基于图形处理器的大规模结构计算研究
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摘要
图形处理器（graphics processing units, GPUs）有较强的并行数值运算功能，基于图形处理器的通用计算（genera l purpose GPU computations, GPGPU computations）正应用于更广泛的领域。本文使用OpenGL着色语言编写GPU片元处理程序，应用GPU加速大规模结构计算。
对基于GPU的通用计算进行了叙述。描述了GPU的硬件结构、处理管线、基于GPU通用计算的概念与方法、以及OpenGL着色语言。简述了影射与缩减等流操作，提出了纹理边长不是2的幂的新纹理缩减算法。研究了应用GPU实现密集矩阵运算的数据结构、矩阵与矩阵乘法和矩阵与向量乘法的GPU算法。应用GPU实现求解线性方程组的高斯消元法和共轭梯度法。并研究稀疏矩阵与向量乘法的GPU实现算法。
本文应用GPU实现了以下结构计算问题：①结构的动响应。分别用幂法和QR法计算结构的第一和所有自振频率。使用GPU加速Wilson-θ法和Newmark法计算，以计算结构的动响应。②大规模有限元法计算。应用基于GPU的不完全Cholesk y分解预处理共轭梯度法求解大型稀疏有限元线性方程组。③无网格法计算。应用GPU加速无单元Galerkin法计算，并用于计算线弹性问题、弹塑性问题和几何非线性问题。④线性分子结构力学方法计算。计算分析了扶手型和锯齿型碳纳米管的杨氏模量随直径变化的变化规律，并自由度超过10万的碳纳米管的拉伸变形，比较基于GPU和基于CPU的运算时间。⑤非线性分子结构力学方法计算。研究了非线性分子结构力学方法的基本原理和计算步骤，推导了碳原子键拉伸或压缩、弯曲和扭转力系数的非线性计算公式，给出了增量Newton-Raphson法求解非线性分子结构力学方法问题的算法。并给出了应用GPU加速非线性分子结构力学方法计算的算法，使用该算法，计算碳纳米管的拉伸和扭转非线性响应，比较基于GPU和基于CPU的计算的运算时间。以上计算表明，基于GPU的计算有良好的计算精度，随着自由度的增大，GPU所耗费的运算时间比CPU少，证实GPU能加速中、大自由度的结构计算。
虽然基于GPU的通用计算正应用于更多的领域，但基于GPU的计算有只提供单精度实数运算、程序编写较繁琐、数据传输效率不高等不足之处。
在应用GPU加速分子动力学方法计算，并应用于碳纳米管的计算分析；用GPU搜索无网格法影响域内结点；把GPU作为存储器，分块存储大规模有限元问题的刚度矩阵，使用PC机求解大规模有限元问题等方面有待进一步的研究。
Graphics processing units (GPUs) are powerful parallel processors, and genera l purposeGPU computations (GPGPU computations) are implemented in ma ny fields. In thisdissertation, OpenGL shading la nguage is used to compile GPU fragment programs, andGPUs are employed to accelera te the computations for large scale structures.
GPGPU computations are described. GPU hardware architectures, processing pipelines,the concept and procedure for GPGPU computations, and the OpenGL shading la nguage arediscussed. The stream operations such as map and reduction are reviewed, and a new texturereduction algorithm not requiring the texture size be the order of 2 is presented. The datastructure, matrix-matrix multiplication, and matrix-vector multiplication for dense matrixoperations are studied. The Gauss elimina tion method and conjugate gradient method forsolving linear equations are implemented by the GPU. The sparse matrix-vector multiplicationis also implemented by the GPU.
In this dissertation, the GPU is used to compute the following structural computations:①Structural dynamic response. The first natural frequency and all natural frequencies areobtained by the power method and the QR method, respectively. The GPU is used toaccelera te the Wilson-θmethod and Newmark method for computing the structural dynamicresponse.②Large scale finite element method computations. The incomplete Cholesk ydecomposition conjugate gradient method on the GPU is used to solve the larger scale sparsefinite element method equations.③Meshless method computations. The GPU is used toaccelera te the element free Galerkin method, which is used to compute the linear elasticproblems, elastic-plastic problems, and geometrica l nonlinear problems.④Linear molecularstructural mecha nics approach computations. This algorithm is used to analyze changes for theYoung’s moduli while the dia meters of the carbon nanotubes increasing. The tension of thecarbon nanotubes whose degree of freedom are over 100,000 are computed, and the runningtimes on the GPU and on the CPU are compared.⑤Nonlinear molecular structuralmecha nics approach computations . The theorem and computationa l procedure for thenonlinear molecular mecha nics structural mecha nics approach are studied. The nonlinear formula of the carbon atom bond force for stretching, bending and torsion are deduced. Theincrement Newton-Raphson method for solving the nonlinear molecular structural mecha nicsapproach is given, and the GPU implementation for nonlinear molecular structural mecha nicsapproach is also given. These algorithms are used to analyze the carbon nanotubes’nonlinearstretching and torsiona l responses, and the running times on the GPU algorithms and on theCPU algorithms are compared. These computationa l results show that the computations on theGPU have good computationa l accuracy, while the degree of freedom increasing, the runningtimes on the GPU are less tha n those on the CPU, proving that the GPU can accelera te thecomputations for med ium and large structures.
Although GPGPU computations are being applied in ma ny fields, the computations on theGPU have such deficiencies as only single precision real being provided, the GPU programsbeing complex and trivia l, data transfer between the GPU and the CPU being deficient, etc.
