Development of Finite Element Analysis System for Unsteady, Incompressible Viscous Fluid Flows

本研究的目的是建立一个求解时变不可压缩Navier-Stokes方程的数值程序，并将该程序实现为一个适用于各种非定常流问题的程序系统。以下是目前项目的推广和成果总结。(1)有限元解法的发展：利用研究者提出的一种独特的直接时间积分方法，结合连续条件和边界条件的特殊过程，从原始变量公式的有限元方程中导出了一组表示速度和压力在不可压缩约束下演化的非线性递归方程。在本推导中没有引入过多的人工技术。在每一步积分中对非线性对流项进行迭代处理。(2)有限元程序系统的实现：由于导出的非线性递推方程的系统系数矩阵是非对称的，并且可能具有带型，因此采用利用天际线存储的LDU分解方法作为方程求解器。在时间积分间隔固定的情况下，只需要执行一次因式分解，之后只需要重复因式分解后的数组的前向约简/后向代入。为了图形化地表示速度、压力、流线、涡度、流体力等的空间分布或时间历史，准备了几种类型的后置处理器。(3)对本分析系统性能的验证：将需要求解大系统矩阵方程的涡脱落问题的数值结果与实验观察和测量结果进行比较，结果表明，随时间变化的解与实验可视化流场吻合较好，可以预测作用在障碍物上的流体力。少

The aim of the present research is to develop a numerical procedure for solving the time-dependent, incompressible Navier-Stokes equations, and to implement the procedure as a program system which is applicable to various unsteady flow problems.The followings are the summary of the promotion and the results of the present project.(1) Development of finite element solution procedure: A set of non-linear recurrence equations, which represent evolution of the velocities and the pressures under the incompressibility constraint, are derived from the finite element equations of the primitive variable formulation, by using a unique direct time integration method developed by the investigators and particular processes regarding the continuity conditions and the boundary conditions. Excessively artificial techniques have not been introduced into the present derivation. An iteration process with respect to the non-linear convection terms is performed in every integration step.(2) Implementation … More of finite element program system: Since the system coefficient matrix of the derived non-linear recurrence equations is non-symmetric and may possess a band-profile, the LDU decomposition method using skyline storage is adopted as the equation solver. As far as the time integration interval is fixed, only one factorization need be performed and, after that, only the forward reduction/back substitution of the factorized array need be repeated. Several types of post-processors are prepared in order to express graphically the spatial distribution or the time history of velocity, pressure, streamlines, vorticity, fluid forces, or so on.(3) Verification of the performance of the present analysis system: Comparisons of the numerical results, including vortex shedding problems which require solving large system matrix equations, with experimental observation and measurements show that, the time-dependent solutions agree fairly well with the experimentally visualized flow field, and the fluid forces acting on the obstacles can be predicted. Less