Mathematical Problems in Engineering

Volume 2015, Article ID 914207, 9 pages

http://dx.doi.org/10.1155/2015/914207

## Use of Finite Point Method for Wave Propagation in Nonhomogeneous Unbounded Domains

Department of Civil Engineering, EMU, Famagusta, Northern Cyprus, Mersin 10, Turkey

Received 21 July 2014; Revised 16 January 2015; Accepted 9 February 2015

Academic Editor: Hung Nguyen-Xuan

Copyright © 2015 S. Moazam 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

Wave propagation in an unbounded domain surrounding the stimulation resource is one of the important issues for engineers. Past literature is mainly concentrated on the modelling and estimation of the wave propagation in partially layered, homogeneous, and unbounded domains with harmonic properties. In this study, a new approach based on the Finite Point Method (FPM) has been introduced to analyze and solve the problems of wave propagation in any nonhomogeneous unbounded domain. The proposed method has the ability to use the domain properties by coordinate as an input. Therefore, there is no restriction in the form of the domain properties, such as being periodical as in the case of existing similar numerical methods. The proposed method can model the boundary points between phases with trace of errors and the results of this method satisfy both conditions of decay and radiation.

#### 1. Introduction

Wave propagation in the unbounded domains is one of the important engineering issues. To solve this problem, many researches have been carried out and the ones that are most relevant to the proposed method are referred to in this study. Wang et al. [1] tried to simulate wave propagation in domains with nonhomogeneous cross-anisotropic properties. In another research, Wang et al. [2] simulated the wave parameters, such as stress and displacement, in a nonhomogeneous transversely isotropic half-space subjected to a uniform vertical circular load. Daros [3] presented a solution for SH-waves in a nonhomogeneous anisotropic media. Ke et al. [4] worked on simulations of Love waves in a nonhomogeneous, saturated, porous layered half-space with linearly varying properties. In 2006, Bazyar and Song [5] tried to simulate time-harmonic response of nonhomogeneous elastic unbounded domain using Scaled Boundary Finite Element Method. However, the aforementioned studies could not lead to a general method that can simulate unbounded domain with any kind of properties. Furthermore, each one of the mentioned methods has its own limitations in the shape and form of stimulating function problems, which are avoided in the proposed method.

In recent years, due to the rapid development in computer power, researchers from various disciplines have developed a special interest in numerical solution to many problems. Numerical modelling of problems has been the major focus of the researchers and undoubtedly wave propagation has been an important part of this numerical research approach. Over the years, physicists also acknowledged the nature of masses not just as particles but also as waves [6] and this emphasizes the importance of the wave propagation modelling.

The research by Boroomand and Mossaiby [7] introduced a new method, which is based on Finite Element Method, to estimate the wave propagation in a homogeneous unbounded domain. Lately, Moazam [8] used Finite Point Method (FPM) and Finite Difference Method (FDM) to develop the method of Boroomand and Mossaiby [7] to a meshless version to avoid problems caused by element mesh in simulating the wave propagation in unbounded domain. Then, Moazam et al. [9] further developed the method by Moazam [8] so that it could be used for homogeneous domains due to arbitrary stimulation and also it can reduce the estimation errors.

The aim of this study is to investigate the effect of wave propagation due to the vibration of heavy machinery on the surrounding region of the nonhomogeneous arbitrary domain using a new FPM-based numerical method.

The assumptions and concepts of them will be explained in the following parts of this study.

#### 2. Methodology

##### 2.1. Elastic Wave Propagation in Unbounded Domain

The Finite Point Method was used with specific form of generalized coordinate that was developed by Moazam et al. [9] and will be explained as follows. The assumptions and concepts which are used in this study will be explained in the following parts of this study.

The wave equations are given below:in which and are the magnitude of displacement of wave and its second derivative in respect to time, respectively; is a differential equation that signifies the relative deformation; is a matrix of material properties; is the unit weight of the domain; and finally, is the stimulation function of the domain (a Dirac-Delta function in the specified direction and time with sinusoidal form).

One of the uses of the above formula is the elastic wave propagation in which all functions and operators are written in vector format. To solve this equation, a Cartesian coordinate system is adopted, where the center of this coordinate system is used as the stimulation point. If is considered asthen is a Fourier transformation of and is the value of the stimulation frequency. Consequently, these values are substituted in (1) to obtain where is a Fourier transformation of stimulation function of .

According to the stimulation function shape, to solve this problem, symmetric and antisymmetric displacement condition can be used in the domain. Then, the equation is given as follows:This study considers the importance of the reliability of the domain properties which can solve the problem. Therefore, the stimulation can be applied as a boundary condition and thereby (4) can be classified as part of homogeneous equations group with constant coefficients. As a result, one of the significant properties of the differential equations with constant coefficients, such as proportionality, is given in where is a constant vector and and are two constant undefined scalars. Equation (6) is derived from the exponential function properties in the and direction:where and are arbitrary specified values in and direction and and are positive numbers.

By substituting (5) into (4), (7) can be obtained:where is a matrix including values based on and . The nullspace of a matrix is equivalent to the matrix when it reaches to zero:According to the characteristic of (7), and are the main factors relating to the issue which will be discussed in Results and Discussions of this paper.

One of these variables can be calculated in terms of the other one: or . It must be noted that, depending on the degree of characteristic of equation, there may be more than one answer to each of these equations.

The homogenous solution of this equation may be obtained by using the superposition of spectral solutions. For example, makes [7]The inner sigma in the overall nullspace of matrix and the overall integration gives the possible values of .

##### 2.2. Decay and Radiation Condition

Decay condition of amplitude means decreasing the amplitude while increasing the distance from the stimulation point ( (6)); that is,One should note that and can have complex values; thus, (10) is a circle with the radius of one in Gaussian coordinate as shown in Figure 1.