| Adaptive finite element method is an efficient technique used for improving computational accuracy with minimal computational cost,especially for problems with singularities.However,for most large scale problems the computational cost is still too expensive for ordinary computer system.Recently,much attentions have been concentrated on parallel computing.The existing parallel adaptive finite element method primarily focuses on each function block such as mesh refinement,stiffness matrix assembly and solving systems of linear equations,respectively.The serial character seriously restricts the parallel efficiency and becomes one of the bottlenecks in large-scale parallel adaptive finite element analysis.Similar to the meshless method in which a row of stiffness matrix can be generated independently around each node,the node-based parallel finite element method with seamless connection has been proposed.The key of the parallel node-based finite element method is node-based local mesh generation.In this thesis,we study the adaptive mesh refinement method based on bubble-type local mesh generation method.The main research results are shown as follows:1.The node placement by bubble simulation method is briefly introduced and four acceleration strategies are proposed.Firstly the bubbles at the new refinement level are added in advance according to the estimated bubble distribution and the bubble insertion scheme is well designed.Secondly multilevel time step is adopted according to the a posteriori error estimator.The scheme for setting cell length adaptively is proposed and the cell searching method is modified to improve the efficiency of establishing adjacent list.Finally the localization operation is also enhanced.Numerical examples illustrate that the computational cost can be significantly decreased by 70% via adopting the strategies above,while keeping the mesh quality unchanged compared with the traditional method.2.The anisotropic adaptive finite element method based on anisotropic bubble-type local mesh generation has been presented.A new metric tensor is obtained based on the a posteriori error estimator to guide the mesh refining and coarsening in which the direction of error is approximated by recovery technique.Under the new metric,the error is equidistributed in the directions of maximum and minimum stretching on an element,and the mesh size is reduced/coarsend in regions with large/small errors.The mesh size function is defined to modify the mesh size and two threshold values are introduced to avoid excessive refining and coarsening in adaptive mesh refinement.At last,Numerical results in two-dimensions are presented to verify the ability of our metric tensor to generate anisotropic mesh with correct concentration and stretching direction.3.The anisotropic adaptive mesh refinement algorithm based on anisotropic bubble-type local mesh generation for convection-dominated problems is designed to optimize the mesh alignment,size and aspect ratio.Three main techniques are used.First,the streamline upwind Petrov-Galerkin(SUPG)method is used to stabilize the numerical scheme.Second,the a posteriori error estimator is computed and a new metric tensor is deduced.Third,optimal anisotropic meshes are generated by the anisotropic bubble-type local mesh generation(ABLMG)method.Compared with other mesh generation strategies,high-quality mesh can be obtained efficiently.Our algorithm is tested on several examples and the numerical results show that the algorithm is robust.4.Since the node placement method by bubble simulation is suitable for parallel computing,parallel adaptive mesh refinement method based on bubble-type local mesh generation is proposed.The node-based distributed mesh structure is designed to avoid the communication amount spent in mesh generation and finite element calculation.To gain the load balance,a new load balancing algorithm for the new mesh structure is devised and includes three key steps:node distribution estimation,domain re-decomposition and mesh migration.In this study,the domain is divided into uniform cells to construct the graph model and then a parallel multilevel adaptive graph repartitioning library,Par METIS,is used as the domain repartition tool in the parallel adaptive method.In the mesh migration step the elements and nodes are sent to the destination processor.Numerical results show that the method can achieve high efficiency,and obtain a high-quality mesh,especially the mesh near the interface of subdomains. |
