Numerical Solution of an Axisymmetric Eddy Current Model with Current and Voltage Excitations

Abstract

The aim of this paper is to study the numerical approximation of an axisymmetric time-harmonic eddy current problem involving an in-plane current. The analysis of the problem restricts to the conductor. The source of the problem is given in terms of boundary data currents and/or voltage drops defined in the so-called electric ports, which are parts of the boundary connected to exterior sources. This leads to an elliptic problem written in terms of the magnetic field with nonlocal boundary conditions. First, we prove the existence and uniqueness of the solution for a weak formulation written in terms of Sobolev spaces with appropriate weights. We show that the magnetic field is not the most appropriate variable to impose the boundary conditions when Lagrangian finite elements are used to discretize the problem. We propose an alternative weak formulation of the problem which allows us to avoid this drawback. We compute the numerical solution of the problem by using Lagrangian finite elements ad hoc modified on the vicinity of the symmetry axis. We provide a convergence result under rather general conditions. Moreover, we prove quasi-optimal order error estimates under additional regularity assumptions. Finally, we report numerical results which allow us to confirm the theoretical estimates and to assess the performance of the proposed method in a physical application which is the motivation of this paper: the computation of the current density distribution in a steel cylindrical bar submitted to electric-upsetting.

A Appendix. A Mixed Formulation

For the implementation of Problem 3, it is necessary to impose the boundary condition $\widetilde{G}_h^k=\widetilde{G}_h^{k-1}+\frac{I_k}{2\pi },\ k\in N_{{}_{\mathrm I}}$. This can be easily obtained by a static condensation procedure applied to the matrix arising from the sesquilinear form $a(\cdot ,\cdot )$. However, depending on the computational tool, this boundary constraint can be difficult to implement. In order to propose an alternative for the computational implementation, in this appendix we introduce a mixed finite element approximation of Problem 3 which avoids static condensation. This new formulation could be solved, for instance, by using general purpose finite element software packages like FEniCS [18] and FreeFEM [16].

To begin with, let us introduce a finite element space. Let $\mathcal {E}$ be the set of edges in $\mathcal {T}_h$ lying on $\varGamma _{{}_{\mathrm N}}$. Let $e^1_h\in \mathcal {E}$ be the unique edge of $\varGamma _{{}_{\mathrm N}}^0$ intersecting $\varGamma _{{}_{\mathrm D}}$ (see Fig. 13).

Fig. 13

Notations used in the proof of Theorem 2

We define

$$\begin{aligned} \mathcal {P}_h:=\{ \xi _h\in \mathrm {L}^2(\varGamma _{{}_{\mathrm N}})\; :\ \xi _h= 0\ \text { in } e^1_h,\ \xi _h|_e \in \mathbb {P}_0(e),\ \forall e\in \mathcal {E} \}. \end{aligned}$$

A mixed Galerkin approximation of Problem 2 reads as follows:

Problem 4

Find $\widetilde{H}_h\in \mathcal {Y}_h$, $V_h\in \mathcal {P}_h$ and $V_k\in \mathbb {C}$, $k\in N_{{}_{\mathrm I}}$ such that

$$\begin{aligned} a(\widetilde{H}_h,\widetilde{G}_h) +\sum _{k=0}^N\int _{\varGamma _{{}_{\mathrm N}}^k}V_h\frac{\partial {\bar{\widetilde{G}_h}}}{\partial \tau } +\sum _{k\in N_{{}_{\mathrm I}}}\int _{\varGamma _{{}_{\mathrm J}}^k} V_k\frac{\partial {\bar{\widetilde{G}_h}}}{\partial \tau }&=-\sum _{k\in N_{{}_{\mathrm V}}}V_k({\bar{\widetilde{G}_h}}^k-{\bar{\widetilde{G}_h}}^{k-1}) \\ \sum _{k=0}^N\int _{\varGamma _{{}_{\mathrm N}}^k}\frac{\partial \widetilde{H}_h}{\partial \tau }{\bar{\xi }}_h&=0 \\ \sum _{k\in N_{{}_{\mathrm I}}}\int _{\varGamma _{{}_{\mathrm J}}^k} \frac{\partial \widetilde{H}_h}{\partial \tau } \bar{W}_k&=\sum _{k\in N_{{}_{\mathrm I}}}\frac{I_k}{2\pi }\bar{W}_k \end{aligned}$$

for all $(\widetilde{G}_h,\xi _h,W_k)\in \mathcal {Y}_h\times \mathcal {P}_h\times \mathbb {C}$, $k\in N_{{}_{\mathrm I}}$.

Since $\mathcal {X}_h(\varvec{0}) \subset \mathcal {Y}_h$ and $\displaystyle \int _{\varGamma _{{}_{\mathrm J}}^k} \frac{\partial \widetilde{H}_h}{\partial \tau }=\widetilde{H}_h^k-\widetilde{H}_h^{k-1}$, it is straightforward to show that if $\widetilde{H}_h$ is solution to Problem 4 then is also a solution to Problem 3. Moreover, from the following theorem we obtain the equivalence between problems 3 and 4.

Theorem 2

Problem 4 has a unique solution.

