Numerical Methods for the Estimation of the Impact of Geometric Uncertainties on the Performance of Electromagnetic Devices

This work addresses the application of Isogeometric Analysis to the simulation of particle accelerator cavities and other electromagnetic devices whose performance is mainly determined by their geometry. By exploiting the properties of B-Spline and Non-Uniform B-Spline basis functions, the Isogeometric approximation allows for the correct discretisation of the spaces arising from Maxwell's equations and for the exact representation of the computational domain. This choice leads to substantial improvements in both the overall accuracy and computational effort. The suggested framework is applied to the evaluation of the sensitivity of these devices with respect to geometrical changes using Uncertainty Quantification methods and to shape optimisation processes. The particular choice of basis functions simplifies the construction of the geometry deformations significantly. Finally, substructuring methods are proposed to further reduce the computational cost due to matrix assembly and to allow for hybrid coupling of Isogeometric Analysis and more classical Finite Element Methods. Considerations regarding the stability of such methods are addressed. The methods are illustrated by simple numerical tests and real world device simulations with particular emphasis on particle accelerator cavities

Uncertainty Quantification for Maxwell's Eigenproblem based on Isogeometric Analysis and Mode Tracking

The electromagnetic field distribution as well as the resonating frequency of various modes in superconducting cavities used in particle accelerators for example are sensitive to small geometry deformations. The occurring variations are motivated by measurements of an available set of resonators from which we propose to extract a small number of relevant and independent deformations by using a truncated Karhunen-Lo\`eve expansion. The random deformations are used in an expressive uncertainty quantification workflow to determine the sensitivity of the eigenmodes. For the propagation of uncertainty, a stochastic collocation method based on sparse grids is employed. It requires the repeated solution of Maxwell's eigenvalue problem at predefined collocation points, i.e., for cavities with perturbed geometry. The main contribution of the paper is ensuring the consistency of the solution, i.e., matching the eigenpairs, among the various eigenvalue problems at the stochastic collocation points. To this end, a classical eigenvalue tracking technique is proposed that is based on homotopies between collocation points and a Newton-based eigenvalue solver. The approach can be efficiently parallelized while tracking the eigenpairs. In this paper, we propose the application of isogeometric analysis since it allows for the exact description of the geometrical domains with respect to common computer-aided design kernels, for a straightforward and convenient way of handling geometrical variations and smooth solutions

Isogeometric Analysis and Harmonic Stator-Rotor Coupling for Simulating Electric Machines

This work proposes Isogeometric Analysis as an alternative to classical finite elements for simulating electric machines. Through the spline-based Isogeometric discretization it is possible to parametrize the circular arcs exactly, thereby avoiding any geometrical error in the representation of the air gap where a high accuracy is mandatory. To increase the generality of the method, and to allow rotation, the rotor and the stator computational domains are constructed independently as multipatch entities. The two subdomains are then coupled using harmonic basis functions at the interface which gives rise to a saddle-point problem. The properties of Isogeometric Analysis combined with harmonic stator-rotor coupling are presented. The results and performance of the new approach are compared to the ones for a classical finite element method using a permanent magnet synchronous machine as an example

Hierarchical B-spline complexes of discrete differential forms

In this paper, we introduce the hierarchical B-spline complex of discrete differential forms for arbitrary spatial dimension. This complex may be applied to the adaptive isogeometric solution of problems arising in electromagnetics and uid mechanics. We derive a sufficient and necessary condition guaranteeing exactness of the hierarchical B-spline complex for arbitrary spatial dimension, and we derive a set of local, easy-to-compute, and sufficient exactness conditions for the two-dimensional setting. We examine the stability properties of the hierarchical B-spline complex, and we find that it yields stable approximations of both the Maxwell eigenproblem and Stokes problem provided that the local exactness conditions are satisfied. We conclude by providing numerical results showing the promise of the hierarchical B-spline complex in an adaptive isogeometric solution framework