Geometric solution strategy of Laplace problems with free boundary

Abstract

This paper introduces a geometric solution strategy for Laplace problems. Our main interest and emphasis is on efficient solution of the inverse problem with a boundary with Cauchy condition and with a free boundary. This type of problem is known to be sensitive to small errors. We start from the standard Laplace problem and establish the geometric solution strategy on the idea of deforming equipotential layers continuously along the field lines from one layer to another. This results in exploiting ordinary differential equations to solve any boundary value problem that belongs to the class of Laplace's problem. Interpretation in terms of a geometric flow will provide us with stability considerations. The approach is demonstrated with several examples.Here, the Laplace problem has become formulated as a system of ODEs. The field lines 'begin' on @ and they are forced to 'end' to @ † with condition ' D 1. Evidently, such a solution strategy for the Laplace problem is not that competitive compared with finite element kind of methods. However, our aim is not to suggest a new approach to solve the standard Laplace problem but instead to highlight the common geometric properties behind Laplace type of boundary value problems. § An example of such routine is MATLAB function bvp4c. Figure 17. The modeled phone connector in cylindrical coordinates in meters. We assume that the field lines are tangential to the surface Z D 0, which corresponds to a case when two connectors glued together through this surface are electroplated simultaneously. Figure 18. The solved field lines and equipotential surfaces.

Towards design of electrical machines from given air gap field

“…For brevity and simplicity, here, we will model electrical machines in two dimensions. However, we would like to emphasize that the same approach is possible also in three dimensions (Poutala et al, 2016). For convenience, we choose cylindrical coordinates (r, , z) and select the coordinate axis such that H z ϭ 0, B z ϭ 0 hold.…”