coordinate_transformation.py¶

Compute the coordinate transformation related variables for a mapping,

$\Phi : \Omega_{\mathrm{ref}}\to\Omega$

We get an instance of class CoordinateTransformation. The transformations are its methods.

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class coordinate_transformation.CoordinateTransformation(Phi, d_Phi)[source]¶

The coordinate transformation class.

Parameters:

Phi (function) –
The mapping $\Phi$ . It should be a function which returns three components of the mapping, i.e.,

$\Phi = (\Phi_x, \Phi_y, \Phi_z).$
d_Phi (function) –
The Jacobian matrix of $\Phi$ , $\mathcal{J}$ . It should be a function return the 9 components of the Jacobian matrix, i.e.,

( $\dfrac{\partial x}{\partial\xi}$ , $\dfrac{\partial x}{\partial\eta}$ , $\dfrac{\partial x}{\partial\varsigma}$ ), ( $\dfrac{\partial y}{\partial\xi}$ , $\dfrac{\partial y}{\partial\eta}$ , $\dfrac{\partial y}{\partial\varsigma}$ ), ( $\dfrac{\partial z}{\partial\xi}$ , $\dfrac{\partial z}{\partial\eta}$ , $\dfrac{\partial z}{\partial\varsigma}$ ).

Example:

>>> ct = CoordinateTransformation(Phi, d_Phi) # generate a CT instance
>>> xi = np.linspace(-0.9, 0.6, 3)
>>> et = np.linspace(-0.7, 0.9, 3)
>>> sg = np.linspace(-0.8, 0.7, 3)
>>> xi, et, sg = np.meshgrid(xi, et, sg, indexing='ij')
>>> x, y, z = ct.mapping(xi, et, sg)
>>> x 
array([[[0.06469463, 0.05391086, 0.02977458],...
>>> y 
array([[[0.16469463, 0.15391086, 0.12977458],...
>>> z 
array([[[0.11469463, 0.47891086, 0.82977458],...
>>> J = ct.Jacobian_matrix(xi, et, sg)
>>> J[0][0] # J_{1,1} 
array([[[0.64207986, 0.53781345, 0.30444385],...
>>> J[1][1] # J_{2,2} 
array([[[0.53354051, 0.50892654, 0.45383545],...
>>> detJ = ct.Jacobian(xi, et, sg)
>>> detJ 
array([[[0.1847901 , 0.11729178, 0.07611096],...
>>> iJ = ct.inverse_Jacobian_matrix(xi, et, sg)
>>> iJ[2][2] # J^{-1}_{3,3} 
array([[[1.82807508, 2.33068327, 1.69672867],...
>>> iJ[1][2] # J^{-1}_{2,3} 
array([[[-0.17192492,  0.33068327, -0.30327133],...
>>> G = ct.metric_matrix(xi, et, sg)
>>> G[0][1] # g_{1,2} 
array([[[ 0.10210648,  0.02438263, -0.09377707],...
>>> G[1][2] # g_{2,3} 
array([[[ 0.05493377, -0.03640053, -0.0063935 ],...
>>> g = ct.metric(xi, et, sg)
>>> g 
array([[[0.03414738, 0.01375736, 0.00579288],...
>>> iG = ct.inverse_metric_matrix(xi, et, sg)
>>> iG[1][2] # g^{2,3} 
array([[[-0.33977063,  0.72204401,  1.83434454],...

Jacobian(xi, et, sg)[source]¶

Compute the Jacobian $\mathrm{det}(\mathcal{J})$ of the mapping.

Returns:: The Jacobian $\mathrm{det}(\mathcal{J})$ .

Jacobian_matrix(xi, et, sg)[source]¶

A wrap of the input Jacobian matrix, d_Phi, i.e., $\mathcal{J}$ .

Returns:

Return the Jacobian matrix $\mathcal{J}$ :

$\mathcal{J} = \begin{bmatrix} \dfrac{\partial x}{\partial \xi} & \dfrac{\partial x}{\partial \eta} & \dfrac{\partial x}{\partial \varsigma} \\ \dfrac{\partial y}{\partial \xi} & \dfrac{\partial y}{\partial \eta} & \dfrac{\partial y}{\partial \varsigma} \\ \dfrac{\partial z}{\partial \xi} & \dfrac{\partial z}{\partial \eta} & \dfrac{\partial z}{\partial \varsigma} \end{bmatrix}\,.$

inverse_Jacobian(xi, et, sg)[source]¶: Compute the inverse Jacobian, the determinant of the inverse Jacobian matrix, $\mathrm{det}(\mathcal{J}^{-1})$ .

inverse_Jacobian_matrix(xi, et, sg)[source]¶

Compute the Jacobian matrix of the inverse mapping $\Phi^{-1}$ ,

$(\xi,\eta,\varsigma) = \Phi^{-1}(x,y,z).$

Returns:

Return the inverse Jacobian matrix : $\mathcal{J}^{-1}$ :

$\mathcal{J}^{-1} = \begin{bmatrix} \dfrac{\partial \xi}{\partial x} & \dfrac{\partial \xi}{\partial y} & \dfrac{\partial \xi}{\partial z} \\ \dfrac{\partial \eta}{\partial x} & \dfrac{\partial \eta}{\partial y} & \dfrac{\partial \eta}{\partial z} \\ \dfrac{\partial \varsigma}{\partial x} & \dfrac{\partial \varsigma}{\partial y} & \dfrac{\partial \varsigma}{\partial z} \end{bmatrix}\,.$