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.
Highlights
The solution of the modified Helmholtz equation plays a crucial role in the science and engineering fields, such as for boundary detection problems [1], water wave problems [2], Cauchy problems [3,4], diffusion equations [5], topological sensitivity analyses [6], and boundary value problems [7]
The solution of the modified Helmholtz equation is approximated by the superposition theorem using particular nonsingular functions by means of multiple sources located within the domain
We presented a meshfree approach for solving the modified Helmholtz equation bounded by doubly, and multiply connected regions using the multiple source meshfree approach (MSMA) in two dimensions
Summary
The solution of the modified Helmholtz equation plays a crucial role in the science and engineering fields, such as for boundary detection problems [1], water wave problems [2], Cauchy problems [3,4], diffusion equations [5], topological sensitivity analyses [6], and boundary value problems [7]. Over the past 10 years, conventional approaches have been widely adopted for solving the modified Helmholtz equation [8,9,10]. In contrast to the conventional approaches, several boundary collocation meshfree methods that have the basis functions satisfying the partial differential equation have been proposed, for instance, the boundary knot method (BKM) [11,12], the boundary node method [13,14], the singular boundary method (SBM) [15,16,17], and the regularized meshless method [18,19]. In 2003, Chen and Hon [12] adopted the BKM to solve Helmholtz, modified Helmholtz, and the combination of diffusion and convection problems. Chen et al [15] applied the SBM to solve the modified Helmholtz equation
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