ReferenceFiniteElements

ReferenceFiniteElements.jl is meant to serve as an educational and lightweight package to define finite elements in the reference (quadrature space) configuration. The goal is to provide a very simple to utilize interface so researchers can play with new finite element formulations without having to learn an entire large code or package base.

Installation

To use ReferenceFiniteElements first install it via the package manager via the following command

pkg> add ReferenceFiniteElements

Quckstart

To setup a finite element you can do the following with a 8 node hexahedral element with a second order quadrature rule (8 integration points) used as an example below

using ReferenceFiniteElements
re = ReferenceFE(Hex8{Lagrange, 2}())

# output

ReferenceFE:
  Element Type                                  = Hex8{Lagrange, 2}
  Surface Element Type                          = Quad4{Lagrange, 2}
  Interpolation Type                            = Lagrange
  Polynomial Degree                             = 1
  Quadrature Degree                             = 2
  Array Backend Type                            = ReferenceFiniteElements.ArrayBackend{StaticArraysCore.SArray}()
  Edge Nodes Storage Type                       = Nothing
  Face Nodes Storage Type                       = Matrix{Int64}
  Interior Nodes Storage Type                   = Vector{Int64}
  Nodal Coordinates Storage Type                = Vector{StaticArraysCore.SVector{3, Float64}}
  CellInterpolants:
    Quadrature Point Storage Type               = Vector{StaticArraysCore.SVector{3, Float64}}
    Quadrature Weights Storage Type             = Vector{Float64}
    Shape Function Values Storage Type          = Vector{StaticArraysCore.SVector{8, Float64}}
    Shape Function Gradients Storage Type       = Vector{StaticArraysCore.SMatrix{8, 3, Float64, 24}}
    Shape Function Hessians Storage Type        = Vector{StaticArraysCore.SArray{Tuple{8, 3, 3}, Float64, 3, 72}}
  Surface Nodal Coordinates Storage Type        = Nothing
  CellInterpolants:
    Quadrature Point Storage Type               = Matrix{Vector{Float64}}
    Quadrature Weights Storage Type             = Matrix{Float64}
    Shape Function Values Storage Type          = Matrix{StaticArraysCore.SVector{8, Float64}}
    Shape Function Gradients Storage Type       = Matrix{StaticArraysCore.SMatrix{8, 3, Float64, 24}}
    Shape Function Hessians Storage Type        = Matrix{StaticArraysCore.SArray{Tuple{8, 3, 3}, Float64, 3, 72}}

To do something useful with our finite element we can look at the quadrature points. To get all the quadrature points, use the following method

quadrature_points(re)
8-element Vector{StaticArraysCore.SVector{3, Float64}}:
 [-0.5773502691896258, -0.5773502691896258, -0.5773502691896258]
 [0.5773502691896258, -0.5773502691896258, -0.5773502691896258]
 [-0.5773502691896258, 0.5773502691896258, -0.5773502691896258]
 [0.5773502691896258, 0.5773502691896258, -0.5773502691896258]
 [-0.5773502691896258, -0.5773502691896258, 0.5773502691896258]
 [0.5773502691896258, -0.5773502691896258, 0.5773502691896258]
 [-0.5773502691896258, 0.5773502691896258, 0.5773502691896258]
 [0.5773502691896258, 0.5773502691896258, 0.5773502691896258]

To get a specific quadrature point, say the first point, we can use an analogous method as follows

ξ = quadrature_point(re, 1)

# output

3-element StaticArraysCore.SVector{3, Float64} with indices SOneTo(3):
 -0.5773502691896258
 -0.5773502691896258
 -0.5773502691896258

There are similar methods for the quadrature weights. See below.

ws = quadrature_weights(re)

# output

8-element Vector{Float64}:
 1.0
 1.0
 1.0
 1.0
 1.0
 1.0
 1.0
 1.0

w = quadrature_weights(re, 1)

# output

1.0

For the shape function values we can access these via the following methods

Ns = shape_function_values(re)

# output

8-element Vector{StaticArraysCore.SVector{8, Float64}}:
 [0.4905626121623441, 0.13144585576580212, 0.035220810900864506, 0.13144585576580212, 0.13144585576580212, 0.035220810900864506, 0.009437387837655926, 0.035220810900864506]
 [0.13144585576580212, 0.4905626121623441, 0.13144585576580212, 0.035220810900864506, 0.035220810900864506, 0.13144585576580212, 0.035220810900864506, 0.009437387837655926]
 [0.13144585576580212, 0.035220810900864506, 0.13144585576580212, 0.4905626121623441, 0.035220810900864506, 0.009437387837655926, 0.035220810900864506, 0.13144585576580212]
 [0.035220810900864506, 0.13144585576580212, 0.4905626121623441, 0.13144585576580212, 0.009437387837655926, 0.035220810900864506, 0.13144585576580212, 0.035220810900864506]
 [0.13144585576580212, 0.035220810900864506, 0.009437387837655926, 0.035220810900864506, 0.4905626121623441, 0.13144585576580212, 0.035220810900864506, 0.13144585576580212]
 [0.035220810900864506, 0.13144585576580212, 0.035220810900864506, 0.009437387837655926, 0.13144585576580212, 0.4905626121623441, 0.13144585576580212, 0.035220810900864506]
 [0.035220810900864506, 0.009437387837655926, 0.035220810900864506, 0.13144585576580212, 0.13144585576580212, 0.035220810900864506, 0.13144585576580212, 0.4905626121623441]
 [0.009437387837655926, 0.035220810900864506, 0.13144585576580212, 0.035220810900864506, 0.035220810900864506, 0.13144585576580212, 0.4905626121623441, 0.13144585576580212]

N = shape_function_values(re, 1)

# output

8-element StaticArraysCore.SVector{8, Float64} with indices SOneTo(8):
 0.4905626121623441
 0.13144585576580212
 0.035220810900864506
 0.13144585576580212
 0.13144585576580212
 0.035220810900864506
 0.009437387837655926
 0.035220810900864506