Highly accurate calculation of electric field for transcranial magnetic stimulation using hybridizable discontinuous galerkin method.

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.