Highly accurate calculation of electric field for transcranial magnetic stimulation using hybridizable discontinuous galerkin method
Abstract 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 superio...
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Nature Portfolio
2024-12-01
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| Online Access: | https://doi.org/10.1038/s41598-024-76867-z |
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| author | Lianyong Wei Jun Zou |
| author_facet | Lianyong Wei Jun Zou |
| author_sort | Lianyong Wei |
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| description | Abstract 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 2 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 3 in three dimensions to p 2 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. |
| format | Article |
| id | doaj-art-d520b7f11d5041c183a6925dd3ef88c1 |
| institution | Kabale University |
| issn | 2045-2322 |
| language | English |
| publishDate | 2024-12-01 |
| publisher | Nature Portfolio |
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| spelling | doaj-art-d520b7f11d5041c183a6925dd3ef88c12024-12-08T12:28:07ZengNature PortfolioScientific Reports2045-23222024-12-0114111610.1038/s41598-024-76867-zHighly accurate calculation of electric field for transcranial magnetic stimulation using hybridizable discontinuous galerkin methodLianyong Wei0Jun Zou1Dept. of Electrical Engineering, Tsinghua UniversityDept. of Electrical Engineering, Tsinghua UniversityAbstract 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 2 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 3 in three dimensions to p 2 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.https://doi.org/10.1038/s41598-024-76867-zContinuous Galerkin finite element method (CG-FEM)Hybridizable discontinuous Galerkin finite element method (HDG-FEM)Hybrid orderTranscranial magnetic stimulation (TMS)Parameter mutationThin layer |
| spellingShingle | Lianyong Wei Jun Zou Highly accurate calculation of electric field for transcranial magnetic stimulation using hybridizable discontinuous galerkin method Scientific Reports Continuous Galerkin finite element method (CG-FEM) Hybridizable discontinuous Galerkin finite element method (HDG-FEM) Hybrid order Transcranial magnetic stimulation (TMS) Parameter mutation Thin layer |
| title | Highly accurate calculation of electric field for transcranial magnetic stimulation using hybridizable discontinuous galerkin method |
| title_full | Highly accurate calculation of electric field for transcranial magnetic stimulation using hybridizable discontinuous galerkin method |
| title_fullStr | Highly accurate calculation of electric field for transcranial magnetic stimulation using hybridizable discontinuous galerkin method |
| title_full_unstemmed | Highly accurate calculation of electric field for transcranial magnetic stimulation using hybridizable discontinuous galerkin method |
| title_short | Highly accurate calculation of electric field for transcranial magnetic stimulation using hybridizable discontinuous galerkin method |
| title_sort | highly accurate calculation of electric field for transcranial magnetic stimulation using hybridizable discontinuous galerkin method |
| topic | Continuous Galerkin finite element method (CG-FEM) Hybridizable discontinuous Galerkin finite element method (HDG-FEM) Hybrid order Transcranial magnetic stimulation (TMS) Parameter mutation Thin layer |
| url | https://doi.org/10.1038/s41598-024-76867-z |
| work_keys_str_mv | AT lianyongwei highlyaccuratecalculationofelectricfieldfortranscranialmagneticstimulationusinghybridizablediscontinuousgalerkinmethod AT junzou highlyaccuratecalculationofelectricfieldfortranscranialmagneticstimulationusinghybridizablediscontinuousgalerkinmethod |