mirror of
https://github.com/JuliaFEM/JuliaFEM.jl.git
synced 2026-09-17 09:12:09 +00:00
Jukka Aho
161 lines
3.5 KiB
Julia
161 lines
3.5 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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# some preliminary code for constructing hierarchical elements
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abstract Hierarchical <: Element
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bin(n, k) = prod([(n + 1 - i)/i for i=1:k])
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"""
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Return Legendgre polynomial of order n to inverval ξ ∈ [-1, 1]
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Parameters
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----------
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n :: Int
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order of polynomial
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Returns
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-------
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function
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Legendgre polynomial of order n in interval ξ ∈ [-1, 1]
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"""
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function get_legendre_polynomial(n::Int)
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P(xi) = 2^n*sum([xi.^k*bin(n, k)*bin(1/2*(n+k-1), n) for k=0:n])
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P
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end
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"""
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Return derivative of Legendgre polynomial of order n to inverval ξ ∈ [-1, 1]
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"""
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function get_legendre_polynomial_derivative(n::Int)
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dP(xi) = 2^n*sum([k*xi.^(k-1)*bin(n, k)*bin(1/2*(n+k-1), n) for k=1:n])
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dP
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end
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"""
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Return Legendgre polynomial of order n to inverval ξ ∈ [1, 1].
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Parameters
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----------
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n :: Int
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order of polynomial
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Returns
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-------
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function
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Legendgre polynomial of order n in interval ξ ∈ [-1, 1]
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Notes
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-----
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Uses Bonnet's recursion formula. See
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https://en.wikipedia.org/wiki/Legendre_polynomials
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"""
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function get_legendre_polynomial_recursive(n)
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if n == 0
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P(xi) = 1
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elseif n == 1
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P(xi) = xi
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else
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Pm1 = get_legendre_polynomial(n-1)
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Pm2 = get_legendre_polynomial(n-2)
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P(xi) = 1/n*((2*n-1)*xi*Pm1(xi) - (n-1)*Pm2(xi))
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end
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return P
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end
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"""
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Return derivative of Legendgre polynomial of order n to inverval ξ ∈ [-1, 1]
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"""
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function get_legendre_polynomial_derivative_recursive(n)
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if n == 0
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P(xi) = 0
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elseif n == 1
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P(xi) = 1
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else
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Pm1 = get_legendre_polynomial_derivative(n-1)
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Pm2 = get_legendre_polynomial_derivative(n-2)
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P(xi) = 1/(n-1)*( (2*(n-1)+1)*xi.*Pm1(xi) - (n+1-1)*Pm2(xi))
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end
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return P
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end
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"""
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Return hierarchical shape function of order N
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"""
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function get_hierarchial_basis(n)
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if n == 1
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N(xi) = 1/2*(1 - xi)
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elseif n == 2
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N(xi) = 1/2*(1 + xi)
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else
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j = n-1
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Pj = get_legendre_polynomial(j)
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Pjm2 = get_legendre_polynomial(j-2)
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N(xi) = 1/sqrt(2*(2*j-1))*(Pj(xi) - Pjm2(xi))
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end
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return N
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end
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"""
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Return derivative of hierarchical shape function of order N
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"""
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function get_hierarchial_basis_derivative(n)
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if n == 1
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dN(xi) = -1/2
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elseif n == 2
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dN(xi) = 1/2
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else
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j = n-1
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Pj = get_legendre_polynomial_derivative(j)
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Pjm2 = get_legendre_polynomial_derivative(j-2)
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dN(xi) = 1/sqrt(2*(2*j-1))*(Pj(xi) - Pjm2(xi))
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end
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return dN
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end
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"""
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Set degree of hierarchical element
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"""
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function set_degree(el::Hierarchical, degree)
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el.degree = degree
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end
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"""
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Get degree of hierarchical element
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"""
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function get_degree(el::Hierarchical)
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el.degree
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end
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"""
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Hierarchical 1d segment element.
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"""
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type PSeg <: Hierarchical
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connectivity :: Array{Int, 1}
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fields :: Dict{Any, Any}
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degree :: Int
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end
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PSeg(connectivity) = PSeg(connectivity, Dict{Any,Any}(), 1)
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get_number_of_basis_functions(el::Type{PSeg}) = 2
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get_number_of_basis_functions(el::PSeg) = 2 + el.degree - 1
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get_element_dimension(el::Type{PSeg}) = 1
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function get_basis(el::PSeg, xi)
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m = get_number_of_basis_functions(el)
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out = zeros(m)
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for n=1:m
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N = get_hierarchial_basis(n)
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out[n] = N(xi[1])
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end
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return out
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end
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function get_dbasisdxi(el::PSeg, xi)
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m = get_number_of_basis_functions(el)
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out = zeros(m)
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for n=1:m
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dN = get_hierarchial_basis_derivative(n)
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out[n] = dN(xi[1])
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end
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return out
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end
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