Proof

Let $\widetilde{H}_h\in \mathcal {Y}_h$, $V_h\in \mathcal {P}_h$ and $V_k\in \mathbb {C}$, $k\in N_{{}_{\mathrm I}}$ be a solution of the corresponding Problem 4 with $V_k=0$, $k\in N_{{}_{\mathrm V}}$ and $I_k=0$, $k\in N_{{}_{\mathrm I}}$, that is,

$$\begin{aligned} a(\widetilde{H}_h,\widetilde{G}_h) +\sum _{k=0}^N\int _{\varGamma _{{}_{\mathrm N}}^k}V_h\frac{\partial {\bar{\widetilde{G}_h}}}{\partial \tau } +\sum _{k\in N_{{}_{\mathrm I}}}\int _{\varGamma _{{}_{\mathrm J}}^k} V_k\frac{\partial {\bar{\widetilde{G}_h}}}{\partial \tau }&=0&\forall \widetilde{G}_h\in \mathcal {Y}_h\,,\\ \sum _{k=0}^N\int _{\varGamma _{{}_{\mathrm N}}^k}\frac{\partial \widetilde{H}_h}{\partial \tau }{\bar{\xi }}_h&=0&\forall \xi _h\in \mathcal {P}_h\,,\\ \sum _{k\in N_{{}_{\mathrm I}}}\int _{\varGamma _{{}_{\mathrm J}}^k} \frac{\partial \widetilde{H}_h}{\partial \tau } \bar{W}_k&=0&\forall W_k\in \mathbb {C},\ k\in N_{{}_{\mathrm I}}\,. \end{aligned}$$

Taking $\widetilde{G}_h:=\widetilde{H}_h$, $\xi _h:=V_h$ and $W_k:=V_k$, $k\in N_{{}_{\mathrm I}}$, we obtain

$$\begin{aligned} a(\widetilde{H}_h,\widetilde{H}_h) +\sum _{k=0}^N\int _{\varGamma _{{}_{\mathrm N}}^k}V_h\frac{\partial {\bar{\widetilde{H}_h}}}{\partial \tau } +\sum _{k\in N_{{}_{\mathrm I}}}\int _{\varGamma _{{}_{\mathrm J}}^k} V_k\frac{\partial {\bar{\widetilde{H}_h}}}{\partial \tau }&=0\,,\\ \sum _{k=0}^N\int _{\varGamma _{{}_{\mathrm N}}^k}\frac{\partial \widetilde{H}_h}{\partial \tau }{\bar{V_h}}&=0\,,\\ \sum _{k\in N_{{}_{\mathrm I}}}\frac{\partial \widetilde{H}_h}{\partial \tau }\bar{V_k}&=0. \end{aligned}$$

Therefore, $a(\widetilde{H}_h,\widetilde{H}_h)=0$. We recall that the sesquilinear form $a(\cdot ,\cdot )$ is elliptic in $\mathrm {H}^1_{-1}(\varOmega )$ then $\widetilde{H}_h=0$. Thus, we have

$$\begin{aligned} \sum _{k=0}^N\int _{\varGamma _{{}_{\mathrm N}}^k}V_h\frac{\partial {\bar{\widetilde{G}_h}}}{\partial \tau } +\sum _{k\in N_{{}_{\mathrm I}}}\int _{\varGamma _{{}_{\mathrm J}}^k} V_k\frac{\partial {\bar{\widetilde{G}_h}}}{\partial \tau }=0\qquad \forall \widetilde{G}_h\in \mathcal {Y}_h\ . \end{aligned}$$

(32)

By proceeding as in Lemma 3 it is straightforward to show that, for each $k\in N_{{}_{\mathrm I}}$ there exists $\widetilde{G}_{k,h} \in \mathcal {Y}_h$ such that

$$\begin{aligned} \frac{\partial \widetilde{G}_{k,h}}{\partial \tau } ={\left\{ \begin{array}{ll} V_k &{} \text {on }\varGamma _{{}_{\mathrm J}}^k\,,\\ 0 &{} \text {on }\partial \varOmega \setminus \varGamma _{{}_{\mathrm J}}^k\,. \end{array}\right. } \end{aligned}$$

By taking $\widetilde{G}_h=\widetilde{G}_{k,h}$ in (32) we get $|\varGamma _{{}_{\mathrm J}}^k||V_k|^2=0$ and then $V_k=0$, $k\in N_{{}_{\mathrm I}}$. Next, we prove that $V_h$ also vanishes. With this aim, for each $k\in N$, we introduce the curve $\gamma _k({\varvec{x}})$ with end points ${\varvec{x}}_k$ and ${\varvec{x}}$ and lying in $\varGamma _{{}_{\mathrm N}}^k$ (see Fig. 13). For any $k\in N$ we define $\widetilde{G}_{k,h}\in \mathcal {Y}_h$ such that, on $\varGamma _{{}_{\mathrm N}}\cup \varGamma _{{}_{\mathrm J}}\cup \varGamma _{{}_{\mathrm E}}$, satisfies

$$\begin{aligned} \widetilde{G}_{k,h}({\varvec{x}}):={\left\{ \begin{array}{ll} 0 &{} {\varvec{x}}\in e_h^0,\quad {\varvec{x}}\in \varGamma _{{}_{\mathrm J}}^i,\, 1\le i\le k,\quad {\varvec{x}}\in \varGamma _{{}_{\mathrm N}}^i, \,0\le i< k,\\ \displaystyle \int _{\gamma _k({\varvec{x}})}V_h&{}{\varvec{x}}\in \varGamma _{{}_{\mathrm N}}^k,\\ \displaystyle \int _{\varGamma _{{}_{\mathrm N}}^k} V_h&{} {\varvec{x}}\in \varGamma _{{}_{\mathrm J}}^i,\, k< i< N, \quad {\varvec{x}}\in \varGamma _{{}_{\mathrm N}}^i, \,k \le i< N. \end{array}\right. } \end{aligned}$$

On $\varGamma _{{}_{\mathrm E}}$ we define $\widetilde{G}_{k,h}$ as in Remark 3. Then, by taking $\widetilde{G}_h=\widetilde{G}_{k,h}$ in (32) we get $V_h=0$ on $\varGamma _{{}_{\mathrm N}}^k$, $\forall k \in N$, and thus $V_h=0$ on $\varGamma _{{}_{\mathrm N}}$. $\square$