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Combination of Measurement Data and Domain Knowledge for Simulation of Halbach Arrays with Bayesian Inference

Luisa Fleig, Melvin Liebsch, Stephan Russenschuck, Sebastian Schöps

Abstract

Accelerator magnets made from blocks of permanent magnets in a zero-clearance configuration are known as Halbach arrays. The objective of this work is the fusion of knowledge from different measurement sources (material and field) and domain knowledge (magnetostatics) to obtain an updated magnet model of a Halbach array. From Helmholtz-coil measurements of the magnetized blocks, a prior distribution of the magnetization is estimated. Measurements of the magnetic flux density are used to derive, by means of Bayesian inference, a posterior distribution. The method is validated on simulated data and applied to measurements of a dipole of the FASER detector. The updated magnet model of the FASER dipole describes the magnetic flux density one order of magnitude better than the prior magnet model.

Combination of Measurement Data and Domain Knowledge for Simulation of Halbach Arrays with Bayesian Inference

Abstract

Accelerator magnets made from blocks of permanent magnets in a zero-clearance configuration are known as Halbach arrays. The objective of this work is the fusion of knowledge from different measurement sources (material and field) and domain knowledge (magnetostatics) to obtain an updated magnet model of a Halbach array. From Helmholtz-coil measurements of the magnetized blocks, a prior distribution of the magnetization is estimated. Measurements of the magnetic flux density are used to derive, by means of Bayesian inference, a posterior distribution. The method is validated on simulated data and applied to measurements of a dipole of the FASER detector. The updated magnet model of the FASER dipole describes the magnetic flux density one order of magnitude better than the prior magnet model.
Paper Structure (6 sections, 20 equations, 5 figures)

This paper contains 6 sections, 20 equations, 5 figures.

Table of Contents

  1. Introduction
  2. Problem statement
  3. Bayesian inference
  4. Validation
  5. Application
  6. Conclusion

Figures (5)

  • Figure 1: FASER dipole magnet.
  • Figure 2: Left: Domain $D$ with the cross section of the FASER Halbach dipole with PM blocks $D_i$ and their nominal magnetization $\overline{\mathbf{M}}_i$. Right: Absolute value of the Gateaux derivative $\mathbf{B}'$ of the mapping \ref{['eq:mapping13']}.
  • Figure 3: Validation of posterior derivation algorithm in linear case on 3D simulation model. Comparison of ground truth (red), prior $\mathcal{N}(\bm\mu_0,\mathbf{C}_0)$ (black), posterior based on magnetic flux density observation $\mathcal{N}(\bm\mu^{\mathbf{B}}_1,\mathbf{C}^{\mathbf{B}}_1)$ (blue) and posterior based on magnetic Fourier coefficient observation $\mathcal{N}(\bm\mu^{\mathrm{F}}_1,\mathbf{C}^{\mathrm{F}}_1)$ (green).
  • Figure 4: Validation of posterior derivation algorithm in non linear case on 2D simulation model. Comparison of ground truth (red), prior $\mathcal{N}(\bm\mu_0,\mathbf{C}_0)$ (black) and posterior (blue) sample mean and variance.
  • Figure 5: Relative error of prior and posterior simulation model compared to magnetic flux density measurements of the first FASER dipole. Area of fringe field marked in grey.