Improving The Computational Efficiency Of Isogeometric Analysis Via Parallel Computing And Multigrid Methods

Isogeometric analysis is a new numerical method for partial differential equations. In order to make isogeometric analysis practical on various engineering fields, it is nec-essary to study the methods to improve the effeciency of isogeometric analysis and implement effecient solvers for isogeometric analysis. In this dissertation, the com-putational efficiency of isogeometric analysis is improved via parallel computing and multigrid methods. The main work and contributions of this dissertation are as follows.(1) A parallel algorithm for isogeometric analysis which is based on computational domain decompozation is proposed. The computational domains are uniformly divided into a few parts, each part is assigned to1processor. After the computation on the computational domains is completed, the linear system is constructed by using a fast multi-way merge algorithm to assemble the global stiffness matrix.(2) The parallism of isogeometric analysis is divided into the computational do-main level and the global stiffness matrix level. The computation on the computational domain level is parallel via computational domain decompozation. The construction of the linear system is parallel via global stiffness matrix decompozation. The global stiffness matrix decompozation regards each row as a unit, uniformly decomposes the stiffness matrix into a few parts. Each part is assigned to1processor. Based on the global stiffness matrix decompozation, a parallel conjugate gradient method is imple-mented to solve the linear system.(3) A parallel isogeometric analysis framework is implemented on Intel Single Chip Cloud system (SCC). The parallel method is global stiffness matrix decompoza-tion. The computation on computational domain level is computed via lazy computa-tion. Each core uses a hash table to save the values on computational domains. During the construction, the core finds the computational domain in the hash table. If no result is found, the computation on the computational domain is performed and the result will be added to the hash table. A conjuage gradient method is implemented on SCC to solve the linear system.(4) Two multigrid methods for isogeometric analysis are proposed to accelerate the convergence of the solver of the linear system, which are denoted as the correction scheme and the nested iteration. The correction scheme accelerates the convergence the linear system on the finer grid via solving the residual equation exactly on coarse grid. The nested iteration uses the solution on coarse grid as the initial guess for the linear system on finer grid. Based on the nested iteration, a multigrid numerical simulation method for isogeometric analysis is proposed.The experiments show that the computational domain decompozation based algo-rithm achieves a speed up of6.17on an8-core system. The global stiffness matrix de-compozation based algorithm achieves nearly linear speed-up. The multigrid methods proposed in this dissertation effectively accelerate the convergence of iteration solver of the linear system. The efficiency of the multigrid based numerical simulation is much better than the normal numerical simulation method for isogeometric analysis.In this dissertation, the construction of the linear system is accelerated by using parallel computation techniques. The solver of the linear system is accelerated by multi-grid methods. By using the algorithms proposed in this dissertion, efficient solvers for isogeometric analysis can be implemented.