Maths Directory
Only the integration and Funcs modules are used for the BES diagnostics.
The first one contains some useful quadrature formulas and the second one contains some function for the
computation of the solid angle.
Integration Module
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FPSDP.Maths.Integration.integration_points(dim, meth, obj='', size=-1)[source]
Defines a few quadrature formula (in any number of dimension)
| Parameters: |
- dim (int) – Dimension of the integration
- obj (str) – Type of domain of integration (for dim>1, e.g. ‘disk’ )
- meth (str) – Method of integration (e.g ‘GL4’ for Gauss-Legendre accuracy order 2)
- size – Object describing the geometry of the problem (e.g. radius for a disk)
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| Returns: | Points and weights of the quadrature formula
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| Return type: | Named tuple (.pts and .w)
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Funcs Module
simple math functions used for debugging and/or productive runs
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FPSDP.Maths.Funcs.heuman(phi, m)[source]
Compute the Heuman’s lambda function
\(\Lambda_0 (\xi,k) = \frac{2}{\pi}\left[E(k)F(\xi,k') + K(k)E(\xi,k')- K(k)F(\xi,k')\right]\)
where \(k' = \sqrt{(1-k^2)}\)
| Parameters: |
- phi (np.array[N]) – The amplitude of the elliptic integrals
- m (np.array[N]) – The parameter of the elliptic integrals
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| Returns: | Evaluation of the Heuman’s lambda function
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| Return type: | np.array[N]
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FPSDP.Maths.Funcs.solid_angle_disk(pos, r)[source]
Compute the solid angle of a disk on/off-axis from the pos
the center of the circle should be in (0,0,0)
\[\begin{split}\Omega = \left\{\begin{array}{lr}
2\pi-\frac{2L}{R_\text{max}}K(k)-\pi\Lambda_0(\xi,k) & r_0 < r_m \\
\phantom{2}\pi-\frac{2L}{R_\text{max}}K(k) & r_0 = r_m \\
\phantom{2\pi}-\frac{2L}{R_\text{max}}K(k)+\pi\Lambda_0(\xi,k) & r_0 > r_m \\
\end{array}\right.\end{split}\]
Read the paper of Paxton “Solid Angle Calculation for a
Circular Disk” in 1959 for the exact computation.
| Parameters: |
- pos (np.array[N,3]) – Position from which computing the solid angle
- r (float) – Radius of the disk (the disk is centered in (0,0,0) and the perpendicular is along the z-axis)
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| Returns: | Solid angle for each positions
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| Return type: | np.array[N]
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Runge-Kutta Module
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FPSDP.Maths.RungeKutta.runge_kutta_explicit(order, alpha=None)[source]
Coefficient of the explicit Runge-Kutta methods.
| Parameters: | order (int) – Order of the Runge-Kutta method |
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| Returns: | Coefficient of the Butcher table |
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| Return type: | tuple(a,b,c) |
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