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Jukka Aho
2015-10-23 01:11:45 +03:00
parent 6258fd00ba
commit 9b5b88116b
2 changed files with 183 additions and 126 deletions
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notebooks/2015-06-25-elasticity-solver-example.ipynb
+178 -126
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@@ -4,45 +4,39 @@
"cell_type": "markdown",
"metadata": {},
"source": [
"# Solving elasticity problems using JuliaFEM\n",
"# Solving elasticity equations using JuliaFEM\n",
"\n",
"Author(s): Jukka Aho\n",
"\n",
"**Abstract**: Elasticity equations design notes."
"**Abstract**: Elasticity equations design notes.\n",
"\n",
"##Weak form\n",
"\n",
"Given function spaces\n",
"\\begin{align}\n",
"\\boldsymbol{\\mathcal{U}} & =\\left\\{ \\boldsymbol{u}\\in H^{1}\\left(\\Omega\\right)|\\boldsymbol{u}\\left(\\boldsymbol{X},t\\right)=\\hat{\\boldsymbol{u}}\\left(\\boldsymbol{X},t\\right)\\text{ on }\\Gamma_{\\mathrm{u}}\\right\\} ,\\\\\n",
"\\boldsymbol{\\mathcal{V}} & =\\left\\{ \\delta\\boldsymbol{u}\\in H^{1}\\left(\\Omega\\right)|\\delta\\boldsymbol{u}\\left(\\boldsymbol{X}\\right)=0\\text{ on }\\Gamma_{\\mathrm{u}}\\right\\} ,\n",
"\\end{align}\n",
"find $\\boldsymbol{u}\\in\\boldsymbol{\\mathcal{U}}$ such that\n",
"\\begin{equation}\n",
"\\delta\\mathcal{W}:=\\int_{\\Omega_{0}}\\rho_{0}\\ddot{\\boldsymbol{u}}\\cdot\\delta\\boldsymbol{u}\\,\\mathrm{d}V_{0}+\\int_{\\Omega_{0}}\\boldsymbol{S}:\\delta\\boldsymbol{E}\\,\\mathrm{d}V_{0}-\\int_{\\Omega_{0}}\\hat{\\boldsymbol{b}}_{0}\\cdot\\delta\\boldsymbol{u}\\,\\mathrm{d}V_{0}-\\int_{\\Gamma_{\\sigma}}\\hat{\\boldsymbol{t}}_{0}\\cdot\\delta\\boldsymbol{u}\\,\\mathrm{d}A_{0} =0 \\qquad\\forall\\delta\\boldsymbol{u}\\in\\boldsymbol{\\mathcal{V}}\n",
"\\end{equation}"
]
},
{
"cell_type": "code",
"execution_count": 1,
"execution_count": 2,
"metadata": {
"collapsed": false
},
"outputs": [
{
"name": "stderr",
"output_type": "stream",
"text": [
"WARNING: Base.String is deprecated, use AbstractString instead.\n",
" likely near /home/jukka/.julia/v0.4/Logging/src/Logging.jl:24\n",
"WARNING: Base.String is deprecated, use AbstractString instead.\n",
" likely near /home/jukka/.julia/v0.4/Logging/src/Logging.jl:24\n",
"WARNING: Base.String is deprecated, use AbstractString instead.\n",
" likely near /home/jukka/.julia/v0.4/Logging/src/Logging.jl:24\n",
"WARNING: Base.String is deprecated, use AbstractString instead.\n",
" likely near /home/jukka/.julia/v0.4/Logging/src/Logging.jl:40\n",
"WARNING: Base.String is deprecated, use AbstractString instead.\n",
" likely near /home/jukka/.julia/v0.4/Logging/src/Logging.jl:40\n",
"WARNING: Base.String is deprecated, use AbstractString instead.\n",
" likely near /home/jukka/.julia/v0.4/Logging/src/Logging.jl:40\n"
]
},
{
"data": {
"text/plain": [
"Logger(root,DEBUG,Base.PipeEndpoint(open, 0 bytes waiting),root)"
]
},
"execution_count": 1,
"execution_count": 2,
"metadata": {},
"output_type": "execute_result"
}
@@ -58,7 +52,7 @@
},
{
"cell_type": "code",
"execution_count": 2,
"execution_count": 3,
"metadata": {
"collapsed": false
},
@@ -69,7 +63,7 @@
"CPS4"
]
},
"execution_count": 2,
"execution_count": 3,
"metadata": {},
"output_type": "execute_result"
}
@@ -107,7 +101,7 @@
},
