Computational Methods in Conformal and Harmonic Mappings

Conformal geometry and the theory of harmonic mappings have a vast number of applications in engineering and physics. Conformal mappings have been used classically, e.g., in electrostatics, fluid dynamics, and potential flows, where the governing partial differential equation is Laplacian. These applications rely on the conformal invariance property of the harmonic solution of a Dirichlet problem as well as the Carathéodory boundary extension theorem. Recently conformal mappings have gained more popularity, e.g., in electrical impedance tomography and in computer graphics, where computational modelling is studied in the context of Riemann surfaces, which includes a medical application to the brain imaging of the cortex. Harmonic mappings can be used in studying minimal surfaces, which arise from many interesting phenomena in natural science and engineering, ranging from mathematical models of soap bubble surfaces, to topics in molecular engineering, and tensile structures. In this thesis a new method, conjugate function method, of constructing a conformal mappings from domains of interest onto a rectangle is developed. This algorithm makes use of the harmonic conjugate function as well as properties of modulus of quadrilaterals, and it is suitable for a very general class of domains, which may have curved boundaries and even cusps. The elaborated method is also suitable for multiply connected domains, where connectivity is greater than two. The second part of this thesis deals with the harmonic shearing method of obtaining harmonic mappings, and its application to minimal surfaces. Harmonic shearing involves integration of predetermined analytic function, which is the complex dilatation of the mapping being constructed, and a conformal mapping, which has to be convex in the direction of the real axis. This shearing can be done in numerically as well, thus, in particular, the conformal mappings do not need to be given in a closed form.