H. De Gersem

Solution strategies for transient, field-circuit coupled systems

Transient simulation time for field-circuit coupled models of realistic electromagnetic devices becomes unacceptably high. A magnetodynamic formulation is coupled to an electric circuit analysis, yielding a sparse, symmetric and indefinite matrix. The unknown circuit currents correspond to negative eigenvalues in the matrix spectrum. The Quasi-Minimal Residual method performs better than the Minimal Residual approach that is restricted to positive definite preconditioners. The positive definite variant is solved by the Conjugate Gradient method without explicitly building the dense coupled matrix. As an example, both approaches are applied to an induction motor.

A topological method used for field-circuit coupling

A lumped parameter model describing the outside electric connection of conductors is linked with a finite element model to compute the magnetic field distribution. A well established method for circuit analysis is adapted for this purpose. A suitable analysis to consider solid conductors and stranded conductors simultaneously as parts of the external electric circuit is stated. This approach yields a general coupling mechanism that keeps symmetric if both the magnetic and the electric problems are symmetric. The generality of the method makes the implementation straightforward and powerful. The maturity of the method is proved by the computation of different eddy current problems.

A parametric finite element environment tuned for numerical optimization

Nowadays, numerical optimization in combination with finite element (FE) analysis plays an important role in the design of electromagnetic devices. To apply any kind of optimization algorithm, a parametric description of the FE problem is required and the optimization task must be formulated. Most optimization tasks described in the literature, feature either specially developed algorithms for a specific optimization task, or extensions to standard finite element packages. Here, a 2D parametric FE environment is presented, which is designed to be best suited for numerical optimization while maintaining its general applicability. Particular attention is paid to the symbolic description of the model, minimized computation time and the user friendly definition of the optimization task.

Algebraic multigrid for complex symmetric systems

The two dimensional quasistatic time-harmonic Maxwell formulations yield complex Helmholtz equations. Multigrid techniques are known to be efficient for solving the discretization of real valued diffusion equations. In this paper these multigrid techniques are extended to handle the complex equation. The implementation of geometric multigrid techniques can be cumbersome for practical engineering problems. Algebraic multigrid (AMG) techniques on the other hand automatically construct a hierarchy of coarser discretizations without user intervention given the matrix on the finest level. In the linear calculation of an induction motor the use of AMG as preconditioner for a Krylov subspace solver resulted in a six-fold reduction of the CPU time compared to an optimized incomplete LU factorization and in a twenty-fold reduction compared to symmetric successive overrelaxation.

Field-circuit coupling for time-harmonic models discretized by the finite integration technique

This paper develops the coupling between a three-dimensional modified magnetic vector potential formulation discretized by the finite integration technique and a circuit including solid and stranded conductors. The conductor models relate local quantities defined at primary edges or dual facets and the global circuit unknowns. Two possible coupling matrices are derived which either affect the edges perpendicular to a reference cross section or all edges inside the conductor. To preserve the sparsity and the structure of the discrete field system, the circuit analysis is based on a circuit tree, incorporates both voltage drops and currents as additional unknowns, and results in a symmetric coupling. The numerical efficiency of both coupling approaches is compared for technical applications.

An algebraic multigrid method for solving very large electromagnetic systems

Although most finite element programs have quite effective iterative solvers such as an incomplete Cholesky (IC) or symmetric successive overrelaxation (SSOR) preconditioned conjugate gradient (CG) method, the solution time may still become unacceptably long for very large systems. Convergence and thus total solution time can be shortened by using better preconditioners such as geometric multigrid methods. Algebraic multigrid methods have the supplementary advantage that no geometric information is needed and can thus be used as black box equation solvers. In the case of a finite element solution of a non-linear magnetostatic problem, the algebraic multigrid method reduces the overall computation time by a factor of 6 compared to a SSOR-CG solver.

Impact of the displacement current on low-frequency electromagnetic fields computed using high-resolution anatomy models

We present a comparison of simulated low-frequency electromagnetic fields in the human body, calculated by means of the electro-quasistatic formulation. The geometrical data in these simulations were provided by an anatomically realistic, high-resolution human body model, while the dielectric properties of the various body tissues were modelled by the parametric Cole-Cole equation. The model was examined under two different excitation sources and various spatial resolutions in a frequency range from 10 Hz to 1 MHz. An analysis of the differences in the computed fields resulting from a neglect of the permittivity was carried out. On this basis, an estimation of the impact of the displacement current on the simulated low-frequency electromagnetic fields in the human body is obtained.

Embedded Runge-Kutta methods for field-circuit coupled problems with switching elements

The coupling between a three-dimensional modified magnetic vector potential formulation discretized by the finite integration technique and an electrical circuit containing switches is presented. The changes in the states of the switching elements are introduced as changes in the topology of the circuit which results in a structural change of the differential-algebraic system describing the coupled field-circuit problem. This transient formulation is integrated by embedded Runge-Kutta methods. As soon as state changes are detected, the evaluated stage values of the corresponding Runge-Kutta step are used to build the interpolation polynomial with its roots approximating the switching time instants.

Harmonic weighting functions at the sliding interface of a finite-element machine model incorporating angular displacement

The formulation presented in this paper couples the magnetic vector potential and the tangential component of the magnetic field strength at a sliding interface in the air gap of a two-dimensional finite-element machine model using harmonic weighting functions which enable the use of fast Fourier transforms in combination with a simple, diagonal operator to account for the angular displacement between stator and rotor. The method applies to models where the stator and rotor meshes are nonequidistant and nonmatching at the interface. The approach substantially reduces the cost of system assembly and preconditioner setup during a transient simulation and offers convenient ways for dealing with skewing and for torque computation.

Skew interface conditions in 2-D finite-element machine models

In two-dimensional finite-element machine models, the skewing of the stator or the rotor is commonly taken into account by considering several cross sections at different axial positions, assembled by electrical circuit relations. Since the problem size scales with the number of slices, the computational cost rises significantly. In this paper, skew is modeled more accurately and more conveniently by imposing spectral interface conditions incorporating skew factors at a circle or an arc in the air gap.