Region

class felupe.Region(mesh, element, quadrature, grad=True)

A numeric region as a combination of a mesh, an element and a numeric integration scheme (quadrature). The gradients of the element shape functions are evaluated at all integration points of each cell in the region if the optional gradient argument is True.

\[ \begin{align}\begin{aligned}\frac{\partial X^I}{\partial r^J} &= X_a^I \frac{\partial h_a}{\partial r^J}\\\frac{\partial h_a}{\partial X^J} &= \frac{\partial h_a}{\partial r^I} \frac{\partial r^I}{\partial X^J}\\dV &= \det\left(\frac{\partial X^I}{\partial r^J}\right) w\end{aligned}\end{align} \]

Parameters

• mesh (Mesh) – A mesh with points and cells.

• element (Element) – The finite element formulation to be applied on the cells.

• quadrature (Quadrature) – An element-compatible numeric integration scheme with points and weights.

• grad (bool, optional) – A flag to invoke gradient evaluation (default is True).

mesh

A mesh with points and cells.

Type

Mesh

element

The finite element formulation to be applied on the cells.

Type

Finite element

quadrature

An element-compatible numeric integration scheme with points and weights.

Type

Quadrature scheme

h

Element shape function array h_ap of shape function a evaluated at quadrature point p.

Type

ndarray

dhdr

Partial derivative of element shape function array dhdr_aJp with shape function a w.r.t. natural element coordinate J evaluated at quadrature point p for every cell c (geometric gradient or Jacobian transformation between X and r).

Type

ndarray

dXdr

Geometric gradient dXdr_IJpc as partial derivative of undeformed coordinate I w.r.t. natural element coordinate J evaluated at quadrature point p for every cell c (geometric gradient or Jacobian transformation between X and r).

Type

ndarray

drdX

Inverse of dXdr.

Type

ndarray

dV

Numeric Differential volume element as product of determinant of geometric gradient dV_pc = det(dXdr)_pc w_p and quadrature weight w_p, evaluated at quadrature point p for every cell c.

Type

ndarray

dhdX

Partial derivative of element shape functions dhdX_aJpc of shape function a w.r.t. undeformed coordinate J evaluated at quadrature point p for every cell c.

Type

ndarray

class felupe.RegionBoundary(mesh, element, quadrature, grad=True, only_surface=True, mask=None, ensure_3d=False)

A numeric boundary-region as a combination of a mesh, an element and a numeric integration scheme (quadrature). The gradients of the element shape functions are evaluated at all integration points of each cell in the region if the optional gradient argument is True.

\[ \begin{align}\begin{aligned}\frac{\partial X^I}{\partial r^J} &= X_a^I \frac{\partial h_a}{\partial r^J}\\\frac{\partial h_a}{\partial X^J} &= \frac{\partial h_a}{\partial r^I} \frac{\partial r^I}{\partial X^J}\\dV &= \det\left(\frac{\partial X^I}{\partial r^J}\right) w\end{aligned}\end{align} \]

Parameters

• mesh (Mesh) – A mesh with points and cells.

• element (Element) – The finite element formulation to be applied on the cells.

• quadrature (Quadrature) – An element-compatible numeric integration scheme with points and weights.

• grad (bool, optional) – A flag to invoke gradient evaluation (default is True).

• only_surface (bool, optional) – A flag to use only the enclosing outline of the region (default is True).

• mask (ndarray or None, optional) – A boolean array to select a specific set of points (default is None).

• ensure_3d (bool, optional) – A flag to enforce 3d area normal vectors.

mesh

A mesh with points and cells.

Type

Mesh

element

The finite element formulation to be applied on the cells.

Type

Finite element

quadrature

An element-compatible numeric integration scheme with points and weights.

Type

Quadrature scheme

h

Element shape function array h_ap of shape function a evaluated at quadrature point p.

Type

ndarray

dhdr

Partial derivative of element shape function array dhdr_aJp with shape function a w.r.t. natural element coordinate J evaluated at quadrature point p for every cell c (geometric gradient or Jacobian transformation between X and r).

Type

ndarray

dXdr

Geometric gradient dXdr_IJpc as partial derivative of undeformed coordinate I w.r.t. natural element coordinate J evaluated at quadrature point p for every cell c (geometric gradient or Jacobian transformation between X and r).

Type

ndarray

drdX

Inverse of dXdr.

Type

ndarray

dA

Numeric Differential area vectors.

Type

ndarray

normals

Area unit normal vectors.

Type

ndarray

dV

Numeric Differential volume element as norm of Differential area vectors.

Type

ndarray

dhdX

Partial derivative of element shape functions dhdX_aJpc of shape function a w.r.t. undeformed coordinate J evaluated at quadrature point p for every cell c.

Type

ndarray

class felupe.RegionQuad(mesh)

Bases: Region

A region with a quad element.

class felupe.RegionQuadBoundary(mesh, only_surface=True, mask=None, ensure_3d=False)

Bases: RegionBoundary

A region with a quad element.

class felupe.RegionHexahedronBoundary(mesh, only_surface=True, mask=None)

Bases: RegionBoundary

A region with a hexahedron element.

class RegionHexahedron(mesh)

A region with a hexahedron element.

Members

Undoc-members

Show-inheritance

class felupe.RegionTriangle(mesh)

Bases: Region

A region with a triangle element.

class felupe.RegionTetra(mesh)

Bases: Region

A region with a tetra element.

class felupe.RegionConstantQuad(mesh, offset=0, npoints=None)

Bases: Region

A region with a constant quad element.

class felupe.RegionConstantHexahedron(mesh, offset=0, npoints=None)

Bases: Region

A region with a constant hexahedron element.

class felupe.RegionQuadraticHexahedron(mesh)

Bases: Region

A region with a (serendipity) quadratic hexahedron element.

class felupe.RegionTriQuadraticHexahedron(mesh)

Bases: Region

A region with a tri-quadratic (lagrange) hexahedron element.

class felupe.RegionQuadraticTriangle(mesh)

Bases: Region

A region with a quadratic triangle element.

class felupe.RegionQuadraticTetra(mesh)

Bases: Region

A region with a quadratic tetra element.

class felupe.RegionTriangleMINI(mesh)

Bases: Region

A region with a triangle-MINI element.

class felupe.RegionTetraMINI(mesh)

Bases: Region

A region with a tetra-MINI element.

class felupe.RegionLagrange(mesh, order, dim)

Bases: Region

A region with an arbitrary order lagrange element.