simsopt.field package

class simsopt.field.BiotSavart(coils)

Bases: BiotSavart, MagneticField

Computes the MagneticField induced by a list of closed curves \(\Gamma_k\) with electric currents \(I_k\). The field is given by

\[B(\mathbf{x}) = \frac{\mu_0}{4\pi} \sum_{k=1}^{n_\mathrm{coils}} I_k \int_0^1 \frac{(\Gamma_k(\phi)-\mathbf{x})\times \Gamma_k'(\phi)}{\|\Gamma_k(\phi)-\mathbf{x}\|^3} d\phi\]

where \(\mu_0=4\pi 10^{-7}\) is the magnetic constant.

Parameters

coils – A list of simsopt.field.coil.Coil objects.

A_and_dA_vjp(v, vgrad)

Same as simsopt.geo.biotsavart.BiotSavart.A_vjp but returns the vector Jacobian product for \(A\) and \(\nabla A\), i.e. it returns

\[\{ \sum_{i=1}^{n} \mathbf{v}_i \cdot \partial_{\mathbf{c}_k} \mathbf{A}_i \}_k, \{ \sum_{i=1}^{n} {\mathbf{v}_\mathrm{grad}}_i \cdot \partial_{\mathbf{c}_k} \nabla \mathbf{A}_i \}_k.\]

A_vjp(v)

Assume the field was evaluated at points \(\mathbf{x}_i, i\in \{1, \ldots, n\}\) and denote the value of the field at those points by \(\{\mathbf{A}_i\}_{i=1}^n\). These values depend on the shape of the coils, i.e. on the dofs \(\mathbf{c}_k\) of each coil. This function returns the vector Jacobian product of this dependency, i.e.

\[\{ \sum_{i=1}^{n} \mathbf{v}_i \cdot \partial_{\mathbf{c}_k} \mathbf{A}_i \}_k.\]

B_and_dB_vjp(v, vgrad)

Same as simsopt.geo.biotsavart.BiotSavart.B_vjp but returns the vector Jacobian product for \(B\) and \(\nabla B\), i.e. it returns

\[\{ \sum_{i=1}^{n} \mathbf{v}_i \cdot \partial_{\mathbf{c}_k} \mathbf{B}_i \}_k, \{ \sum_{i=1}^{n} {\mathbf{v}_\mathrm{grad}}_i \cdot \partial_{\mathbf{c}_k} \nabla \mathbf{B}_i \}_k.\]

B_vjp(v)

Assume the field was evaluated at points \(\mathbf{x}_i, i\in \{1, \ldots, n\}\) and denote the value of the field at those points by \(\{\mathbf{B}_i\}_{i=1}^n\). These values depend on the shape of the coils, i.e. on the dofs \(\mathbf{c}_k\) of each coil. This function returns the vector Jacobian product of this dependency, i.e.

\[\{ \sum_{i=1}^{n} \mathbf{v}_i \cdot \partial_{\mathbf{c}_k} \mathbf{B}_i \}_k.\]

as_dict() → dict

A JSON serializable dict representation of an object.

d2A_by_dXdcoilcurrents(compute_derivatives=1)

d2B_by_dXdcoilcurrents(compute_derivatives=1)

d3A_by_dXdXdcoilcurrents(compute_derivatives=2)

d3B_by_dXdXdcoilcurrents(compute_derivatives=2)

dA_by_dcoilcurrents(compute_derivatives=0)

dB_by_dcoilcurrents(compute_derivatives=0)

classmethod from_dict(d)

Parameters

d – Dict representation.

Returns

MSONable class.

class simsopt.field.BoozerAnalytic(etabar, B0, N, G0, psi0, iota0, Bbar=1.0, I0=0.0, G1=0.0, I1=0.0, K1=0.0)

Bases: BoozerMagneticField

Computes a BoozerMagneticField based on a first-order expansion in distance from the magnetic axis (Landreman & Sengupta, Journal of Plasma Physics 2018). Here the magnetic field strength is expressed as,

\[B(s,\theta,\zeta) = B_0 \left(1 + \overline{\eta} \sqrt{2s\psi_0/\overline{B}}\cos(\theta - N \zeta)\right),\]

the covariant components are,

\[ \begin{align}\begin{aligned}G(s) = G_0 + \sqrt{2s\psi_0/\overline{B}} G_1\\I(s) = I_0 + \sqrt{2s\psi_0/\overline{B}} I_1\\K(s,\theta,\zeta) = \sqrt{2s\psi_0/\overline{B}} K_1 \sin(\theta - N \zeta),\end{aligned}\end{align} \]

and the rotational transform is,

\[\iota(s) = \iota_0.\]

While formally \(I_0 = I_1 = G_1 = K_1 = 0\), these terms have been included in order to test the guiding center equations at finite beta.

