ReferenceFiniteElements.jl

Author Cthonios
Popularity
3 Stars
Updated Last
3 Months Ago
Started In
June 2023

ReferenceFiniteElements

Stable Dev Build Status Coverage

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(2))

# output

ReferenceFE
  Element type                = Hex8{8}
  Dimension                   = 3
  Number of nodes             = 8
  Number of quadrature points = 8
  Integer type                = Int64
  Float type                  = Float64
  Nodal coordinates type      = Matrix{Float64}
  Face nodes type             = Matrix{Int64}
  Interior nodes type         = Vector{Int64}
  Interpolants type           = StructArray{Interpolants}

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

ξs = quadrature_points(re)

# output

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_points(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

For shape function gradients, the method calls are as follows

∇N_ξ = shape_function_gradients(re)

# output

8-element Vector{StaticArraysCore.SMatrix{8, 3, Float64, 24}}:
 [-0.3110042339640731 -0.3110042339640731 -0.3110042339640731; 0.3110042339640731 -0.08333333333333331 -0.08333333333333331;  ; 0.022329099369260218 0.022329099369260218 0.022329099369260218; -0.022329099369260218 0.08333333333333331 0.08333333333333331]
 [-0.3110042339640731 -0.08333333333333331 -0.08333333333333331; 0.3110042339640731 -0.3110042339640731 -0.3110042339640731;  ; 0.022329099369260218 0.08333333333333331 0.08333333333333331; -0.022329099369260218 0.022329099369260218 0.022329099369260218]
 [-0.08333333333333331 -0.3110042339640731 -0.08333333333333331; 0.08333333333333331 -0.08333333333333331 -0.022329099369260218;  ; 0.08333333333333331 0.022329099369260218 0.08333333333333331; -0.08333333333333331 0.08333333333333331 0.3110042339640731]
 [-0.08333333333333331 -0.08333333333333331 -0.022329099369260218; 0.08333333333333331 -0.3110042339640731 -0.08333333333333331;  ; 0.08333333333333331 0.08333333333333331 0.3110042339640731; -0.08333333333333331 0.022329099369260218 0.08333333333333331]
 [-0.08333333333333331 -0.08333333333333331 -0.3110042339640731; 0.08333333333333331 -0.022329099369260218 -0.08333333333333331;  ; 0.08333333333333331 0.08333333333333331 0.022329099369260218; -0.08333333333333331 0.3110042339640731 0.08333333333333331]
 [-0.08333333333333331 -0.022329099369260218 -0.08333333333333331; 0.08333333333333331 -0.08333333333333331 -0.3110042339640731;  ; 0.08333333333333331 0.3110042339640731 0.08333333333333331; -0.08333333333333331 0.08333333333333331 0.022329099369260218]
 [-0.022329099369260218 -0.08333333333333331 -0.08333333333333331; 0.022329099369260218 -0.022329099369260218 -0.022329099369260218;  ; 0.3110042339640731 0.08333333333333331 0.08333333333333331; -0.3110042339640731 0.3110042339640731 0.3110042339640731]
 [-0.022329099369260218 -0.022329099369260218 -0.022329099369260218; 0.022329099369260218 -0.08333333333333331 -0.08333333333333331;  ; 0.3110042339640731 0.3110042339640731 0.3110042339640731; -0.3110042339640731 0.08333333333333331 0.08333333333333331]
∇N_ξ = shape_function_gradients(re, 1)

# output

8×3 StaticArraysCore.SMatrix{8, 3, Float64, 24} with indices SOneTo(8)×SOneTo(3):
 -0.311004   -0.311004   -0.311004
  0.311004   -0.0833333  -0.0833333
  0.0833333   0.0833333  -0.0223291
 -0.0833333   0.311004   -0.0833333
 -0.0833333  -0.0833333   0.311004
  0.0833333  -0.0223291   0.0833333
  0.0223291   0.0223291   0.0223291
 -0.0223291   0.0833333   0.0833333

For shape function hessians, the method calls are as follows

∇∇N_ξ = shape_function_hessians(re)

