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analytical tests, hollow sphere and radial displacement + longitudinal vibration of rod (modal analysis)
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Jukka Aho
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# 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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using JuliaFEM
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using JuliaFEM.Preprocess
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using JuliaFEM.Testing
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#=
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test subjects:
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- surface pressure load in curved surface
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- verification of elements wedge6 and wedge15
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from Code Aster:
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N38 -8.85861895037377E-01 -3.46944695195361E-18 -3.46944695195361E-18
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coords of N38 = (1.0, 0.0, 0.0)
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Analytical solution
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uᵣ(r) = b³p/(2Er²(a³-b³)) * (a³(ν+1) + r³(-4ν+2)), where
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a = inner surface radial distance, b = outer surface ...
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if a=0.9, b=1.0, ν=1/3, E = 24580 and p = 7317 equation yields
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-9/10 for radial displacement
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=#
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@testset """1/8 hollow sphere with surface load""" begin
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mesh_file = Pkg.dir("JuliaFEM") * "/test/testdata/primitives.med"
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mesh = aster_read_mesh(mesh_file, "HOLLOWSPHERE8_WEDGE6")
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body = Problem(Elasticity, "hollow sphere 1/8 model", 3)
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body.elements = create_elements(mesh, "HOLLOWSPHERE8")
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update!(body, "youngs modulus", 24580.0)
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update!(body, "poissons ratio", 1/3)
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bc = Problem(Dirichlet, "symmetry bc", 3, "displacement")
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el1 = create_elements(mesh, "FACE1")
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update!(el1, "displacement 3", 0.0)
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el2 = create_elements(mesh, "FACE2")
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update!(el2, "displacement 2", 0.0)
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el3 = create_elements(mesh, "FACE3")
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update!(el3, "displacement 1", 0.0)
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bc.elements = [el1; el2; el3]
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lo = Problem(Elasticity, "pressure load", 3)
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lo.elements = create_elements(mesh, "OUTER")
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update!(lo, "surface pressure", 7317.0)
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solver = LinearSolver(body, bc, lo)
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solver()
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X = lo("geometry")
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u = lo("displacement")
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nids = sort(collect(keys(X)))
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umag = Float64[norm(u[id]) for id in nids]
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um = mean(umag)
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info("mean umag = $um")
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info("std umag = ", std(umag))
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rtol = norm(um - 0.9) / max(norm(um), 0.9) * 100.0
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info("rtol = $rtol")
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@test rtol < 1.5 # percents
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u_CA = [-8.85861895037377E-01, -3.46944695195361E-18, -3.46944695195361E-18]
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rtol = norm(u[38] - u_CA) / max(norm(u[38]), norm(u_CA)) * 100.0
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info("rel diff to CA = $rtol %")
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@test isapprox(u[38], u_CA)
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end
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@@ -0,0 +1,142 @@
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# 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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using JuliaFEM
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using JuliaFEM.Preprocess
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using JuliaFEM.Postprocess
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using JuliaFEM.Testing
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@testset "calculate cross-sectional properties" begin
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mesh_file = Pkg.dir("JuliaFEM") * "/test/testdata/primitives.med"
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mesh = aster_read_mesh(mesh_file, "CYLINDER_20_TET4")
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# calculate cross-sectional properties A and Iₓ
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fixed1 = Problem(Dirichlet, "left support", 3, "displacement")
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fixed1.elements = create_elements(mesh, "FACE1")
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A = calculate_area(fixed1)
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info("cross-section area: $A")
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# real area is π
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@test isapprox(A, pi; rtol=0.1)
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Xc = calculate_center_of_mass(fixed1)
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info("center of mass: $Xc")
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@test isapprox(Xc, [0.0, 0.0, 0.0]; atol=1.0e-12)
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I = calculate_second_moment_of_mass(fixed1)
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info("moments:")
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info(I)
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I_expected = zeros(3, 3)
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I_expected[2,2] = I_expected[3,3] = pi/4
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rtol = norm(I[2,2]-I_expected[2,2]) / max(I[2,2],I_expected[2,2])
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info("I rtol = $rtol")
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@test isapprox(I, I_expected; rtol = 0.2)
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end
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#=
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test subjects:
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- calculate cross-sectional properties
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- modal analysis with known solution
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Fixed-fixed solution is ωᵢ = λᵢ²√(EI/ρA) , where λᵢ = cosh(λᵢℓ)cos(λᵢℓ)
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1: 4.730040744862704
