scholarly journals On Solving Modified Helmholtz Equation in Layered Materials Using the Multiple Source Meshfree Approach

Mathematics ◽  
2019 ◽  
Vol 7 (11) ◽  
pp. 1114 ◽  
Author(s):  
Ku ◽  
Xiao ◽  
Yeih ◽  
Liu
Keyword(s):  
Helmholtz Equation ◽  
Domain Decomposition Method ◽  
Domain Boundary ◽  
Layered Materials ◽  
Fundamental Solutions ◽  
Multiple Source ◽  
Method Of Fundamental Solutions ◽  
Multiple Sources ◽  
Trefftz Method ◽  
Modified Helmholtz Equation

This paper presents a study for solving the modified Helmholtz equation in layered materials using the multiple source meshfree approach (MSMA). The key idea of the MSMA starts with the method of fundamental solutions (MFS) as well as the collocation Trefftz method (CTM). The multiple source collocation scheme in the MSMA stems from the MFS and the basis functions are formulated using the CTM. The solution of the modified Helmholtz equation is therefore approximated by the superposition theorem using particular nonsingular functions by means of multiple sources located within the domain. To deal with the two-dimensional modified Helmholtz equation in layered materials, the domain decomposition method was adopted. Numerical examples were carried out to validate the method. The results illustrate that the MSMA is relatively simple because it avoids a complicated procedure for finding the appropriate position of the sources. Additionally, the MSMA for solving the modified Helmholtz equation is advantageous because the source points can be collocated on or within the domain boundary and the results are not sensitive to the location of source points. Finally, compared with other methods, highly accurate solutions can be obtained using the proposed method.

Fast Solution for Solving the Modified Helmholtz Equation withthe Method of Fundamental Solutions

2015 ◽  
Vol 17 (3) ◽  
pp. 867-886 ◽  
Author(s):  
C. S. Chen ◽  
Xinrong Jiang ◽  
Wen Chen ◽  
Guangming Yao
Keyword(s):  
Helmholtz Equation ◽  
Large Scale ◽  
Solution Process ◽  
Homogeneous Solution ◽  
Fundamental Solutions ◽  
Method Of Fundamental Solutions ◽  
Dense Matrix ◽  
Modified Helmholtz Equation ◽  
Large Scale Problems ◽  
Particular Solution

AbstractThe method of fundamentalsolutions (MFS)is known as aneffective boundary meshless method. However, the formulation of the MFS results in a dense and extremely ill-conditioned matrix. In this paper we investigate the MFS for solving large-scale problems for the nonhomogeneous modified Helmholtz equation. The key idea is to exploit the exponential decay of the fundamental solution of the modified Helmholtz equation, and consider a sparse or diagonal matrix instead of the original dense matrix. Hence, the homogeneous solution can be obtained efficiently and accurately. A standard two-step solution process which consists of evaluating the particular solution and the homogeneous solution is applied. Polyharmonic spline radial basis functions are employed to evaluate the particular solution. Five numerical examples in irregular domains and a large number of boundary collocation points are presented to show the simplicity and effectiveness of our approach for solving large-scale problems.

scholarly journals On Solving Two-Dimensional Inverse Heat Conduction Problems Using the Multiple Source Meshless Method

2019 ◽  
Vol 9 (13) ◽  
pp. 2629 ◽  
Author(s):  
Ku ◽  
Xiao ◽  
Huang ◽  
Yeih ◽  
Liu
Keyword(s):  
Heat Conduction ◽  
Meshless Method ◽  
Domain Decomposition Method ◽  
Domain Boundary ◽  
Addition Theorem ◽  
Two Dimensions ◽  
Multiple Source ◽  
Inverse Heat Conduction ◽  
Trefftz Method ◽  
Inverse Heat Conduction Problems

In this article, a newly developed multiple-source meshless method (MSMM) capable of solving inverse heat conduction problems in two dimensions is presented. Evolved from the collocation Trefftz method (CTM), the MSMM approximates the solution by using many source points through the addition theorem such that the ill-posedness is greatly reduced. The MSMM has the same superiorities as the CTM, such as the boundary discretization only, and is advantageous for solving inverse problems. Several numerical examples are conducted to validate the accuracy of solving inverse heat conduction problems using boundary conditions with different levels of noise. Moreover, the domain decomposition method is adopted for problems in the doubly-connected domain. The results demonstrate that the proposed method may recover the unknown data with remarkably high accuracy, even though the over-specified boundary data are assigned a portion that is less than 1/10 of the overall domain boundary.

A Boundary Meshless Method for Solving Heat Transfer Problems Using the Fourier Transform

2011 ◽  
Vol 3 (5) ◽  
pp. 572-585 ◽  
Author(s):  
A. Tadeu ◽  
C. S. Chen ◽  
J. António ◽  
Nuno Simões
Keyword(s):  
Fourier Transform ◽  
Helmholtz Equation ◽  
Initial Conditions ◽  
Inverse Fourier Transform ◽  
Fundamental Solutions ◽  
Method Of Fundamental Solutions ◽  
Static Response ◽  
Particular Solution ◽  
The Fourier Transform ◽  
Transfer Problems

AbstractFourier transform is applied to remove the time-dependent variable in the diffusion equation. Under non-harmonic initial conditions this gives rise to a non-homogeneous Helmholtz equation, which is solved by the method of fundamental solutions and the method of particular solutions. The particular solution of Helmholtz equation is available as shown in [4, 15]. The approximate solution in frequency domain is then inverted numerically using the inverse Fourier transform algorithm. Complex frequencies are used in order to avoid aliasing phenomena and to allow the computation of the static response. Two numerical examples are given to illustrate the effectiveness of the proposed approach for solving 2-D diffusion equations.

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