# output

8-element Vector{StaticArraysCore.SArray{Tuple{8, 3, 3}, Float64, 3, 72}}:
 [0.0 0.19716878364870322 0.19716878364870322; 0.0 -0.19716878364870322 -0.19716878364870322;  ; 0.0 0.05283121635129677 0.05283121635129677; 0.0 -0.05283121635129677 -0.05283121635129677;;; 0.19716878364870322 0.0 0.19716878364870322; -0.19716878364870322 0.0 0.05283121635129677;  ; 0.05283121635129677 0.0 0.05283121635129677; -0.05283121635129677 0.0 0.19716878364870322;;; 0.19716878364870322 0.19716878364870322 0.0; -0.19716878364870322 0.05283121635129677 0.0;  ; 0.05283121635129677 0.05283121635129677 0.0; -0.05283121635129677 0.19716878364870322 0.0]
 [0.0 0.19716878364870322 0.19716878364870322; 0.0 -0.19716878364870322 -0.19716878364870322;  ; 0.0 0.05283121635129677 0.05283121635129677; 0.0 -0.05283121635129677 -0.05283121635129677;;; 0.19716878364870322 0.0 0.05283121635129677; -0.19716878364870322 0.0 0.19716878364870322;  ; 0.05283121635129677 0.0 0.19716878364870322; -0.05283121635129677 0.0 0.05283121635129677;;; 0.19716878364870322 0.05283121635129677 0.0; -0.19716878364870322 0.19716878364870322 0.0;  ; 0.05283121635129677 0.19716878364870322 0.0; -0.05283121635129677 0.05283121635129677 0.0]
 [0.0 0.19716878364870322 0.05283121635129677; 0.0 -0.19716878364870322 -0.05283121635129677;  ; 0.0 0.05283121635129677 0.19716878364870322; 0.0 -0.05283121635129677 -0.19716878364870322;;; 0.19716878364870322 0.0 0.19716878364870322; -0.19716878364870322 0.0 0.05283121635129677;  ; 0.05283121635129677 0.0 0.05283121635129677; -0.05283121635129677 0.0 0.19716878364870322;;; 0.05283121635129677 0.19716878364870322 0.0; -0.05283121635129677 0.05283121635129677 0.0;  ; 0.19716878364870322 0.05283121635129677 0.0; -0.19716878364870322 0.19716878364870322 0.0]
 [0.0 0.19716878364870322 0.05283121635129677; 0.0 -0.19716878364870322 -0.05283121635129677;  ; 0.0 0.05283121635129677 0.19716878364870322; 0.0 -0.05283121635129677 -0.19716878364870322;;; 0.19716878364870322 0.0 0.05283121635129677; -0.19716878364870322 0.0 0.19716878364870322;  ; 0.05283121635129677 0.0 0.19716878364870322; -0.05283121635129677 0.0 0.05283121635129677;;; 0.05283121635129677 0.05283121635129677 0.0; -0.05283121635129677 0.19716878364870322 0.0;  ; 0.19716878364870322 0.19716878364870322 0.0; -0.19716878364870322 0.05283121635129677 0.0]
 [0.0 0.05283121635129677 0.19716878364870322; 0.0 -0.05283121635129677 -0.19716878364870322;  ; 0.0 0.19716878364870322 0.05283121635129677; 0.0 -0.19716878364870322 -0.05283121635129677;;; 0.05283121635129677 0.0 0.19716878364870322; -0.05283121635129677 0.0 0.05283121635129677;  ; 0.19716878364870322 0.0 0.05283121635129677; -0.19716878364870322 0.0 0.19716878364870322;;; 0.19716878364870322 0.19716878364870322 0.0; -0.19716878364870322 0.05283121635129677 0.0;  ; 0.05283121635129677 0.05283121635129677 0.0; -0.05283121635129677 0.19716878364870322 0.0]
 [0.0 0.05283121635129677 0.19716878364870322; 0.0 -0.05283121635129677 -0.19716878364870322;  ; 0.0 0.19716878364870322 0.05283121635129677; 0.0 -0.19716878364870322 -0.05283121635129677;;; 0.05283121635129677 0.0 0.05283121635129677; -0.05283121635129677 0.0 0.19716878364870322;  ; 0.19716878364870322 0.0 0.19716878364870322; -0.19716878364870322 0.0 0.05283121635129677;;; 0.19716878364870322 0.05283121635129677 0.0; -0.19716878364870322 0.19716878364870322 0.0;  ; 0.05283121635129677 0.19716878364870322 0.0; -0.05283121635129677 0.05283121635129677 0.0]
 [0.0 0.05283121635129677 0.05283121635129677; 0.0 -0.05283121635129677 -0.05283121635129677;  ; 0.0 0.19716878364870322 0.19716878364870322; 0.0 -0.19716878364870322 -0.19716878364870322;;; 0.05283121635129677 0.0 0.19716878364870322; -0.05283121635129677 0.0 0.05283121635129677;  ; 0.19716878364870322 0.0 0.05283121635129677; -0.19716878364870322 0.0 0.19716878364870322;;; 0.05283121635129677 0.19716878364870322 0.0; -0.05283121635129677 0.05283121635129677 0.0;  ; 0.19716878364870322 0.05283121635129677 0.0; -0.19716878364870322 0.19716878364870322 0.0]
 [0.0 0.05283121635129677 0.05283121635129677; 0.0 -0.05283121635129677 -0.05283121635129677;  ; 0.0 0.19716878364870322 0.19716878364870322; 0.0 -0.19716878364870322 -0.19716878364870322;;; 0.05283121635129677 0.0 0.05283121635129677; -0.05283121635129677 0.0 0.19716878364870322;  ; 0.19716878364870322 0.0 0.19716878364870322; -0.19716878364870322 0.0 0.05283121635129677;;; 0.05283121635129677 0.05283121635129677 0.0; -0.05283121635129677 0.19716878364870322 0.0;  ; 0.19716878364870322 0.19716878364870322 0.0; -0.19716878364870322 0.05283121635129677 0.0]
∇∇N_ξ = shape_function_hessians(re, 1)

# output

8×3×3 StaticArraysCore.SArray{Tuple{8, 3, 3}, Float64, 3, 72} with indices SOneTo(8)×SOneTo(3)×SOneTo(3):
[:, :, 1] =
 0.0   0.197169    0.197169
 0.0  -0.197169   -0.197169
 0.0   0.197169   -0.0528312
 0.0  -0.197169    0.0528312
 0.0   0.0528312  -0.197169
 0.0  -0.0528312   0.197169
 0.0   0.0528312   0.0528312
 0.0  -0.0528312  -0.0528312

[:, :, 2] =
  0.197169   0.0   0.197169
 -0.197169   0.0   0.0528312
  0.197169   0.0  -0.0528312
 -0.197169   0.0  -0.197169
  0.0528312  0.0  -0.197169
 -0.0528312  0.0  -0.0528312
  0.0528312  0.0   0.0528312
 -0.0528312  0.0   0.197169

[:, :, 3] =
  0.197169    0.197169   0.0
 -0.197169    0.0528312  0.0
 -0.0528312  -0.0528312  0.0
  0.0528312  -0.197169   0.0
 -0.197169   -0.197169   0.0
  0.197169   -0.0528312  0.0
  0.0528312   0.0528312  0.0
 -0.0528312   0.197169   0.0