Coupling of adaptive refinement with variational multiscale element free Galerkin method for high gradient problems

Abstract

In this thesis, a new adaptive refinement coupled with variational multiscale element free Galerkin method (EFGM) is developed for solving high gradient problems. The aim of this thesis is to propose a new framework of moving least squares (MLS) approximation with coupling method based on the variational multiscale concept. Additional new nodes will be inserted automatically at high gradient regions by adaptive algorithm based on refinement criteria. An enrichment function is embedded in the MLS approximation for the fine scale part of the problem. Besides, this new technique will be parallelized by using OpenMP which is based on shared memory architecture. The proposed new approach is first applied in two-dimensional large localized gradient problem, transient heat conduction problem as well as Burgers' equation in order to analyze the accuracy of the proposed method and validated with an available analytic solutions. The obtained numerical results show a very good agreement with the analytic solutions and is able to obtain more accurate results than the standard EFGM. It is found that the average relative error of this new method is reduced in the range of 15% to 70%. Besides, this new method is also extended to solve two-dimensional sine-Gordon solitons. The results obtained show good agreement with the published results. Moreover, the parallelization of adaptive variational multiscale EFGM can improve the computational efficiency by reducing the execution time without loss of accuracy. Therefore, the capability and robustness of this new method has the potential to investigate more complicated problems in order to produce higher precision solutions with shorter computational time.