Development of Meshfree Strong-Form Methods

Abstract

This thesis focuses on the development of meshfree strong-form methods. Asmeshfree strong-form methods possess many attractive and distinguished features thatfacilitate the implementation of adaptive analysis, the meshfree strong-form methodsproposed in this thesis are extended to the adaptive analysis. The first part of my workis to provide several stabilization procedures to stabilize the solution of a conventionalstrong-form method, radial point collocation method (RPCM), that uses local radialbasis functions (RBFs). These stabilization procedures have shown great effectivenessto stabilize the unstable solution of RPCM. Without the stabilization procedures, it isimpossible to extend RPCM to adaptive analysis. The second part of my work is todevelop a robust and effective error estimator that feasible for strong-form methods. Anovel residual based error estimator is proposed and demonstrated not only effectivebut also versatile to be used for both weak-form and strong-form methods in theadaptive analysis. Besides the strong-form methods, a very classical subdomainmethod that employs meshfree shape functions is also presented in my work.Excellence numerical performance is observed in this thesis. As approximationformulated based on the local RBFs is adopted in all the proposed meshfree methodsin this thesis, a comprehensive study on the effects of the number of local nodes formeshfree methods is conducted. Strong evidences and reasons of using local nodesare shown in the study.