三维电阻抗成像的体积元方法的数值模拟和分析

电阻抗成像是一类椭圆方程反问题,本文在三维区域上对其进行数值模拟和分析.对于椭圆方程Neumann边值正问题,本文提出了四面体单元上的一类对称体积元格式,并证明了格式的半正定性及解的存在性;引入单元形状矩阵的概念,简化了系数矩阵的计算;提出了对电阻率进行拼接逼近的方法来降低反问题求解规模,使之与正问题的求解规模相匹配;导出了误差泛函的Jacobi矩阵的计算公式,利用体积元格式的对称性和特殊的电流基向量,将每次迭代中需要求解的正问题的个数降到最低.一系列数值实验的结果验证了数学模型的可靠性和算法的可行性.本文所提出的这些方法,已成功应用于三维电阻抗成像的实际数值模拟.

SIMULATION AND ANALYSIS OF FINITE VOLUME METHOD FOR 3-DIMENSIONAL ELECTRICAL IMPEDANCE TOMOGRAPHY

Electrical impedance tomography is an inverse problem of elliptic differential equations, numerical simulation and analysis for it in 3-dimensional domain are presented.In this paper a modified symmetric finite volume element method is proposed,positive semi-definiteness and existence of solution for this scheme are proved;element geometry matrix is introduced, which is helpful for simplifying the calculation of coefficient matrix;patch approximation for electrical resistivity is present to lower the scale of this inverse problem;the computational formula of Jacobian matrix of error functional is obtained,a class of electrical current pat- terns is proposed,under which the number of direct problems to be solved at each iteration can be reduced to the least.A series of numerical experiments verify the reliability of its mathematical model and the feasibility of the algorithm.These methods have been applied successfully in practical simulation of electrical impedance tomography.