PhD thesis: Meshless Adaptive Solution Procedure for Efficient Solving of Partial Differential Equations

Abstract

Meshless methods are becoming increasingly popular in computational mechanics and engineering. Their main feature is the ability to manage complex geometries while avoiding the often tedious process of mesh generation required by the traditional methods. Various meshless approximations of linear differential operators appearing in the governing problem have been proposed over the years. However, even with the state-of-the-art methods and immense computational power at our disposal, efficient solution procedures for solving systems of partial differential equations (PDEs) are being actively studied, as the complexity of the problems under consideration continues to increase.

In this thesis, we study different approaches towards efficient PDE-solving solution procedures employing meshless approximation methods. Our study is limited to three commonly used approximation methods: the Radial Basis Function-generated Finite Differences (RBF-FD), the Diffuse Approximation Method (DAM), referred to as Weighted Least Squares (WLS) in this thesis, and the simplest collocation method, referred to as MON throughout this thesis.

The first part of the thesis is devoted to a discussion of the fundamentals of meshless approximation. The effects of monomial augmentation on the accuracy of the numerical solution, as well as on the stability and the computational complexity of the solution procedure are investigated. Our understanding of the meshless approximation is then enhanced with a study of the stencil size impact on the accuracy of the numerical solution. We then compare the performance of the RBF-FD and the WLS approximation methods in terms of the accuracy of the obtained solution and the stability of the solution procedure.

After demonstrating the stability of the high-order RBF-FD approximations, we focus on the development and implementation of the hp-adaptive solution procedure in the second part of the dissertation. We first employ p-refinement to demonstrate the effects of spatially-variable approximation order on the efficiency of the solution procedure. We then develop an original error indicator, which subsequently enables the implementation of a fully adaptive strong-form meshless method employing both h- and p-refinement procedures. This method simultaneously adjusts the spatial discretization resolution (h-adaptivity) and the approximation order (p-adaptivity) to allocate the available computational resources in the domain regions where they are most needed.

The last part of the thesis deals with yet another attempt to improve the efficiency of PDE-solving solution procedures, namely by spatially varying the approximation method. Specifically, the advantages of local approximation and properties of different approximation methods are exploited to propose a hybrid WLS–RBF-FD approximation method. A step further is done by introducing spatially variable node regularity in conjunction with spatially variable approximation method, employing RBF-FD on scattered nodes and MON on uniform nodes. The performance of both spatially-varying approximation methods is demonstrated on a set of fluid-flow and linear elasticity problems to illustrate the computational efficiency gains offered by such solution procedures.