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2026-09-23 02:59:52 +00:00 · 2015-09-14 23:32:46 +03:00
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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
+
+# Lagrange (Continous Galerkin) finite elements
+
+abstract CG <: Element
+
+"""
+Create new element with element_name to family element_family
+
+Examples
+--------
+>>> @create_element(Seg2, CG, "2 node linear segment")
+"""
+macro create_element(element_name, element_family, element_description)
+#   Logging.debug("Creating element ", element_name, ": ", element_description, "\n")
+    eltype = esc(element_name)
+    elfam = esc(element_family)
+    quote
+        global get_element_description
+        type $eltype <: $elfam
+            connectivity :: Array{Int, 1}
+            fields :: Dict{Any, Any}
+        end
+        $eltype(connectivity) = $eltype(connectivity, Dict{Any, Any}())
+        get_element_description(el::Type{$eltype}) = $element_description
+    end
+end
+
+"""
+Given polynomial P and coordinates of reference element, calculate
+Lagrange basis function and partial derivatives.
+"""
+function calculate_lagrange_basis(P, X)
+    dim, nbasis = size(X)
+    A = zeros(nbasis, nbasis)
+    for i=1:nbasis
+        A[i,:] = P(X[:, i])
+    end
+#   Logging.debug("Calculating inverse of A")
+    invA = inv(A)'
+    basis(xi) = invA*P(xi)
+    dbasisdxi = ForwardDiff.jacobian(basis)
+    basis, dbasisdxi
+end
+
+"""
+Assign Lagrange basis for element.
+"""
+macro create_lagrange_basis(element_name, X, P)
+
+#   Logging.debug("Creating Lagrange basis for element ", element_name, ". ")
+    eltype = esc(element_name)
+
+    quote
+
+        global get_number_of_basis_functions, get_element_dimension
+        global get_basis, get_dbasisdxi
+
+        dim = size($X, 1)
+        nbasis = size($X, 2)
+#       Logging.debug("Number of basis functions: ", nbasis, ". ")
+#       Logging.debug("Element dimension: ", dim)
+
+        get_number_of_basis_functions(el::Type{$(esc(element_name))}) = nbasis
+        get_number_of_basis_functions(el::$(esc(element_name))) = nbasis
+        get_element_dimension(el::$(esc(element_name))) = dim
+
+        basis, dbasisdxi = calculate_lagrange_basis($P, $X)
+        get_basis(el::$eltype, xi) = basis(xi)
+        get_dbasisdxi(el::$eltype, xi) = dbasisdxi(xi)
+#       Logging.debug("Element ", $element_name, " created.")
+    end
+
+end
+
+# 0d Lagrange element
+
+@create_element(Point1, CG, "1 node point element")
+
+# 1d Lagrange elements
+
+@create_element(Seg2, CG, "2 node linear line element")
+@create_lagrange_basis(Seg2, [-1.0 1.0], (xi) -> [1.0, xi[1]])
+
+@create_element(Seg3, CG, "3 node quadratic line element")
+@create_lagrange_basis(Seg3, [-1.0 1.0 0.0], (xi) -> [1.0, xi[1], xi[1]^2])
+
+# 2d Lagrange elements
+
+@create_element(Quad4, CG, "4 node bilinear quadrangle element")
+@create_lagrange_basis(Quad4,
+    [-1.0  1.0 1.0 -1.0
+     -1.0 -1.0 1.0  1.0],
+    (xi) -> [1.0, xi[1], xi[2], xi[1]*xi[2]])
+
+# 3d Lagrange elements
+
+@create_element(Tet10, CG, "10 node quadratic tetrahedron")
+@create_lagrange_basis(Tet10,
+    [0.0 1.0 0.0 0.0 0.5 0.5 0.0 0.0 0.5 0.0
+     0.0 0.0 1.0 0.0 0.0 0.5 0.5 0.0 0.0 0.5
+     0.0 0.0 0.0 1.0 0.0 0.0 0.0 0.5 0.5 0.5],
+    (xi) -> [    1.0,   xi[1],       xi[2],       xi[3],     xi[1]^2,
+             xi[2]^2, xi[3]^2, xi[1]*xi[2], xi[2]*xi[3], xi[3]*xi[1]])
+