spatial–temporal radial Trefftz collocation method is proposed without differential approximation for the temporal derivatives, which may cause the accumulated error to solve the 3D transient wave equations, and the composite multiple reciprocity method is extended from the space domain to the spatial–temporal domain to treat the time-dependent source term. Due to the use of the related spatial–temporal radial Trefftz functions, the proposed STRTCM requires fewer node discretizations to obtain more accurate results. A brief outline of this paper is as follows. In
Section 2, the numerical procedure of the spatial–temporal radial Trefftz collocation method for solving 3D transient wave equations with specified sound source excitations is introduced. The efficiency of the proposed method is numerically verified by four benchmark examples in
Section 3. In
Section 4, several conclusions are drawn based on the present study.
2. Methodology
Considering a transient wave equation in 3D finite domain
bounded by
, the governing equation of 3D transient wave propagation problem with sound source excitations is stated as follows:
subjected to the initial conditions
and the Dirichlet boundary condition
where
is the Laplace operator,
denotes the wave speed,
represents the final time instant,
and
are the known functions, and
is the known sound source function.
If the sound source function
, then the homogeneous type of Equation (1) is obtained as
By using the derived spatial–temporal radial Trefftz function [
41],
the approximate solution of Equation (5) by using the spatial–temporal radial Trefftz collocation method can be represented as follows
where
denotes the Euclidean distance between collocation nodes
and source nodes
,
and
are the time variables corresponding to the collocation nodes
and source nodes
, respectively.
are the unknown coefficients and
represents the number of the source node pair
, in which
stands for the boundaries of the considered spatial–temporal domain
. Substituting expression (7) into the initial conditions (2) (3) and boundary conditions (4), one may have
To determine the unknown coefficient
,
collocation node pairs
are placed on the boundaries of the considered spatial-temporal domain, and then the discretized formulation can be represented as follows
which can be also written as the following matrix form
where
, in which
and
represent the number of the collocation nodes inside the spatial domain
and on the boundary
of spatial domain
, respectively. The total node number in the spatial domain
is
and
represents the number of the collocation nodes along the time axis. If the same set of nodes to the collocation node discretization is used in the source node discretization inside the spatial domain
and on the boundary
of spatial domain
, the resultant matrix in Equation (14) is square due to
.
Next, consider 3D transient wave propagation problems (1–4) with the non-zero sound source function
. The approximate solution can be first divided into two parts, homogeneous solution
and particular solution
, i.e.,
where the particular solution
is constructed to satisfy the following equation
and then the following updated homogeneous problem can be represented by substituting Equations (15) and (16) into the original transient wave propagation problems (1–4),
where the homogeneous solution
of Equation (17) can be obtained by using the spatial–temporal radial Trefftz collocation method with node discretization on the boundaries of the considered spatial–temporal domain.
To evaluate the particular solution
, the composite multiple reciprocity method (CMRM) is extended from the spatial domain to the spatial–temporal domain. The key issue is to introduce the different spatial/spatial–temporal differential operators
to eliminate the related non-zero sound source function
in Equation (16), namely,
It should be pointed out that the commonly used differential operators (Laplace, Helmholtz, modified Helmholtz, diffusion equation and wave equation operators) can be chosen as according to the form of , which could be the polynomial, exponential and trigonometric functions, or a combination of these functions. For complex functions, e.g., non-smooth functions, a set of discrete measured data or large-gradient functions in the source term , we can express the complex functions or a set of discrete measured data by a series representation of polynomial or trigonometric functions, and then Laplace and Helmholtz operators can be chosen as to satisfy Equation (18). In the numerical implementation, the order M is usually finite, or can be determined by a specified truncation error.
Then the particular solution
can be obtained by solving the following
m-order homogeneous equation with
constraint conditions
where
denotes the governing differential operator. Then the particular solution
can be represented by a linear combination of high-order spatial–temporal radial Trefftz functions
, namely,
where the high-order spatial–temporal radial Trefftz functions are derived by satisfying the following equations:
Table 1 lists the related radial Trefftz functions for several commonly used spatial/spatial–temporal differential operators.
By combining Equations (7), (15) and (20), the approximate solution of 3D transient wave propagation problems (1–4) can be expressed as follows:
By employing Equation (22) to discretize Equations (17) and (19), the unknown coefficients can be obtained. After that, the numerical solution at any node pair can be calculated by using Equation (22).
3. Numerical Results and Discussions
This section presents four benchmark examples of 3D transient wave propagation problems with specified sound source excitations to verify the efficiency of the proposed spatial–temporal radial Trefftz collocation method (STRTCM). To assess the performance of the proposed solver, the following
relative error
, relative error
and maximum relative error
are adopted as follows:
where
and
stand for the analytical solution and the numerical solution on the test nodes
at time instant
t, respectively.
denotes the number of test nodes
. Unless otherwise specified, the test nodes
are chosen as the same set of the collocation nodes inside 3D spatial domain
and
in this study.
Example 1. Transient wave equation with specified sound source excitation under a unit cube.
In this example, the efficiency and accuracy of the proposed spatial–temporal radial Trefftz collocation method (STRTCM) in the solution of transient wave equations with the sound source
under the unit cubic domain
are investigated. The geometry and node distribution are depicted in
Figure 1. The governing equation is represented as
subjected to the initial conditions
and the Dirichlet boundary condition
Its analytical solution of Example 1 is
Table 2 presents
relative errors obtained by the proposed STRTCM and COMSOL at different time instants in Example 1. It can be found that under the same node discretization, the proposed STRTCM produces more accurate results with a slight longer computational time than the COMSOL (FEM).
