非结构网格的并行多重网格算法研究
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摘要
非结构网格作为计算流体力学的一个重要研究分支，被广泛地应用于复杂外形的实验测试和工程计算中。受几何形体和空间位置的影响，非结构网格中的流体求解需要更多的存储空间和计算时间，对数值算法的有效性和高效性提出了更高的要求。多重网格方法作为非结构网格的高效解算器，其串行和并行实现，在时空上都具有优良的特性，成为近年来备受研究者关注的计算方法。然而，多重网格的应用，对非结构网格计算格式的适应能力存在一定差别。有限体积方法中，单元中心型计算格式的多重网格方法，远远不如在节点中心型计算格式中应用得自然流畅。
本文的研究工作围绕单元中心型计算格式的多重网格方法展开。首先，对多重网格方法的应用研究进行了简单的回顾，介绍了其高效性的原理、误差校正格式、分类方法及其并行化策略。然后，从流体计算的控制方程出发，使用有限体积方法离散，引入多重网格的计算过程，研究了传统的插值、限制算子的计算公式，分析了经典收敛理论对插值、限制算子的精度要求，为提高相应数值传递算子的精度提供理论依据。接着，针对单元中心型计算格式不能有效地处理各种网格类型，形成质量较好、适合计算的粗网格层次的问题，重点研究开发了三种不同的聚合多重网格方法，分别用于处理高延展性网格单元、提高粗网格质量、改进树结构粗化过程，并通过相关算例验证了各自的有效性和高效性。最后，基于消息传递模型，实现了三种不同多重网格方法的并行化应用，对大规模的流体问题开展了并行计算研究和测试。
本文取得的主要研究成果包括：
1．提出了强耦合型网格聚合策略。借鉴代数多重网格能有效处理各向异性矩阵的相关经验，将非结构网格的几何特征转化为数值关系，通过数值大小区分几何网格的各向异性程度，形成单元强弱耦合关系，用于聚合生成粗网格层次，弥补了单元中心型计算格式不能有效处理高延展性网格的缺陷。为减小粗网格层的计算复杂度以及适应不同维度的计算需求，对高延展性网格单元使用多次强耦合聚合，其它网格单元使用单元中心型聚合。通过对不同算例的串行测试，对比了现有几种聚合方法的计算性能，验证了本方法的有效性和高效性。
2．提出了三种一阶精度的插值算子。根据对经典收敛理论和改进原则的分析，采用几何重构的思想，扩展插值算子的计算节点模板使其具有一阶数值精度，并在此基础上分别以控制体积、控制面积和相对距离为依据，建立了相应的数值平均插值方程。通过对二维粘性流体的数值求解，对比了七种不同的插值算子，对其适用的问题类型及其范围进行了测试和总结。
3．提出了基于纵横比的组合型网格聚合策略。由于聚合生成的网格质量决定了粗网格层的方程离散以及误差校正精度，对多重网格方法的收敛加速效果具有重要影响，纵横比就是用于衡量网格质量的重要参数之一。通过纵横比标示各向异性几何特征，同时作为单元聚合的判定标准，修改标准的波前聚合队列，使其能够顺利处理不同类型单元的聚合，对具有各向异性特征的网格单元使用基于单元的聚合方式，其它单元使用基于节点的聚合方式。在多个算例的测试中，对比了MGridGen软件的生成结果，验证了本方法生成的粗网格具有较高的质量，提供了比较理想的收敛加速效果。
4．提出了改进型的树结构网格粗化方法。利用非结构网格的全局划分建立的树形数据结构，具备层次化存储网格单元的能力，为建立粗网格层次提供了另一种思路。针对层次结构的建立受限于单元中心的空间位置，不能准确捕捉网格的各向异性特征，导致粗网格质量低下的问题，采用不同的数据结构处理不同的单元聚合过程，修正了各向异性网格单元的聚合过程，通过对算例数值求解，同时纵向对比了强耦合型聚合算法的计算结果，验证了本方法的有效性。
5．实现了三种多重网格聚合算法的并行化应用。基于消息传递模型的MPI标准，对三种聚合策略进行了并行化优化实现，发展了适合非结构网格的大规模并行计算方法，在高性能并行计算平台上，对多个算例进行了测试，计算结果表明开发的三种多重网格算法具有较高的计算精度、并行效率性和良好的可扩展性，为工程计算提供重要的实现方法。
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.
