Analysis of electric field induced by ELF magnetic field utilizing fast‐multipole surface‐charge simulation method for voxel data

Abstract

SUMMARYThis paper presents a fast-multipole surface-chargesimulation method for calculating three-dimensional Laplacian fields in voxel models. This method treats a surface of a voxel that has different inside and outside conductivities as a surface element of the indirect BEM (boundary element method). The main features of the proposed method are as follows. (1) An O(D 2 ) performance in the memory capacity and operation cost is provided by applying the diagonal form fast multipole method (FMM), when the number of voxels is about D 3 . (2) The boundary matching is imposed by the continuity of the total flux passing through each element, which guarantees the solution globally satisfying Gauss's law; therefore the solution is globally stabilized.This method is successfully applied to calculate the electric field induced by an applied homogeneous ELF (extremely low frequency) magnetic field in a human head model that has 1 mm × 1 mm × 1 mm voxel size.

Performance comparison of three types of GPU‐accelerated indirect boundary element method for voxel model analysis

“…By contrast, Equation (16) has O.p 2 / complexity because it requires no summation and it depends on , Â, and . Therefore, after M m n is translated into W j k , the W j k is repeatedly available for a lot of translations using Equation (16). Therefore, the most significant part of the M2L operation is reduced to O.p 2 /.…”

“…After selecting a translation direction from˙x,˙y, and˙´, the following operations are sequentially performed: rotate all related M toward the selected direction using Equation (19); translate all M to exponential expansion coefficients W using Equations (14) and (15); translate W to W 0 that corresponds to L using Equation (16); translate W 0 to L with Equations (17) and (18); reversely rotate all related L using Equation (20). D-M2L is completed after performing all of these processes in all directions.…”

“…The parameter k max used in D-M2L [9] was set to 18. Parameters M.k/, k D 1 to k max , were set to 6,8,12,16,20,26,30,34,38,44, 48, 52, 56, 60, 60, 52, 4, and 2. Odd numbers were avoided for M.k/ to improve the calculation efficiency [16].…”

“…Therefore, the CUDA kernels of these processes can share the template in Table II. However, this required a modification in the template for diagonal form translation based on Equation (16). Lines 05 and 06 were exchanged irregularly, that is, 'For loop 2' specified source boxes, whereas CUDA threads handled 568 pairs of target and source coefficients that correspond to the one-to-one mapping in the diagonal-form translation.…”

“…The BEM is geared to electrostatic field analysis in voxel models, and it considers square walls on cubic voxels as boundary surface elements [16]. Using the BEM, three-dimensional fields were analyzed in human voxel models derived from anatomical images.…”