isoparam

dg_maxwell.isoparam.isoparam_1D(x_nodes, xi)

Maps points in \(\xi\) space to \(x\) space using the formula \(x = \frac{1 - \xi}{2} x_0 + \frac{1 + \xi}{2} x_1\)

Parameters:

x_nodes : arrayfire.Array [2 1 1 1]

Element nodes.

xi : arrayfire.Array [N 1 1 1]

Value of \(\xi\) coordinate for which the corresponding \(x\) coordinate is to be found.

Returns:

x : arrayfire.Array

\(x\) value in the element corresponding to \(\xi\).

dg_maxwell.isoparam.isoparam_x_2D(x_nodes, xi, eta)

Finds the \(x\) coordinate using isoparametric mapping of a \(2^{nd}\) order element with \(8\) nodes

\[(P_0, P_1, P_2, P_3, P_4, P_5, P_6, P_7)\]

Here \(P_i\) corresponds to \((\xi_i, \eta_i)\) coordinates, \(i \in \{0, 1, ..., 7\}\) respectively, where,

\[\begin{split}(\xi_0, \eta_0) &\equiv (-1, 1) \\ (\xi_1, \eta_1) &\equiv (-1, 0) \\ (\xi_2, \eta_2) &\equiv (-1, -1) \\ (\xi_3, \eta_3) &\equiv ( 0, -1) \\ (\xi_4, \eta_4) &\equiv ( 1, -1) \\ (\xi_5, \eta_5) &\equiv ( 1, 0) \\ (\xi_6, \eta_6) &\equiv ( 1, 1) \\ (\xi_7, \eta_7) &\equiv ( 0, 1)\end{split}\]

Parameters:

x_nodes : np.ndarray [8]

\(x\) nodes.

xi : float

\(\xi\) coordinate for which \(x\) has to be found.

eta : float

\(\eta\) coordinate for which \(x\) has to be found.

Returns:

x : float

\(x\) coordinate corresponding to \((\xi, \eta)\) coordinate.

dg_maxwell.isoparam.isoparam_y_2D(y_nodes, xi, eta)

This function allows isoparametric mapping of a \(2^{nd}\) order element with \(8\) nodes

\[(P_0, P_1, P_2, P_3, P_4, P_5, P_6, P_7)\]

Here \(P_i\) corresponds to \((\xi_i, \eta_i)\) coordinates, \(i \in \{0, 1, ..., 7\}\) respectively, where,

\[\begin{split}(\xi_0, \eta_0) &\equiv (-1, 1) \\ (\xi_1, \eta_1) &\equiv (-1, 0) \\ (\xi_2, \eta_2) &\equiv (-1, -1) \\ (\xi_3, \eta_3) &\equiv ( 0, -1) \\ (\xi_4, \eta_4) &\equiv ( 1, -1) \\ (\xi_5, \eta_5) &\equiv ( 1, 0) \\ (\xi_6, \eta_6) &\equiv ( 1, 1) \\ (\xi_7, \eta_7) &\equiv ( 0, 1)\end{split}\]

Parameters:

y_nodes : np.ndarray [8]

\(y\) nodes.

xi : float

\(\xi\) coordinate for which \(y\) has to be found.

eta : float

\(\eta\) coordinate for which \(y\) has to be found.

Returns:

float

\(y\) coordinate corresponding to \((\xi, \eta)\) coordinate.