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Modal Decomposition in Numerical Computation of Eddy Current Transients

Salvatore Ventre, Andrea Chiariello, Nicola Isernia, Vincenzo Mottola, Antonello Tamburrino

TL;DR

This work introduces a modal decomposition framework to reduce the computational burden of long-time eddy current simulations, including cases with currents injected through boundary electrodes. By splitting the current space into an interior solenoidal subspace and an electrode-crossing subspace, the method decouples the dynamics into eigenmodes obtained from a generalized eigenproblem, enabling independent, parallelizable time integration. The discrete formulation preserves the physics with reduced incidence matrices and yields a set of decoupled scalar ODEs whose solution is far cheaper per time step. In a large NdT benchmark, the modal approach outperforms conventional Cholesky-based time stepping and is competitive with or faster than frequency-domain methods, especially when many time steps or multiple excitations are required, despite higher initial setup costs.

Abstract

A methodology to reduce the computational cost of time domain computations of eddy currents problems is proposed and implemented in a parallel computing environment. It is based on the modal decomposition of the current density and it is applicable even in presence of injected currents into the electrodes of a conducting domain. Using a theta-method integration algorithm, the performances of the the proposed approach are compared against those of a classical method based on the Cholesky factorization, for a case of interest from eddy current nondestructive testing. For this large eddy current problem (number of unknowns greater than 100k, number of time steps of interest equal to 100k) the proposed solution method is shown to be much faster than those based on standard time integration schemes.

Modal Decomposition in Numerical Computation of Eddy Current Transients

TL;DR

This work introduces a modal decomposition framework to reduce the computational burden of long-time eddy current simulations, including cases with currents injected through boundary electrodes. By splitting the current space into an interior solenoidal subspace and an electrode-crossing subspace, the method decouples the dynamics into eigenmodes obtained from a generalized eigenproblem, enabling independent, parallelizable time integration. The discrete formulation preserves the physics with reduced incidence matrices and yields a set of decoupled scalar ODEs whose solution is far cheaper per time step. In a large NdT benchmark, the modal approach outperforms conventional Cholesky-based time stepping and is competitive with or faster than frequency-domain methods, especially when many time steps or multiple excitations are required, despite higher initial setup costs.

Abstract

A methodology to reduce the computational cost of time domain computations of eddy currents problems is proposed and implemented in a parallel computing environment. It is based on the modal decomposition of the current density and it is applicable even in presence of injected currents into the electrodes of a conducting domain. Using a theta-method integration algorithm, the performances of the the proposed approach are compared against those of a classical method based on the Cholesky factorization, for a case of interest from eddy current nondestructive testing. For this large eddy current problem (number of unknowns greater than 100k, number of time steps of interest equal to 100k) the proposed solution method is shown to be much faster than those based on standard time integration schemes.
Paper Structure (20 sections, 1 theorem, 54 equations, 6 figures, 2 tables)

This paper contains 20 sections, 1 theorem, 54 equations, 6 figures, 2 tables.

Table of Contents

  1. Introduction
  2. Modal Decomposition of the Eddy Current Density
  3. Notations and functional spaces
  4. Mathematical Model for the Eddy Current Problem
  5. Modal Decomposition
  6. Modal Decomposition in the discrete model
  7. Discrete model
  8. Modal decomposition
  9. Time integration scheme
  10. Cholesky decomposition
  11. Modal decomposition
  12. Results
  13. Proposed NdT method
  14. Defects definition
  15. Excitation current
  16. ...and 5 more sections

Key Result

Proposition 2.3

If $\Omega$ is a domain and on its boundary $\partial \Omega$ there are $N_e$ electrodes/ports $\Sigma_1, \ldots, \Sigma_{N_e}$, then $\dim \left( H_D \right)=N_e-1$.

Figures (6)

  • Figure 1: Left: Geometry of the sample, in red the position of the analyzed cracks. Right: Injection of the exciting current. In yellow the two pairs of electrodes, the blue arrow represents the current density (not in scale).
  • Figure 2: Simulated cracks
  • Figure 3: Injected current. The current has a quite constant envelope, with a maximum value around $48\,\text{A}$ and a minimum value around $-43\,\text{A}$.
  • Figure 4: Left: position of sensing points. Right: zoom showing the density current perturbation due to the crack
  • Figure 5: Characterization of the cracks response. Top: outer crack, bottom: inner crack.
  • ...and 1 more figures

Theorems & Definitions (7)

  • Remark 2.1
  • Remark 2.2
  • Proposition 2.3
  • proof
  • Remark 2.4
  • Remark 3.1
  • Remark 3.2