NSSC/Exercise_04/fem.jl

104 lines
2.1 KiB
Julia

module FEM
export Node, Element, stiffness, tri_tiles, gradient, center, normalize, p
include("./Parameters.jl")
using GeometryBasics
using GeometryBasics.LinearAlgebra
struct Node
x::Float64
y::Float64
index::UInt
end
p(n::Node) = Point2f(n.x, n.y)
struct Element
nodes::Vector{Node}
a::Vector{Float64}
b::Vector{Float64}
c::Vector{Float64}
Δ::Float64
function Element(n::Vector{Node})
xs = map(nd->nd.x,n)
ys = map(nd->nd.y,n)
a = cross(xs, ys)
b = cross(ys, ones(3))
c = cross(ones(3), xs)
Δ = dot(xs,b) / 2
new(n,a,b,c,Δ)
end
end
function stiffness(e::Element)::Matrix{Float64}
# tensor product
Hₑ = e.b .* e.b'
Hₑ += e.c .* e.c'
Hₑ *= Parameters.hz*Parameters.k/4e.Δ
return Hₑ
end
function tri_tiles(L::Float64, divisions::Int, trapezoidal::Bool=false, biased::Bool=false, ring::Bool=false)::Tuple{Vector{Node}, Vector{Element}}
nodes = Matrix{Node}(undef, divisions+1, divisions+1)
i = 1
for y in 0:divisions
for x in 0:divisions
xₑ = L*x/divisions
yₑ = L*y/divisions
if biased
B = yₑ/2Parameters.L
xₑ = xₑ*(xₑ*B/Parameters.L - B + 1)
end
if trapezoidal
xₑ *= 1 - 0.5*(y/divisions)
end
if ring
r = Parameters.L*(1 + x/divisions)
θ = (π/4)*(y/divisions)
xₑ = r*cos(θ)
yₑ = r*sin(θ)
xₑ -= Parameters.L
end
nodes[x+1,y+1] = Node(xₑ,yₑ,i)
i += 1
end
end
elements = []
for y in 1:divisions
for x in 1:divisions
# lower/upper triangle
push!(elements, Element([nodes[x,y], nodes[x+1,y], nodes[x,y+1]]))
push!(elements, Element([nodes[x+1,y+1], nodes[x,y+1], nodes[x+1,y]]))
end
end
return vec(nodes), elements
end
function gradient(e::Element, T::Vector{Float64})::Vec2f
return Vec2f([
e.b[1] e.b[2] e.b[3] ;
e.c[1] e.c[2] e.c[3]
] * [
T[e.nodes[1].index]
T[e.nodes[2].index]
T[e.nodes[3].index]
] / 2e.Δ)
end
center(e::Element)::Point2f = Point2f(sum([n.x for n in e.nodes])/3.0, sum([n.y for n in e.nodes])/3.0)
normalize(v::Vec2f)::Vec2f = v / sqrt(v[1]^2 + v[2]^2)
end