Logo
ScienceStack
Table of Contents
Fetching ...
Sign In

Regularization and passivity-preserving model reduction of quasilinear magneto-quasistatic coupled problems

Johanna Kerler-Back, Timo Reis, Tatjana Stykel

Abstract

We consider the quasilinear magneto-quasistatic field equations that arise in the simulation of low-frequency electromagnetic devices coupled to electrical circuits. Spatial discretization of these equations on 3D domains using the finite element method results in a singular system of differential-algebraic equations (DAEs). First, we analyze the structural properties of this system and present a novel regularization approach based on projecting out the singular state components. Next, we explore the passivity of the variational magneto-quasistatic problem and its discretization by defining suitable storage functions. For model reduction of the magneto-quasistatic system, we employ the proper orthogonal decomposition (POD) technique combined with the discrete empirical interpolation method (DEIM), to facilitate efficient evaluation of the system's nonlinearities. Our model reduction approach involves the transformation of the regularized DAE into a system of ordinary differential equations, leveraging a special block structure inherent in the problem, followed by applying standard model reduction techniques to the transformed system. We prove that the POD-reduced model preserves passivity, and for the POD-DEIM-reduced model, we propose to enforce passivity by perturbing the output in a way that accounts for DEIM errors. Numerical experiments illustrate the effectiveness of the presented model reduction methods and the passivity enforcement technique.

Regularization and passivity-preserving model reduction of quasilinear magneto-quasistatic coupled problems

Abstract

We consider the quasilinear magneto-quasistatic field equations that arise in the simulation of low-frequency electromagnetic devices coupled to electrical circuits. Spatial discretization of these equations on 3D domains using the finite element method results in a singular system of differential-algebraic equations (DAEs). First, we analyze the structural properties of this system and present a novel regularization approach based on projecting out the singular state components. Next, we explore the passivity of the variational magneto-quasistatic problem and its discretization by defining suitable storage functions. For model reduction of the magneto-quasistatic system, we employ the proper orthogonal decomposition (POD) technique combined with the discrete empirical interpolation method (DEIM), to facilitate efficient evaluation of the system's nonlinearities. Our model reduction approach involves the transformation of the regularized DAE into a system of ordinary differential equations, leveraging a special block structure inherent in the problem, followed by applying standard model reduction techniques to the transformed system. We prove that the POD-reduced model preserves passivity, and for the POD-DEIM-reduced model, we propose to enforce passivity by perturbing the output in a way that accounts for DEIM errors. Numerical experiments illustrate the effectiveness of the presented model reduction methods and the passivity enforcement technique.

Paper Structure

This paper contains 12 sections, 8 theorems, 115 equations, 5 figures, 1 table.

Table of Contents

  1. Introduction
  2. Model problem
  3. Weak formulation
  4. Passivity
  5. Properties of the FEM model
  6. FEM discretization
  7. Regularization of the FEM model
  8. Passivity of the FEM model
  9. Passivity-preserving POD model reduction
  10. Enforcing passivity for the POD-DEIM-reduced model
  11. Numerical results
  12. Conclusion

Key Result

Theorem 5

Consider the DAE eq:MQS3Dreg, eq:MQS3Dregmat, where $M_{11}$ and $M_\nu(C_d\,a)$ are symmetric and positive definite, $X_2$ has full column rank, and the columns of $\hat{Y}_{C_2}$ form a basis of $\hbox{\rm im}(C_2^T)$. This system has tractability index one.

Figures (5)

  • Figure 1: Absolute error in the state and error bounds
  • Figure 2: Absolute error in the output and error bounds
  • Figure 3: Components of the perturbation $\delta(t)u_{\rm test}(t)$
  • Figure 4: Output components of the original and POD-DEIM-reduced models
  • Figure 5: Relative errors in the output for the POD-DEIM and perturbed POD-DEIM systems

Theorems & Definitions (25)

  • Remark 1
  • Remark 2
  • Definition 1: Passivity
  • Definition 2
  • Remark 3
  • Remark 4
  • Theorem 5
  • proof
  • Theorem 6
  • proof
  • ...and 15 more