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

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.

FAST SOLUTION OF PARAMETRIZED PARTIAL DIFFERENTIAL EQUATIONS BY USING REDUCED BASIS FINITE ELEMENT METHOD

High-fidelity simulation based on numerical solution of governing partial differential equations has been widely applied in scientific research and engineering design. However, even by exploiting the modern high-performance computation, fast solution is still a challenging task when the simulations have to be run many times such as in optimization problems or in nonlinear coupling problems. For the class of problems which can be described by parametrized partial differential equations, the reduced basis finite element constructs the basis functions on top of typical high-fidelity solutions and thus greatly reduces the number of unknowns. The principles of the method is introduced in the present study and the favorable features are demonstrated through solving the heat conduction problem and the neutron diffusion problem. It is shown a speedup of three orders of magnitude is achieved during the online stage.