Parameters

• etabar – magnitude of first order correction to magnetic field strength

• B0 – magnetic field strength on the axis

• N – helicity of symmetry (integer)

• G0 – lowest order toroidal covariant component

• psi0 – (toroidal flux)/ (2*pi) on the boundary

• iota0 – lowest order rotational transform

• Bbar – normalizing magnetic field strength (defaults to 1)

• I0 – lowest order poloidal covariant component (defaults to 0)

• G1 – first order correction to toroidal covariant component (defaults to 0)

• I1 – first order correction to poloidal covariant component (defaults to 0)

• K1 – first order correction to radial covariant component (defaults to 0)

set_B0(B0)

set_Bbar(Bbar)

set_G0(G0)

set_G1(G1)

set_I0(I0)

set_I1(I1)

set_K1(K1)

set_N(N)

set_etabar(etabar)

set_iota0(iota0)

set_psi0(psi0)

class simsopt.field.BoozerMagneticField(psi0)

Bases: BoozerMagneticField

Generic class that represents a magnetic field in Boozer coordinates \((s,\theta,\zeta)\). Here \(s = \psi/\psi_0\) is the normalized toroidal flux where \(2\pi\psi_0\) is the toroidal flux at the boundary. The magnetic field in the covariant form is,

\[\textbf B(s,\theta,\zeta) = G(s) \nabla \zeta + I(s) \nabla \theta + K(s,\theta,\zeta) \nabla \psi,\]

and the contravariant form is,

\[\textbf B(s,\theta,\zeta) = \frac{1}{\sqrt{g}} \left(\frac{\partial \mathbf r}{\partial \zeta} + \iota(s)\frac{\partial \mathbf r}{\partial \theta}\right),\]

where,

\[\sqrt{g}(s,\theta,\zeta) = \frac{G(s) + \iota(s)I(s)}{B^2}.\]

Here \(\iota(s) = \psi_P'(\psi)\) where \(2\pi\psi_P\) is the poloidal flux and \(2\pi\psi\) is the toroidal flux. Each subclass of BoozerMagneticField implements functions to compute \(B\), \(G\), \(I\), \(\iota\), \(\psi_P\), and their derivatives. The cylindrical coordinates \(R(s,\theta,\zeta)\) and \(Z(s,\theta,\zeta)\) in addition to \(K(s,\theta,\zeta)\) and \(\nu\) where \(\zeta = \phi + \nu(s,\theta,\zeta)\) and \(\phi\) is the cylindrical azimuthal angle are also implemented by BoozerRadialInterpolant and InterpolatedBoozerField. The usage is similar to the MagneticField class.

The usage of BoozerMagneticField` is as follows:

booz = BoozerAnalytic(etabar,B0,N,G0,psi0,iota0) # An instance of BoozerMagneticField
points = ... # points is a (n, 3) numpy array defining :math:`(s,\theta,\zeta)`
booz.set_points(points)
modB = bfield.modB() # returns the magnetic field strength at `points`

BoozerMagneticField has a cache to avoid repeated calculations. To clear this cache manually, call the clear_cached_properties() function. The cache is automatically cleared when set_points() is called or one of the dependencies changes.

clear_cached_properties()

Clear the cache.

recompute_bell(parent=None)

class simsopt.field.BoozerRadialInterpolant(equil, order, mpol=32, ntor=32, N=None, enforce_vacuum=False, rescale=False, ns_delete=0, no_K=False)

Bases: BoozerMagneticField

Given a Vmec instance, performs a Boozer coordinate transformation using BOOZXFORM. The magnetic field can be computed at any point in Boozer coordinates using radial spline interpolation (scipy.interpolate.InterpolatedUnivariateSpline) and an inverse Fourier transform in the two angles. Throughout stellarator symmetry is assumed.

Parameters

• equil – instance of simsopt.mhd.vmec.Vmec or simsopt.mhd.boozer.Boozer. If it is an instance of simsopt.mhd.boozer.Boozer, the compute_surfs needs to include all of the grid points in the half-radius grid of the corresponding Vmec equilibrium.

• order – (int) order for radial interpolation. Must satisfy 1 <= order <= 5.

• mpol – (int) number of poloidal mode numbers for BOOZXFORM (defaults to 32)

• ntor – (int) number of toroidal mode numbers for BOOZXFORM (defaults to 32)

• N – Helicity of quasisymmetry to enforce. If specified, then the non-symmetric Fourier harmonics of \(B\) and \(K\) are filtered out. Otherwise, all harmonics are kept. (defaults to None)

• enforce_vacuum – If True, a vacuum field is assumed, \(G\) is set to its mean value, \(I = 0\), and \(K = 0\).

• rescale –

  If True, use the interpolation method in the DELTA5D code. Here, a few of the first radial grid points or (bmnc, rmnc, zmns, numns, kmns) are deleted (determined by ns_delete). The Fourier harmonics are then rescaled as:

  bmnc(s)/s^(1/2) for m = 1

  bmnc(s)/s for m even and >= 2

  bmnc(s)/s^(3/2) for m odd and >=3

  before performing interpolation and spline differentiation to obtain dbmncds. If False, interpolation of the unscaled Fourier harmonics and its finite-difference derivative wrt s is performed instead (defaults to False)

• ns_delete – (see rescale) (defaults to 0)

compute_K()

init_splines()

class simsopt.field.CircularCoil(r0=0.1, center=[0, 0, 0], I=159154.94309189534, normal=[0, 0])

Bases: MagneticField

Magnetic field created by a single circular coil evaluated using analytical functions, including complete elliptic integrals of the first and second kind. As inputs, it takes the radius of the coil (r0), its center, current (I) and its normal vector [either spherical angle components (normal=[theta,phi]) or (x,y,z) components of a vector (normal=[x,y,z])]). The (theta,phi) angles are related to the (x,y,z) components of the normal vector via theta = np.arctan2(normal[1], normal[0]) and phi = np.arctan2(np.sqrt(normal[0]**2+normal[1]**2), normal[2]). Sign convention: CircularCoil with a positive current produces a magnetic field vector in the same direction as the normal when evaluated at the center of the coil.a

Parameters

• r0 – radius of the coil

• center – point at the coil center

• I – current of the coil in Ampere’s

• normal – if list with two values treats it as spherical angles theta and phi of the normal vector to the plane of the coil centered at the coil center, if list with three values treats it a vector

as_dict()

A JSON serializable dict representation of an object.

classmethod from_dict(d)

Parameters

d – Dict representation.

