自定义边元FEM求解器及其在Eddy-Current中的应用
Custom Edge-Element FEM Solver and its Application to Eddy-Current Simulation of Realistic 2M-Element Human Brain Phantom
摘要：
三维计算技术的大量研究论文被广泛用于人类大脑病理生理学的研究。涡流分析可以提供生物体内电导率变化的指示。当前趋势分析的一个重大障碍是发展一种数值稳定和高效的有限元方案，该方案在低频下表现良好，并且不需要大量的自由度。在这里，一个定制的有限元法（FEM）求解器的基础上，提出了使用弱耦合理论，它分为两个步骤的解决方案。首先，计算并存储背景场（每个边缘上的磁矢量位）。然后，通过有限元法，基于伽辽金公式，得到了每个节点上的标量电势。因此，可以获得物体中的电场和涡电流分布。这种求解器比典型的商业求解器更有效，因为它减少了矢量涡流方程的标量之一，并减少了网格域只是涡流区域。因此，它可以解决复杂的涡流计算的模型与大量的元素，如在生物组织中的涡流计算中遇到的。一个例子是一个现实的人脑网格的200万个元素。此外，利用该求解器，施加从励磁线圈感应的等效磁场，因此不需要网格化励磁线圈。结合起来，这些显著提高了求解器的效率。
英文摘要：
Extensive research papers of three-dimensional computational techniques are widely used for the investigation of human brain pathophysiology. Eddy current analyzing could provide an indication of conductivity change within a biological body. A significant obstacle to current trend analyses is the development of a numerically stable and efficiency-finite element scheme that performs well at low frequency and does not require a large number of degrees of freedom. Here, a custom finite element method (FEM) solver based on edge elements is proposed using the weakly coupled theory, which separates the solution into two steps. First, the background field (the magnetic vector potential on each edge) is calculated and stored. Then, the electric scalar potential on each node is obtained by FEM based on Galerkin formulations. Consequently, the electric field and eddy current distribution in the object can be obtained. This solver is more efficient than typical commercial solvers since it reduces the vector eddy current equation to a scalar one, and reduces the meshing domain to just the eddy current region. It can therefore tackle complex eddy current calculations for models with much larger numbers of elements, such as those encountered in eddy current computation in biological tissues. An example is presented with a realistic human brain mesh of 2 million elements. In addition, with this solver, the equivalent magnetic field induced from the excitation coil is applied, and therefore there is no need to mesh the excitation coil. In combination, these significantly increase the efficiency of the solver.
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