Model Order Reduction: Application to Electromagnetic Problems

With the increase in computational resources, numerical modeling has grown expo- nentially these last two decades. From structural analysis to combustion modeling and electromagnetics, discretization methods–in particular the finite element method–have had a tremendous impact. Their main advantage consists in a correct representation of dynamical and nonlinear behaviors by solving equations at local scale, however the spatial discretization inherent to such approaches is also its main drawback. In- deed, it usually leads to (very) large systems of equations—requiring abundance of computational resources, usually far too much for quasi-real time simulations.
In this dissertation, model order reduction of numerical models from finite element discretization is analyzed to efficiently and accurately downsize the number of degrees of freedom in static and dynamic, linear and nonlinear electromagnetic applications. In particular, an in-depth review of state of the art model order reduction methods is performed in view of the aforementioned problems. To this end, the proper orthogonal decomposition is considered to limit the number of unknowns in the resolution process. Nonlinear sampling methods such as: the missing point estimation approach and discrete empirical interpolation method, are compared to reduce the assembly phase. The parametric dependencies are taken into account by resorting to global reduced basis and nonlinear interpolation on manifolds techniques. Finally, a novel decoupled approach for the reduction of a coupled nonlinear magnetodynamic three-phase energy converter with external electric circuits is proposed and analyzed by combining all the aforementioned methods—impressively reducing the computational cost by 95%.
This dissertation is genuinely geared towards the application of a priori known meth- ods on a variety of different numerical models of electromagnetic devices. Additional automatic algorithms which eliminate the arbitrary choices of numerical reduction parameters are proposed and compared to reference methods proposed in the litera- ture. The following applications have been considered: a 2D inductor-core system to first illustrate and provide understanding of the proposed methods, a 2D single phase transformer, a 2D three-phase transformer and a 3D microwave antenna.