User Reference¶
This page documents the eight classes, and its members, corresponding to the eight analytical solutions provided in the access python package.
Assess is a python package that implements analytical solutions to the Stokes equations in cylindrical and spherical domains.
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class
assess.
CylindricalStokesSolutionSmoothFreeSlip
(n, k, Rp=2.22, Rm=1.22, nu=1.0, g=1.0)¶ Bases:
assess.cylindrical.CylindricalStokesSolutionSmooth
Analytical Solution in cylindrical domain with smooth r^k forcing and free-slip boundary conditions
Parameters: - n – wave number
- k – polynomial order of forcing
- Rp – outer radius
- Rm – inner radius
- nu – viscosity
- g – forcing strength
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delta_rho
(r, phi)¶ Perturbation density \(\rho'\) in forcing term: \(g\rho'\hat r\)
Parameters: - r – radius
- phi – angle with x-axis
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delta_rho_cartesian
(X)¶ Perturbation density \(\rho'\) in forcing term: \(g\rho'\hat r\)
Parameters: X – 2D Cartesian coordinate
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dpsi_rdr
(r)¶ Radial derivative of radial part of streamfunction
Parameters: r – radius
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dpsi_rdr2
(r)¶ Second radial derivative of radial part of streamfunction
Parameters: r – radius
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p
(r, phi)¶ Pressure solution
Parameters: - r – radius
- phi – angle with x-axis
-
pressure_cartesian
(X)¶ Return pressure solution at Cartesian location.
Parameters: X – 2D Cartesian location
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psi_r
(r)¶ Radial part of streamfunction
Parameters: r – radius
-
radial_stress
(r, phi)¶ Return radial component of stress.
Parameters: - r – radius
- phi – angle with x-axis.
-
radial_stress_cartesian
(X)¶ Return radial component of stress at Cartesian location.
Parameters: X – 2D Cartesian location
-
tau_rphi
(r, phi)¶ Return shear stress \(\tau_{r\varphi}\).
Parameters: - r – radius
- phi – angle with x-axis.
-
tau_rr
(r, phi)¶ Return radial component of deviatoric stress.
Parameters: - r – radius
- phi – angle with x-axis.
-
u_phi
(r, phi)¶ Return tangential component of velocity.
Parameters: - r – radius
- phi – angle with x-axis.
-
u_r
(r, phi)¶ Return radial component of velocity.
Parameters: - r – radius
- phi – angle with x-axis.
-
velocity_cartesian
(X)¶ Return Cartesian velocity at Cartesian location.
Parameters: X – 2D Cartesian location
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class
assess.
CylindricalStokesSolutionSmoothZeroSlip
(n, k, Rp=2.22, Rm=1.22, nu=1.0, g=1.0)¶ Bases:
assess.cylindrical.CylindricalStokesSolutionSmooth
Analytical Solution in cylindrical domain with smooth r^k forcing and zero-slip boundary conditions
Parameters: - n – wave number
- k – polynomial order of forcing
- Rp – outer radius
- Rm – inner radius
- nu – viscosity
- g – forcing strength
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delta_rho
(r, phi)¶ Perturbation density \(\rho'\) in forcing term: \(g\rho'\hat r\)
Parameters: - r – radius
- phi – angle with x-axis
-
delta_rho_cartesian
(X)¶ Perturbation density \(\rho'\) in forcing term: \(g\rho'\hat r\)
Parameters: X – 2D Cartesian coordinate
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dpsi_rdr
(r)¶ Radial derivative of radial part of streamfunction
Parameters: r – radius
-
dpsi_rdr2
(r)¶ Second radial derivative of radial part of streamfunction
Parameters: r – radius
-
p
(r, phi)¶ Pressure solution
Parameters: - r – radius
- phi – angle with x-axis
-
pressure_cartesian
(X)¶ Return pressure solution at Cartesian location.
Parameters: X – 2D Cartesian location
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psi_r
(r)¶ Radial part of streamfunction
Parameters: r – radius
-
radial_stress
(r, phi)¶ Return radial component of stress.
Parameters: - r – radius
- phi – angle with x-axis.
-
radial_stress_cartesian
(X)¶ Return radial component of stress at Cartesian location.
Parameters: X – 2D Cartesian location
-
tau_rphi
(r, phi)¶ Return shear stress \(\tau_{r\varphi}\).
Parameters: - r – radius
- phi – angle with x-axis.
-
tau_rr
(r, phi)¶ Return radial component of deviatoric stress.
Parameters: - r – radius
- phi – angle with x-axis.
-
u_phi
(r, phi)¶ Return tangential component of velocity.
