Title: New advances in dual reciprocity and boundary-only RBF methods Show Chinese title

Title: 双互易和仅边界径向基函数方法的新进展

Abstract: This paper made some significant advances in the dual reciprocity and boundary-only RBF techniques. The proposed boundary knot method (BKM) is different from the standard boundary element method in a number of important aspects. Namely, it is truly meshless, exponential convergence, integration-free (of course, no singular integration), boundary-only for general problems, and leads to symmetric matrix under certain conditions (able to be extended to general cases after further modified). The BKM also avoids the artificial boundary in the method of fundamental solution. An amazing finding is that the BKM can formulate linear modeling equations for nonlinear partial differential systems with linear boundary conditions. This merit makes it circumvent all perplexing issues in the iteration solution of nonlinear equations. On the other hand, by analogy with Green's second identity, this paper also presents a general solution RBF (GSR) methodology to construct efficient RBFs in the dual reciprocity and domain-type RBF collocation methods. The GSR approach first establishes an explicit relationship between the BEM and RBF itself on the ground of the weighted residual principle. This paper also discusses the RBF convergence and stability problems within the framework of integral equation theory. Show Chinese abstract

Abstract: 本文在对偶乘法和仅边界径向基函数技术方面取得了一些重要的进展。 所提出的边界节点法（BKM）在多个重要方面与标准边界元法不同。 即，它是真正的无网格方法，具有指数收敛性，无需积分（当然，没有奇异积分），对于一般问题仅需要边界条件，并且在某些条件下会导致对称矩阵（在进一步修改后可以扩展到一般情况）。 BKM还避免了基本解法中的人工边界。 一个令人惊讶的发现是，BKM可以为具有线性边界条件的非线性偏微分系统建立线性建模方程。 这一优点使其能够避开非线性方程迭代求解中的所有复杂问题。 另一方面，通过类比格林第二恒等式，本文还提出了一种通用解径向基函数（GSR）方法，以在对偶乘法和域型径向基函数配置方法中构建高效的径向基函数。 GSR方法首先基于加权残差原理，建立了边界元法和径向基函数本身之间的显式关系。 本文还讨论了在积分方程理论框架下的径向基函数收敛性和稳定性问题。