高维热流耦合方程正反演的数值计算方法

发布时间：2018-05-19 16:17

本文选题：高维问题 + 热流耦合方程　；参考：《东华理工大学》2017年硕士论文

【摘要】：热流耦合问题在地热资源开发利用、放射性废物处置、天然气开发、地下水开采等方面都广泛存在.本文对高维热流耦合模型进行分析,推导出方程的离散格式,对所得离散格式方程组进行了数值分析,对耦合模型参数反演问题进行了数值求解.本文首先对几种热流耦合模型进行分析,详细讨论了一种热流耦合模型的控制方程.用差分法对耦合方程进行数值离散,构造该方程的交替方向迭代(ADI)差分格式,并对该差分格式进行了截断误差分析以及稳定性分析,得到差分格式解的条件稳定性.热流耦合方程直接离散后得到非线性方程组.由于非线性方程组求解困难,本文先求出稳态渗流方程在离散点处的渗流速度的数值解,之后代入离散热传导方程,将非线性方程组转化为线性方程组.在高维的情况下,空间域与时间域网格离散,节点多,从而导致离散方程组的阶数很大.求解线性方程组系数矩阵阶数会很大,采用Thomas算法和克劳特方法对线性方程组进行求解分析.为了提高计算精度,进一步采用广义极小残余(GMRES)算法求解分析,并探讨该算法的收敛性.进行了数值模拟,验证该数值方法的可行性.本文最后讨论了热流耦合方程参数反问题,给出了二维热流耦合方程参数反演的计算格式,对参数反演格式进行了分析,并结合模拟退火算法进行反演计算和数值模拟.
[Abstract]:Heat flow coupling problem exists widely in geothermal resources exploitation, radioactive waste disposal, natural gas development, groundwater exploitation and so on. In this paper, the coupling model of high dimensional heat flux is analyzed, the discrete scheme of the equation is deduced, the equations of the discrete form are numerically analyzed, and the parameter inversion problem of the coupled model is solved numerically. In this paper, several heat flux coupling models are analyzed, and the governing equations of a heat flux coupled model are discussed in detail. The coupled equations are discretized numerically by the difference method, and the alternating direction iterative ADI difference scheme is constructed. The truncation error and stability analysis of the scheme are carried out, and the conditional stability of the solution of the difference scheme is obtained. The heat flux coupling equation is discretized directly and the nonlinear equations are obtained. Due to the difficulty of solving the nonlinear equations, the numerical solution of the seepage velocity of the steady state seepage equation at the discrete point is obtained first, and then the discrete heat conduction equation is replaced, and the nonlinear equations are transformed into the linear equations. In the case of high dimensional space and time domain grids are discrete and there are many nodes resulting in the large order of discrete equations. The order of coefficient matrix for solving linear equations is very large. The Thomas algorithm and the Crowter method are used to solve the linear equations. In order to improve the calculation accuracy, the generalized minimal residual algorithm (GMRES) is used to solve the problem, and the convergence of the algorithm is discussed. Numerical simulation is carried out to verify the feasibility of the numerical method. In the end, the inverse problem of the parameters of the coupled heat flux equation is discussed, and the calculating format of the parameter inversion of the two-dimensional heat flux coupling equation is given. The inversion scheme of the parameters is analyzed, and the inversion calculation and numerical simulation are carried out in combination with the simulated annealing algorithm.
【学位授予单位】：东华理工大学
【学位级别】：硕士
【学位授予年份】：2017
【分类号】：O241.8

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