Abstract
Inhomogeneous elliptical inclusions with partial differential equations have aroused appreciable concern in many disciplines. In this paper, a pseudo-spectral collocation method, based on Fourier basis functions, is proposed for the numerical solutions of two- (2D) and three-dimensional (3D) inhomogeneous elliptic boundary value problems. We describe how one can improve the numerical accuracy by making some extra “reconstruction techniques” before applying the traditional Fourier series approximation. After the particular solutions have been obtained, the resulting homogeneous equation can then be calculated using various boundary-type methods, such as the method of fundamental solutions (MFS). Using Fourier basis functions, one does not need to use large matrices, making accrual computations relatively fast. Three benchmark numerical examples involving Poisson, Helmholtz, and modified-Helmholtz equations are presented to illustrate the applicability and accuracy of the proposed method.
Highlights
After the pioneering work of stress analyses considering elliptical region, inhomogeneous elliptical inclusions with partial differential equations have aroused appreciable concern, such as two-dimensional problems [1–3] and three-dimensional problems [4–6].It is worth noting that the term ‘inclusions’ here refers to a medium whose properties are different from surrounding media inside which the eigenstrain occurs [7–9]
The applicability of the classical boundary element method (BEM) [10–14] and method of fundamental solutions (MFS) [13,15–24] depends heavily on how we evaluate the particular solution of the given problem [25–38]
Once the particular solutions have been obtained, the solution of the original problem can be converted to a homogeneous one which can be solved by using the BEM/MFS-based methods [11,13,14,19,46–64]
Summary
After the pioneering work of stress analyses considering elliptical region, inhomogeneous elliptical inclusions with partial differential equations have aroused appreciable concern, such as two-dimensional problems [1–3] and three-dimensional problems [4–6]. The applicability of the classical boundary element method (BEM) [10–14] and method of fundamental solutions (MFS) [13,15–24] depends heavily on how we evaluate the particular solution of the given problem [25–38]. Many types of “basis functions” are available, a good choice for most of all applications is the Fourier series [39–43]. Another popular used “basis function” is the well-known Chebyshev series, which is just a Fourier cosine expansion with a change of variable [25,40,42,44,45]. Once the particular solutions have been obtained, the solution of the original problem can be converted to a homogeneous one which can be solved by using the BEM/MFS-based methods [11,13,14,19,46–64].
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