Returns

MSONable class.

class simsopt.field.Coil(curve, current)

Bases: Coil, Optimizable

A Coil combines a Curve and a Current and is used as input for a BiotSavart field.

as_dict() → dict

A JSON serializable dict representation of an object.

classmethod from_dict(d)

Parameters

d – Dict representation.

Returns

MSONable class.

plot(**kwargs)

Plot the coil’s curve. This method is just shorthand for calling the plot() function on the underlying Curve. All arguments are passed to simsopt.geo.curve.Curve.plot()

vjp(v_gamma, v_gammadash, v_current)

class simsopt.field.Current(current, **kwargs)

Bases: Current, CurrentBase

An optimizable object that wraps around a single scalar degree of freedom. It represents the electric current in a coil, or in a set of coils that are constrained to use the same current.

as_dict() → dict

A JSON serializable dict representation of an object.

classmethod from_dict(d)

Parameters

d – Dict representation.

Returns

MSONable class.

vjp(v_current)

class simsopt.field.Dommaschk(mn=[[0, 0]], coeffs=[[0, 0]])

Bases: MagneticField

Vacuum magnetic field created by an explicit representation of the magnetic field scalar potential as proposed by W. Dommaschk (1986), Computer Physics Communications 40, 203-218. As inputs, it takes the arrays for the harmonics m, n and its corresponding coefficients.

Parameters

• m – first harmonic array

• n – second harmonic array

• coeffs – coefficient for Vml for each of the ith index of the harmonics m and n

as_dict() → dict

A JSON serializable dict representation of an object.

classmethod from_dict(d)

Parameters

d – Dict representation.

Returns

MSONable class.

class simsopt.field.InterpolatedBoozerField(field, degree, srange, thetarange, zetarange, extrapolate=True, nfp=1, stellsym=True)

Bases: InterpolatedBoozerField, BoozerMagneticField

This field takes an existing BoozerMagneticField and interpolates it on a regular grid in \(s,\theta,\zeta\). This resulting interpolant can then be evaluated very quickly. This is modeled after InterpolatedField.

__init__(field, degree, srange, thetarange, zetarange, extrapolate=True, nfp=1, stellsym=True)

Parameters

• field – the underlying simsopt.field.boozermagneticfield.BoozerMagneticField to be interpolated.

• degree – the degree of the piecewise polynomial interpolant.

• srange – a 3-tuple of the form (smin, smax, ns). This mean that the interval [smin, smax] is split into ns many subintervals.

• thetarange – a 3-tuple of the form (thetamin, thetamax, ntheta). thetamin must be >= 0, and thetamax must be <=2*pi.

• zetarange – a 3-tuple of the form (zetamin, zetamax, nzeta). zetamin must be >= 0, and thetamax must be <=2*pi.

• extrapolate – whether to extrapolate the field when evaluate outside the integration domain or to throw an error.

• nfp – Whether to exploit rotational symmetry. In this case any toroidal angle is always mapped into the interval \([0, 2\pi/\mathrm{nfp})\), hence it makes sense to use zetamin=0 and zetamax=2*np.pi/nfp.

• stellsym – Whether to exploit stellarator symmetry. In this case theta is always mapped to the interval \([0, \pi]\), hence it makes sense to use thetamin=0 and thetamax=np.pi.

class simsopt.field.InterpolatedField(field, degree, rrange, phirange, zrange, extrapolate=True, nfp=1, stellsym=False, skip=None)

Bases: InterpolatedField, MagneticField

This field takes an existing field and interpolates it on a regular grid in \(r,\phi,z\). This resulting interpolant can then be evaluated very quickly.

__init__(field, degree, rrange, phirange, zrange, extrapolate=True, nfp=1, stellsym=False, skip=None)

Parameters

• field – the underlying simsopt.field.magneticfield.MagneticField to be interpolated.

• degree – the degree of the piecewise polynomial interpolant.

• rrange – a 3-tuple of the form (rmin, rmax, nr). This mean that the interval \([rmin, rmax]\) is split into nr many subintervals.

• phirange – a 3-tuple of the form (phimin, phimax, nphi).

• zrange – a 3-tuple of the form (zmin, zmax, nz).

• extrapolate – whether to extrapolate the field when evaluate outside the integration domain or to throw an error.

• nfp – Whether to exploit rotational symmetry. In this case any angle is always mapped into the interval \([0, 2\pi/\mathrm{nfp})\), hence it makes sense to use phimin=0 and phimax=2*np.pi/nfp.

• stellsym – Whether to exploit stellarator symmetry. In this case z is always mapped to be positive, hence it makes sense to use zmin=0.

• skip –

  a function that takes in a point (in cylindrical (r,phi,z) coordinates) and returns whether to skip that location when building the interpolant or not. The signature should be

  def skip(r: double, phi: double, z: double) -> bool:
      ...
  

