Research On Parallel Multigrid Solver For Unstructured Grid

As an important branch of research in computational fluid dynamics, unstructuredgrid has been widely applied to the complex shape of the experimental tests andengineering calculations. The numerical solution of the fluid on the unstructured gridrequires more storage space and computing time, which is affected by the geometry andspatial location, and puts forward higher requirements on the effectiveness andefficiency. As an efficient unstructured grid solver, multigrid has excellentcharacteristics both in time and space for serial and parallel implementation, which hasattracted researchers much more attentions for this method in recent years. However,there are some differences in multigrid application for different unstructured gridcomputing format. The application of multigrid in Cell-Centered scheme is far lesssmoothly than in the Vertex-Centered scheme with finite volume method.This thesis details our researches on multigrid used in Cell-Centered scheme.Firstly, we conduct a simple review on multigrid application, and then introduce theprinciple of efficiency, error correction format, classification, parallelization strategies.Secondly, starting with discretization of governing equations of fluid, we indtroducehow to use the multigrid for finite volume method on unstructured grid, and study thecalculations of traditional interpolation and restriction operators, then we analyzetheclassical convergence theory on these operators, which will be useful in improving thecorresponding numerical accuracy. Thirdly, we research on the multigrid coarseningproblems for Cell-Centered scheme, which can’t provide suitable complexity and goodquality coarsen grid with variety of grid types. Three different agglomeration multigridmethods have been developed for using to handle the highly stretched grids, improvethe quality of coarse grid and amend the tree-structured coarsening process respectively.The effectiveness and efficiency of the three different agglomeration methods have beenverified by many examples. Finally, the three different multigrid methods have beenparallelized with message passing model, and conducted parallel computing researchand testing on large-scale fluid problems.The primarily innovative works in this thesis are as follows.1．Strongly coupled agglomeration multigrid is developed. The algebric multigridcan effectively deal with anisotropic matrix compution. We use numerical relationshipform geometric characteristics of unstructured grid to denote degree of anisotropy, andform strong or weak coupling relations between cells for generating coarse grid levels.It is useful to deal with highly stretched grid for Cell-Centered scheme. In order toreduce the computational complexity of coarse grid level and meet the needs ofdifferent dimensions, several times strongly coupled agglomeration are conducted onhighly stretched grid for Cell-Centered scheme. Through serial different examples, comparing the performance of several exiting agglomeration methods, those resultsverify what the method we present with effecitiveness and efficiency.2．Three second-order accuracy interpolation operators are developed. Accordingto classic convergence theory and principle, we use geometric reconstruction andextension computing node stencil for interpolation operator, so it has a second-ordernumerical accuracy. On this basis, we apply control volume, the control area and therelative distance for average value interpolation equation respectively. When comparingseven different interpolation operators, it is applicable to the numerical solution fortwo-dimensional viscous fluid.3．Aspect ratio based combination agglomeration method is developed. As thequality of coarse grid generated by agglomeration has important effect on multigridconvergence acceleration methods, aspect ratio is one of the most important charactersto measure the quality. We apply aspect ratio as a standard of anisotropic grid andagglomeration, and modify the front queue to enable to deal with different types of gridcells. This method use cell-based agglomeration for anisotropic cells and vertex-basedagglomeration for the others. It is providing the high quality coarse grid and idealconvergence ratio for this method in many testing cases, comparing with MGridGen.4．We propose improved tree-based data structure grid coarsening method. Theglobal division of unstructured grid established tree data structure with hierarchicalstorage capacity of the grid cell, which can be used for coarse grid level. It can notaccurately capture the anisotropic characteristics for the space location of cell center,which leads to low-quality coarse grid. We use different data structure to process cellagglomeration process, and verify the effectiveness of the method by comparing withstrongly coupled agglomeration on numerically solving the numerical examples.5． We conduct three agglomeration multigrid methods for parallelizationapplication. The parallelization is used for massive unstructured grid based on messagepassing model with MPI standard. The computation results indicate that the threeagglomeration multigrid methods are acceptable, and could be used for followingmassive parallel computation and engineering application.