| This thesis proposes a novel way to find the electromagnetic fields when the computational domain is defined by a fine grid of pixels (2D) or voxels (3D). This happens quite often in bioelectromagnetic problems, since tissue shapes are usually obtained by tomography.;The proposed method is a finite element method in which, in 3D, each element is simply a set of p x p x p voxels, where p is an integer. It therefore avoids the heavy burden of surface extraction and meshing. Since there may be multiple materials within one element, conventional basis functions are not suitable. Instead, basis functions are computed using the voxel grid, so that the internal discontinuities are respected.;The idea is first tested on problems consisting of nested squares (2D) and cubes (3D) of dielectric, with a charge pair placed inside. The results obtained by using different element sizes p agree well with those obtained by commercial software: when p = 4, the root-mean-square (RMS) difference is 1.5 % of the maximum potential.;Then the new method is applied to solve an electroencephalography (EEG) problem, in which the head is modelled as a volume conductor and neural activity by current dipoles. The head model consists of 180x217x181 voxels. The computed electric potential is sampled along a contour on the outer side of the scalp, for different element sizes p. These results, again, agree well with a reference solution: for p = 4, the RMS difference is about 1% of the maximum potential. Solving one FE problem with p = 4 is 4.7 times faster than when using each voxel as an element, i.e., p = 1. When the solution is required for multiple righthand sides, as is common, the speedup is greater. For example, with 24 righthand sides, the p = 4 solution is 40 times faster than when p = 1. |
