20180115
Quasistatic electromagnetic field models are considered for situations, where the shortest wave length of a problem exceeds the spatial dimensions of the problem by at least one order of magnitude. A taxonomy of such quasistatic field models distinguishes between the electroquasistatic, the magnetoquasistic and the socalled Darwin field models. These models have in common, that the additional assumptions made within the Maxwell equations changes the describing set of partial differential equations from hyperbolic to parabolic, or even to elliptic in the static limit case.
In electroquasistatic fields problems the electric energy density exceeds the magnetic field energy of the problem. These fields are modelled commonly using a scalar electric potential formulation within the continuity equation. This field formulation allows to describe capacitive and resistive field effects and is used to simulate electric fields mostly in electric power distribution systems and high voltage technology, but also has applications in microelectronics or in the simulation of biological cells. After spatial discretization of the governing equations with schemes e.g. as the Finite Integration Technique (FIT) or the Finite Element Method (FEM), the resulting electroquasistatic system of ordinary differential equations (ODE) is discretized in time with either implicit or (semi)explicit time integration schemes. Implementations using advanced multiple right hand side, GPU acceleration and both linear/nonlinear model order reduction techniques, respectively, allow e.g. to efficiently simulate components of electric power supply with nonlinear electric field stress grading materials. Alternative variants of the electroquasistatic field model are used for simulations of the quasistatic evolution of electric fields and space charges within highvoltage direct current cables.
Many of these numerical techniques are directly applicable also to magnetoquasistatic field problems. Such field are considered when the magnetic field energy of a quasistatic problem exceeds the electric energy density. Often also dubbed as eddy current problems, magnetoquasistatic field models are used to describe electromechanical energy conversion systems, nondestructive testing and inductive energy transport systems. Based on a variety of applicable vector and combined vector and scalar potential formulations the spatially discretized resulting systems of differentialalgebraic equations (DAE) are commonly time integrated using implicit schemes. Recent research efforts concentrate on the applicability of (semi)explicit time integration schemes. These become applicable after reformulating the magnetoquasistatic DAE systems into ODE systems using generalized Schur complements, and also on multigridtype spaceandtime parallel time integration schemes using the Parareal framework.
For the electromagnetic environmental analysis of magnetoquasistatic fields of inductive power transport systems involving highresolution body phantoms of biological organisms recently several twostep and flexible cosimulation schemes have been developed using either frequencyscaled full wave solution schemes or a modified scalar potential finite difference scheme.
Finally, the socalled Darwin field models include both the electro and the magnetoquasistatic field description and describe quasistatic field models for problems, where inductive, resistive and capacitive field effects are considered as e.g. in the electromagnetic compatibility analyses of electric and magnetic fields emanating from electric power inverter systems as e.g. traction inverters in hybrid electric cars. Again, the applicability of implicit as well as semiexplicit time integration schemes for the resulting Darwin DAE systems is of practical interest, but faces additional problems due to the nonsymmetry and illconditioning of the system matrices.
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