基于特大增量步算法的网络并行计算
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摘要
论文第一部分工作对特大增量步算法的体系进行了比较全面完整的阐述。主要成果是:(1)从力学概念出发详细阐述了特大增量步算法源于经典力法的推导思路,给出了完整的特大增量步算法的格式,包括非线性弹性问题、弹塑性问题和有限元格式一般表达的公式;(2)对比了与特大增量步算法具有一定相似性的LATIN 算法,指出基于力法和广义逆矩阵理论的LIM 算法和LATIN 算法虽然在方程分组和迭代形式上有相似之处,但在力学概念和数学理论基础上都有根本的不同;(3)给出了特大增量步算法完整的数学上的证明,包括特大增量步算法解的唯一性、解的收敛性和解的稳定性的证明,由此证明了特大增量步算法在数学上的完备性;(4)针对解的收敛性问题讨论了算法在弹塑性问题下样本点选取,指出在保证问题的收敛性的前提下LIM 算法在样本点的选择上比逐步增量法具有更大的自由性。
论文第二部分工作也是本论文更主要的工作是特大增量步算法的并行化研究工作。主要内容及成果为:(1)深入分析了特大增量步算法的并行性,指出算法在力学概念层面上所具有的时间上的并行和空间上的并行的内在并行性,以及算法在数值计算层面上在矩阵向量运算方面的并行性;(2)设计了特大增量步算法的分布式存储异步并行算法模型,提出了LIM 算法主从式的并行程序结构;(3)以平面应力问题为例,设计并给出了特大增量步算法基于广播的并行方案和基于非阻塞通信的并行方案;(4)描述了本论文所使用的并行环境,其硬件环境为机群系统,软件环境为Linux 操作系统加上基于消息传递MPI 编程环境;(5)对特大增量步算法并行程序的两个方案的并行性能进行了测试,分析了方案中的问题,进而给出了针对通信的程序优化方案,在清华大学计算机系高性能计算研究所的MPI环境下机群系统上的运行结果给出了接近10的加速比和95%以上的节点资源占用率,求解的规模可以达到上千单元网格,表现出良好的并行效率。
The Large Increment Method (LIM) is a new method presented by Prof. Xila Liu’s group in recent years, which is based on the force method and the generalized inverse matrix (GIM) theory for material nonlinearity problems. The whole theory system and the deduction of LIM are presented and discussed in detail. And the formulations of LIM are given. After the comparison between LIM and LATIN ( Large Time Increment Method presented by Ladeveze from Paris ), a conclusion is made that LIM and LATIN are basically two different iteration method in both mechanical concept and mathematical background.
The mathematical proof of LIM is also given for the unique of the solution, the astringency of the solution and the stability of iteration, which means that LIM is self-contained in mathematics. Furthermore, the chosen of sample points in LIM shows more optional without break the stability of iteration according to the step by step method. This is also an advantage of LIM. In a word, LIM brings the force method a new light in the computer calculation field.
The parallel computation of structural mechanics is a hot research field in western counties, while in China what has been done is far from enough. LIM has some natural parallel-calculating characteristics which are different with the classical sub-structural algorithm. These characteristics are discussed detailedly. LIM can be executed parallel in time field and space field. The calculation of matrix and vector in LIM can also be parallelized. A master-slave parallel model based on the distributed memory is given. Taking the plane stress problem for example, two parallel schemes are given, based on the broadcast and noblocking communication respectively.
The parallel environment used by LIM is a cluster system installed with LINUX and MPI, which belongs to the Computation Science Department of
Tsinghua University. The two given parallel schemes are tested and evaluated in this environment. Then the optimized program is given and shows a good efficiency with an approximate accelerated ratio of 8 and an occupancy of calculation joint up to 95%. While the ratio between the calculation and the communication is the key for a cluster system using massage passing mode. In order to get higher efficiency, one must pay more attention to it. Besides, the load-balancing is another important factor in parallel computation. The parallelization of LIM can bring some fresh air into the parallel computation field of structural mechanics.
论文第二部分工作也是本论文更主要的工作是特大增量步算法的并行化研究工作。主要内容及成果为:(1)深入分析了特大增量步算法的并行性,指出算法在力学概念层面上所具有的时间上的并行和空间上的并行的内在并行性,以及算法在数值计算层面上在矩阵向量运算方面的并行性;(2)设计了特大增量步算法的分布式存储异步并行算法模型,提出了LIM 算法主从式的并行程序结构;(3)以平面应力问题为例,设计并给出了特大增量步算法基于广播的并行方案和基于非阻塞通信的并行方案;(4)描述了本论文所使用的并行环境,其硬件环境为机群系统,软件环境为Linux 操作系统加上基于消息传递MPI 编程环境;(5)对特大增量步算法并行程序的两个方案的并行性能进行了测试,分析了方案中的问题,进而给出了针对通信的程序优化方案,在清华大学计算机系高性能计算研究所的MPI环境下机群系统上的运行结果给出了接近10的加速比和95%以上的节点资源占用率,求解的规模可以达到上千单元网格,表现出良好的并行效率。
The Large Increment Method (LIM) is a new method presented by Prof. Xila Liu’s group in recent years, which is based on the force method and the generalized inverse matrix (GIM) theory for material nonlinearity problems. The whole theory system and the deduction of LIM are presented and discussed in detail. And the formulations of LIM are given. After the comparison between LIM and LATIN ( Large Time Increment Method presented by Ladeveze from Paris ), a conclusion is made that LIM and LATIN are basically two different iteration method in both mechanical concept and mathematical background.
The mathematical proof of LIM is also given for the unique of the solution, the astringency of the solution and the stability of iteration, which means that LIM is self-contained in mathematics. Furthermore, the chosen of sample points in LIM shows more optional without break the stability of iteration according to the step by step method. This is also an advantage of LIM. In a word, LIM brings the force method a new light in the computer calculation field.
The parallel computation of structural mechanics is a hot research field in western counties, while in China what has been done is far from enough. LIM has some natural parallel-calculating characteristics which are different with the classical sub-structural algorithm. These characteristics are discussed detailedly. LIM can be executed parallel in time field and space field. The calculation of matrix and vector in LIM can also be parallelized. A master-slave parallel model based on the distributed memory is given. Taking the plane stress problem for example, two parallel schemes are given, based on the broadcast and noblocking communication respectively.
The parallel environment used by LIM is a cluster system installed with LINUX and MPI, which belongs to the Computation Science Department of
Tsinghua University. The two given parallel schemes are tested and evaluated in this environment. Then the optimized program is given and shows a good efficiency with an approximate accelerated ratio of 8 and an occupancy of calculation joint up to 95%. While the ratio between the calculation and the communication is the key for a cluster system using massage passing mode. In order to get higher efficiency, one must pay more attention to it. Besides, the load-balancing is another important factor in parallel computation. The parallelization of LIM can bring some fresh air into the parallel computation field of structural mechanics.
引文
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