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Shape uncertainty quantification of Maxwell eigenvalues and -modes with application to TESLA cavities

Jürgen Dölz, David Ebert, Sebastian Schöps, Anna Ziegler

TL;DR

This work develops a perturbation-based uncertainty quantification framework for Maxwell eigenvalue problems on randomly deformed cavity domains, with the TESLA cavity serving as a driving example. By mapping geometric perturbations to coefficient perturbations through domain transformations, and deriving Fréchet-like derivatives for both the coefficients and eigenpairs, the authors enable efficient estimation of the mean and covariance of eigenvalues and eigenmodes for small perturbations. The approach extends to eigenpairs with finite multiplicity and potential crossings, and leverages isogeometric analysis to obtain closed-form deformation coefficients and accurate derivatives. Numerical experiments on a 9-cell TESLA cavity with real-world deformation data demonstrate accurate variance and mode-sensitivity insights while achieving substantial computational speedups over surrogate models like polynomial chaos expansions. This framework offers a practical, scalable route for shape-uncertainty quantification in high-frequency electromagnetic cavities used in accelerator technology.

Abstract

We consider Maxwell eigenvalue problems on uncertain shapes with perfectly conducting TESLA cavities being the driving example. Due to the shape uncertainty the resulting eigenvalues and eigenmodes are also uncertain and it is well known that the eigenvalues may exhibit crossings or bifurcations under perturbation. We discuss how the shape uncertainties can be modelled using the domain mapping approach and how the deformation mapping can be expressed as coefficients in Maxwell's equations. Using derivatives of these coefficients and derivatives of the eigenpairs, we follow a perturbation approach to compute approximations of mean and covariance of the eigenpairs. For small perturbations these approximations are faster and more accurate than sampling or surrogate model strategies. For the implementation we use an approach based on isogeometric analysis, which allows for straightforward modelling of the domain deformations and computation of the required derivatives. Numerical experiments for a three-dimensional 9-cell TESLA cavity are presented.

Shape uncertainty quantification of Maxwell eigenvalues and -modes with application to TESLA cavities

TL;DR

This work develops a perturbation-based uncertainty quantification framework for Maxwell eigenvalue problems on randomly deformed cavity domains, with the TESLA cavity serving as a driving example. By mapping geometric perturbations to coefficient perturbations through domain transformations, and deriving Fréchet-like derivatives for both the coefficients and eigenpairs, the authors enable efficient estimation of the mean and covariance of eigenvalues and eigenmodes for small perturbations. The approach extends to eigenpairs with finite multiplicity and potential crossings, and leverages isogeometric analysis to obtain closed-form deformation coefficients and accurate derivatives. Numerical experiments on a 9-cell TESLA cavity with real-world deformation data demonstrate accurate variance and mode-sensitivity insights while achieving substantial computational speedups over surrogate models like polynomial chaos expansions. This framework offers a practical, scalable route for shape-uncertainty quantification in high-frequency electromagnetic cavities used in accelerator technology.

Abstract

We consider Maxwell eigenvalue problems on uncertain shapes with perfectly conducting TESLA cavities being the driving example. Due to the shape uncertainty the resulting eigenvalues and eigenmodes are also uncertain and it is well known that the eigenvalues may exhibit crossings or bifurcations under perturbation. We discuss how the shape uncertainties can be modelled using the domain mapping approach and how the deformation mapping can be expressed as coefficients in Maxwell's equations. Using derivatives of these coefficients and derivatives of the eigenpairs, we follow a perturbation approach to compute approximations of mean and covariance of the eigenpairs. For small perturbations these approximations are faster and more accurate than sampling or surrogate model strategies. For the implementation we use an approach based on isogeometric analysis, which allows for straightforward modelling of the domain deformations and computation of the required derivatives. Numerical experiments for a three-dimensional 9-cell TESLA cavity are presented.
Paper Structure (32 sections, 4 theorems, 57 equations, 9 figures, 1 table)

This paper contains 32 sections, 4 theorems, 57 equations, 9 figures, 1 table.

Table of Contents

  1. Introduction
  2. Motivation
  3. Related Work
  4. Uncertainty quantification for Eigenvalue problems
  5. Uncertainty Quantification for Cavities
  6. Contributions
  7. Outline
  8. Shape perturbations of Maxwell's eigenvalue problem
  9. Maxwell's Eigenvalue Problem
  10. Shape Perturbation Model
  11. Maxwell's Eigenvalue Problem on Perturbed Domains
  12. Derivatives of Domain Transformation Coefficients
  13. Derivatives of Eigenpairs
  14. Shape Uncertainty Quantification of Eigenvalues and -modes
  15. Maxwell's Eigenvalue Problem on Randomly Deformed Shapes
  16. ...and 17 more sections

Key Result

Lemma 1

Assume that $\mathrm{d}_t {\mathbf{G}}_{t}(\boldsymbol{x};\boldsymbol{z})$ is linear in $\boldsymbol{z}$ and $\boldsymbol{z}$ is centered, i.e. $\mathop{\mathrm{\mathbb{E}}}\nolimits[\boldsymbol{z}] = \boldsymbol{0}$, then the mean of the derivatives of the deformation coefficients evaluated at the

Figures (9)

  • Figure 1: The 9-cell TESLA cavity with attached elongated beampipes.
  • Figure 2: Nine smallest eigenvalues of a 9-cell TESLA cavity and respective eigenmodes (cross-section along the longitudal direction, color map according to the normalized magnitude of the electric field strength).
  • Figure 3: Physical reference domain of the TESLA cavity (top) and domain according to a sampled deformation direction $\boldsymbol{z}$, ($t = \{0,.5,1\}$). On the left, the deformation in the $yz$-plane. On the right, the deformation in the $xz$-plane. The deformations are displayed increased by factor $250$. The red line indicates the deformation of the central axis.
  • Figure 4: The TESLA cavity and the manufacturing from dumbbells (DB) and endgroups (EGL = endgroup left, EGR = endgroup right).
  • Figure 5: The mean deformation and the seven deformation modes as identified in the Karhunen-Loève decomposition displayed for the cavity axis. The red crosses mark the cell centers when the axis is ideal and undeformed. The blue and black lines indicate the deformed axes in the mean geometry and for the deformation modes. The blue lines correspond to the deformation seen in the $yz$ plane and the black lines to the deformation seen in the $xz$ plane.
  • ...and 4 more figures

Theorems & Definitions (10)

  • Lemma 1
  • proof
  • Lemma 2
  • proof
  • Lemma 3
  • proof
  • Remark 4
  • Remark 5
  • Lemma 6
  • proof