  See also here https://github.com/hiddenSymmetries/simsopt/pull/227 for a graphical illustration.

to_vtk(filename)

Export the field evaluated on a regular grid for visualisation with e.g. Paraview.

class simsopt.field.IterationStoppingCriterion

Bases: IterationStoppingCriterion

Stop the iteration once the maximum number of iterations is reached.

class simsopt.field.LevelsetStoppingCriterion(classifier)

Bases: LevelsetStoppingCriterion

Based on a scalar function \(f:R^3\to R\), this criterion checks whether \(f(x, y, z) < 0\) and stops the iteration once this is true.

The idea is to use this for example with signed distance functions to a surface.

class simsopt.field.MagneticField(**kwargs)

Bases: MagneticField, Optimizable

Generic class that represents any magnetic field from which each magnetic field class inherits. The usage of MagneticField is as follows:

bfield = BiotSavart(coils) # An instance of a MagneticField
points = ... # points is a (n, 3) numpy array
bfield.set_points(points)
B = bfield.B() # returns the Magnetic field at `points`
dA = bfield.dA_by_dX() # returns the gradient of the potential of the field at `points`

MagneticField has a cache to avoid repeated calculations. To clear this cache manually, call the clear_cached_properties() function. The cache is automatically cleared when set_points is called or one of the dependencies changes.

__add__(other)

Add two magnetic fields.

__mul__(other)

Multiply a field with a scalar.

__rmul__(other)

Multiply a field with a scalar.

clear_cached_properties()

Clear the cache.

recompute_bell(parent=None)

Function to be called whenever new DOFs input is given or if the parent Optimizable’s data changed, so the output from the current Optimizable object is invalid.

This method gets called by various DOF setters. If only the local DOFs of an object are being set, the recompute_bell method is called in that object and also in the descendent objects that have a dependency on the object, whose local DOFs are being changed. If gloabl DOFs of an object are being set, the recompute_bell method is called in the object, ancestors of the object, as well as the descendents of the object.

Need to be implemented by classes that provide a dof_setter for external handling of DOFs.

set_points(self: simsoptpp.MagneticField, arg0: numpy.ndarray[numpy.float64]) → simsoptpp.MagneticField

Shorthand for set_points_cart.

set_points_cart(self: simsoptpp.MagneticField, arg0: numpy.ndarray[numpy.float64]) → simsoptpp.MagneticField

Set the points where to evaluate the magnetic fields, in cartesian coordinates.

set_points_cyl(self: simsoptpp.MagneticField, arg0: numpy.ndarray[numpy.float64]) → simsoptpp.MagneticField

Set the points where to evaluate the magnetic fields, in cylindrical coordinates (the order is \((r, \phi, z)\)).

to_vtk(filename, nr=10, nphi=10, nz=10, rmin=1.0, rmax=2.0, zmin=- 0.5, zmax=0.5)

Export the field evaluated on a regular grid for visualisation with e.g. Paraview.

class simsopt.field.MagneticFieldMultiply(scalar, Bfield)

Bases: MagneticField

Class used to multiply a magnetic field by a scalar. It takes as input a MagneticField class and a scalar and multiplies B, A and their derivatives by that value.

as_dict() → dict

A JSON serializable dict representation of an object.

classmethod from_dict(d)

Parameters

d – Dict representation.

Returns

MSONable class.

class simsopt.field.MagneticFieldSum(Bfields)

Bases: MagneticField

Class used to sum two or more magnetic field together. It can either be called directly with a list of magnetic fields given as input and outputing another magnetic field with B, A and its derivatives added together or it can be called by summing magnetic fields classes as Bfield1 + Bfield1

B_vjp(v)

as_dict() → dict

A JSON serializable dict representation of an object.

classmethod from_dict(d)

Parameters

d – Dict representation.

Returns

MSONable class.

class simsopt.field.MaxToroidalFluxStoppingCriterion

Bases: MaxToroidalFluxStoppingCriterion

Stop the iteration once a particle falls above a critical value of s, the normalized toroidal flux. This should only be used when tracing trajectories in a flux coordinate system (i.e., trace_particles_boozer).

Usage:

stopping_criteria=[MaxToroidalFluxStopingCriterion(s)]

where s is the value of the maximum normalized toroidal flux.

class simsopt.field.MinToroidalFluxStoppingCriterion

Bases: MinToroidalFluxStoppingCriterion

Stop the iteration once a particle falls below a critical value of s, the normalized toroidal flux. This StoppingCriterion is important to use when tracing particles in flux coordinates, as the poloidal angle becomes ill-defined at the magnetic axis. This should only be used when tracing trajectories in a flux coordinate system (i.e., trace_particles_boozer).

Usage:

stopping_criteria=[MinToroidalFluxStopingCriterion(s)]

where s is the value of the minimum normalized toroidal flux.

class simsopt.field.PoloidalField(R0, B0, q)

Bases: MagneticField

Magnetic field purely in the poloidal direction, that is, in the theta direction of a poloidal-toroidal coordinate system. Its modulus is given by B = B0 * r / (R0 * q) so that, together with the toroidal field, it creates a safety factor equals to q

Parameters

• B0 – modulus of the magnetic field at R0

• R0 – major radius of the magnetic axis

• q – safety factor/pitch angle of the magnetic field lines

as_dict() → dict

A JSON serializable dict representation of an object.

classmethod from_dict(d)

Parameters

d – Dict representation.

