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Describe time-averaged joule heating in the readme
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Co-authored-by: Josh Williams <[email protected]>
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nmnobre and jvwilliams23 committed Sep 11, 2024
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# HIVE: Heating by Induction to Verify Extremes

_Nuno Nobre and Karthikeyan Chockalingam_
_Nuno Nobre, Josh Williams and Karthikeyan Chockalingam_

###

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Solved for the magnetic vector potential $\mathbf{A} \in \mathcal{N}^0_I$
[<sup>(*)</sup>](https://defelement.com/elements/examples/tetrahedron-nedelec1-lagrange-1.html),
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, $\mathbf{∇} × \mathbf{A} × \mathbf{n} = 0$.
everywhere in space and for each time step $\Delta t_\mathrm{EM}$ of only a
reduced selection of cycles of the voltage source in (1), 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
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Solved for the temperature $T \in \mathcal{P}^1$
[<sup>(*)</sup>](https://defelement.com/elements/examples/tetrahedron-lagrange-equispaced-1.html),
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, $T = T_\mathrm{room}$.
everywhere in space and for each time step $\Delta t_\mathrm{HT}$, 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.
The right-hand side is the Joule heating term which we compute only on the
target and is time-averaged over the simulated time interval in (2). This
enables quicker simulations by solving for the temperature $T$ on a larger
time scale than the magnetic vector potential $\mathbf{A}$, i.e.
$\Delta t_\mathrm{HT} >> \Delta t_\mathrm{EM}$.

See [input/Parameters.i](input/Parameters.i) for the set of parameters
influencing the simulation.
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### 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.

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