Mathematical Problems in Engineering

Volume 2015, Article ID 640305, 13 pages

http://dx.doi.org/10.1155/2015/640305

## Generalized Finite Difference Time Domain Method and Its Application to Acoustics

^{1}School of Computer Software, Tianjin University, Tianjin 300072, China^{2}Tianjin Key Laboratory of Cognitive Computing and Application, Tianjin University, Tianjin 300072, China^{3}Japan Advanced Institute of Science and Technology, Ishikawa 923-1292, Japan

Received 28 August 2014; Revised 11 January 2015; Accepted 11 January 2015

Academic Editor: Shaofan Li

Copyright © 2015 Jianguo Wei et al. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

#### Abstract

A meshless generalized finite difference time domain (GFDTD) method is proposed and applied to transient acoustics to overcome difficulties due to use of grids or mesh. Inspired by the derivation of meshless particle methods, the generalized finite difference method (GFDM) is reformulated utilizing Taylor series expansion. It is in a way different from the conventional derivation of GFDM in which a weighted energy norm was minimized. The similarity and difference between GFDM and particle methods are hence conveniently examined. It is shown that GFDM has better performance than the modified smoothed particle method in approximating the first- and second-order derivatives of 1D and 2D functions. To solve acoustic wave propagation problems, GFDM is used to approximate the spatial derivatives and the leap-frog scheme is used for time integration. By analog with FDTD, the whole algorithm is referred to as GFDTD. Examples in one- and two-dimensional domain with reflection and absorbing boundary conditions are solved and good agreements with the FDTD reference solutions are observed, even with irregular point distribution. The developed GFDTD method has advantages in solving wave propagation in domain with irregular and moving boundaries.

#### 1. Introduction

Partial differential equations (PDEs) modeling problems in science and engineering, such as electromagnetics, acoustics, and hydrodynamics, are usually solved by numerical methods that discretize the computational domain with mesh or grids. Grid-based methods such as finite difference method (FDM), finite element method (FEM), and boundary element method (BEM) [1, 2] have had much achievements and still dominate the field of scientific computing. However, numerical difficulties originating from usage of grids often emerge. For complicated and irregular geometry, implementation of boundary conditions could be a big challenge for FDM. Generation of grids with high quality is not an easy task in FEM and BEM. Moreover, when free surface and moving boundary/interface have to be treated, the transformation of grids will turn the conventional grid-based methods into a difficult, time-consuming process. Numerical accuracy often degenerates and divergence problem occurs.

In recent 20 years, to overcome numerical difficulties due to use of grids or mesh, meshless methods (MMs) based on different techniques have been proposed and widely used in many fields such as hydrodynamics [3], astrophysics [4], and solid mechanics [3, 5]. Among the MMs, generalized finite difference method (GFDM) is the one that evolved from traditional FDM [6, 7] and many different forms have been developed [8]. Benito and his coauthors made great contribution to its recent development [9–11]. For heat conduction problem, it has been compared with the element-free Garlerkin (EFG) method (one of the most used MMs in solid mechanics) and better performance has been observed [10]. Recently, GFDM was used to solve the wave equations [11] and Burgers’ equations [12] and simulate seismic wave propagation problems in heterogeneous media [13]. An application to the detonation shock dynamics [14] was also carried out. Nevertheless, few work on computational acoustics has been reported.

For acoustic wave propagation problems, the concentration is on the ones in confined domain, for which grid-based methods like FDTD and TDFEM (time-domain finite-element methods) [15], are mostly used. However, moving boundary exists in many acoustic problems like sound wave propagation inside a deforming vocal tract. This problem is hardly solved by conventional grid-based methods and MMs provide a possibility. As one of the MMs, GFDM is extended to transient acoustics in this paper, which is helpful to solve wave propagation problems with moving boundary in the future.

Inspired by the derivation of meshless particle methods, we firstly formulated the GFDM in a way different from the original one that minimizes an energy norm. Such that the relationship between GFDM and meshless particle methods like smoothed particle hydrodynamics (SPH) and its improvements can be conveniently examined. Comparison with the modified dmoothed particle hydrodynamics (MSPH) method, which has better performances than SPH and its corrections [16], shows higher approximation accuracy of the GFDM, especially at the boundary region. By analog with FDTD, a method referred to as generalized finite difference time domain (GFDTD) is proposed, in which GFDM is used to discretize the spatial operators and the leap-frog algorithm is used for time integration. To show its good performance and efficiency, the GFDTD method is applied to transient acoustics. Comparison with conventional FDTD solutions is presented and discussed.

#### 2. Generalized Finite Difference Method (GFDM)

Other than conventional derivation of GFDM by minimizing an energy norm [10], a different derivation of GFDM is presented in this section. Taylor series expansion of around point remaining up to second-order terms yieldswhere , , , and .

By multiplying both sides of (1) with and ( is a weighting function with compact support) and integrating the resulted equations over the support domain , we get two equations, and the following, as an example, is the result for :where is a volume measure.

Repeating the same procedure with , , and instead of and , we get other three equations and the following is the result for : To approximate the integrations by Riemann sum, the volume of the support domain is divided into points with associated volumes , . Equations (2), (3), and the other three constitute a system of five equations written in matrix form aswithwhere with , , and , and is a measure of the support size.

The conventional derivation of GFDM is presented in the appendix. It is clear that the difference between the conventional and the current derivation is not only the procedure but also the final form. The conventional derivation loses term (see (A.5)). If all the points in the domain have the same volume, at both sides of (4) will be cancelled, and the two final forms will be the same. However, can hardly be the same when points are irregularly spaced. From this point of view, our derived final form is more general and takes point irregularity into account.

#### 3. Modified Smoothed Particle Hydrodynamics (MSPH)

As a modification to SPH, the MSPH method improves the accuracy of the approximations especially at points near the boundary of the domain [16]. It uses Taylor series expansion of function as in (1). Similar to the derivations of (2) and (3), but with different weight functions , , , , and , the following equations, as examples, for and , are obtained:Again the Riemann sum over the support domain is used to approximate the integrations and a system of five equations is obtained asCompared with formula (4), the only difference is the terms multiplied to both sides of (1). In GFDM, , , , , and are used instead of , , , , and in MSPH. As a result, GFDM avoids computing the derivatives of the weight function and hence saves computational efforts and leads to more choice of the weight function.

#### 4. Numerical Tests for Approximation of Derivatives

In previous sections the deviation of GFDM and MSPH is presented. In this section, to compare the performance of the two methods, they are used to approximate the derivatives of certain 1D and 2D functions. For the convenience of evaluation, a global error measure is defined as follows: where can be , , , and and the superscripts and refer to the exact and numerical solutions, respectively.

The quartic spline function is used as the weight function : where is the kernel radius taken as ( is the space interval) which is usually used in meshless methods.

##### 4.1. One-Dimensional Case

Consider the following function:Figure 1 shows the first- and second-order derivatives estimated by GFDM and MSPH and the exact results when the domain is discretized into 21 equally spaced points. It is seen that GFDM has better performance in both derivatives especially for the points near boundaries. When the number of points increases to 51, the results are similar as exhibited in Figure 2. Error analysis shown in Table 1 indicates that GFDM has higher accuracy. With increasing number of points, the global error decreases.