FEM.Elements.E2D package

Collection of 2D Elements

Submodules

FEM.Elements.E2D.Element2D module

Defines a general 2D element

class FEM.Elements.E2D.Element2D.Element2D(coords: ndarray, _coords: ndarray, gdl: ndarray, **kargs)

Bases: Element

Create a 2D element

Parameters:

• coords (np.ndarray) – Element coordinate matrix

• _coords (np.ndarray) – Element coordinate matrix for graphical interface purposes

• gdl (np.ndarray) – Degree of freedom matrix

draw() → None

Create a graph of element

isInside(x: ndarray) → ndarray

Test if a given points is inside element domain

Parameters:

x (np.ndarray) – Point to be tested

Returns:

Bolean array of test result

Return type:

np.ndarray

jacobianGraph() → None

Create the determinant jacobian graph

FEM.Elements.E2D.LTriangular module

TRIANGULAR ELEMENT

Defines the lagrange first order triangular element

3 nodes triangular element. (Reddy, 2005)

Shape Functions

\[\begin{split}\Psi_i=\left[\begin{matrix}- x - y + 1.0\\x\\y\end{matrix}\right]\end{split}\]

Shape Functions Derivatives

\[\begin{split}\frac{\partial \Psi_i}{\partial x_j}=\left[\begin{matrix}-1 & -1\\1 & 0\\0 & 1\end{matrix}\right]\end{split}\]

class FEM.Elements.E2D.LTriangular.LTriangular(coords: ndarray, gdl: ndarray, n: int = 2, **kargs)

Bases: Element2D, TriangularScheme

Creates a lagrangian triangular element of order 1

Parameters:

• coords (np.ndarray) – Element coordinates matrix

• gdl (np.ndarray) – Element gdl matrix

• n (int, optional) – Number of Gauss Points. Defaults to 2.

dpsis(z: ndarray) → ndarray

Calculates the shape functions derivatives of a given natural coordinates

Parameters:

z (np.ndarray) – Natural coordinates matrix

Returns:

Shape function derivatives evaluated in Z points

Return type:

np.ndarray

psis(z: ndarray) → ndarray

Calculates the shape functions of a given natural coordinates

Parameters:

z (np.ndarray) – Natural coordinates matrix

Returns:

Shape function evaluated in Z points

Return type:

np.ndarray

FEM.Elements.E2D.QTriangular module

TRIANGULAR SECOND ORDER ELEMENT

Defines the lagrange second order triangular element

6 nodes triangular element. (Reddy, 2005)

Shape Functions

\[\begin{split}\Psi_i=\left[\begin{matrix}\left(x + y - 0.5\right) \left(2.0 x + 2.0 y - 2.0\right)\\2.0 x \left(x - 0.5\right)\\2.0 y \left(y - 0.5\right)\\x \left(- 4.0 x - 4.0 y + 4.0\right)\\4.0 x y\\- 4.0 y \left(x + y - 1.0\right)\end{matrix}\right]\end{split}\]

Shape Functions Derivatives

\[\begin{split}\frac{\partial \Psi_i}{\partial x_j}=\left[\begin{matrix}4.0 x + 4.0 y - 3.0 & 4.0 x + 4.0 y - 3.0\\4.0 x - 1.0 & 0\\0 & 4.0 y - 1.0\\- 8.0 x - 4.0 y + 4.0 & - 4.0 x\\4.0 y & 4.0 x\\- 4.0 y & - 4.0 x - 8.0 y + 4.0\end{matrix}\right]\end{split}\]

class FEM.Elements.E2D.QTriangular.QTriangular(coords: ndarray, gdl: ndarray, n: int = 3, **kargs)

Bases: Element2D, TriangularScheme

Creates a lagrangian element of order 2

Parameters:

• coords (np.ndarray) – Element coordinates matrix

• gdl (np.ndarray) – Element gdl matrix

• n (int, optional) – Number of Gauss Points. Defaults to 2.

dpsis(z: ndarray) → ndarray

Calculates the shape functions derivatives of a given natural coordinates

Parameters:

z (np.ndarray) – Natural coordinates matrix

Returns:

Shape function derivatives evaluated in Z points

Return type:

np.ndarray

psis(z: ndarray) → ndarray

Calculates the shape functions of a given natural coordinates

Parameters:

z (np.ndarray) – Natural coordinates matrix

Returns:

Shape function evaluated in Z points

Return type:

np.ndarray

FEM.Elements.E2D.Quadrilateral module

QUADRILATERAL ELEMENT

Defines the lagrange first order quadrilateral element

4 nodes quadrilateral element. (Reddy, 2005)

Shape Functions

\[\begin{split}\Psi_i=\left[\begin{matrix}\left(0.25 - 0.25 x\right) \left(1.0 - y\right)\\\left(1.0 - y\right) \left(0.25 x + 0.25\right)\\\left(0.25 x + 0.25\right) \left(y + 1.0\right)\\\left(0.25 - 0.25 x\right) \left(y + 1.0\right)\end{matrix}\right]\end{split}\]

Shape Functions Derivatives

\[\begin{split}\frac{\partial \Psi_i}{\partial x_j}=\left[\begin{matrix}0.25 y - 0.25 & 0.25 x - 0.25\\0.25 - 0.25 y & - 0.25 x - 0.25\\0.25 y + 0.25 & 0.25 x + 0.25\\- 0.25 y - 0.25 & 0.25 - 0.25 x\end{matrix}\right]\end{split}\]

class FEM.Elements.E2D.Quadrilateral.Quadrilateral(coords: ndarray, gdl: ndarray, n: int = 2, **kargs)

