Wei Lianyong, Zou Jun
Dept. of Electrical Engineering, Tsinghua University, Beijing, 100084, China.
Sci Rep. 2024 Dec 2;14(1):29936. doi: 10.1038/s41598-024-76867-z.
The computation of electric field in transcranial magnetic stimulation (TMS) is essentially a problem of gradient calculation for thin layers. This paper introduces a hybrid-order hybridizable discontinuous Galerkin finite element method (HDG-FEM) and systematically demonstrates its superiority in TMS computations. The discrete format of HDG-FEM employing hybrid orders for TMS is derived and, from a fundamental numerical principle perspective, this study provides the elucidation of why HDG-FEM exhibits superior gradient computation capabilities compared to the widely used CG-FEM. Furthermore, the exceptional performance of HDG-FEM in thin layer calculation is demonstrated on both modified head models and realistic head models, focusing on three aspects: calculation errors, utilization of hybrid order, and computational cost. For the calculation of E-field in thin-layer regions with parameter mutation, the L norm error of the first-order HDG-FEM with the same tetrahedral mesh is comparable to the L norm error of the second-order CG-FEM. The L norm error of the same-order HDG-FEM is smaller than that of the same-order CG-FEM. By utilizing the hybrid order, HDG-FEM achieves a rapid reduction in errors of thin layers without a significant increase in the computational cost. This study transforms the three-dimensional TMS problem into a special two-dimensional problem for computation, reducing computational complexity from p in three dimensions to p in two dimensions, while achieving significantly higher accuracy compared to the commonly used CG-FEM. The utilization of hybrid orders in thin layers of the head demonstrates significant flexibility, making HDG-FEM a new alternative choice for TMS computations.
经颅磁刺激（TMS）中电场的计算本质上是一个薄层梯度计算问题。本文介绍了一种混合阶可杂交间断伽辽金有限元方法（HDG-FEM），并系统地论证了其在TMS计算中的优越性。推导了用于TMS的采用混合阶的HDG-FEM离散格式，并且从基本数值原理的角度，本研究阐明了为什么HDG-FEM与广泛使用的CG-FEM相比具有卓越的梯度计算能力。此外，在改进的头部模型和真实头部模型上，从计算误差、混合阶的利用和计算成本三个方面展示了HDG-FEM在薄层计算中的优异性能。对于参数突变的薄层区域中的电场计算，相同四面体网格下的一阶HDG-FEM的L范数误差与二阶CG-FEM的L范数误差相当。同阶HDG-FEM的L范数误差小于同阶CG-FEM的L范数误差。通过利用混合阶，HDG-FEM在不显著增加计算成本的情况下实现了薄层误差的快速降低。本研究将三维TMS问题转化为特殊的二维问题进行计算，将计算复杂度从三维的p降低到二维的p，同时与常用的CG-FEM相比实现了显著更高的精度。在头部薄层中混合阶的利用展示了显著的灵活性，使得HDG-FEM成为TMS计算的一种新的替代选择。