Returns

MSONable class.

class simsopt.field.Reiman(iota0=0.15, iota1=0.38, k=[6], epsilonk=[0.01], m0=1)

Bases: MagneticField

Magnetic field model in section 5 of Reiman and Greenside, Computer Physics Communications 43 (1986) 157—167. This field allows for an analytical expression of the magnetic island width that can be used for island optimization. However, the field is not completely physical as it does not have nested flux surfaces.

Parameters

• iota0 – unperturbed rotational transform

• iota1 – unperturbed global magnetic shear

• k – integer array specifying the Fourier modes used

• epsilonk – coefficient of the Fourier modes

• m0 – toroidal symmetry parameter (normally m0=1)

as_dict()

A JSON serializable dict representation of an object.

classmethod from_dict(d)

Parameters

d – Dict representation.

Returns

MSONable class.

class simsopt.field.ScalarPotentialRZMagneticField(phi_str)

Bases: MagneticField

Vacuum magnetic field as a solution of B = grad(Phi) where Phi is the magnetic field scalar potential. It takes Phi as an input string, which should contain an expression involving the standard cylindrical coordinates (R, phi, Z) Example: ScalarPotentialRZMagneticField(“2*phi”) yields a magnetic field B = grad(2*phi) = (0,2/R,0). In order for the analytical derivatives to be performed by sympy, a term 1e-30*Phi*R*Z is added to every entry. Note: this function needs sympy.

Parameters

phi_str – string containing vacuum scalar potential expression as a function of R, Z and phi

as_dict() → dict

A JSON serializable dict representation of an object.

classmethod from_dict(d)

Parameters

d – Dict representation.

Returns

MSONable class.

class simsopt.field.SurfaceClassifier(surface, p=1, h=0.05)

Bases: object

Takes in a toroidal surface and constructs an interpolant of the signed distance function \(f:R^3\to R\) that is positive inside the volume contained by the surface, (approximately) zero on the surface, and negative outisde the volume contained by the surface.

__init__(surface, p=1, h=0.05)

Parameters

• surface – the surface to contruct the distance from.

• p – degree of the interpolant

• h – grid resolution of the interpolant

evaluate_rphiz(rphiz)

evaluate_xyz(xyz)

to_vtk(filename, h=0.01)

class simsopt.field.ToroidalField(R0, B0)

Bases: MagneticField

Magnetic field purely in the toroidal direction, that is, in the phi direction with (R,phi,Z) the standard cylindrical coordinates. Its modulus is given by B = B0*R0/R where R0 is the first input and B0 the second input to the function.

Parameters

• B0 – modulus of the magnetic field at R0

• R0 – radius of normalization

as_dict() → dict

A JSON serializable dict representation of an object.

classmethod from_dict(d)

Parameters

d – Dict representation.

Returns

MSONable class.

class simsopt.field.ToroidalTransitStoppingCriterion

Bases: ToroidalTransitStoppingCriterion

Stop the iteration once the maximum number of toroidal transits is reached.

Usage:

stopping_criteria=[ToroidalTransitStoppingCriterion(ntransits,flux)]

where ntransits is the maximum number of toroidal transits and flux is a boolean indicating whether tracing is being performed in a flux coordinate system.

simsopt.field.apply_symmetries_to_currents(base_currents, nfp, stellsym)

Take a list of n Current`s and return ``n * nfp * (1+int(stellsym))` Current objects obtained by copying (for nfp rotations) and sign-flipping (optionally for stellarator symmetry).

simsopt.field.apply_symmetries_to_curves(base_curves, nfp, stellsym)

Take a list of n simsopt.geo.curve.Curve`s and return ``n * nfp * (1+int(stellsym))` simsopt.geo.curve.Curve objects obtained by applying rotations and flipping corresponding to nfp fold rotational symmetry and optionally stellarator symmetry.

simsopt.field.coils_via_symmetries(curves, currents, nfp, stellsym)

Take a list of n curves and return n * nfp * (1+int(stellsym)) Coil objects obtained by applying rotations and flipping corresponding to nfp fold rotational symmetry and optionally stellarator symmetry.

simsopt.field.compute_fieldlines(field, R0, Z0, tmax=200, tol=1e-07, phis=[], stopping_criteria=[], comm=None)

Compute magnetic field lines by solving

\[[\dot x, \dot y, \dot z] = B(x, y, z)\]

Parameters

• field – the magnetic field \(B\)

• R0 – list of radial components of initial points

• Z0 – list of vertical components of initial points

• tmax – for how long to trace. will do roughly |B|*tmax/(2*pi*r0) revolutions of the device

• tol – tolerance for the adaptive ode solver

• phis – list of angles in [0, 2pi] for which intersection with the plane corresponding to that phi should be computed

• stopping_criteria – list of stopping criteria, mostly used in combination with the LevelsetStoppingCriterion accessed via simsopt.field.tracing.SurfaceClassifier.

Returns: 2 element tuple containing

• res_tys:

  A list of numpy arrays (one for each particle) describing the solution over time. The numpy array is of shape (ntimesteps, 4). Each row contains the time and the position, i.e.`[t, x, y, z]`.

