some code for hierarchical basis construction

src/hierarchical.jl

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+# This file is a part of JuliaFEM.
+# License is MIT: see https://github.com/JuliaFEM/JuliaFEM.jl/blob/master/LICENSE.md
+
+# some preliminary code for constructing hierarchical elements
+
+"""
+Return Legendgre polynomial of order n to inverval ξ ∈  [-1, 1]
+
+Maybe slow version?
+"""
+function get_legendre_polynomial_2(n)
+    bin(n, k) = prod([(n + 1 - i)/i for i=1:k])
+    P(xi) = sum([2^n*xi.^k*bin(n, k)*bin(1/2*(n+k-1), n) for k=0:n])
+    P
+end
+
+"""
+Return Legendgre polynomial of order n to inverval ξ ∈ [1, 1].
+
+Parameters
+----------
+n :: Int
+    order of polynomial
+
+Returns
+-------
+function
+    Legendgre polynomial of order n in interval ξ ∈ [-1, 1]
+
+Notes
+-----
+Uses Bonnet's recursion formula. See
+https://en.wikipedia.org/wiki/Legendre_polynomials
+"""
+function get_legendre_polynomial(n)
+    if n == 0
+        P(xi) = 1
+    elseif n == 1
+        P(xi) = xi
+    else
+        Pm1 = get_legendre_polynomial(n-1)
+        Pm2 = get_legendre_polynomial(n-2)
+        P(xi) = 1/n*((2*n-1)*xi*Pm1(xi) - (n-1)*Pm2(xi))
+    end
+    return P
+end
+
+"""
+Return derivative of Legendgre polynomial of order n to inverval ξ ∈  [-1, 1]
+"""
+function get_legendre_polynomial_derivative(n)
+    if n == 0
+        P(xi) = 0*xi
+    elseif n == 1
+        P(xi) = 0*xi + 1
+    else
+        Pm1 = get_legendre_polynomial_derivative(n-1)
+        Pm2 = get_legendre_polynomial_derivative(n-2)
+        P(xi) = 1/(n-1)*( (2*(n-1)+1)*xi.*Pm1(xi) - (n+1-1)*Pm2(xi))
+    end
+    return P
+end
+
+"""
+Return hierarchical shape function of order N
+"""
+function get_hierarchial_basis(n)
+    if n == 1
+        N(xi) = 1/2*(1 - xi)
+    elseif n == 2
+        N(xi) = 1/2*(1 + xi)
+    else
+        j = n-1
+        Pj = get_legendre_polynomial(j)
+        Pjm2 = get_legendre_polynomial(j-2)
+        N(xi) = 1/sqrt(2*(2*j-1))*(Pj(xi) - Pjm2(xi))
+    end
+    return N
+end
+
+"""
+Return derivative of hierarchical shape function of order N
+"""
+function get_hierarchial_basis_derivative(n)
+    if n == 1
+        dN(xi) = 1/2*(0*xi - 1)
+    elseif n == 2
+        dN(xi) = 1/2*(0*xi + 1)
+    else
+        j = n-1
+        Pj = get_legendre_polynomial_derivative(j)
+        Pjm2 = get_legendre_polynomial_derivative(j-2)
+        dN(xi) = 1/sqrt(2*(2*j-1))*(Pj(xi) - Pjm2(xi))
+    end
+    return dN
+end
+