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2: 7.853204624095838
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3: 10.995607838001671
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[1] De Silva, Clarence W. Vibration: fundamentals and practice. CRC press, 2006, p.355
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=#
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@testset "long rod under point load" begin
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mesh_file = Pkg.dir("JuliaFEM") * "/test/testdata/primitives.med"
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mesh = aster_read_mesh(mesh_file, "CYLINDER_20_TET10")
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# for (id, coords) in mesh.nodes
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# mesh.nodes[id][1] *= 5.0
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# end
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body = Problem(Elasticity, "rod", 3)
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body.elements = create_elements(mesh, "CYLINDER")
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E = 50475.44814745859
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rho = 1.0
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update!(body.elements, "youngs modulus", E)
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update!(body.elements, "poissons ratio", 0.3)
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update!(body.elements, "density", rho)
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# calculate cross-sectional properties A and Iₓ
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fixed1 = Problem(Dirichlet, "left support", 3, "displacement")
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fixed1.elements = create_elements(mesh, "FACE1")
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update!(fixed1.elements, "displacement 1", 0.0)
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update!(fixed1.elements, "displacement 2", 0.0)
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update!(fixed1.elements, "displacement 3", 0.0)
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fixed2 = Problem(Dirichlet, "right support", 3, "displacement")
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fixed2.elements = create_elements(mesh, "FACE2")
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update!(fixed2.elements, "displacement 1", 0.0)
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update!(fixed2.elements, "displacement 2", 0.0)
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update!(fixed2.elements, "displacement 3", 0.0)
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A = calculate_area(fixed1)
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info("cross-section area: $A")
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# using SALOME / SMESH, A = 2.82843
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# real area is π
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@test isapprox(A, pi; rtol=0.1)
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Xc = calculate_center_of_mass(fixed1)
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info("center of mass: $Xc")
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@test isapprox(Xc, [0.0, 0.0, 0.0]; atol=1.0e-5)
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I = calculate_second_moment_of_mass(fixed1)
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info("moments:")
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info(I)
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I_expected = zeros(3, 3)
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r = 1.0
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I_expected[2,2] = I_expected[3,3] = pi/4*r^2
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rtol = norm(I[2,2]-I_expected[2,2]) / max(I[2,2],I_expected[2,2])
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info("I rtol = $rtol")
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@test isapprox(I, I_expected; rtol = 0.2)
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#=
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# apply transform Tx + b, in this case move cross-section to
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# xy-plane from yz-plane, i.e.
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# x₁ = y₂
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# y₁ = z₂
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T = [
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0.0 1.0 0.0
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0.0 0.0 1.0]
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b = [0.0, 0.0]
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X 1 = first(cross_section)("geometry", [1/3, 1/3], 0.0)
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apply_affine_transform!(cross_section, T, b)
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X2 = first(cross_section)("geometry", [1/3, 1/3], 0.0)
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info("X1 = $X1, X2 = $X2")
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@test isapprox(T*X1+b, X2)
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=#
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c = sqrt(E*I[2,2]/(rho*A))
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info("c = $c")
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# analytical solution is
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l = 20.0
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r = 1.0
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la = 4.730040744862704/l
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# semi-analytical (c numerical)
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freq_sa = (c*la^2)/(2*pi)
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info("freq_sa = $freq_sa")
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A = pi*r^2
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I = pi/4*r^4
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c = sqrt(E*I/(rho*A))
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info("c analytical = $c")
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freq_a = (c*la^2)/(2*pi)
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info("freq_a = $freq_a")
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solver = Solver(Modal, body, fixed1, fixed2)
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solver()
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freqs = keys(body["displacement"])
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rtol1 = norm(freq_sa - freqs[2])/max(freq_sa, freqs[2])
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rtol2 = norm(freq_a - freqs[2])/max(freq_a, freqs[2])
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info("rtol 1 = $rtol1, rtol 2 = $rtol2")
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@test rtol2 < 1.0e-2
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#=
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result = XDMF()
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for (i, freq) in enumerate(freqs)
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isapprox(freq, 0.0) && continue
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info("$i freq: $freq")
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xdmf_new_result!(result, body, freq)
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xdmf_save_field!(result, body, freq, "displacement"; field_type="Vector")
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end
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xdmf_save!(result, "/tmp/rod_nf.xmf")
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=#
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end
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