Added alternate Jacobian matrix derivation for reference

PyNite/Quad3D.py

@@ -53,10 +53,10 @@ def __init__(self, name, i_node, j_node, m_node, n_node, t, material_name, model
             raise KeyError('Please define the material ' + str(material_name) + ' before assigning it to plates.')
 
     # def _local_coords(self):
-    #     '''
+    #     """
     #     Calculates or recalculates and stores the local (x, y) coordinates for each node of the
     #     quadrilateral.
-    #     '''
+    #     """
 
     #     # Get the global coordinates for each node
     #     X1, Y1, Z1 = self.i_node.X, self.i_node.Y, self.i_node.Z
@@ -98,10 +98,9 @@ def __init__(self, name, i_node, j_node, m_node, n_node, t, material_name, model
     #     self.y4 = np.dot(vector_04, y_axis)
 
     def _local_coords(self):
-        '''
-        Calculates or recalculates and stores the local (x, y) coordinates for each node of the
-        quadrilateral.
-        '''
+        """
+        Calculates or recalculates and stores the local (x, y) coordinates for each node of the quadrilateral.
+        """
 
         # Get the global coordinates for each node
         X1, Y1, Z1 = self.i_node.X, self.i_node.Y, self.i_node.Z
@@ -205,6 +204,62 @@ def P_k(self, k, xi, eta):
             return 1/2*(1 - xi)*(1 - eta**2)
         else:
             raise Exception('Unable to calculate shape function. Invalid value specified for k.')
+
+    def Co(self, xi, eta):
+        """This alternate calculation of the Jacobian matrix follows "The development of DKMQ plate bending element for thick to thin shell analysis based on the Naghdi/Reissner/Mindlin shell theory" by Katili, Batoz, Maknun and Hamdouni (2015). In the reference "C^o" is used instead of "J" to refer to the Jacobian. This method does not seem to produce correct results, but will be kept for future reference. It is helpful for understanding this plate element and may prove a useful simplification to the code base if implemented correctly someday.
+        """
+
+        # Nodal global coordinates (i=1, j=2, m=3, n=4)
+        x_1 = np.array([[self.i_node.X, self.i_node.Y, self.i_node.Z]])
+        x_2 = np.array([[self.j_node.X, self.j_node.Y, self.j_node.Z]])
+        x_3 = np.array([[self.m_node.X, self.m_node.Y, self.m_node.Z]])
+        x_4 = np.array([[self.n_node.X, self.n_node.Y, self.n_node.Z]])
+
+        # Derivatives of the bilinear interpolation functions
+        N1_xi = 0.25*(eta - 1)
+        N2_xi = -0.25*(eta - 1)
+        N3_xi = 0.25*(eta + 1)
+        N4_xi = -0.25*(eta + 1)
+        N1_eta = 0.25*(xi - 1)
+        N2_eta = -0.25*(xi + 1)
+        N3_eta = 0.25*(xi + 1)
+        N4_eta = -0.25*(xi - 1)
+
+        # Equation 4 - Katili 2015
+        a_1 = N1_xi*x_1 + N2_xi*x_2 + N3_xi*x_3 + N4_xi*x_4
+        a_2 = N1_eta*x_1 + N2_eta*x_2 + N3_eta*x_3 + N4_eta*x_4
+
+        # Normal vector
+        n = np.cross(a_1, a_2)/np.linalg.norm(np.cross(a_1, a_2))
+
+        # Global unit vectors
+        i = np.array([[1.0, 0.0, 0.0]])
+        k = np.array([[0.0, 0.0, 1.0]])
+
+        if np.array_equal(n, k) or np.array_equal(n, -k):
+            t_1 = i
+        else:
+            t_1 = np.cross(n, k)
+
+        t_2 = np.cross(n, t_1)
+
+        a_11 = np.dot(a_1, a_1.T)[0, 0]
+        a_12 = np.dot(a_1, a_2.T)[0, 0]
+        a_21 = np.dot(a_2, a_1.T)[0, 0]
+        a_22 = np.dot(a_2, a_2.T)[0, 0]
+
+        a = np.array([[a_11, a_12],
+                      [a_21, a_22]])
+
+        a_det = np.linalg.det(a)
+
+        a1 = 1/a_det*(a_22*a_1 - a_12*a_2)
+        a2 = 1/a_det*(-a_21*a_1 + a_11*a_2)
+
+        Co = np.array([[np.dot(a1, t_1.T)[0, 0], np.dot(a1, t_2.T)[0, 0]],
+                       [np.dot(a2, t_1.T)[0, 0], np.dot(a2, t_2.T)[0, 0]]])
+
+        return Co
 
     def J(self, xi, eta):
         """