- Isogeometric Analysis was born, less than a decade ago, with the goal of bridging the gap between Computer Aided Design (CAD) and Finite Element Method (FEM). The main distinctive feature of IGA is that CAD geometries, commonly defined in terms of Non-Uniform Rational B-splines (NURBS), are represented exactly throughout the analysis, regardless of the level of mesh refinement, while in standard FEM the computational domain needs to be remeshed when performing h-refinement and its geometry approaches the exact one only in the limit of vanishing mesh size h.
Moreover, in addition to h-refinement and p-refinement, k-refinement was introduced as a combination of degree elevation and mesh refinement, yielding approximation spaces with higher regularity properties. k-refinement has the advantage of not increasing the number of degrees of freedom of the problem, but produces matrices with larger bandwidth.
- Low-frequency fields and circuit simulation. Often, devices can be simulated sufficiently accurate as electrical circuits, i.e., by network models. However, the increasing frequencies and the decreasing size forces designers to account for wave propagation effects, eddy-currents, ferromagnetic saturation and hysteresis. Some effects can be represented by order-reduced equivalent models that are embedded in an electrical circuit. However, the representation of field-dependent nonlinearities and hysteresis effects is not straightforward.
- Cosimulation of coupled problems. Classical time-integration methods solve in each time step for all unknowns at once. This may become very inefficient or impossible for large systems of equations, in particular if the system stems from a coupled problem. This is typically a system of (partial-differential-algebraic) equations that consists of subproblems with different properties. Often the subproblem describe different physical effects (multiphysics), for example electromagnetic and heat distribution. The subproblems are mutually connected by coupling conditions (connecting ‘inputs’ and ‘outputs’). Often, the various phenomenae evolve at different time and spatial scales (multiscale).
- Differential algebraic equations. Differential Algebraic equations (DAEs) are a combination of ordinary differential equations (ODEs) and algebraic constraints. They occur often in space-discrete multiphysics problems due to coupling conditions.
The algebraic equations create severe numerical difficulties because the computation necessitates not only integration but also differentiation.
- Uncertainty quantification. The design of efficient electrical machines requires deep insight into the device’s electromagnetic field distribution. Today, this is typically obtained by computer simulations instead of physical prototypes. On the other hand, the available input data, e.g. material curves, include uncertainties, e.g. unknown errors due to measurements. The influence of these errors can be characterized by uncertainty quantification. In the mathematical models, the corresponding parameters are substituted by random variables to describe the uncertainties.
Current research topics include