mirror of
https://github.com/JuliaFEM/JuliaFEM.jl.git
synced 2026-08-31 08:16:23 +00:00
Jukka Aho
261 lines
7.4 KiB
Julia
261 lines
7.4 KiB
Julia
# This file is a part of JuliaFEM.
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# License is MIT: see https://github.com/JuliaFEM/JuliaFEM.jl/blob/master/LICENSE.md
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# Let's drop here all integration schemes and some defaults for different element types maybe parse from txt file ..?
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### Gauss quadrature rules for one dimension
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function get_integration_points(::Type{Val{1}})
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return [2.0], [0.0]
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end
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function get_integration_points(::Type{Val{2}})
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return [1.0, 1.0], sqrt(1.0/3.0)*[-1.0, 1.0]
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end
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function get_integration_points(::Type{Val{3}})
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return 1.0/9.0*[5.0, 8.0, 5.0], sqrt(3.0/5.0)*[-1.0, 0.0, 1.0]
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end
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function get_integration_points(::Type{Val{4}})
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weights = 1.0/36.0*[
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18.0-sqrt(30.0),
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18.0+sqrt(30.0),
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18.0+sqrt(30.0),
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18.0-sqrt(30.0)]
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points = [
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-sqrt(3.0/7.0 + 2.0/7.0*sqrt(6.0/5.0)),
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-sqrt(3.0/7.0 - 2.0/7.0*sqrt(6.0/5.0)),
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sqrt(3.0/7.0 - 2.0/7.0*sqrt(6.0/5.0)),
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sqrt(3.0/7.0 + 2.0/7.0*sqrt(6.0/5.0))]
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return weights, points
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end
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function get_integration_points(::Type{Val{5}})
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weights = [
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1.0/900.0*(322.0-13.0*sqrt(70.0)),
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1.0/900.0*(322.0+13.0*sqrt(70.0)),
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128.0/225.0,
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1.0/900.0*(322.0+13.0*sqrt(70.0)),
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1.0/900.0*(322.0-13.0*sqrt(70.0))]
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points = [
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-1.0/3.0*sqrt(5.0 + 2.0*sqrt(10.0/7.0)),
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-1.0/3.0*sqrt(5.0 - 2.0*sqrt(10.0/7.0)),
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0.0,
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1.0/3.0*sqrt(5.0 - 2.0*sqrt(10.0/7.0)),
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1.0/3.0*sqrt(5.0 + 2.0*sqrt(10.0/7.0))]
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return weights, points
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end
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function get_integration_points(order::Int64)
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if order <= 5
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return get_integration_points(Val{order})
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else
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points, weights = Base.QuadGK.gauss(Float64, order)
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return weights, points
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end
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end
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### "cartesian" elements, integration rules comes from tensor product
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### 0d elements
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function get_integration_points(element::Poi1)
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[ (1.0, [] ) ]
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end
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### 1d elements
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typealias CartesianLineElement Union{Seg2, Seg3, NSeg}
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typealias CartesianSurfaceElement Union{Quad4, NSurf}
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typealias CartesianVolumeElement Union{Hex8, NSolid}
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function get_integration_points(element::CartesianLineElement, order::Int64)
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w, xi = get_integration_points(order)
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[ (w[i], [xi[i]]) for i=1:order ]
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end
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function get_integration_points(element::CartesianSurfaceElement, order::Int64)
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w, xi = get_integration_points(order)
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[ (w[i]*w[j], [xi[i], xi[j]]) for i=1:order, j=1:order ]
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end
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function get_integration_points(element::CartesianVolumeElement, order::Int64)
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w, xi = get_integration_points(order)
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[ (w[i]*w[j]*w[k], [xi[i], xi[j], xi[k]]) for i=1:order, j=1:order, k=1:order ]
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end
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### triangular and tetrahedral elements
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# http://math2.uncc.edu/~shaodeng/TEACHING/math5172/Lectures/Lect_15.PDF
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# http://libmesh.github.io/doxygen/quadrature__gauss__2D_8C_source.html
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typealias TriangularElement Union{Tri3, Tri6}
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function get_integration_points(element::TriangularElement, ::Type{Val{1}})
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weights = [0.5]
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points = Vector{Float64}[1.0/3.0*[1.0, 1.0]]
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return zip(weights, points)
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end
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function get_integration_points(element::TriangularElement, ::Type{Val{2}})
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weights = 1.0/6.0*[1.0, 1.0, 1.0]
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points = Vector{Float64}[
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[2.0/3.0, 1.0/6.0],
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[1.0/6.0, 2.0/3.0],
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[1.0/6.0, 1.0/6.0]]
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return zip(weights, points)
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end
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function get_integration_points(element::TriangularElement, ::Type{Val{3}})
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weights = 0.5*[
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-0.5625,
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0.5208333333333333,
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0.5208333333333333,
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0.5208333333333333]
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points = Vector{Float64}[
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[1.0/3.0, 1.0/3.0],
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[0.2, 0.2],
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[0.2, 0.6],
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[0.6, 0.2]]
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return zip(weights, points)
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end
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function get_integration_points(element::TriangularElement, ::Type{Val{4}})
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weights = 0.5*[
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0.22338158967801,
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0.22338158967801,
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0.22338158967801,
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0.10995174365532,
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0.10995174365532,