Further research is necessary in the following: The GPU is used to accelera te moleculardynamics computations, and employ this algorithm to analyze carbon nanotubes; The GPU isused to search the nodes within the influence doma in in meshless method; As a storageelement, the GPU is used to store the blocks of the stiffness matrix for large scale finiteelement computations; etc.
对基于GPU的通用计算进行了叙述。描述了GPU的硬件结构、处理管线、基于GPU通用计算的概念与方法、以及OpenGL着色语言。简述了影射与缩减等流操作，提出了纹理边长不是2的幂的新纹理缩减算法。研究了应用GPU实现密集矩阵运算的数据结构、矩阵与矩阵乘法和矩阵与向量乘法的GPU算法。应用GPU实现求解线性方程组的高斯消元法和共轭梯度法。并研究稀疏矩阵与向量乘法的GPU实现算法。
本文应用GPU实现了以下结构计算问题：①结构的动响应。分别用幂法和QR法计算结构的第一和所有自振频率。使用GPU加速Wilson-θ法和Newmark法计算，以计算结构的动响应。②大规模有限元法计算。应用基于GPU的不完全Cholesk y分解预处理共轭梯度法求解大型稀疏有限元线性方程组。③无网格法计算。应用GPU加速无单元Galerkin法计算，并用于计算线弹性问题、弹塑性问题和几何非线性问题。④线性分子结构力学方法计算。计算分析了扶手型和锯齿型碳纳米管的杨氏模量随直径变化的变化规律，并自由度超过10万的碳纳米管的拉伸变形，比较基于GPU和基于CPU的运算时间。⑤非线性分子结构力学方法计算。研究了非线性分子结构力学方法的基本原理和计算步骤，推导了碳原子键拉伸或压缩、弯曲和扭转力系数的非线性计算公式，给出了增量Newton-Raphson法求解非线性分子结构力学方法问题的算法。并给出了应用GPU加速非线性分子结构力学方法计算的算法，使用该算法，计算碳纳米管的拉伸和扭转非线性响应，比较基于GPU和基于CPU的计算的运算时间。以上计算表明，基于GPU的计算有良好的计算精度，随着自由度的增大，GPU所耗费的运算时间比CPU少，证实GPU能加速中、大自由度的结构计算。
虽然基于GPU的通用计算正应用于更多的领域，但基于GPU的计算有只提供单精度实数运算、程序编写较繁琐、数据传输效率不高等不足之处。
在应用GPU加速分子动力学方法计算，并应用于碳纳米管的计算分析；用GPU搜索无网格法影响域内结点；把GPU作为存储器，分块存储大规模有限元问题的刚度矩阵，使用PC机求解大规模有限元问题等方面有待进一步的研究。
Graphics processing units (GPUs) are powerful parallel processors, and genera l purposeGPU computations (GPGPU computations) are implemented in ma ny fields. In thisdissertation, OpenGL shading la nguage is used to compile GPU fragment programs, andGPUs are employed to accelera te the computations for large scale structures.
GPGPU computations are described. GPU hardware architectures, processing pipelines,the concept and procedure for GPGPU computations, and the OpenGL shading la nguage arediscussed. The stream operations such as map and reduction are reviewed, and a new texturereduction algorithm not requiring the texture size be the order of 2 is presented. The datastructure, matrix-matrix multiplication, and matrix-vector multiplication for dense matrixoperations are studied. The Gauss elimina tion method and conjugate gradient method forsolving linear equations are implemented by the GPU. The sparse matrix-vector multiplicationis also implemented by the GPU.
In this dissertation, the GPU is used to compute the following structural computations:①Structural dynamic response. The first natural frequency and all natural frequencies areobtained by the power method and the QR method, respectively. The GPU is used toaccelera te the Wilson-θmethod and Newmark method for computing the structural dynamicresponse.②Large scale finite element method computations. The incomplete Cholesk ydecomposition conjugate gradient method on the GPU is used to solve the larger scale sparsefinite element method equations.③Meshless method computations. The GPU is used toaccelera te the element free Galerkin method, which is used to compute the linear elasticproblems, elastic-plastic problems, and geometrica l nonlinear problems.④Linear molecularstructural mecha nics approach computations. This algorithm is used to analyze changes for theYoung’s moduli while the dia meters of the carbon nanotubes increasing. The tension of thecarbon nanotubes whose degree of freedom are over 100,000 are computed, and the runningtimes on the GPU and on the CPU are compared.⑤Nonlinear molecular structuralmecha nics approach computations . The theorem and computationa l procedure for thenonlinear molecular mecha nics structural mecha nics approach are studied. The nonlinear formula of the carbon atom bond force for stretching, bending and torsion are deduced. Theincrement Newton-Raphson method for solving the nonlinear molecular structural mecha nicsapproach is given, and the GPU implementation for nonlinear molecular structural mecha nicsapproach is also given. These algorithms are used to analyze the carbon nanotubes’nonlinearstretching and torsiona l responses, and the running times on the GPU algorithms and on theCPU algorithms are compared. These computationa l results show that the computations on theGPU have good computationa l accuracy, while the degree of freedom increasing, the runningtimes on the GPU are less tha n those on the CPU, proving that the GPU can accelera te thecomputations for med ium and large structures.
Although GPGPU computations are being applied in ma ny fields, the computations on theGPU have such deficiencies as only single precision real being provided, the GPU programsbeing complex and trivia l, data transfer between the GPU and the CPU being deficient, etc.
Further research is necessary in the following: The GPU is used to accelera te moleculardynamics computations, and employ this algorithm to analyze carbon nanotubes; The GPU isused to search the nodes within the influence doma in in meshless method; As a storageelement, the GPU is used to store the blocks of the stiffness matrix for large scale finiteelement computations; etc.
引文
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