A novel numerical method for infinite domain potential problems

Abstract

The infinite domain potential problems arise in many branches of scientific and engineering fields, which by now still pose a great challenge to scientific computing community. This study proposes a novel meshless singular boundary method (SBM) to solve infinite domain potential problems. The SBM is mathematically simple, easy-to-program, meshless and integration-free. To guarantee the uniqueness of numerical solutions, this article adds a constant term into the SBM approximate representation. The efficiency and accuracy of the proposed technique are tested to the three infinite domain potential problems.infinite domain, potential problem, singularity, fundamental solution, singular boundary method, meshless Citation:Chen W, Fu Z J. A novel numerical method for infinite domain potential problems.The infinite potential problems [1−3] have been observed in a wide variety of scientific and engineering fields, such as the ideal potential flow around a body, the electrostatic field and the steady temperature field, etc. However, the numerical solution to such problems still presents a great challenge so far to the communities of engineering simulation and scientific computing. Most of popular numerical methods such as finite element and finite volume methods need to truncate infinite domain to an artificial finite region with subtle artificial boundary conditions [4] or absorbing layers [5]. This truncation can be arbitrary largely based on trial-error experiences. On the other hand, the boundary element method (BEM) [6−8] appears very attractive to handle the unbounded domain problem because it applies the fundamental solution as the basis function, which satisfies the governing equation and the boundary condition at infinity. And no domain truncation is required. However, the singular or hyper-singular integrals [8] in the BEM are not mathematically simple and require additional computing costs.To avoid the singularities of fundamental solutions, the method of fundamental solutions (MFS) [9−11] distributes the boundary knots on fictitious boundary which is outside the physical domain, and the location of fictitious boundary is vital for the accuracy and reliability of the MFS solution. However, despite great efforts for decades, the determination of fictitious boundary is still arbitrary and tricky, largely based on experiences. Recently, Young et al. [12] proposed an alternative meshless method, namely regularized meshless method (RMM) [13] to remedy this drawback of the MFS. By employing the desingularization of subtracting and adding-back technique, the RMM can place the source points on the real physical boundary. In addition, the ill-conditioned interpolation matrix of BEM and MFS is also circumvented in the RMM. However, the original RMM requires the uniform distribution of nodes and severely reduces its applicability to complex-shaped boundary problems. Similar to the RMM, Sarler [14] proposed the modified method of fundamental solution (MMFS) to solve the potential flow problems. However, the MMFS demands a ...

Boundary-Type RBF Collocation Methods

“…By now the two techniques have been developed to evaluate the origin intensity factors. The first one is called the inverse interpolation technique (IIT) [7,8,[56][57][58], which numerically evaluates the origin intensity factors. The second approach is to derive the analytical formula for calculating the origin intensity factors [59][60][61][62][63][64][65].…”

“…The above SBM formulation (4.35) has successfully been applied to interior and exterior Laplace [7,8,56], Helmholtz [67], and elastostatic [57] problems.…”

Numerical investigation on three treatments for eliminating the singularities of acoustic fundamental solutions in the singular boundary method

“…This section will introduce a simple numerical technique, called the inverse interpolation technique (IIT) [3,6], to determine the source intensity factors for Laplace equations. Then we can use the relationships (4) to determine the source intensity factors for Helmholtz equations.…”

“…This SBM formulation has been successfully applied to interior and exterior Laplace [2][3][4], Poisson [5], Helmholtz [6] and elastostatic [7] problems. Later, Gu et al [8] introduced the desingularization of the subtracting and adding-back technique and proposed an improved singular boundary method (ISBM) for interior and exterior potential problems.…”

Intervention-point principle of meshless method

“…Some impressive contributions have been reported and applied, such as the development on the radial basis functions (RBF) by Wu [6], the least square collocation meshless method (LSC) by Zhang et al [7], the hybrid boundary node method (HBNM) by Zhang et al [8], the boundary knot method (BKM) and the singular boundary method (SBM) by Chen et al [9,10], the complex variable moving least-square approximation method (CVMLS) by Cheng et al [11], the application research on MLPG method by Long et al [12], the Galerkin boundary node method (GBN) by Li et al [13], and so on.…”