三维电阻抗成像的体积元方法的数值模拟和分析  

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

作　　者：李久平[1] 袁益让[1] 

机构地区：[1]山东大学,数学与系统科学学院,济南250100

出　　处：《计算数学》2008年第1期59-74,共16页Mathematica Numerica Sinica

基　　金：国家重点基础研究专项经费(G1999032803);国家自然科学基金(10372052;10271066);教育部博士点基金(20030422047)资助项目

摘　　要：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.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 patterns 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.

关 键 词：三维电阻抗成像 修正体积元 稳定性 单元形状矩阵 数值模拟 

分 类 号：O241[理学—计算数学]