快速求解参数化偏微分方程的缩减基有限元方法及其在核工程中的应用

发布时间：2018-09-07 16:53

【摘要】：基于求解偏微分方程的高保真数值模拟已经广泛应用于科学研究和工程设计.然而,即使借助超级计算机的并行计算能力,经典的有限元方法和其它数值方法在面对需要多次求解或需要快速甚至实时求解的问题时仍然面临效率的挑战.针对可用参数化微分方程表示的问题,缩减基有限元方法利用少数代表性的经典有限元解构造基函数,同时通过仿射分解使得系统矩阵和载荷向量的组装变为简单的代数叠加,因此该方法可以大幅度地提高这类问题的求解效率.本文介绍了这种方法的原理,并以固体热传导和中子扩散的快速求解为例,展示了这种方法的优良特性.结果表明,在线阶段的求解效率可以实现两到三个数量级的提升.基于高保真模拟的缩减基模型是将高性能计算应用于工程优化设计、应急指挥以及复杂问题的反分析等工作的有效手段.
[Abstract]:High fidelity numerical simulation based on solving partial differential equations has been widely used in scientific research and engineering design. However, even with the parallel computing power of supercomputers, classical finite element methods and other numerical methods still face the challenge of efficiency in the face of problems that require multiple solutions or need to be solved quickly or in real time. In order to solve the problem of parameterized differential equations, the reduced-base finite element method uses a few representative classical finite element solutions to construct basis functions, and by affine decomposition, the assembly of system matrices and load vectors becomes a simple algebraic superposition. Therefore, this method can greatly improve the efficiency of solving this kind of problems. In this paper, the principle of this method is introduced, and the excellent properties of this method are demonstrated by taking the fast solution of solid heat conduction and neutron diffusion as examples. The results show that the efficiency of online solution can be improved by two to three orders of magnitude. The reduced base model based on high fidelity simulation is an effective method to apply high performance computing to engineering optimization design emergency command and back analysis of complex problems.
【作者单位】：中山大学中法核工程与技术学院;
【基金】：NSFC-广东联合基金超级计算科学应用专项项目(20144500031650003)
【分类号】：O241.82

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