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A Study on Accuracy, Convergence and Stability of Galerkin and Collocation Meshfree Methods

Abstract

In the past two decades substantial advancements of meshfree methods have been made for various engineering and scientific applications. Meshfree methods can be categorized in two branches, methods based on a Galerkin weak form and on a strong form with collocation. For example, the element-free Galerkin (EFG) method and the reproducing kernel particle method (RKPM) are formulated under the Galerkin framework, and the Radial Basis collocation method (RBCM) and reproducing kernel collocation method (RKCM) are formulated directly on strong form with collocation. While much success has been made in meshfree Galerkin methods and collocation methods, the in-depth comparison of numerical performance of these two classes of methods is scarce. This dissertation investigates the accuracy, convergence, stability, and effectiveness of Galerkin and strong form collocation meshfree methods, with focus on RKPM, RBCM and RKCM. The numerical performance comparison between these methods as well as their advantages and disadvantages are discussed. In this work, the convergence and accuracy of RKCM, RBCM and RKPM are first studied with varying approximation parameters. Different arrangements of collocation points are used for RKCM and RBCM to investigate how the convergence behaviors are affected by the number and location of collocation points. Meanwhile, RKPM with various quadrature rules for domain integration techniques has also been studied, and the results show that it requires well-formulated domain integration techniques to reach optimal convergence. On the other hand, strong form collocation methods such as RBCM and RKCM can achieve similar accuracy as that in RKPM without encountering complexity involved in advanced quadrature rules. Next, the performance of collocation methods under non-uniform discretization has also been investigated. The results indicate that both RKCM and RBCM solutions are relatively insensitive to the non-uniformity in discretization compared to RKPM with direct nodal integration (DNI) or Gauss integration under the same domain discretization. RKPM, on the other hand, requires the employment of advanced quadrature rules such as the Stabilized conforming nodal integration (SCNI) and the Variational Consistent Integration (VCI) to overcome solution sensitivity to discretization non-uniformity. The comparison of these three methods shows that while RBCM generates best accuracy in coarse discretization, the method is suffered from ill-conditioning in the model refinement. RKCM and RBCM, on the other hand, are more stable under model refinement but offer less accurate results compared to RBCM in coarse discretization. RKCM appears to be an attractive alternative of RKPM due to its ability in gaining stability and good convergence without the need of well-formulated quadrature rules

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