This dissertation studies numerical methods for solving time-harmonic Maxwell's equations. In this work, boundary element and a least squares type finite element method for time-harmonic Maxwell's equations is analysed.The boundary element method is studied from the point of view of shape optimization and shape sensitivity analysis. The main result of the dissertation is the description how to calculate numerically the derivative of electromagnetic properties of a metallic scatterer with respect to variation of its shape. The motivation of the result is the adjoint variable method aimed at computing the derivative of a cost function of a generic optimization problem. The method has the property that, essentially, it suffices to differentiate the system matrix with respect to shape if the load functional or the cost functional do not depend on the shape explicitly. The boundary element method is also applied in as wide as possible characterisation of the shapes of wire dipoles and that way in shape optimisation. The least-squares finite-element method for time-harmonic Maxwell's equations results in a method where the system matrix is positive definite despite the underlying physics those of time-harmonic fields usually lead to strictly indefinite systems. The positive definiteness of the system matrix is inherited from the ellipticity of the sesquilinear form of the variational method. The implementation of a conformal discretization is not, however, always reasonable for the reason that in certain geometries the finite element solution vanishes on the boundary when only tangential, or normal, component is required to vanish. Therefore, the discrete solution space is widened in such a manner that on mesh dense enough the sesquilinear form is still elliptic and that the tangential, or normal, component vanishes in suitable weak sense.
|Translated title of the contribution||Luotettavia ja tehokkaita numeerisia menetelmiä aikaharmonisiin sähkömagneettisiin suunnittelutehtäviin|
|Publication status||Published - 2014|
|MoE publication type||G5 Doctoral dissertation (article)|
- boundary element method
- finite element method