Figure 2 shows the numerical errors along with the time evolution by using the proposed STRTCM in Example 1. From
Table 2 and
Figure 2, it can be observed that the numerical errors may slightly increase with the time evolution.
Table 3 presents
relative errors obtained by the proposed STRTCM with different total node numbers, it can be found that with the increasing total node number, the
L2 relative error decreases rapidly and then remains at the same level.
For ease of comparison, the same set of discretized nodes are used in both the proposed STRTCM and COMSOL, in which the number of collocation nodes is , the boundary nodes number is and the interior nodes number is , and the number of total nodes distributed on the boundaries of spatial–temporal domain is . It should be mentioned that, based on these collocation nodes, 4892 four-node tetrahedral elements are used in the COMSOL simulation. In addition, some other parameters are set as follows: the wave speed v = 1.0 m/s, the final time T = 1.0 s, the time interval dt = 0.1 s. The annihilation spatial–temporal differential operator is employed to vanish the specified sound source excitation in Equation (26) by using the extended CMRM.
Example 2. Transient wave equation with specified sound source excitation under a circular tube.
This example considers the transient wave equations with the sound source
under the circular tube domain as shown in
Figure 3a. The distribution of boundary nodes and interior nodes is depicted in
Figure 3b. The governing equation is represented as
subjected to the initial conditions
and the Dirichlet boundary condition
Its analytical solution of Example 2 is
In the proposed STRTCM implementation, the number of collocation nodes is , the boundary nodes number is , the interior nodes number is , the wave speed is , the final time instant is , and the annihilation spatial–temporal differential operator is employed to vanish the specified sound source excitation in Equation (31) by using the extended CMRM.
Table 4 presents numerical errors at final time instant
T = 10 s obtained by using the proposed STRTCM with different time intervals
in Example 2. From
Table 3, it can be found that the proposed STRTCM with different time intervals
can provide equally accurate results, which reveals that the time interval
has a slight influence on the numerical accuracy.
Figure 4 plots the absolute and relative error distributions at two time instants (t = 5 s and 10 s) by using the proposed STRTCM with a large time interval
dt = 5.0 s. Numerical results given in
Figure 4 show that the proposed STRTCM performs very accurate results, even with large time interval
dt = 5.0 s.
Example 3. Transient wave equation with specified sound source excitation under a room model.
This example considers the transient wave equations with the sound source
under the room model [
42] with principal dimensions being 5.0 m in length, 4.0 m in width and 3.0 m in height (see
Figure 5a). The distributions of boundary nodes and interior nodes are depicted in
Figure 5b. The governing equation is represented as
subjected to the initial conditions
and the Dirichlet boundary condition
Its analytical solution of Example 3 is
In the proposed STRTCM implementation, the number of collocation nodes is , the boundary nodes number is , the interior nodes number is , the number of total nodes distributed on the boundaries of the spatial–temporal domain is N = 17,305, the wave speed is v = 1.0 m/s, the final time instant is , the time interval is , and the annihilation spatial–temporal differential operator is employed to vanish the specified sound source excitation in Equation (36) by using the extended CMRM.
Figure 6 shows the relative error distributions on the plane
x3 = 1.5 at different time instants (t = 250, 500, 750, 1000 s) by using the proposed STRTCM with time interval
in Example 3. It can be observed that the proposed STRTCM with a large time interval can still obtain very accurate results in the solution of the transient wave problem under a complicated geometry domain (room model). This is because the semi-analytical spatial–temporal radial Trefftz functions are employed as the basis functions in the proposed STRTCM, which allows the few temporal discretizations to simulate the long-term evolution of the wave propagation.
Example 4. Transient wave equation with specified sound source excitation under a submarine model.
The final example considers the transient wave equations with the sound source
under the submarine model with the principal dimensions being 15.0 m in length, 4.0 m in width and 6.0 m in height (see
Figure 7a). The distribution of boundary nodes and interior nodes are depicted in
Figure 7b.
The governing equation is represented as
subjected to the initial conditions
and the Dirichlet boundary condition
Its analytical solution of Example 4 is
In the proposed STRTCM implementation, the number of collocation nodes is , the boundary nodes number is , and the interior nodes number is , the wave speed is v = 1.0 m/s, the time interval is , the final time is , and the annihilation spatial–temporal differential operator is employed to vanish the specified sound source excitation in Equation (41) by using the extended CMRM.
By using the proposed STRTCM for Example 4, very accurate results with
can be obtained in 1 min. However, it requires about 9 GB memory storage for getting the results in a large time instant
. To enhance the ability of the proposed STRTCM for long-time evolution simulation, the entire time interval [0, 10] is divided into
NP sub-time intervals (
), and the STRTCM is used to solve the problems (41–44) in each sub-time interval in sequence. For each problem in the considered sub-time interval (
np), the initial conditions are updated by using the final solution at the previous sub-time interval (
np − 1),
.
Table 5 gives the numerical results of Example 4 by using the proposed STRTCM with different numbers of sub-time intervals. It can be found from
Table 5 that with the increasing
NP, the proposed STRTCM can perform enough accurate results with less computational cost (CPU time and memory storage).