• res_phi_hits:

  A list of numpy arrays (one for each particle) containing information on each time the particle hits one of the phi planes or one of the stopping criteria. Each row of the array contains [time, idx, x, y, z], where idx tells us which of the phis or stopping_criteria was hit. If idx>=0, then phis[int(idx)] was hit. If idx<0, then stopping_criteria[int(-idx)-1] was hit.

simsopt.field.compute_poloidal_transits(res_tys, ma=None, flux=True)

Computes the number of poloidal transits of an orbit. For the case of particles traced in a MagneticField (not a BoozerMagneticField), the poloidal angle is computed using the arctangent angle in the poloidal plane with respect to the coordinate axis, ma,

\[\theta = \tan^{-1} \left( \frac{R(\phi)-R_{\mathrm{ma}}(\phi)}{Z(\phi)-Z_{\mathrm{ma}}(\phi)} \right),\]

where \((R,\phi,Z)\) are the cylindrical coordinates of the trajectory and \((R_{\mathrm{ma}}(\phi),Z_{\mathrm{ma}(\phi)})\) is the position of the coordinate axis.

Parameters

• res_tys – trajectory solution computed from trace_particles() or trace_particles_boozer() with forget_exact_path=False.

• ma – an instance of Curve representing the coordinate axis with respect to which the poloidal angle is computed. If orbit is computed in Boozer coordinates, ma should be None.

• flux – if True, res_tys represents the position in flux coordinates (should be True if computed from trace_particles_boozer()). If True, ma is not used.

Returns

array with length len(res_tys). Each element contains the

number of poloidal transits of the orbit.

Return type

ntransits

simsopt.field.compute_resonances(res_tys, res_phi_hits, ma=None, delta=0.01)

Computes resonant particle orbits given the output of either trace_particles() or trace_particles_boozer(), res_tys and res_phi_hits/res_zeta_hits, with forget_exact_path=False. Resonance indicates a trajectory which returns to the same position at the \(\zeta = 0\) plane after mpol poloidal turns and ntor toroidal turns. For the case of particles traced in a MagneticField (not a BoozerMagneticField), the poloidal angle is computed using the arctangent angle in the poloidal plane with respect to the coordinate axis, ma,

\[\theta = \tan^{-1} \left( \frac{R(\phi)-R_{\mathrm{ma}}(\phi)}{Z(\phi)-Z_{\mathrm{ma}}(\phi)} \right),\]

where \((R,\phi,Z)\) are the cylindrical coordinates of the trajectory and \((R_{\mathrm{ma}}(\phi),Z_{\mathrm{ma}(\phi)})\) is the position of the coordinate axis.

Parameters

• res_tys – trajectory solution computed from trace_particles() or trace_particles_boozer() with forget_exact_path=False

• res_phi_hits – output of trace_particles() or trace_particles_boozer() with phis/zetas = [0]

• ma – an instance of Curve representing the coordinate axis with respect to which the poloidal angle is computed. If the orbit is computed in flux coordinates, ma should be None. (defaults to None)

• delta – the distance tolerance in the poloidal plane used to compute a resonant orbit. (defaults to 1e-2)

Returns

list of 7d arrays containing resonant particle orbits. The

elements of each array is [s0, theta0, zeta0, vpar0, t, mpol, ntor] if ma=None, and [R0, Z0, phi0, vpar0, t, mpol, ntor] otherwise. Here (s0, theta0, zeta0, vpar0)/(R0, Z0, phi0, vpar0) indicates the initial position and parallel velocity of the particle, t indicates the time of the resonance, mpol is the number of poloidal turns of the orbit, and ntor is the number of toroidal turns.

Return type

resonances

simsopt.field.compute_toroidal_transits(res_tys, flux=True)

Computes the number of toroidal transits of an orbit.

Parameters

• res_tys – trajectory solution computed from trace_particles() or trace_particles_boozer() with forget_exact_path=False.

• flux – if True, res_tys represents the position in flux coordinates (should be True if computed from trace_particles_boozer())

Returns

array with length len(res_tys). Each element contains the

number of toroidal transits of the orbit.

Return type

ntransits

simsopt.field.particles_to_vtk(res_tys, filename)

Export particle tracing or field lines to a vtk file. Expects that the xyz positions can be obtained by xyz[:, 1:4].

simsopt.field.plot_poincare_data(fieldlines_phi_hits, phis, filename, mark_lost=False, aspect='equal', dpi=300)

Create a poincare plot. Usage:

phis = np.linspace(0, 2*np.pi/nfp, nphis, endpoint=False)
res_tys, res_phi_hits = compute_fieldlines(
    bsh, R0, Z0, tmax=1000, phis=phis, stopping_criteria=[])
plot_poincare_data(res_phi_hits, phis, '/tmp/fieldlines.png')

Requires matplotlib to be installed.

simsopt.field.trace_particles(field: MagneticField, xyz_inits: Union[Sequence[Real], NDArray[Any, float64]], parallel_speeds: Union[Sequence[Real], NDArray[Any, float64]], tmax=0.0001, mass=6.69509884346e-27, charge=3.204353268e-19, Ekin=5.639661751679999e-13, tol=1e-09, comm=None, phis=[], stopping_criteria=[], mode='gc_vac', forget_exact_path=False, phase_angle=0)

Follow particles in a magnetic field.

In the case of mod='full' we solve

\[[\ddot x, \ddot y, \ddot z] = \frac{q}{m} [\dot x, \dot y, \dot z] \times B\]

in the case of mod='gc_vac' we solve the guiding center equations under the assumption \(\nabla p=0\), that is

\[\begin{split}[\dot x, \dot y, \dot z] &= v_{||}\frac{B}{|B|} + \frac{m}{q|B|^3} (0.5v_\perp^2 + v_{||}^2) B\times \nabla(|B|)\\ \dot v_{||} &= -\mu (B \cdot \nabla(|B|))\end{split}\]

where \(v_\perp = 2\mu|B|\). See equations (12) and (13) of [Guiding Center Motion, H.J. de Blank, https://doi.org/10.13182/FST04-A468].

