Boundary element method for quaternion valued Laplace equation in three dimensions

In this thesis, a quaternion boundary element method (BEM) for solving three-dimensional problems governed by scalar, vector and quaternion Laplace equations is developed. To derive quaternion valued boundary integral equations (BIEs) for both the domain point and the out-of-domain point, the quaternion valued Stokes’ theorem is utilized. For smooth boundary points and points at corners and edges of nonsmooth boundary, the singular quaternion valued BIEs are all obtained; the integrals are noted for singularity, which exists in the sense of the Cauchy principal value (CPV). Here, we develop an analytical scheme to evaluate the CPV by introducing a simple quaternion valued harmonic function. For the domain points close to the boundary, some sorts of analogous, nearly singular, so-called “boundary layer” phenomena appear and are remedied by a similar analytic evaluation. The quaternion BEM features the oriented surface element, combining the unit outward normal vector with the ordinary surface element. Finally, several numerical examples including the problems of magnetostatics, heat conduction in functionally graded materials and Green’s function, are considered to demonstrate the validity of the present approach.