title: Isogeometric Modeling, Simulation and Optimization of Rotating Electric Machines
Creators: name: Merkel, Anna Melina
Publisher: name: UNSPECIFIED
Year: 2024

description: Given the ongoing the transition to renewable energy and the concurrent rise in e-mobility, the efficient and robust design and optimization of electric machines has become increasingly important. Simulations are a crucial tool to obtain improved designs. This thesis focuses on the development of suitable simulation and optimization methodologies for electric machines, based on isogeometric analysis (IGA). IGA allows for a direct use of computer aided design (CAD) data, offering an exact geometry representation and easy shape optimization through straightforward geometry modification via shifting the control points. Furthermore, when compared to the conventional finite element method (FEM), IGA provides a higher accuracy per degree of freedom (DoF). Moreover, when employing IGA for multiphysics simulations, a remeshing or mesh exchange between multiple physics is not necessary. However, several challenges arise when using IGA, e.g., the incorporation of rotation and the resulting non-conformity of the discretization. This thesis addresses these difficulties, proposing approaches to facilitate the simulation, including the integration of rotation in the simulation, ensuring stability of the formulation, regularization of the formulation, and freeform shape optimization of the machine. 

When applying IGA to machine problems, one encounters multi-patch spline spaces that have incompatible discretizations between rotor and stator, particularly in the case of rotation. We utilize a variety of methods to couple non-conforming discretizations to address these challenges. In the two-dimensional case, harmonic stator-rotor coupling offers a promising approach to interconnect these non-conforming subdomains. However, this method results in a saddle-point system, which is stable only with an appropriate choice of the Lagrange multiplier space. In this work, we derive a general criterion to ensure the inf-sup stability of this problem. The criterion applies to a variety of discretization approaches, including FEM and IGA. A Schur complement is used to reduce the system to a low-dimensional interface problem, significantly decreasing the computational cost during the simulation of rotation. For the three-dimensional problem, the Nitsche-type coupling method and the mortar method are examined in the context of IGA for the realization of rotation. The established theory for the choice of the Lagrange multiplier space for isogeometric mortaring is extended to be applicable for multi-patch domains. Convergence analysis shows that both methods converge with the degree of the IGA basis functions. 

The simulation of three-dimensional magneto(quasi)static problems using the vector potential formulation poses the problem of a non-uniqueness of the solution. A method which can be employed to remove the discrete kernel from the system is the tree-cotree decomposition. The application of this gauging to IGA is proposed, where the tree-cotree decomposition is performed based on the control mesh. The method works for non-contractible domains and can be straightforwardly applied independently of the degree of the B-spline bases. To allow for the gauging of mortared domains, a modified tree-cotree decomposition is derived and applied to the model of a three-dimensional permanent magnet synchronous machine (PMSM). 

One important quantity of interest in electric machines is the torque. The classical FEM, often suffers from inaccuracies in torque computation. This work proposes methods for efficient torque computations based on the Lagrange multipliers of the harmonic coupling and the mortar method. In the harmonic coupling method, we derive explicit formulas for torque computation using energy considerations. The harmonic nature of the Lagrange multiplier allows for an elegant treatment of rotation, eliminating the need for additional numerical evaluations of integrals. These proposed methods are compared to the Arkkio method and are found to be similarly accurate and significantly more efficient in terms of computational cost. 

Finally, an optimization method is introduced, based on shape calculus. Shape sensitivity analysis is used to determine the shape derivative of the objective functional, i.e., the total harmonic distortion (THD) of the electromotive force (EMF). This method allows for a freeform shape optimization which is not restricted by the choice of a set of optimization parameters. This freeform shape optimization is applied for the first time to an isogeometric model of a rotating PMSM, minimizing the THD of the EMF, where a reduction of 75% is achieved. 

The proposed methods are successfully applied to two- and three-dimensional models of a PMSM. They are ready to be employed in the design process of electric machines, facilitating the workflow by operating directly on CAD geometries, alleviating the need to generate a computational mesh, and therefore pave the way for multiphysical simulations. While this thesis focuses on the simulation of electric machines, the proposed methods can be applied to different applications, e.g., eddy current brakes or magnetocaloric cooling devices, and are particularly beneficial when considering non-conforming or moving subdomains.