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

Cheng-Yu Ku, Jing-En Xiao, Weichung Yeih and Chih-Yu Liu
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Cheng-Yu Ku: Department of Harbor and River Engineering, National Taiwan Ocean University, Keelung 20224, Taiwan
Jing-En Xiao: Department of Harbor and River Engineering, National Taiwan Ocean University, Keelung 20224, Taiwan
Weichung Yeih: Department of Harbor and River Engineering, National Taiwan Ocean University, Keelung 20224, Taiwan
Chih-Yu Liu: Department of Harbor and River Engineering, National Taiwan Ocean University, Keelung 20224, Taiwan

Mathematics, 2019, vol. 7, issue 11, 1-22

Abstract: 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.

Keywords: the modified Helmholtz equation; meshfree; domain decomposition; layered materials; source point (search for similar items in EconPapers)
JEL-codes: C (search for similar items in EconPapers)
Date: 2019
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