Applications of the Clifford algebra valued boundary element method to electromagnetic scattering problems

Abstract

Electromagnetic problems governed by Maxwell's equations are solved by using a Clifford algebra valued boundary element method (BEM). The well-known Maxwell's equations consist of eight pieces of scalar partial differential equations of the first order. They can be rewritten in terms of the language of Clifford analysis as a nonhomogeneous k-Dirac equation with a Clifford algebra valued function. It includes three-component electric fields and three-component magnetic fields. Furthermore, we derive Clifford algebra valued boundary integral equations (BIEs) with Cauchy-type kernels and then develop a Clifford algebra valued BEM to solve electromagnetic scattering problems. To deal with the problem of the Cauchy principal value, we use a simple Clifford algebra valued k-monogenic function to exactly evaluate the Cauchy principal value. Free of calculating the solid angle for the boundary point is gained. The remaining boundary integral is easily calculated by using a numerical quadrature except the part of Cauchy principal value. This idea can also preserve the flexibility of numerical method, hence it is suitable for any geometry shape. In the numerical implementation, we introduce an oriented surface element instead of the unit outward normal vector and the ordinary surface element. In addition, we adopt the Dirac matrices to express the bases of Clifford algebra Cl3(ℂ). We also use an orthogonal matrix to transform global boundary densities into local boundary densities for satisfying boundary condition straightforward. Finally, two electromagnetic scattering problems with a perfect spherical conductor and a prolate spheroidal conductor are both considered to examine the validity of the Clifford algebra valued BEM with Cauchy-type kernels.