Parameters: - r – radius
- phi – angle with x-axis.
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u_r
(r, phi)¶ Return radial component of velocity.
Parameters: - r – radius
- phi – angle with x-axis.
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velocity_cartesian
(X)¶ Return Cartesian velocity at Cartesian location.
Parameters: X – 2D Cartesian location
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class
assess.
CylindricalStokesSolutionDeltaFreeSlip
(n, sign, Rp=2.22, Rm=1.22, rp=1.72, nu=1.0, g=1.0)¶ Bases:
assess.cylindrical.CylindricalStokesSolutionDelta
Analytical Solution in cylindrical domain with delta(r-r’) forcing and free-slip boundary conditions
Parameters: - n – wave number
- sign – +1 for upper half solution r’<r<Rp -1 for lower half solution Rm<r<r’
- Rp – outer radius
- Rm – inner radius
- nu – viscosity
- g – forcing strength
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dpsi_rdr
(r)¶ Radial derivative of radial part of streamfunction
Parameters: r – radius
-
dpsi_rdr2
(r)¶ Second radial derivative of radial part of streamfunction
Parameters: r – radius
-
p
(r, phi)¶ Pressure solution
Parameters: - r – radius
- phi – angle with x-axis
-
pressure_cartesian
(X)¶ Return pressure solution at Cartesian location.
Parameters: X – 2D Cartesian location
-
psi_r
(r)¶ Radial part of streamfunction
Parameters: r – radius
-
radial_stress
(r, phi)¶ Return radial component of stress.
Parameters: - r – radius
- phi – angle with x-axis.
-
radial_stress_cartesian
(X)¶ Return radial component of stress at Cartesian location.
Parameters: X – 2D Cartesian location
-
tau_rphi
(r, phi)¶ Return shear stress \(\tau_{r\varphi}\).
Parameters: - r – radius
- phi – angle with x-axis.
-
tau_rr
(r, phi)¶ Return radial component of deviatoric stress.
Parameters: - r – radius
- phi – angle with x-axis.
-
u_phi
(r, phi)¶ Return tangential component of velocity.
Parameters: - r – radius
- phi – angle with x-axis.
-
u_r
(r, phi)¶ Return radial component of velocity.
Parameters: - r – radius
- phi – angle with x-axis.
-
velocity_cartesian
(X)¶ Return Cartesian velocity at Cartesian location.
Parameters: X – 2D Cartesian location
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class
assess.
CylindricalStokesSolutionDeltaZeroSlip
(n, sign, Rp=2.22, Rm=1.22, rp=1.72, nu=1.0, g=1.0)¶ Bases:
assess.cylindrical.CylindricalStokesSolutionDelta
Analytical Solution in cylindrical domain with delta(r-r’) forcing and zero-slip boundary conditions
Parameters: - n – wave number
- sign – +1 for upper half solution r’<r<Rp -1 for lower half solution Rm<r<r’
- Rp – outer radius
- Rm – inner radius
- nu – viscosity
- g – forcing strength
-
dpsi_rdr
(r)¶ Radial derivative of radial part of streamfunction
Parameters: r – radius
-
dpsi_rdr2
(r)¶ Second radial derivative of radial part of streamfunction
Parameters: r – radius
-
p
(r, phi)¶ Pressure solution
Parameters: - r – radius
- phi – angle with x-axis
-
pressure_cartesian
(X)¶ Return pressure solution at Cartesian location.
Parameters: X – 2D Cartesian location
-
psi_r
(r)¶ Radial part of streamfunction
Parameters: r – radius
-
radial_stress
(r, phi)¶ Return radial component of stress.
Parameters: - r – radius
- phi – angle with x-axis.
-
radial_stress_cartesian
(X)¶ Return radial component of stress at Cartesian location.
Parameters: X – 2D Cartesian location
-
tau_rphi
(r, phi)¶ Return shear stress \(\tau_{r\varphi}\).
Parameters: - r – radius
- phi – angle with x-axis.
-
tau_rr
(r, phi)¶ Return radial component of deviatoric stress.
Parameters: - r – radius
- phi – angle with x-axis.
-
u_phi
(r, phi)¶ Return tangential component of velocity.
Parameters: - r – radius
- phi – angle with x-axis.
-
u_r
(r, phi)¶ Return radial component of velocity.
Parameters: - r – radius
- phi – angle with x-axis.
-
velocity_cartesian
(X)¶ Return Cartesian velocity at Cartesian location.
Parameters: X – 2D Cartesian location
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class
assess.
