Available Quantities

Below is a list of all quantities QUADCOIL supports. Scalar quantities can be used as both objectives and constraints. Fields (with array output) can only be used as constraints. To use these quantities as objectives, constraints or metrics, simply pass their names into quadcoil.quadcoil. quadcoil.quadcoil, like:

out_dict, qp, dofs_opt, solve_results = quadcoil(
    ...
    objective_name=('f_B',),
    constraint_name=('K_theta',),
    constraint_type=('>=',),
    constraint_value=(0.,),
    constraint_unit=(None,),
    metric_name=('f_B',),
    ...
)

If needed, these quantities can also be directly imported as functions from quadcoil.quantity:

from quadcoil.quantity import K_theta
print(K_theta(qp, dofs_opt))

All members of quadcoil.quantity require the same inputs:

qp : QuadcoilParams - Stores the plasma and winding surface information.
dofs_opt : dict - The optimized degrees of freedom produced by quadcoil.quadcoil.

Notation

\(\mathbf{B}\): The magnetic field on the plasma boundary.
\(\mathbf{K}\): The sheet current representing the coil set.
\(\hat{\mathbf{n}}\): The unit normal of the plasma boundary.
\(n_{FP}\): The number of field periods.
\(n_\phi^P\): The plasma toroidal resolution, len(plasma_quadpoints_phi).
\(n_\theta^P\): The plasma poloidal resolution, len(plasma_quadpoints_theta).
\(n_\phi^E\): The toroidal evaluation resolution on the winding surface, len(quadpoints_phi).
\(n_\theta^E\): The poloidal evaluation resolution on the winding surface, len(quadpoints_theta).

Magnetic Field

These objectives are related to the magnetic field on the plasma surface:

Name	Formula	Output Shape	Description
`'winding_surface_B'`	\(\mathbf{B}\)	\((n_\phi^P, n_\theta^P, 3)\)	The \((x, y, z)\) components of the magnetic field on the plasma surface.
`'Bnormal'`	\(\mathbf{B}\cdot\hat{\mathbf{n}}\)	\((n_\phi^P, n_\theta^P)\)	The normal magnetic field on the plasma surface.
`'Bnormal2'`	\((\mathbf{B}\cdot\hat{\mathbf{n}})^2\)	\((n_\phi^P, n_\theta^P)\)	The squared normal field error on the plasma surface.
`'f_B'`	\(f_B\equiv\frac{n_{FP}}{2}\oint_\text{plasma} da \\|\mathbf{B}\cdot\hat{\mathbf{n}}\\|^2\)	Scalar	The integrated normal field error. Also the NESCOIL objective.
`'f_B_normalized_by_Bnormal_IG'`	\(\frac{f_B}{f_B(\Phi_{sv}=0)}\)	Scalar	\(f_B\), normalized by its value with only the net toroidal and poloidal currents.
`'f_max_Bnormal'`	\(\max_\text{plasma surface} \\|\mathbf{B}\cdot\hat{\mathbf{n}}\\|\)	Scalar	The maximum normal field error.
`'f_max_Bnormal2'`	\(\max_\text{plasma surface} \\|\mathbf{B}\cdot\hat{\mathbf{n}}\\|^2\)	Scalar	The maximum normal field error squared. A convex quadratic constraint may behave better than a linear constraint.

Current Magnitude and Sign

These objectives are related to the magnitude and sign of the sheet current \(\mathbf{K}\) that represents a coil set:

Name	Formula	Output Shape	Description
`'K'`	\(\mathbf{K}\)	\((n_\phi^E, n_\theta^E, 3)\)	The \((x, y, z)\) components of the current on the winding surface.
`'K2'`	\(\\|\mathbf{K}\\|^2\)	\((n_\phi^E, n_\theta^E)\)	The current density squared on the winding surface.
`'K_theta'`	\(K_\theta\)	\((n_\phi^E, n_\theta^E)\)	The poloidal current distribution on the winding surface.
`'f_K'`	\(\frac{n_{FP}}{2}\oint_\text{WS} da \\|\mathbf{K}\\|^2\)	Scalar	The integrated current density squared on the winding surface. Also the REGCOIL regularization factor.
`'f_max_K2'`	\(\max_\text{WS}\\|K\\|_2^2\)	Scalar	The maximum current density squared on the winding surface.
`'f_huber_K'`	Pseudo Huber penalty on \(\mathbf{K}\)	Scalar	A smooth approximation of the surface L-1 norm of \(\|\mathbf{K}\|\). Promotes sparsity in \(\mathbf{K}\).

