Effective Material Transformation for Ferromagnetic Sheets

Minimizing the losses in electrical devices is becoming increasingly important due to the growing electromobility and increasing energy efficiency requirements. The iron core is not made of one piece but is composed of many very fine ferromagnetic iron sheets to minimize the eddy currents that cause the losses. Therefore, an accurate and efficient calculation of the eddy currents in laminated iron cores is of enormous practical importance in the design of electrical devices. The finite element method is preferably applied in case of problems with complex geometry and highly nonlinear materials. However, the overall dimensions of a single sheet are in the range of meters, whereas the thickness and the penetration depth are essentially smaller than one millimeter. Thus, a detailed finite element model of large electrical devices would yield extremely large equation systems impossible to solve with reasonable computational effort. Therefore, multiscale and homogenization methods have been developed using a very coarse finite element mesh and thus leading to much smaller systems of equations. However, specific problems still lead to rising computational costs. For instance, to cope with a small penetration depth of electromagnetic fields, the number of unknowns grows linearly with the order of the multiscale approach. Strong field variations across a sheet require many integration points which makes the assembling of finite element equation systems expensive especially in case of hysteresis. The aim is to accurately compute eddy currents in thin highly nonlinear ferromagnetic sheets and associated magnetic stray fields by novel methods requiring radically less computational costs. For example, finite element methods working with a single scalar potential at the best instead of multiscale approaches with several different potentials will be developed. To avoid expensive finite element system assembling equivalent effective material parameters will be determined at negligible costs, which combine the very fine structure, for instance, present in laminated iron cores and the highly nonlinear material properties also including hysteresis. A homogenization of the problem is performed. Homogenization replaces a given fine-scale heterogeneous structure with a homogeneous one such that certain quantities, e.g., reaction fields and losses, remain approximately the same. Preliminary investigations have shown that the requirement of equal losses and reactive power in the homogenization with the aid of a unit cell problem yield nonlinear complex-valued magnetization curves. At the same time the nature of physics changes, the basic eddy current problem becomes a complex-valued static magnetic field problem.

The demand for electrical energy is constantly increasing, not least due to the rapid growth of electric mobility. To avoid reaching the limits of supply capacity, the requirements for the energy efficiency of electrical devices are consequently becoming significantly greater. In the design of electrical devices, the aim is therefore to minimize losses as far as possible. A large proportion of these losses is caused by eddy currents in the devices' iron cores. To minimize the eddy currents - and thus the losses - the iron cores are composed of very thin ferromagnetic sheets by stacking them. An accurate numerical calculation of the eddy currents in laminated iron cores, using as little computing power as possible, is therefore of enormous practical importance for the optimal design of electrical devices. The finite element method is the preferred approach for the numerical calculation of problems involving complex geometries and nonlinear materials (hysteresis). The overall dimensions of an iron core are in the order of meters, while the thickness of the laminations is less than one millimeter. A detailed finite element model of large devices therefore results in very large nonlinear systems of equations. These nonlinear systems of equations must be solved iteratively - that is, several times - which requires an unacceptably high level of computing power. The aim of this project was to drastically reduce the computation time and memory requirements in finite element simulations to provide the best possible support for the efficient design of the devices. To this end, homogenization methods using effective materials were developed. With the aid of suitable effective materials, the fine structure of the very thin sheets is "homogenized". The originally fine structure no longer needs to be modelled using finite elements. For the homogenized problem, a very coarse finite-element mesh with, in some cases, simpler material models is now sufficient, resulting in significantly smaller systems of equations. Although these smaller systems of equations are still nonlinear, they can be solved efficiently using much less computing power. In some cases, entire metallic components of the devices can also be omitted from the simulation by applying suitable boundary conditions (effective surface impedance) on their surfaces. Depending on the specific problem, finite element simulations using homogenization with effective materials can reduce memory requirements to as little as one-hundredth and computation time to as little as one-thousandth of that required for brute-force simulations. This greatly facilitates the optimal design of electrical devices.