SphericalStokesSolutionSmoothFreeSlip
(l, m, k, Rp=2.22, Rm=1.22, nu=1.0, g=1.0)¶ Bases:
assess.spherical.SphericalStokesSolutionSmooth
Analytical Solution in cylindrical domain with smooth r^k forcing and free-slip boundary conditions
Parameters: - l – degree of the harmonic
- m – order of the harmonic
- k – polynomial order of forcing
- Rp – outer radius
- Rm – inner radius
- nu – viscosity
- g – forcing strength
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Pl
(r)¶ Radial part of poloidal function
Parameters: r – radius
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dPldr
(r)¶ Radial derivative of radial part of poloidal function
Parameters: r – radius
-
dPldr2
(r)¶ Second radial derivative of radial part of poloidal function
Parameters: r – radius
-
delta_rho
(r, theta, phi)¶ Perturbation density \(\rho'\) in forcing term: \(g\rho'\hat r\)
Parameters: - r – radius
- theta – co-latitude in [0, pi]
- phi – longitude in [0, 2*pi]
-
delta_rho_cartesian
(X)¶ Perturbation density \(\rho'\) in forcing term: \(g\rho'\hat r\)
Parameters: X – 3D Cartesian coordinate
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p
(r, theta, phi)¶ Pressure solution
Parameters: - r – radius
- theta – co-latitude in [0, pi]
- phi – longitude in [0, 2*pi]
-
pressure_cartesian
(X)¶ Return pressure solution at Cartesian location.
Parameters: X – 3D Cartesian location
-
radial_stress
(r, theta, phi)¶ Return radial component of stress.
Parameters: - r – radius
- theta – co-latitude in [0, pi]
- phi – longitude in [0, 2*pi]
-
radial_stress_cartesian
(X)¶ Return radial component of stress at Cartesian location.
Parameters: X – 3D Cartesian location
-
tau_rphi
(r, theta, phi)¶ Return longitudinal component of shear stress.
Parameters: - r – radius
- theta – co-latitude in [0, pi]
- phi – longitude in [0, 2*pi]
-
tau_rr
(r, theta, phi)¶ Return radial component of deviatoric stress.
Parameters: - r – radius
- theta – co-latitude in [0, pi]
- phi – longitude in [0, 2*pi]
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tau_rtheta
(r, theta, phi)¶ Return colatitudinal component of shear stress.
Parameters: - r – radius
- theta – co-latitude in [0, pi]
- phi – longitude in [0, 2*pi]
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u_phi
(r, theta, phi)¶ Return longitudinal (eastward) component of velocity.
Parameters: - r – radius
- theta – co-latitude in [0, pi]
- phi – longitude in [0, 2*pi]
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u_r
(r, theta, phi)¶ Return radial component of velocity.
Parameters: - r – radius
- theta – co-latitude in [0, pi]
- phi – longitude in [0, 2*pi]
-
u_theta
(r, theta, phi)¶ Return colatitudinal (southward) component of velocity.
Parameters: - r – radius
- theta – co-latitude in [0, pi]
- phi – longitude in [0, 2*pi]
-
velocity_cartesian
(X)¶ Return Cartesian velocity at Cartesian location.
Parameters: X – 3D Cartesian location
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class
assess.
SphericalStokesSolutionSmoothZeroSlip
(l, m, k, Rp=2.22, Rm=1.22, nu=1.0, g=1.0)¶ Bases:
assess.spherical.SphericalStokesSolutionSmooth
Analytical Solution in cylindrical domain with smooth r^k forcing and zero-slip boundary conditions
Parameters: - l – degree of the harmonic
- m – order of the harmonic
- k – polynomial order of forcing
- Rp – outer radius
- Rm – inner radius
- nu – viscosity
- g – forcing strength
-
Pl
(r)¶ Radial part of poloidal function
Parameters: r – radius
-
dPldr
(r)¶ Radial derivative of radial part of poloidal function
Parameters: r – radius
-
dPldr2
(r)¶ Second radial derivative of radial part of poloidal function
Parameters: r – radius
-
delta_rho
(r, theta, phi)¶ Perturbation density \(\rho'\) in forcing term: \(g\rho'\hat r\)
Parameters: - r – radius
- theta – co-latitude in [0, pi]
- phi – longitude in [0, 2*pi]
-
delta_rho_cartesian
(X)¶ Perturbation density \(\rho'\) in forcing term: \(g\rho'\hat r\)
Parameters: X – 3D Cartesian coordinate
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p
(r, theta, phi)¶ Pressure solution
Parameters: - r – radius
- theta – co-latitude in [0, pi]
- phi – longitude in [0, 2*pi]
-
pressure_cartesian
(X)¶ Return pressure solution at Cartesian location.
