Abstract
In this paper, a localized boundary knot method is proposed, based on the local concept in the localized method of fundamental solutions. The localized boundary knot method is formed by combining the classical boundary knot method and the localization approach. The localized boundary knot method is truly free from mesh and numerical quadrature, so it has great potential for solving complicated engineering applications, such as multiply connected problems. In the proposed localized boundary knot method, both of the boundary nodes and interior nodes are required, and the algebraic equations at each node represent the satisfaction of the boundary condition or governing equation, which can be derived by using the boundary knot method at every subdomain. A sparse system of linear algebraic equations can be yielded using the proposed localized boundary knot method, which can greatly reduce the computer time and memory required in computer calculations. In this paper, several cases of simply connected domains and multi-connected domains of the Laplace equation and bi-harmonic equation are demonstrated to evidently verify the accuracy, convergence and stability of this proposed meshless method.
Highlights
With the emergence of various engineering problems, an increasing number of numerical methods have been proposed in the past decades
Is applied to solve large-scale engineering science problems [11,12,13,14,15]. Another well-known meshless method is the method of fundamental solutions (MFS), which was first proposed by Kupradze and Aleksidze [16]
The proposed meshless method is the combination of the convectional boundary knot method (BKM) and the concept of localization from the localized method of fundamental solutions (LMFS) and the local RBFCM (LRBFCM)
Summary
With the emergence of various engineering problems, an increasing number of numerical methods have been proposed in the past decades. The RBFCM has been developed to form the local RBFCM (LRBFCM), which can yield a sparse matrix, so the LRBFCM is applied to solve large-scale engineering science problems [11,12,13,14,15] Another well-known meshless method is the method of fundamental solutions (MFS), which was first proposed by Kupradze and Aleksidze [16]. The localized method of fundamental solutions (LMFS) approximate the numerical solution by implementing the MFS within each local subdomain and the sparse system of linear algebraic equations of the LMFS can be efficiently solved even for problems in complicated domains. Combining the advantages of RBF and MFS, the boundary knot method (BKM) was proposed by Chen [26] in 2002 to avoid the problem of fictitious boundary and singularity in the arrangement of fundamental solutions in the MFS.
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