带源参数的二维热传导反问题的无网格方法
The meshless method for a two-dimensional inverse heat conduction problem with a source parameter
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摘要: 利用无网格有限点法求解带源参数的二维热传导反问题，推导了相应的离散方程. 与其它基于网格的方法相比，有限点法采用移动最小二乘法构造形函数，只需要节点信息，不需要划分网格，用配点法离散控制方程，可以直接施加边界条件，不需要在区域内部求积分.用有限点法求解二维热传导反问题具有数值实现简单、计算量小、可以任意布置节点等优点.最后通过算例验证了该方法的有效性.Abstract: Inverse problems are difficult to be solved in scientific research. And theyare widely applied in the aerospace, nuclear physics, metallurgyand other fields. The finite difference method and the finite element methodare main numerical methods to obtain numerical solutions of inverseproblems. The finite point method is one of meshless methods. Comparing withthe numerical methods based on mesh, such as finite element method andboundary element method, the finite point method only needs the scatterednodes instead of meshing the domain of the problem when the shape functionare formed. For problems with complicated domain which need to be re-meshed,the finite point method has the advantage of that no mesh is needed. In thispaper, the finite point method is used to obtain numerical solutions oftwo-dimensional inverse heat conduction problems with a source parameter,and the corresponding discretized equations are obtained. The collocationmethod is used to discretize the governing partial differential equations,and boundary conditions can be directly enforced without numericalintegration in the problem domain. This reduces the computation costgreatly. A numerical example is presented to show the method in this paperis effective. The finite point method, which is for two-dimensional inverseheat conduction problems with a source parameter, presented in this paperhas some advantages of arbitrary nodes distribution, simple numericalprocedures and low computation cost. The researches in this paper provide anew numerical method for inverse heat conduct problems, and can be appliedto other inverse problems.
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