Logo
ScienceStack
Table of Contents
Fetching ...
Sign In

A numerical scheme for solving an induction heating problem with moving non-magnetic conductor

Van Chien Le, Marián Slodička, Karel Van Bockstal

Abstract

This paper investigates an induction heating problem in a multi-component system containing a moving non-magnetic conductor. The electromagnetic process is described by the eddy current model, and the heat transfer process is governed by the convection-diffusion equation. Both processes are coupled by a restrained Joule heat source. A temporal discretization scheme is introduced to solve the corresponding variational system numerically. With the aid of the Reynolds transport theorem, we prove the convergence of the proposed scheme as well as the well-posedness of the variational problem. Some numerical experiments are also performed to assess the performance of the numerical scheme.

A numerical scheme for solving an induction heating problem with moving non-magnetic conductor

Abstract

This paper investigates an induction heating problem in a multi-component system containing a moving non-magnetic conductor. The electromagnetic process is described by the eddy current model, and the heat transfer process is governed by the convection-diffusion equation. Both processes are coupled by a restrained Joule heat source. A temporal discretization scheme is introduced to solve the corresponding variational system numerically. With the aid of the Reynolds transport theorem, we prove the convergence of the proposed scheme as well as the well-posedness of the variational problem. Some numerical experiments are also performed to assess the performance of the numerical scheme.
Paper Structure (14 sections, 10 theorems, 134 equations, 7 figures, 1 table)

This paper contains 14 sections, 10 theorems, 134 equations, 7 figures, 1 table.

Table of Contents

  1. Introduction
  2. Mathematical model
  3. Geometrical setting
  4. Mathematical model
  5. Functional setting
  6. Standard function spaces
  7. Auxiliary results
  8. Uniqueness
  9. Time discretization
  10. Existence of a solution
  11. Numerical results
  12. Numerical experiments
  13. Numerical simulation
  14. Conclusion

Key Result

Lemma 3.1

The space $\mathop{\mathrm{C}}\nolimits^1([0, T], \mathop{\mathrm{H}}\nolimits^1(\Omega))$ is densely contained in $\mathop{\mathrm{Y}}\nolimits_{\alpha}$.

Figures (7)

  • Figure 1: The domain $\Omega$ consisting of a workpiece $\Sigma$ moving with velocity $\mathbf{v}$, a fixed coil $\Pi$ and the surrounding air $\Xi$. The coil $\Pi$ shares common interfaces $\Gamma_{\textrm{in}}$ and $\Gamma_{\textrm{out}}$ of strictly positive measures with the boundary (see LSV2021b).
  • Figure 2: The circular domain of experiments consisting of an aluminium circular area (red) and the complementary area filled by copper (blue). The domains are rotating with velocity $\mathbf{v}$. Left: the first experiment with a concentric interior circle. Right: the second experiment with an eccentric interior circle.
  • Figure 3: Temperature distribution of the first experiment at different time points. Left:$t = 0.125 \mathrm{s}$. Middle:$t = 8 \mathrm{s}$. Right:$t = 64 \mathrm{s}$.
  • Figure 4: Temperature distribution of the second experiment at different time points. Left:$t = 0.125 \mathrm{s}$. Middle:$t = 8 \mathrm{s}$. Right:$t = 64 \mathrm{s}$.
  • Figure 5: Relative error $\tilde{E}_u$ w.r.t. the time step. Left: the first experiment. Right: the second experiment. Both numerical experiments show the potentially optimal convergence rate of the temporal discretization.
  • ...and 2 more figures

Theorems & Definitions (23)

  • Remark 2.1
  • Remark 2.2
  • Definition 3.1
  • Lemma 3.1
  • proof
  • Lemma 3.2
  • proof
  • Remark 4.1
  • Theorem 4.1: Uniqueness
  • proof
  • ...and 13 more