Kimmo Forsman

FORMULATION OF THE EDDY CURRENT PROBLEM IN MULTIPLY CONNECTED REGIONS IN TERMS OF H

In this paper various formulations for the eddy current problem are presented. The formulations are based on solving directly for the magnetic field h, and they differ from each other mainly by how the field on the boundary is treated. The electromagnetic problem is studied in connection with the fivefold decomposition of the space of square integrable vector fields within a bounded region. This provides us with numerical approaches with clear signposts about how to solve the eddy current problem in multiply connected domains. Besides the fivefold decomposition, another essential tool in our approach is Whitney elements, as they provide the structure needed to retain consistency between the continuous and discrete problems. The paper demonstrates the usefulness of these mathematical tools in solving electromagnetic field problems.

Gauging in Whitney spaces

In this paper gauging is approached as a problem of selecting a representative in classes of equivalent representations. In this light we interpret how different gauging techniques are related to each other, and examine how they can be imposed on the discrete level using Whitney elements.

Variable step size time integration methods for transient eddy current problems

For transient eddy current problems modelled as differential-algebraic equations (DAEs), a time integration method suitable for ordinary differential equations (ODEs) will not necessarily work. We present two Runge-Kutta methods that are suitable for the time integration of the classes of DAEs to which eddy current problems belong. Both methods have error estimators and hence allow variable step sizes. In tests our variable step size integrators were competitive with fixed step size integrators, in particular with Crank-Nicolson.

Discrete spaces for div and curl-free fields

In this paper we construct discrete spaces for fields whose curl or div vanishes but for which one cannot find potentials. In electromagnetism such fields appear when the problem domain is not topologically trivial. The discrete spaces are constructed with Whitney elements in simplicial meshes.

Solution of dense systems of linear equations arising from integral-equation formulations

This paper discusses the efficient solution of dense systems of linear equations, arising from integral-equation formulations. Several preconditioners, in connection with Krylov iterative solvers, are examined, and compared with LU factorization. Results are shown, demonstrating practical aspects and issues we have encountered in implementing iterative solvers, on both parallel and sequential computers, for a magnetostatic volume-integral formulation.

Force and torque computations with hybrid methods

Alternative methods are presented for computing the magnetic forces and torques associated with hybrid solutions of magnetostatic or eddy current problems. The challenge is to retrieve the forces not just accurately but also with as little extra work as possible. The authors analyze several methods and the errors inherent in them. The analysis shows that the equivalent currents method is the best approach for computing forces with hybrid solutions. Force values obtained with test problems by using the proposed methods, including the coupling of the computed forces to the equations of motion, are compared to analytical and measured results.

Properties of b- and h-type integral equation formulations

Various options to establish integral equation formulations for solving nonlinear problems are studied in connection with Whitney forms. Several alternatives to implement the corresponding numerical procedures are presented. Numerical results are given demonstrating typical features of the different options.

Tetrahedral mesh generation in convex primitives by maximizing solid angles

This paper presents a method to generate tetrahedral meshes in three dimensional primitives for finite element computation. Input parameters define the global size of tetrahedra which can be increased or decreased locally. All the new nodes, which are not needed to describe the geometry, are generated automatically. The algorithm first discretizes all arrises of primitives to edges, then all faces of primitives are split to triangles and finally the primitives are filled with tetrahedra. The algorithm tries to generate elements close to regular tetrahedra by maximizing locally the minimum solid angles associated to a set of a few neighbouring tetrahedra. >

Hybrid formulations for eddy current problem with moving objects

In this paper hybrid formulations combining differential and integral operators for the eddy current problem with moving objects are studied. In problems including moving bodies hybrid approaches are appealing as the finite element mesh does not have to connect the stationary and moving parts to each other. Both b- and h-oriented formulations are considered for 2D and 3D problems.

Hybrid and integral formulations for 3D eddy current problems

A set of 3D eddy current formulations enabling one to solve problems with conducting, non-conducting, and magnetic subregions are presented. The formulations are derived, and discretized by adopting Whitney edge elements. In the proposed hybrid and integral formulations, integral operators are utilized such that air regions need not to be discretized. Some test results validating the presented hybrid and integral methods are shown.