用脉冲谱—优化法求解对流—扩散方程边界条件控制反问题

Numerical Solution to an Inverse Problem of Boundary Condition Control for Convection-Diffusion Equations by PST-Optimization

本文综合利用脉冲谱法和优化法求解一类流体力学反问题对流—扩散方程的边界条件控制反问题.作为典型算例,可根据下游河道的环境容量反求上游边界的限制浓度,以作为制定排放标准的依据.首先把对流-扩散方程作Laplace变换,并根据下游部分区域的环境容量建立优化目标泛函,然后应用脉冲谱原理推导出目标泛函对未知边界条件的变分的解析式,利用标准的变尺度优化方法确定上游边界的限制浓度.数值计算结果表明,本文提出的计算方法精度高、速度快、避免了冗繁的"试错"过程,不但在环境水力学领域具有较大的实用价值,而且可应用于泥沙悬移质控制问题.

A new method of PST-Optimization is presented in this paper to solve a class of inverse problems involved in Fluid Mechanics—inverse problems of boundary condition control for convection-diffusion equations. As a typical numerical example, the limited concentration, i. e. the upstream boundary condition, is determined by the partially desired concentration distribution along a river. The limited concentration is then a good referenee for determining discharge standard. At first, the Laplacian transformation is used to convert the space-time problem into the space-spectrum problem. Then an objective function, based on the desired concentration distribution, is formulated. An analytical expression of variation for the objective function to the unknown boundary condition is derived by using PST. Hereafter, the standard variable-metric optimization is used to determine the limited concentration, i. e. the boundary condition which should be specified on the upstream boundary.Numerical results show that the present method is not only high in accuracy but also fast in computing, thus avoiding the traditional nerve-wracking trial and error.