Uncertainty quantification for Maxwell’s eigenproblem based on isogeometric analysis and mode tracking

Abstract

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-Loève 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.

Solving Maxwell's Eigenvalue Problem via Isogeometric Boundary Elements and a Contour Integral Method

“…Proof. Assertion a) has been already shown by the derivation of the boundary integral formulation (6). For assertion b) note that V V V κ j j j is a solution of Maxwell's equation in Ω [21,Sect.…”

“…There are also other, more advanced applications which have such high demands on accuracy. One is presented by Georg et al [6], who show that even higher accuracies than those already achievable are required to simulate eccentricities.…”

Analysis and Implementation of Isogeometric Boundary Elements for Electromagnetism

“…While the tuners still can compensate for manufacturing errors and deformations due to Lorentz detuning, conventional numerical methods struggle to resolve changes of this magnitude, since they happen within a relative margin of roughly 10 −7 , and there are other examples of applications, where such high demands of accuracy are desired of simulations. One other example is presented by Georg et al [55], who explain that one must resolve eccentric deformations of cavities within a relative error margin of 10 −6 w.r.t. the operating frequency.…”

Solving Maxwell's eigenvalue problem via isogeometric boundary elements and a contour integral method

“…There are also other, more advanced applications which have high demands on accuracy. One is presented by Georg et al, 6 who show that even higher accuracies than those already achievable are required to simulate eccentricities.…”