Bases: Element2D, RectangularScheme

Creates a lagrangian rectangular element of order 1

Parameters:

• coords (np.ndarray) – Element coordinates matrix

• gdl (np.ndarray) – Element gdl matrix

• n (int, optional) – Number of Gauss Points. Defaults to 2.

dpsis(z: ndarray) → ndarray

Calculates the shape functions derivatives of a given natural coordinates

Parameters:

z (np.ndarray) – Natural coordinates matrix

Returns:

Shape function derivatives evaluated in Z points

Return type:

np.ndarray

psis(z: ndarray) → ndarray

Calculates the shape functions of a given natural coordinates

Parameters:

z (np.ndarray) – Natural coordinates matrix

Returns:

Shape function evaluated in Z points

Return type:

np.ndarray

FEM.Elements.E2D.RectangularScheme module

Define the rectangular scheme used by rectangular elements

class FEM.Elements.E2D.RectangularScheme.RectangularScheme(n: int, **kargs)

Bases: object

Generates a rectangular integration scheme

Parameters:

n (int) – Number of gauss points

FEM.Elements.E2D.Serendipity module

QUADRILATERAL SERENDIPITY ELEMENT

Defines the serendipity second order quadrilateral element

8 nodes quadrilateral element. (Reddy, 2005)

Shape Functions

\[\begin{split}\Psi_i=\left[\begin{matrix}\left(0.25 - 0.25 x\right) \left(1.0 - y\right) \left(- x - y - 1.0\right)\\\left(1.0 - y\right) \left(0.25 x + 0.25\right) \left(x - y - 1.0\right)\\\left(0.25 x + 0.25\right) \left(y + 1.0\right) \left(x + y - 1.0\right)\\\left(0.25 - 0.25 x\right) \left(y + 1.0\right) \left(- x + y - 1.0\right)\\\left(0.5 - 0.5 x^{2.0}\right) \left(1.0 - y\right)\\\left(1.0 - y^{2.0}\right) \left(0.5 x + 0.5\right)\\\left(0.5 - 0.5 x^{2.0}\right) \left(y + 1.0\right)\\\left(0.5 - 0.5 x\right) \left(1.0 - y^{2.0}\right)\end{matrix}\right]\end{split}\]

Shape Functions Derivatives

\[\begin{split}\frac{\partial \Psi_i}{\partial x_j}=\left[\begin{matrix}- \left(0.25 - 0.25 x\right) \left(1.0 - y\right) + \left(0.25 y - 0.25\right) \left(- x - y - 1.0\right) & - \left(0.25 - 0.25 x\right) \left(1.0 - y\right) + \left(0.25 x - 0.25\right) \left(- x - y - 1.0\right)\\\left(0.25 - 0.25 y\right) \left(x - y - 1.0\right) + \left(1.0 - y\right) \left(0.25 x + 0.25\right) & - \left(1.0 - y\right) \left(0.25 x + 0.25\right) + \left(- 0.25 x - 0.25\right) \left(x - y - 1.0\right)\\\left(0.25 x + 0.25\right) \left(y + 1.0\right) + \left(0.25 y + 0.25\right) \left(x + y - 1.0\right) & \left(0.25 x + 0.25\right) \left(y + 1.0\right) + \left(0.25 x + 0.25\right) \left(x + y - 1.0\right)\\- \left(0.25 - 0.25 x\right) \left(y + 1.0\right) + \left(- 0.25 y - 0.25\right) \left(- x + y - 1.0\right) & \left(0.25 - 0.25 x\right) \left(y + 1.0\right) + \left(0.25 - 0.25 x\right) \left(- x + y - 1.0\right)\\- 1.0 x^{1.0} \left(1.0 - y\right) & 0.5 x^{2.0} - 0.5\\0.5 - 0.5 y^{2.0} & - 2.0 y^{1.0} \left(0.5 x + 0.5\right)\\- 1.0 x^{1.0} \left(y + 1.0\right) & 0.5 - 0.5 x^{2.0}\\0.5 y^{2.0} - 0.5 & - 2.0 y^{1.0} \left(0.5 - 0.5 x\right)\end{matrix}\right]\end{split}\]

class FEM.Elements.E2D.Serendipity.Serendipity(coords: ndarray, gdl: ndarray, n: int = 3, **kargs)

Bases: Element2D, RectangularScheme

Creates a Serendipity element

Parameters:

• coords (np.ndarray) – Coordinate matrix of element

• gdl (np.ndarray) – Coordinate matrix of element GDL’s

• n (int, optional) – Number of gauss points. Defaults to 3.

dpsis(z: ndarray) → ndarray

Calculates the shape functions derivatives of a given natural coordinates

Parameters:

z (np.ndarray) – Natural coordinates matrix

Returns:

Shape function derivatives evaluated in Z points

Return type:

np.ndarray

psis(z: ndarray) → ndarray

Calculates the shape functions of a given natural coordinates

Parameters:

z (np.ndarray) – Natural coordinates matrix

Returns:

Shape function evaluated in Z points

Return type:

np.ndarray

FEM.Elements.E2D.TriangularScheme module

Define the triangular scheme used by triangular elements

class FEM.Elements.E2D.TriangularScheme.TriangularScheme(n: int, **kargs)

Bases: object

Generate a triangular integration scheme

Parameters:

n (int) – Number of gauss points