Parameters: X – 3D Cartesian location
-
radial_stress
(r, theta, phi)¶ Return radial component of stress.
Parameters: - r – radius
- theta – co-latitude in [0, pi]
- phi – longitude in [0, 2*pi]
-
radial_stress_cartesian
(X)¶ Return radial component of stress at Cartesian location.
Parameters: X – 3D Cartesian location
-
tau_rphi
(r, theta, phi)¶ Return longitudinal component of shear stress.
Parameters: - r – radius
- theta – co-latitude in [0, pi]
- phi – longitude in [0, 2*pi]
-
tau_rr
(r, theta, phi)¶ Return radial component of deviatoric stress.
Parameters: - r – radius
- theta – co-latitude in [0, pi]
- phi – longitude in [0, 2*pi]
-
tau_rtheta
(r, theta, phi)¶ Return colatitudinal component of shear stress.
Parameters: - r – radius
- theta – co-latitude in [0, pi]
- phi – longitude in [0, 2*pi]
-
u_phi
(r, theta, phi)¶ Return longitudinal (eastward) component of velocity.
Parameters: - r – radius
- theta – co-latitude in [0, pi]
- phi – longitude in [0, 2*pi]
-
u_r
(r, theta, phi)¶ Return radial component of velocity.
Parameters: - r – radius
- theta – co-latitude in [0, pi]
- phi – longitude in [0, 2*pi]
-
u_theta
(r, theta, phi)¶ Return colatitudinal (southward) component of velocity.
Parameters: - r – radius
- theta – co-latitude in [0, pi]
- phi – longitude in [0, 2*pi]
-
velocity_cartesian
(X)¶ Return Cartesian velocity at Cartesian location.
Parameters: X – 3D Cartesian location
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class
assess.
SphericalStokesSolutionDeltaFreeSlip
(l, m, sign, Rp=2.22, Rm=1.22, rp=1.72, nu=1.0, g=1.0)¶ Bases:
assess.spherical.SphericalStokesSolutionDelta
Analytical Solution in cylindrical domain with delta(r-r’) forcing and free-slip boundary conditions
Parameters: - l – degree of the harmonic
- m – order of the harmonic
- sign – +1 for upper half solution r’<r<Rp -1 for lower half solution Rm<r<r’
- Rp – outer radius
- Rm – inner radius
- nu – viscosity
- g – forcing strength
-
Pl
(r)¶ Radial part of poloidal function
Parameters: r – radius
-
dPldr
(r)¶ Radial derivative of radial part of poloidal function
Parameters: r – radius
-
dPldr2
(r)¶ Second radial derivative of radial part of poloidal function
Parameters: r – radius
-
p
(r, theta, phi)¶ Pressure solution
Parameters: - r – radius
- theta – co-latitude in [0, pi]
- phi – longitude in [0, 2*pi]
-
pressure_cartesian
(X)¶ Return pressure solution at Cartesian location.
Parameters: X – 3D Cartesian location
-
radial_stress
(r, theta, phi)¶ Return radial component of stress.
Parameters: - r – radius
- theta – co-latitude in [0, pi]
- phi – longitude in [0, 2*pi]
-
radial_stress_cartesian
(X)¶ Return radial component of stress at Cartesian location.
Parameters: X – 3D Cartesian location
-
tau_rphi
(r, theta, phi)¶ Return longitudinal component of shear stress.
Parameters: - r – radius
- theta – co-latitude in [0, pi]
- phi – longitude in [0, 2*pi]
-
tau_rr
(r, theta, phi)¶ Return radial component of deviatoric stress.
Parameters: - r – radius
- theta – co-latitude in [0, pi]
- phi – longitude in [0, 2*pi]
-
tau_rtheta
(r, theta, phi)¶ Return colatitudinal component of shear stress.
Parameters: - r – radius
- theta – co-latitude in [0, pi]
- phi – longitude in [0, 2*pi]
-
u_phi
(r, theta, phi)¶ Return longitudinal (eastward) component of velocity.
Parameters: - r – radius
- theta – co-latitude in [0, pi]
- phi – longitude in [0, 2*pi]
-
u_r
(r, theta, phi)¶ Return radial component of velocity.
Parameters: - r – radius
- theta – co-latitude in [0, pi]
- phi – longitude in [0, 2*pi]
-
u_theta
(r, theta, phi)¶ Return colatitudinal (southward) component of velocity.
Parameters: - r – radius
- theta – co-latitude in [0, pi]
- phi – longitude in [0, 2*pi]
-
velocity_cartesian
(X)¶ Return Cartesian velocity at Cartesian location.
Parameters: X – 3D Cartesian location
-
class
assess.
