Structure

Structure of element shape data.

Status

TODO : documentation of ElemshapeData_ getMethods setMethods
TODO : support for H1DIV_
TODO : support for H1CURL_
TODO : support for DG_
TODO : HermitInterpolation
TODO: SerendipityInterpolation
TODO: HierarchyInterpolation
TODO: Change continuityType to conformityType

Introduction

ElemshapeData_ datatype contains the shape-function data related information at the integration points in a finite element. In FEM, generally we need the following information about the shape functions, at the integration points.

Shape functions, $N$
Local gradient of shape functions, $\frac{dN}{d\xi}$
Global gradient of shape functions, $\frac{dN}{dx_{i}}$
Jacobian of mapping from reference element to parent elemen $\frac{dx_{i}}{dX_{J}}$
The value of weight function $W_{s}$
Thickness in the case of plane-stress condition
Barycentric coordinate of physical/parent element

One of the essential parameters for defining the shape functions is continuity of the shape functions, which is described below.

H1_
H1DIV_
H1CURL_
DG_
Another important information to complete the information stored inside an instance of ElemshapeData_ is quadrature point.
Information of the reference element ReferenceElement_ is also required to construct the element shape data
Information about the Interpolation type is also necessary. Following interpolations are possible.
- LagrangeInterpolation
- HermitInterpolation
- SerendipityInterpolation
- HierarchyInterpolation

The structure of this data type is given below.

TYPE :: ElemShapeData_
  REAL(DFP), ALLOCATABLE :: N(:, :)
  REAL(DFP), ALLOCATABLE :: dNdXi(:, :, :)
  REAL(DFP), ALLOCATABLE :: jacobian(:, :, :)
  REAL(DFP), ALLOCATABLE :: js(:)
  REAL(DFP), ALLOCATABLE :: ws(:)
  REAL(DFP), ALLOCATABLE :: dNdXt(:, :, :)
  REAL(DFP), ALLOCATABLE :: thickness(:)
  REAL(DFP), ALLOCATABLE :: coord(:, :)
  REAL(DFP), ALLOCATABLE :: normal(:, :)
  TYPE(ReferenceElement_) :: refElem
  TYPE(QuadraturePoint_) :: quad
END TYPE ElemShapeData_

The information about these variables is given below:

N shape-function value N(I,ips)
dNdXi local derivative of a shape function $\frac{dN^{I}}{d\xi_{j}}$ dNdXi(I,j,ips)
jacobian Jacobian of mapping from reference element to parent element $\frac{\partial x_{i}}{\partial \xi_{j}}$ Jacobian(i,j,ips)
js determinant of Jacobian Js(ips)
ws weight for numerical integration Ws(ips)
dNdXt global derivative of a shape function $\frac{dN^{I}}{dx_{j}}$ dNdx(I,j,ips)
thickness thickness at the integration points
coord barycentric coordinates Coord(i,ips)
normal normal vector in the case of facet element
refelem pointer to the reference element [[ReferenceElement_]]
quad instance of [[QuadraturePoint_]]

Submodules

The main module is ElemshapeData_Method.f90, which contains following submodules.

@ConstructorMethods.F90
@IOMethods.F90
@GetMethods.F90
@StabilizationParamMethods.F90
@ProjectionMethods.F90
@InterpolMethods.F90
@LocalGradientMethods.F90
@GradientMethods.F90
@LocalDivergenceMethods.F90
@DivergenceMethods.F90
@SetMethod.F90
@H1Lagrange.F90
@H1Lagrange@Line.F90
@H1Lagrange@Triangle.F90
@H1Lagrange@Quadrangle.F90
@H1Lagrange@Tetrahedron.F90
@H1Lagrange@Hexahedron.F90
@H1Lagrange@Prism.F90
@H1Lagrange@Pyramid.F90
@H1Serendipity.F90
@H1Hermit.F90
@H1Hierarchy.F90
@H1DivLagrange.F90
@H1DivSerendipity.F90
@H1DivHermit.F90
@H1DivHierarchy.F90
@H1CurlLagrange.F90
@H1CurlSerendipity.F90
@H1CurlHermit.F90
@H1CurlHierarchy.F90
@DGLagrange.F90
@DGSerendipity.F90
@DGHermit.F90
@DGHierarchy.F90

For example, the submodule ElemshapeData_Method@H1Lagrange.f90 implements the shape function which belongs to the H1 Lagrange polynomial family. It has following sub-submodules corresponding to the geometry of the reference element.

@Line.f90 Arbitrary order
@Triangle.f90 upto 3rd order
@Quadrangle.f90 upto 1st order
@Tetrahedron.f90 Todo
@Hexahedron.f90 Todo
@Prism.f90 Todo
@Pyramid.f90 Todo

Status​

Introduction​

Submodules​

Status

Introduction

Submodules