Parameters

• field – The magnetic field \(B\).

• xyz_inits – A (nparticles, 3) array with the initial positions of the particles.

• parallel_speeds – A (nparticles, ) array containing the speed in direction of the B field for each particle.

• tmax – integration time

• mass – particle mass in kg, defaults to the mass of an alpha particle

• charge – charge in Coulomb, defaults to the charge of an alpha particle

• Ekin – kinetic energy in Joule, defaults to 3.52MeV

• tol – tolerance for the adaptive ode solver

• comm – MPI communicator to parallelize over

• phis – list of angles in [0, 2pi] for which intersection with the plane corresponding to that phi should be computed

• stopping_criteria – list of stopping criteria, mostly used in combination with the LevelsetStoppingCriterion accessed via simsopt.field.tracing.SurfaceClassifier.

• mode – how to trace the particles. options are gc: general guiding center equations, gc_vac: simplified guiding center equations for the case \(\nabla p=0\), full: full orbit calculation (slow!)

• forget_exact_path – return only the first and last position of each particle for the res_tys. To be used when only res_phi_hits is of interest or one wants to reduce memory usage.

• phase_angle – the phase angle to use in the case of full orbit calculations

Returns: 2 element tuple containing

• res_tys:

  A list of numpy arrays (one for each particle) describing the solution over time. The numpy array is of shape (ntimesteps, M) with M depending on the mode. Each row contains the time and the state. So for mode=’gc’ and mode=’gc_vac’ the state consists of the xyz position and the parallel speed, hence each row contains [t, x, y, z, v_par]. For mode=’full’, the state consists of position and velocity vector, i.e. each row contains [t, x, y, z, vx, vy, vz].

• res_phi_hits:

  A list of numpy arrays (one for each particle) containing information on each time the particle hits one of the phi planes or one of the stopping criteria. Each row of the array contains [time] + [idx] + state, where idx tells us which of the phis or stopping_criteria was hit. If idx>=0, then phis[int(idx)] was hit. If idx<0, then stopping_criteria[int(-idx)-1] was hit.

simsopt.field.trace_particles_boozer(field: BoozerMagneticField, stz_inits: Union[Sequence[Real], NDArray[Any, float64]], parallel_speeds: Union[Sequence[Real], NDArray[Any, float64]], tmax=0.0001, mass=6.69509884346e-27, charge=3.204353268e-19, Ekin=5.639661751679999e-13, tol=1e-09, comm=None, zetas=[], stopping_criteria=[], mode='gc_vac', forget_exact_path=False)

Follow particles in a BoozerMagneticField. This is modeled after trace_particles().

In the case of mod='gc_vac' we solve the guiding center equations under the vacuum assumption, i.e \(G =\) const. and \(I = 0\):

\[ \begin{align}\begin{aligned}\dot s = -|B|_{,\theta} m(v_{||}^2/|B| + \mu)/(q \psi_0)\\\dot \theta = |B|_{,s} m(v_{||}^2/|B| + \mu)/(q \psi_0) + \iota v_{||} |B|/G\\\dot \zeta = v_{||}|B|/G\\\dot v_{||} = -(\iota |B|_{,\theta} + |B|_{,\zeta})\mu |B|/G,\end{aligned}\end{align} \]

where \(q\) is the charge, \(m\) is the mass, and \(v_\perp^2 = 2\mu|B|\).

In the case of mode='gc' we solve the general guiding center equations for an MHD equilibrium:

\[ \begin{align}\begin{aligned}\dot s = (I |B|_{,\zeta} - G |B|_{,\theta})m(v_{||}^2/|B| + \mu)/(\iota D \psi_0)\\\dot \theta = ((G |B|_{,\psi} - K |B|_{,\zeta}) m(v_{||}^2/|B| + \mu) - C v_{||} |B|)/(\iota D)\\\dot \zeta = (F v_{||} |B| - (|B|_{,\psi} I - |B|_{,\theta} K) m(\rho_{||}^2 |B| + \mu) )/(\iota D)\\\dot v_{||} = (C|B|_{,\theta} - F|B|_{,\zeta})\mu |B|/(\iota D)\\C = - m v_{||} K_{,\zeta}/|B| - q \iota + m v_{||}G'/|B|\\F = - m v_{||} K_{,\theta}/|B| + q + m v_{||}I'/|B|\\D = (F G - C I))/\iota\end{aligned}\end{align} \]

where primes indicate differentiation wrt \(\psi\). In the case mod='gc_noK', the above equations are used with \(K=0\).

Parameters

• field – The BoozerMagneticField instance

• stz_inits – A (nparticles, 3) array with the initial positions of the particles in Boozer coordinates \((s,\theta,\zeta)\).

• parallel_speeds – A (nparticles, ) array containing the speed in direction of the B field for each particle.

• tmax – integration time

• mass – particle mass in kg, defaults to the mass of an alpha particle

• charge – charge in Coulomb, defaults to the charge of an alpha particle

• Ekin – kinetic energy in Joule, defaults to 3.52MeV

• tol – tolerance for the adaptive ode solver

• comm – MPI communicator to parallelize over

• zetas – list of angles in [0, 2pi] for which intersection with the plane corresponding to that zeta should be computed

• stopping_criteria – list of stopping criteria, mostly used in combination with the LevelsetStoppingCriterion accessed via simsopt.field.tracing.SurfaceClassifier.

