HIVE: Heating by Induction to Verify Extremes

Nuno Nobre and Karthikeyan Chockalingam

HIVE, located at UKAEA's Culham campus, is a testing rig designed to stress small prototype cooled and un-cooled fusion reactor components under high induction heat fluxes in a high vacuum environment.

This repository hosts a MOOSE app that aims to replicate the rig's behavior towards full digital twinning capability.

Getting started

Installation is as usual for any MOOSE app:

1. Install MOOSE using the installation method of your choice.

2. Clone this repo alongside MOOSE, git clone https://github.com/farscape-project/HIVE.git.

3. Build the app with a no. of jobs of your choice, cd HIVE && make [-j [jobs]].

4. Run it with a no. of processes of your choice, mpirun [-n processes] ./hive-opt -w -i input/THeat.i.

A brief conceptual introduction

For simulation purposes, we discretize HIVE using a tetrahedral mesh and segment it into three components: a cuboid vacuum chamber, an electromagnetic coil, and a target prototype component. See mesh/ for both a coarse and a fine option, but note the latter is not a uniform refinement of the former and that, in particular, the coil is discretized using the same number of elements in both. Both the coil and the target sit within the vacuum chamber and are assumed to be made of electrically conductive materials. Here we use copper for both, but other conductors can also be used.

HIVE is, quite simply, an expensive induction hob, not unlike the one you might have at home. By applying a time-varying electric potential difference to the coil terminals, a time-varying electric current flows through the coil, creating a time-varying magnetic field which induces a time-varying electric current in the target that gradually warms up via Joule heating.

Recall that the time-varying electric current flowing through the target will self-induce a current, causing the (net) current to flow mostly along the outer perimeter, or skin, of the target. This is known as the skin effect and is more pronounced the higher the frequency. Furthermore, since induced currents lag their sources by one-quarter cycle, i.e. $\pi/2$, the resulting current on the target will, with increasing frequency, tend to lag the current on the coil by one-half cycle, i.e. $\pi$.

App design

This app leverages a linear, but time-dependent, finite element formulation split into three sub-apps. Each sub-app solves a different equation for a different field, and feeds the solution to the next sub-app. This one-way coupling pipeline proceeds as follows:

1. Laplace's equation: $∇^2 V = 0$. See input/VLaplace.i.

   Solved for the electric potential $V \in \mathcal{P}^1$ ^(*), only on the coil and only once, with Dirichlet boundary conditions on both its terminals, $V_\mathrm{in} = V_\mathrm{max}$ and $V_\mathrm{out} = 0$, and Neumann boundary conditions on its $\mathbf{n}$-oriented lateral surface, $\mathbf{∇}V \cdot \mathbf{n} = 0$. This sole solution can then be scaled appropriately for any time step if the time-dependent Dirichlet boundary condition is assumed uniform, i.e. $V_\mathrm{in}(\mathbf{r},t) \equiv V_\mathrm{in}(t)$. Here, we take $V_\mathrm{in}(t)=V_\mathrm{max}\mathrm{sin}(\omega t)$ for some angular frequency $\omega$.

2. The $\mathbf{A}$ formulation: $\mathbf{∇}× \left(ν \mathbf{∇}× \mathbf{A}\right) +σ \partial_t \mathbf{A} = -σ \mathbf{∇}V$. See input/AForm.i.

   Solved for the magnetic vector potential $\mathbf{A} \in \mathcal{N}^0_I$ ^(*), everywhere in space and for each time step, with Dirichlet boundary conditions on the $\mathbf{n}$-oriented plane boundary of the vacuum chamber where the coil terminals sit, $\mathbf{A} × \mathbf{n} = 0$, and Neumann boundary conditions on its remaining $\mathbf{n}$-oriented outer surfaces, $\mathbf{∇} × \mathbf{A} × \mathbf{n} = 0$. $ν$ is the magnetic reluctivity (the reciprocal of the magnetic permeability) and $σ$ is the electrical conductivity. The right-hand side is non-zero only within the coil, see (1), and is simply the current flowing through it.

3. The heat equation: $ρc \partial_t T - \mathbf{∇} \cdot (k \mathbf{∇}T) = σ ||\partial_t \mathbf{A}||^2$. See input/THeat.i.

   Solved for the temperature $T \in \mathcal{P}^1$ ^(*), everywhere in space and for each time step, with Neumann boundary conditions on the $\mathbf{n}$-oriented outer surface of the vacuum chamber, $\mathbf{∇}T \cdot \mathbf{n} = 0$, and initial conditions everywhere in space, $T = T_\mathrm{room}$. $ρ$ is the density, $c$ is the specific heat capacity, and $k$ is the thermal conductivity. The right-hand side is the Joule heating term which, as of this writing, we compute only on the target.

See input/Parameters.i for the set of parameters influencing the simulation. This file is included at the top of the input files for each of the three sub-apps, input/VLaplace.i, input/AForm.i and input/THeat.i. Since none uses the entire parameter set, MOOSE issues a few warnings that can be safely ignored (thus the need for the -w flag above). All material properties are assumed uniform within each of the three components and, as of this writing, also temperature-independent.

Next steps

This app is under active development and is being updated frequently. We are currently looking to improve its capability when it comes to solution accuracy, time-to-solution and general usability.

Solution accuracy

• Add two-way coupling between the heat equation and $\mathbf{A}$ formulation sub-apps so the material properties influencing the latter can be made temperature-dependent.

• Flow a coolant through the target's cooling pipe.

• Add support to MOOSE (+libMesh) for $\mathcal{N}^1_I$ variables on TET14s.

Time-to-solution

• Switch to a time-averaged Joule heating source in the heat equation sub-app to keep the problem tractable when simulating over a long physical time span. The $\mathbf{A}$ formulation sub-app will then sub-cycle, i.e. perform multiple time steps for each time step of the heat equation sub-app.

• Study the potential gains of solving all, but most importantly the $\mathbf{A}$ formulation sub-app, on the GPU simply via PETSc/hypre flags.

• Add support to MOOSE for the hypre AMS preconditioner. This would allow the $\mathbf{A}$ formulation sub-app to drop LU, which is relatively slow, as its preconditioner.

• Add support to MOOSE to impose strong boundary conditions for $H(\mathrm{curl})$-conforming variables. This would allow the $\mathbf{A}$ formulation sub-app to drop the penalty-method boundary conditions it currently uses.

Usability

• Create a GUI for use by the people operating HIVE and looking to conduct virtual testing and qualification to understand operational ranges, plan experiments and quantify uncertainties.