SphericalStokesSolutionDeltaZeroSlip
(l, m, sign, Rp=2.22, Rm=1.22, rp=1.72, nu=1.0, g=1.0)¶ Bases:
assess.spherical.SphericalStokesSolutionDelta
Analytical Solution in cylindrical domain with delta(r-r’) forcing and zero-slip boundary conditions
Parameters: - l – degree of the harmonic
- m – order of the harmonic
- sign – +1 for upper half solution r’<r<Rp -1 for lower half solution Rm<r<r’
- Rp – outer radius
- Rm – inner radius
- nu – viscosity
- g – forcing strength
-
Pl
(r)¶ Radial part of poloidal function
Parameters: r – radius
-
dPldr
(r)¶ Radial derivative of radial part of poloidal function
Parameters: r – radius
-
dPldr2
(r)¶ Second radial derivative of radial part of poloidal function
Parameters: r – radius
-
p
(r, theta, phi)¶ Pressure solution
Parameters: - r – radius
- theta – co-latitude in [0, pi]
- phi – longitude in [0, 2*pi]
-
pressure_cartesian
(X)¶ Return pressure solution at Cartesian location.
Parameters: X – 3D Cartesian location
-
radial_stress
(r, theta, phi)¶ Return radial component of stress.
Parameters: - r – radius
- theta – co-latitude in [0, pi]
- phi – longitude in [0, 2*pi]
-
radial_stress_cartesian
(X)¶ Return radial component of stress at Cartesian location.
Parameters: X – 3D Cartesian location
-
tau_rphi
(r, theta, phi)¶ Return longitudinal component of shear stress.
Parameters: - r – radius
- theta – co-latitude in [0, pi]
- phi – longitude in [0, 2*pi]
-
tau_rr
(r, theta, phi)¶ Return radial component of deviatoric stress.
Parameters: - r – radius
- theta – co-latitude in [0, pi]
- phi – longitude in [0, 2*pi]
-
tau_rtheta
(r, theta, phi)¶ Return colatitudinal component of shear stress.
Parameters: - r – radius
- theta – co-latitude in [0, pi]
- phi – longitude in [0, 2*pi]
-
u_phi
(r, theta, phi)¶ Return longitudinal (eastward) component of velocity.
Parameters: - r – radius
- theta – co-latitude in [0, pi]
- phi – longitude in [0, 2*pi]
-
u_r
(r, theta, phi)¶ Return radial component of velocity.
Parameters: - r – radius
- theta – co-latitude in [0, pi]
- phi – longitude in [0, 2*pi]
-
u_theta
(r, theta, phi)¶ Return colatitudinal (southward) component of velocity.
Parameters: - r – radius
- theta – co-latitude in [0, pi]
- phi – longitude in [0, 2*pi]
-
velocity_cartesian
(X)¶ Return Cartesian velocity at Cartesian location.
Parameters: X – 3D Cartesian location
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assess.
Y
(l, m, theta, phi)¶ Real-valued spherical harmonic function \(Y_{lm}(\theta, \varphi)\)
This is based on the following definition:
\[Y_{lm}(\theta, \varphi) = \sqrt{\frac{(2l+1)}{4\pi}\frac{(l-m)!}{(l+m)!}}P_l^m(\cos(\theta))\cos(m\varphi)\]which is equal to the real part of scipy.special.sph_harm.
Parameters: - l – degree of the harmonic
- m – order of the harmonic
- theta – co-latitude in [0, pi]
- phi – longitude in [0, 2*pi]
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assess.
dYdphi
(l, m, theta, phi)¶ Colatitudinal derivative of spherical harmonic function \(Y_{lm}(\theta, \varphi)\)
Parameters: - l – degree of the harmonic
- m – order of the harmonic
- theta – co-latitude in [0, pi]
- phi – longitude in [0, 2*pi]
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assess.
dYdtheta
(l, m, theta, phi)¶ Longitudinal derivative of spherical harmonic function \(Y_{lm}(\theta, \varphi)\)
Parameters: - l – degree of the harmonic
- m – order of the harmonic
- theta – co-latitude in [0, pi]
- phi – longitude in [0, 2*pi]
-
assess.
to_spherical
(X)¶ Convert Cartesian 3D coordinates X to spherical r, theta, phi
Parameters: X – Cartesian 3D coordinates Returns r, theta, phi: radius, colatitude, longitude
-
assess.
from_spherical
(r, theta, phi)¶ Convert spherical r, theta, phi to 3D Cartesian coordinates.
Parameters: - r, – radius
- theta – co-latitude in [0, pi]
- phi – longitude in [0, 2*pi]
Returns X: Cartesian 3D coordinates