• mode – how to trace the particles. Options are gc: general guiding center equations. gc_vac: simplified guiding center equations for the case \(G\) = const., \(I = 0\), and \(K = 0\). gc_noK: simplified guiding center equations for the case \(K = 0\).

• forget_exact_path – return only the first and last position of each particle for the res_tys. To be used when only res_zeta_hits is of interest or one wants to reduce memory usage.

Returns: 2 element tuple containing

• res_tys:

  A list of numpy arrays (one for each particle) describing the solution over time. The numpy array is of shape (ntimesteps, M) with M depending on the mode. Each row contains the time and the state. So for mode=’gc’ and mode=’gc_vac’ the state consists of the \((s,\theta,\zeta)\) position and the parallel speed, hence each row contains [t, s, t, z, v_par].

• res_zeta_hits:

  A list of numpy arrays (one for each particle) containing information on each time the particle hits one of the zeta planes or one of the stopping criteria. Each row of the array contains [time] + [idx] + state, where idx tells us which of the zetas or stopping_criteria was hit. If idx>=0, then zetas[int(idx)] was hit. If idx<0, then stopping_criteria[int(-idx)-1] was hit.

simsopt.field.trace_particles_starting_on_curve(curve, field, nparticles, tmax=0.0001, mass=6.69509884346e-27, charge=3.204353268e-19, Ekin=5.639661751679999e-13, tol=1e-09, comm=None, seed=1, umin=- 1, umax=1, phis=[], stopping_criteria=[], mode='gc_vac', forget_exact_path=False, phase_angle=0)

Follows particles spawned at random locations on the magnetic axis with random pitch angle. See simsopt.field.tracing.trace_particles for the governing equations.

Parameters

• curve – The simsopt.geo.curve.Curve to spawn the particles on. Uses rejection sampling to sample points on the curve. Warning: assumes that the underlying quadrature points on the Curve are uniformly distributed.

• field – The magnetic field \(B\).

• nparticles – number of particles to follow.

• tmax – integration time

• mass – particle mass in kg, defaults to the mass of an alpha particle

• charge – charge in Coulomb, defaults to the charge of an alpha particle

• Ekin – kinetic energy in Joule, defaults to 3.52MeV

• tol – tolerance for the adaptive ode solver

• comm – MPI communicator to parallelize over

• seed – random seed

• umin – the parallel speed is defined as v_par = u * speed_total where u is drawn uniformly in [umin, umax]

• umax – see umin

• phis – list of angles in [0, 2pi] for which intersection with the plane corresponding to that phi should be computed

• stopping_criteria – list of stopping criteria, mostly used in combination with the LevelsetStoppingCriterion accessed via simsopt.field.tracing.SurfaceClassifier.

• mode – how to trace the particles. options are gc: general guiding center equations, gc_vac: simplified guiding center equations for the case \(\nabla p=0\), full: full orbit calculation (slow!)

• forget_exact_path – return only the first and last position of each particle for the res_tys. To be used when only res_phi_hits is of interest or one wants to reduce memory usage.

• phase_angle – the phase angle to use in the case of full orbit calculations

Returns: see simsopt.field.tracing.trace_particles

simsopt.field.trace_particles_starting_on_surface(surface, field, nparticles, tmax=0.0001, mass=6.69509884346e-27, charge=3.204353268e-19, Ekin=5.639661751679999e-13, tol=1e-09, comm=None, seed=1, umin=- 1, umax=1, phis=[], stopping_criteria=[], mode='gc_vac', forget_exact_path=False, phase_angle=0)

Follows particles spawned at random locations on the magnetic axis with random pitch angle. See simsopt.field.tracing.trace_particles for the governing equations.

Parameters

• surface – The simsopt.geo.surface.Surface to spawn the particles on. Uses rejection sampling to sample points on the curve. Warning: assumes that the underlying quadrature points on the Curve as uniformly distributed.

• field – The magnetic field \(B\).

• nparticles – number of particles to follow.

• tmax – integration time

• mass – particle mass in kg, defaults to the mass of an alpha particle

• charge – charge in Coulomb, defaults to the charge of an alpha particle

• Ekin – kinetic energy in Joule, defaults to 3.52MeV

• tol – tolerance for the adaptive ode solver

• comm – MPI communicator to parallelize over

• seed – random seed

• umin – the parallel speed is defined as v_par = u * speed_total where u is drawn uniformly in [umin, umax].

• umax – see umin

• phis – list of angles in [0, 2pi] for which intersection with the plane corresponding to that phi should be computed

• stopping_criteria – list of stopping criteria, mostly used in combination with the LevelsetStoppingCriterion accessed via simsopt.field.tracing.SurfaceClassifier.

• mode – how to trace the particles. options are gc: general guiding center equations, gc_vac: simplified guiding center equations for the case \(\nabla p=0\), full: full orbit calculation (slow!)

• forget_exact_path – return only the first and last position of each particle for the res_tys. To be used when only res_phi_hits is of interest or one wants to reduce memory usage.

• phase_angle – the phase angle to use in the case of full orbit calculations

Returns: see simsopt.field.tracing.trace_particles