本文的研究工作围绕单元中心型计算格式的多重网格方法展开。首先，对多重网格方法的应用研究进行了简单的回顾，介绍了其高效性的原理、误差校正格式、分类方法及其并行化策略。然后，从流体计算的控制方程出发，使用有限体积方法离散，引入多重网格的计算过程，研究了传统的插值、限制算子的计算公式，分析了经典收敛理论对插值、限制算子的精度要求，为提高相应数值传递算子的精度提供理论依据。接着，针对单元中心型计算格式不能有效地处理各种网格类型，形成质量较好、适合计算的粗网格层次的问题，重点研究开发了三种不同的聚合多重网格方法，分别用于处理高延展性网格单元、提高粗网格质量、改进树结构粗化过程，并通过相关算例验证了各自的有效性和高效性。最后，基于消息传递模型，实现了三种不同多重网格方法的并行化应用，对大规模的流体问题开展了并行计算研究和测试。
本文取得的主要研究成果包括：
1．提出了强耦合型网格聚合策略。借鉴代数多重网格能有效处理各向异性矩阵的相关经验，将非结构网格的几何特征转化为数值关系，通过数值大小区分几何网格的各向异性程度，形成单元强弱耦合关系，用于聚合生成粗网格层次，弥补了单元中心型计算格式不能有效处理高延展性网格的缺陷。为减小粗网格层的计算复杂度以及适应不同维度的计算需求，对高延展性网格单元使用多次强耦合聚合，其它网格单元使用单元中心型聚合。通过对不同算例的串行测试，对比了现有几种聚合方法的计算性能，验证了本方法的有效性和高效性。
2．提出了三种一阶精度的插值算子。根据对经典收敛理论和改进原则的分析，采用几何重构的思想，扩展插值算子的计算节点模板使其具有一阶数值精度，并在此基础上分别以控制体积、控制面积和相对距离为依据，建立了相应的数值平均插值方程。通过对二维粘性流体的数值求解，对比了七种不同的插值算子，对其适用的问题类型及其范围进行了测试和总结。
3．提出了基于纵横比的组合型网格聚合策略。由于聚合生成的网格质量决定了粗网格层的方程离散以及误差校正精度，对多重网格方法的收敛加速效果具有重要影响，纵横比就是用于衡量网格质量的重要参数之一。通过纵横比标示各向异性几何特征，同时作为单元聚合的判定标准，修改标准的波前聚合队列，使其能够顺利处理不同类型单元的聚合，对具有各向异性特征的网格单元使用基于单元的聚合方式，其它单元使用基于节点的聚合方式。在多个算例的测试中，对比了MGridGen软件的生成结果，验证了本方法生成的粗网格具有较高的质量，提供了比较理想的收敛加速效果。
4．提出了改进型的树结构网格粗化方法。利用非结构网格的全局划分建立的树形数据结构，具备层次化存储网格单元的能力，为建立粗网格层次提供了另一种思路。针对层次结构的建立受限于单元中心的空间位置，不能准确捕捉网格的各向异性特征，导致粗网格质量低下的问题，采用不同的数据结构处理不同的单元聚合过程，修正了各向异性网格单元的聚合过程，通过对算例数值求解，同时纵向对比了强耦合型聚合算法的计算结果，验证了本方法的有效性。
5．实现了三种多重网格聚合算法的并行化应用。基于消息传递模型的MPI标准，对三种聚合策略进行了并行化优化实现，发展了适合非结构网格的大规模并行计算方法，在高性能并行计算平台上，对多个算例进行了测试，计算结果表明开发的三种多重网格算法具有较高的计算精度、并行效率性和良好的可扩展性，为工程计算提供重要的实现方法。
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.
引文
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