FEMBasis.jl contains interpolation routines for standard finite element
function spaces. Given ansatz and coordinates of domain, interpolation
functions are calculated symbolically in a very general way to get efficient
code. As a concrete example, to generate basis functions for a standard 10-node
tetrahedron one can write
code = FEMBasis.create_basis(
:Tet10,
"10 node quadratic tetrahedral element",
[
(0.0, 0.0, 0.0), # N1
(1.0, 0.0, 0.0), # N2
(0.0, 1.0, 0.0), # N3
(0.0, 0.0, 1.0), # N4
(0.5, 0.0, 0.0), # N5
(0.5, 0.5, 0.0), # N6
(0.0, 0.5, 0.0), # N7
(0.0, 0.0, 0.5), # N8
(0.5, 0.0, 0.5), # N9
(0.0, 0.5, 0.5), # N10
],
:(1 + u + v + w + u*v + v*w + w*u + u^2 + v^2 + w^2),
)The resulting code is
struct Tet10 <: FEMBasis.AbstractBasis{3}
end
Base.@pure function Base.size(::Type{Tet10})
return (3, 10)
end
function Base.size(::Type{Tet10}, j::Int)
j == 1 && return 3
j == 2 && return 10
end
Base.@pure function Base.length(::Type{Tet10})
return 10
end
function FEMBasis.get_reference_element_coordinates(::Type{Tet10})
return Tensors.Tensor{1,3,Float64,3}[[0.0, 0.0, 0.0], [1.0, 0.0, 0.0], [0.0, 1.0, 0.0], [0.0, 0.0, 1.0], [0.5, 0.0, 0.0], [0.5, 0.5, 0.0], [0.0, 0.5, 0.0], [0.0, 0.0, 0.5], [0.5, 0.0, 0.5], [0.0, 0.5, 0.5]]
end
function FEMBasis.eval_basis!(::Type{Tet10}, N::Vector{<:Number}, xi::Vec)
assert length(N) == 10
(u, v, w) = xi
begin
N[1] = 1.0 + -3.0u + -3.0v + -3.0w + 4.0 * (u * v) + 4.0 * (v * w) + 4.0 * (w * u) + 2.0 * u ^ 2 + 2.0 * v ^ 2 + 2.0 * w ^ 2
N[2] = -u + 2.0 * u ^ 2
N[3] = -v + 2.0 * v ^ 2
N[4] = -w + 2.0 * w ^ 2
N[5] = 4.0u + -4.0 * (u * v) + -4.0 * (w * u) + -4.0 * u ^ 2
N[6] = +(4.0 * (u * v))
N[7] = 4.0v + -4.0 * (u * v) + -4.0 * (v * w) + -4.0 * v ^ 2
N[8] = 4.0w + -4.0 * (v * w) + -4.0 * (w * u) + -4.0 * w ^ 2
N[9] = +(4.0 * (w * u))
N[10] = +(4.0 * (v * w))
end
return N
end
function FEMBasis.eval_dbasis!(::Type{Tet10}, dN::Vector{<:Vec{3}}, xi::Vec)
@assert length(dN) == 10
(u, v, w) = xi
begin
dN[1] = Vec(-3.0 + 4.0v + 4.0w + 2.0 * (2u), -3.0 + 4.0u + 4.0w + 2.0 * (2v), -3.0 + 4.0v + 4.0u + 2.0 * (2w))
dN[2] = Vec(-1 + 2.0 * (2u), 0, 0)
dN[3] = Vec(0, -1 + 2.0 * (2v), 0)
dN[4] = Vec(0, 0, -1 + 2.0 * (2w))
dN[5] = Vec(4.0 + -4.0v + -4.0w + -4.0 * (2u), -4.0u, -4.0u)
dN[6] = Vec(4.0v, 4.0u, 0)
dN[7] = Vec(-4.0v, 4.0 + -4.0u + -4.0w + -4.0 * (2v), -4.0v)
dN[8] = Vec(-4.0w, -4.0w, 4.0 + -4.0v + -4.0u + -4.0 * (2w))
dN[9] = Vec(4.0w, 0, 4.0u)
dN[10] = Vec(0, 4.0w, 4.0v)
end
return dN
end
endAlso more unusual elements can be defined. For example, pyramid element cannot be
descibed with ansatz, but it's still possible to implement by defining shape functions,
Calculus.jl is taking care of defining partial derivatives of function:
code = FEMBasis.create_basis(
:Pyr5,
"5 node linear pyramid element",
[
(-1.0, -1.0, -1.0), # N1
( 1.0, -1.0, -1.0), # N2
( 1.0, 1.0, -1.0), # N3
(-1.0, 1.0, -1.0), # N4
( 0.0, 0.0, 1.0), # N5
],
[
:(1/8 * (1-u) * (1-v) * (1-w)),
:(1/8 * (1+u) * (1-v) * (1-w)),
:(1/8 * (1+u) * (1+v) * (1-w)),
:(1/8 * (1-u) * (1+v) * (1-w)),
:(1/2 * (1+w)),
],
)
eval(code)Basis function can have internal variables if needed, e.g. variable dof basis like hierarchical basis functions or NURBS.
It's also possible to do some very common FEM calculations, like calculate Jacobian or gradient of some variable with respect to some coordinates. For example, to calculate displacement gradient du/dX in unit square [0,1]^2, one could write:
using Tensors
B = Quad4()
X = Vec.([(0.0, 0.0), (1.0, 0.0), (1.0, 1.0), (0.0, 1.0)])
u = Vec.([(0.0, 0.0), (1.0, -1.0), (2.0, 3.0), (0.0, 0.0)])
grad(B, u, X, Vec(0.0, 0.0))Result is
2×2 Tensors.Tensor{2,2,Float64,4}:
1.5 0.5
1.0 2.0