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0.10995174365532]
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points = Vector{Float64}[
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[0.44594849091597, 0.44594849091597],
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[0.44594849091597, 0.10810301816807],
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[0.10810301816807, 0.44594849091597],
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[0.09157621350977, 0.09157621350977],
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[0.09157621350977, 0.81684757298046],
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[0.81684757298046, 0.09157621350977]]
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return zip(weights, points)
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end
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function get_integration_points(element::TriangularElement, ::Type{Val{5}})
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weights = 0.5*[
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0.22500000000000,
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0.13239415278851,
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0.13239415278851,
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0.13239415278851,
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0.12593918054483,
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0.12593918054483,
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0.12593918054483]
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points = Vector{Float64}[
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[0.33333333333333, 0.33333333333333],
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[0.47014206410511, 0.47014206410511],
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[0.47014206410511, 0.05971587178977],
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[0.05971587178977, 0.47014206410511],
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[0.10128650732346, 0.10128650732346],
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[0.10128650732346, 0.79742698535309],
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[0.79742698535309, 0.10128650732346]]
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return zip(weights, points)
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end
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### 3d elements
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typealias TetrahedralElement Union{Tet4, Tet10}
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function get_integration_points(element::TetrahedralElement, ::Type{Val{1}})
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weights = 1.0/6.0*[1.0]
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points = Vector{Float64}[
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1.0/4.0*[1.0, 1.0, 1.0]]
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return zip(weights, points)
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end
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function get_integration_points(element::TetrahedralElement, ::Type{Val{2}})
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a = (5.0+3.0*sqrt(5.0))/20.0
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b = (5.0-sqrt(5.0))/20.0
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w = 1.0/24.0
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weights = [w, w, w, w]
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points = Vector{Float64}[
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[a, b, b],
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[b, a, b],
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[b, b, a],
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[b, b, b]]
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return zip(weights, points)
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end
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function get_integration_points(element::TetrahedralElement, ::Type{Val{3}})
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a = 1.0/4.0
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b = 1.0/6.0
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c = 1.0/2.0
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weights = [-2.0/15.0, 3.0/40.0, 3.0/40.0, 3.0/40.0, 3.0/40.0]
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points = Vector{Float64}[
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[a, a, a],
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[b, b, b],
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[b, b, c],
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[b, c, b],
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[c, b, b]]
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return zip(weights, points)
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end
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function get_integration_points(element::TetrahedralElement, ::Type{Val{4}})
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a = 0.25
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b1 = 1.0/34.0*(7.0 + sqrt(15.0))
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b2 = 1.0/34.0*(7.0 - sqrt(15.0))
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c1 = 1.0/34.0*(13.0 - 3.0*sqrt(15.0))
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c2 = 1.0/34.0*(13.0 + 3.0*sqrt(15.0))
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d = 1.0/20.0*(5.0 - sqrt(15.0))
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e = 1.0/20.0*(5.0 + sqrt(15.0))
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w1 = 8.0/405.0
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w2 = (2665.0 - 14.0*sqrt(15.0))/226800.0
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w3 = (2665.0 + 14.0*sqrt(15.0))/226800.0
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w4 = 5.0/567.0
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weights = [w1, w2, w2, w2, w2, w3, w3, w3, w3, w4, w4, w4, w4, w4]
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points = Vector{Float64}[
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[a, a, a],
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[b1, b1, b1],
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[b1, b1, c1],
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[b1, c1, b1],
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[c1, b1, b1],
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[b2, b2, b2],
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[b2, b2, c2],
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[b2, c2, b2],
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[c2, b2, b2],
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[d, d, e],
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[d, e, d],
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[e, d, d],
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[d, e, e],
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[e, d, e],
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[e, e, d]]
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return zip(weights, points)
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end
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function get_integration_points(element::Union{TriangularElement, TetrahedralElement}, order::Int64)
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return get_integration_points(element, Val{order})
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end
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### default number of integration points for each element
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### 2 for linear elements, 3 for quadratic
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typealias LinearElement Union{Seg2, Tri3, Quad4, Tet4, Hex8}
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typealias QuadraticElement Union{Seg3, Tri6, Tet10}
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function get_integration_order(element::LinearElement)
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return 2
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end
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function get_integration_order(element::QuadraticElement)
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return 3
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end
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function get_integration_points(element::LinearElement)
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order = get_integration_order(element)
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get_integration_points(element, order)
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end
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function get_integration_points(element::QuadraticElement)
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order = get_integration_order(element)
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get_integration_points(element, order)
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end
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