平板电容器问题边界元法奇异性处理及正则化

Treatment of singularity and regularization of the boundary element method in the problem of a planar capacitance sensor

为了研究带有复杂边界条件的平板式电容传感器数学模型,针对利用边界元方法数值求解时所遇到的边界积分方程具有积分奇异性的问题,采用解析积分方法完全消除了边界积分的奇异性.同时在实际仿真过程中,依据正则化理论对病态线性代数方程组进行处理,克服了其不适定性.使用平板电容器检测木材含水率问题的数值计算结果表明,奇异积分和病态矩阵的处理是正确有效的,该结果可应用于解决工程实际问题.文中所提出的方法可以方便地推广到类似的复杂边界问题的数值求解中去.

In order to investigate the mathematical model of a planar capacitance sensor with complex boundary conditions,the analytical integral method was applied in accordance with the singularity of the boundary integral which was the encountered problem for numerically solving the mathematical model by BEM.Meanwhile,the regularization theory was used in treating an ill-conditioned system of linear algebraic equations,and its ill-posed nature was overcome in the process of practical simulation.The numerical results of wood moisture content tested by a planar capacitance sensor indicate that the treatment of singular integral and ill-conditioned matrices is correct and efficient.The results can be applied to resolve practical engineering problems.The presented method is easily extended for numerically solving the similar complex boundary problems.