Current Curvature

These objectives are related to the curvature of the sheet current:

Name	Formula	Output Shape	Description
`'K_dot_grad_K'`	\(\mathbf{K}\cdot\nabla\mathbf{K}\)	\((n_\phi^E, n_\theta^E, 3)\)	The \((x, y, z)\) components of \(\mathbf{K}\cdot\nabla\mathbf{K}\) on the winding surface.
`'K_dot_grad_K_cyl'`	\((\mathbf{K}\cdot\nabla\mathbf{K})_{(R, \phi, Z)}\)	\((n_\phi^E, n_\theta^E, 3)\)	The \((R, \phi, Z)\) components of \(\mathbf{K}\cdot\nabla\mathbf{K}\) on the winding surface.
`'f_max_K_dot_grad_K_cyl'`	\(\max_\text{WS}\\|(\mathbf{K}\cdot\nabla\mathbf{K})_{(R, \phi, Z)}\\|_\infty\)	Scalar	Maximum \((R, \phi, Z)\) component of \(\mathbf{K}\cdot\nabla\mathbf{K}\) over the winding surface.

Dipole

These objectives are related to dipole optimization:

Name	Formula	Output Shape	Description
`'Phi'`	\(\Phi_{sv}\)	\((n_\phi^E, n_\theta^E)\)	The dipole density distribution on the winding surface. Also referred to as the single valued component of the current potential.
`'Phi2'`	\(\\|\Phi_{sv}\\|^2\)	\((n_\phi^E, n_\theta^E)\)	The squared dipole density distribution on the winding surface.
`'f_Phi'`	\(\frac{n_{FP}}{2}\oint_\text{WS} da\ \\|\Phi_{sv}\\|^2\)	Scalar	The integrated squared dipole density on the winding surface.
`'Phi_with_net_current'`	\(\Phi = \Phi_{sv} + \frac{G\phi'}{2\pi} + \frac{I\theta'}{2\pi}\)	\((n_\phi^E, n_\theta^E)\)	The full current potential on the winding surface, with the secular components representing the net poloidal and toroidal currents \(G\) and \(I\).
`'f_max_Phi'`	\(\max_\text{WS}\\|\Phi_{sv}\\|\)	Scalar	The maximum dipole density on the winding surface.
`'f_l1_Phi'`	\(\int_\text{WS}dA\\|\Phi_{sv}\\|\)	Scalar	The sum of the absolute values of dipole density over the winding surface for L1 sparsity regularization.
`'f_max_Phi2'`	\(\max_\text{WS}\\|\Phi_{sv}\\|^2\)	Scalar	The maximum dipole density squared on the winding surface. A convex quadratic constraint may behave better than a linear constraint.
`'f_max_Phi4'`	\(\max_\text{WS}\\|\Phi_{sv}\\|^4\)	Scalar	The maximum fourth power of the dipole density on the winding surface. Experimental. Added to test the convergence behavior of high-order convex terms.

Lorentz Force

These objectives are related to the self-force of the sheet current. The force is reported in \((R, \phi, Z)\) components on the winding surface.

Name	Formula	Output Shape	Description
`'f_max_force_cyl'`	\(\max_\text{WS}\\|\mathbf{F}_{(R, \phi, Z)}\\|_\infty\)	Scalar	Maximum \((R, \phi, Z)\) component of the sheet-current self-force over the winding surface.
`'f_l1_force_cyl'`	\(\int_\text{WS} dA\ \\|\mathbf{F}_{(R, \phi, Z)}\\|_1\)	Scalar	L1 penalty on the \((R, \phi, Z)\) components of the sheet-current self-force.
`'f_max_force2_cyl'`	\(\max_\text{WS}\\|\mathbf{F}_{(R, \phi, Z)}\\|_\infty^2\)	Scalar	Maximum squared \((R, \phi, Z)\) component of the sheet-current self-force over the winding surface. Experimental. Added to test the convergence behavior of high-order non-convex terms.