Paulsen K D, Jia X, Sullivan J M
Thayer School of Engineering, Dartmouth College, Hanover, NH 03755.
IEEE Trans Biomed Eng. 1993 Sep;40(9):933-45. doi: 10.1109/10.245615.
Finite element techniques for three-dimensional specific absorption rate (SAR) computation in anatomically based human models are presented. The formulations center on Helmholtz weak forms which have been shown to be numerically robust and to afford additional sparsity in the resulting system of algebraic equations. Practical solution of these equations depends critically on the realization of an effective sparse matrix solver. Experience with several conjugate gradient-type methods is reported. The findings show that convergence rate (and even convergence in some cases) degrades significantly with increasing matrix rank and decreasing electrical loss for mesh spacings which adequately resolve the physical wavelengths of the electromagnetic wave propagation. However, with proper choice of algorithm and preconditioning, reliable convergence has been achieved for matrix ranks exceeding 2 x 10(5) on domains having sizeable volumes of electrically lossless regions. An automatic grid generation scheme for constructing meshes which consist of variable element sizes that conform to a predefined set of boundaries is discussed. Example meshes of homogeneous and heterogeneous human anatomies, the boundaries of which have been derived from CT-scan information, are shown. These results highlight the fact that 3D finite element mesh generation remains a difficult problem, but usable meshes with this level of complexity can be generated. Integration of the finite element formulation, the sparse matrix solver, and the mesh generation scheme is shown to lead to algorithms that can be implemented on inexpensive reduced instruction set computer (RISC) workstations with run times on the order of hours. An example of hyperthermia device simulation is presented which suggests that the finite element method is a practical alternative that rivals the impressive finite-difference time-domain (FDTD) computations that have appeared.
本文介绍了基于人体解剖模型的三维比吸收率（SAR）计算的有限元技术。公式推导围绕亥姆霍兹弱形式展开，该形式在数值上具有鲁棒性，并且能使所得代数方程组具有额外的稀疏性。这些方程的实际求解严重依赖于有效的稀疏矩阵求解器的实现。报告了几种共轭梯度型方法的经验。研究结果表明，对于能充分解析电磁波传播物理波长的网格间距，随着矩阵秩的增加和电损耗的减小，收敛速度（在某些情况下甚至收敛性）会显著下降。然而，通过适当选择算法和预处理，对于在具有相当大体积电无损区域的域上矩阵秩超过2×10⁵的情况，已实现可靠收敛。讨论了一种自动网格生成方案，用于构建由符合预定义边界集的可变单元尺寸组成的网格。展示了均匀和非均匀人体解剖结构的示例网格，其边界是从CT扫描信息中得出的。这些结果突出了三维有限元网格生成仍然是一个难题，但可以生成具有这种复杂程度的可用网格这一事实。有限元公式、稀疏矩阵求解器和网格生成方案的集成被证明可以产生能够在廉价的精简指令集计算机（RISC）工作站上实现的算法，运行时间约为数小时。给出了一个热疗设备模拟的例子，表明有限元方法是一种实用的替代方法，可与已出现的令人印象深刻的时域有限差分（FDTD）计算相媲美。
