JuliaFEM.jl/test/test_potential_energy.jl

# This file is a part of JuliaFEM.
# License is MIT: see https://github.com/JuliaFEM/JuliaFEM.jl/blob/master/LICENSE.md

module ElementTests

using JuliaFEM.Test

using JuliaFEM
using JuliaFEM: Equation, Quad4, IntegrationPoint, Assembly, assemble!,
                get_element, get_basis, grad, get_unknown_field_name,
                PlaneHeatProblem, Seg2, Problem, solve!,
                get_default_integration_points, Equation

abstract MyEquation <: Equation

function JuliaFEM.get_unknown_field_name(equation::MyEquation)
    return "temperature"
end

""" Diffusive heat transfer for 4-node bilinear element, with a nonlinear source term. """
type DC2D4NL <: MyEquation
    element :: Quad4
    integration_points :: Vector{IntegrationPoint}
end

function DC2D4NL(element::Quad4)
    integration_points = get_default_integration_points(element)
    if !haskey(element, "temperature")
        element["temperature"] = zeros(4)
    end
    DC2D4NL(element, integration_points)
end

function Base.size(equation::DC2D4NL)
    return (1, 4)
end

""" Nonlinear flux term. """
type DC2D2NL <: MyEquation
    element :: Seg2
    integration_points :: Vector{IntegrationPoint}
end

function DC2D2NL(element::Seg2)
    integration_points = JuliaFEM.line5()
    if !haskey(element, "temperature")
        element["temperature"] = zeros(2)
    end
    DC2D2NL(element, integration_points)
end

function Base.size(equation::DC2D2NL)
    return (1, 2)
end

""" Calculate a potential Π = Wint - Wext of system. """
function JuliaFEM.get_potential_energy(equation::DC2D4NL, ip, time; variation=nothing)
    element = get_element(equation)
    basis = get_basis(element)
    k = basis("temperature thermal conductivity", ip, time)
    f = basis("temperature load", ip, time)
    T = basis("temperature", ip, time, variation)
    c = basis("temperature nonlinearity coefficient", ip, time)
    gradT = grad(basis)("temperature", ip, time, variation)
    Wint = (k + c*T) * 1/2*vecdot(gradT, gradT)
    Wext = f*T
    return Wint - Wext
end

function JuliaFEM.get_potential_energy(equation::DC2D2NL, ip, time; variation=nothing)
    element = get_element(equation)
    basis = get_basis(element)
    T = basis("temperature", ip, time, variation)[1]
    T_ext = basis("temperature external", ip, time)[1]
    coeff = basis("temperature coefficient", ip, time)[1]
    q0 = coeff*(T_ext^4 - T^4)
    Wint = 0.0
    Wext = q0*T
    W = Wint - Wext
    return W
end

function test_potential_energy_method()

    # create model -- start
    element = Quad4([1, 2, 3, 4])
    element["geometry"] = Vector[[0.0,0.0], [1.0,0.0], [1.0,1.0], [0.0,1.0]]
    element["temperature thermal conductivity"] = 6.0
    element["temperature load"] = [0.0, 0.0, 0.0, 0.0]
    element["temperature nodal load"] = [3.0, 3.0, 0.0, 0.0]
    element["temperature nonlinearity coefficient"] = 6.0
    equation = DC2D4NL(element)
    # create model -- end

    ass = Assembly()
    info("unknown field name: $(get_unknown_field_name(equation))")

    T = zeros(4) # create workspace for solution vector
    dT = zeros(4) # 
    fd = [1, 2] # free dofs
    # start loops, in principle solve ∂r(u)/∂uΔu = -r(u) and update.
    for i=1:10
        empty!(ass)
        assemble!(ass, equation)  # calculate local matrices
        dT[fd] = full(ass.stiffness_matrix)[fd,fd] \ full(ass.force_vector)[fd]
        T += dT
        push!(element["temperature"], T) # add new increment to model
        @printf("increment %2d, |du| = %8.5f\n", i, norm(dT))
        err = last(element["temperature"])[1] - 2/3
        isapprox(err, 0.0) && break
    end
    err = last(element["temperature"])[1] - 2/3
    info("error: $err")
    @test isapprox(err, 0.0)
end


type TestProblem <: Problem
    unknown_field_name :: ASCIIString
    unknown_field_dimension :: Int
    equations :: Vector{Equation}
    element_mapping :: Dict{DataType, DataType}
end

function TestProblem(equations=[])
    element_mapping = Dict(
        Quad4 => DC2D4NL,
        Seg2 => DC2D2NL)
    TestProblem("temperature", 1, equations, element_mapping)
end

function test_potential_energy_method_2()

    # create model -- start
    N = Vector[[0.0, 0.0], [1.0, 0.0], [1.0, 1.0], [0.0, 1.0]]
    element1 = Quad4([1, 2, 3, 4])
    element1["geometry"] = Vector[N[1], N[2], N[3], N[4]]
    element1["temperature thermal conductivity"] = 6.0
    element1["temperature load"] = [0.0, 0.0, 0.0, 0.0]
    element1["temperature nonlinearity coefficient"] = [0.0, 0.0, 0.0, 0.0]
    element1["temperature"] = ones(4)

    element2 = Seg2([1, 2])
    element2["geometry"] = Vector[N[1], N[2]]
    element2["temperature coefficient"] = 3.0e-8 # ~ 5.7e-8 * 0.5
    element2["temperature external"] = 100.0
    element2["temperature"] = ones(2)
    # create model -- end
    
    equation1 = DC2D4NL(element1)
    equation2 = DC2D2NL(element2)
 
    ass = Assembly()
    info("unknown field name: $(get_unknown_field_name(equation1))")

    T = zeros(4) # create workspace for solution vector
    dT = zeros(4) # 
    fd = [1, 2] # free dofs
    # start loops, in principle solve ∂r(u)/∂uΔu = -r(u) and update.
    for i=1:10
        empty!(ass)
        assemble!(ass, equation1)
        assemble!(ass, equation2)
        dT[fd] = full(ass.stiffness_matrix)[fd,fd] \ full(ass.force_vector)[fd]
        T += dT
        push!(element1["temperature"], T)
        push!(element2["temperature"], T[fd])
        @printf("increment %2d, |du| = %8.5f\n", i, norm(dT))
        err = last(element1["temperature"])[1] - 0.5
        isapprox(err, 0.0) && break
    end

    err = last(element1["temperature"])[1] - 0.5
    info("error: $err")
    @test isapprox(err, 0.0)
    
    # @test isapprox(temp, 2.93509690572300E+00)  # tested using Code Aster
end

end