{
"cell_type": "code",
"execution_count": 40,
"execution_count": 4,
"metadata": {
"collapsed": false
},
@@ -115,79 +109,58 @@
{
"data": {
"text/plain": [
"8-element Array{Float64,1}:\n",
" -189.267 \n",
" 26.1073 \n",
" 5.7554e-13 \n",
" 2.6894e-12 \n",
" -8.73968e-13\n",
" 20.0 \n",
" 189.267 \n",
" -46.1073 "
"has_rhs (generic function with 4 methods)"
]
},
"execution_count": 40,
"execution_count": 4,
"metadata": {},
"output_type": "execute_result"
}
],
"source": [
"function get_lhs_and_rhs(equation::CPS4, ip, time)\n",
" # boilerplate code start\n",
" element = get_element(equation)\n",
" N = FEM.get_basis(element)\n",
" dN = FEM.diff(N)(ip.xi)\n",
" geometry = element[\"geometry\"](time)\n",
" basis = FEM.get_basis(element)\n",
" dbasis = FEM.diff(basis)(ip.xi)\n",
" grad(u) = dbasis*u*inv(dbasis*geometry)\n",
" # boilerplate code end -- replace with a macro?\n",
"\n",
" # fields at time \n",
" X = element[\"geometry\"](time)\n",
" u = element[\"displacement\"](time)\n",
" young = element[\"young\"](time)\n",
" poisson = element[\"poisson\"](time)\n",
" # interpolate fields in temporal dimension\n",
" young = element[\"youngs modulus\"](time)\n",
" poisson = element[\"poissons ratio\"](time)\n",
" displacement = element[\"displacement\"](time)\n",
"\n",
" # material\n",
" young = interpolate(N, young, ip)\n",
" poisson = interpolate(N, poisson, ip)\n",
" # interpolate material in spatial dimension\n",
" young = interpolate(basis, young, ip)\n",
" poisson = interpolate(basis, poisson, ip)\n",
" mu = young/(2*(1+poisson))\n",
" lambda = young*poisson/((1+poisson)*(1-2*poisson))\n",
" lambda = 2*lambda*mu/(lambda + 2*mu)\n",
" lambda = 2*lambda*mu/(lambda + 2*mu) # <- correction for 2d\n",
"\n",
" gradw = dN*inv(dN*X)\n",
" grad(u) = dN*u*inv(dN*(X+u))\n",
" Grad(u) = dN*u*inv(dN*X)\n",
"\n",
" function calc_R(data::Vector)\n",
" # create new field similar to displacement and fill it with data\n",
" f = similar(u, data)\n",
" # kinematics\n",
" F = I + Grad(f) # deformation gradient\n",
" C = F'*F\n",
" E = 1/2*(C - I) # strain\n",
" function W(data::Vector)\n",
" Wint = 0.0\n",
" # create new field u, similar to field displacement, and fill it with data\n",
" u = similar(displacement, data)\n",
" F = I + grad(u) # deformation gradient\n",
" E = 1/2*(F'*F - I) # strain\n",
" S = 2*mu*E + lambda*trace(E)*I # stress\n",
" P = F*S\n",
" return (P*gradw')[:]\n",
" Wint += 1/2*trace(S*E')\n",
"\n",
" Wext = 0.0\n",
" # any volume load?\n",
" if haskey(element, \"displacement volume load\")\n",
" b = interpolate(element, \"displacement volume load\", ip, time)\n",
" δu = interpolate(basis, u, ip)\n",
" Wext += dot(b, δu)\n",
" end\n",
" return Wint - Wext\n",
" end\n",
"\n",
" function calc_Wint(data::Vector)\n",
" # create new field similar to displacement and fill it with data\n",
" δu = similar(u, data)\n",
" # kinematics\n",
" F = I + Grad(δu) # deformation gradient\n",
" C = F'*F\n",
" E = 1/2*(C - I) # strain\n",
" #E = 1/2*(Grad(δu)' + Grad(δu) + Grad(δu)'*Grad(δu))\n",
" S = 2*mu*E + lambda*trace(E)*I # stress\n",
" δF = Grad(δu)\n",
" #δɛ = 1/2*(δF' + δF)\n",
" #δE = F'*δɛ*F\n",
" #δE = 1/2*(F'*δF + δF'*F)\n",
" #δE = 1/2*(Grad(δu)' + Grad(δu) + Grad(δu)'*Grad(δu))\n",
" δE = 1/2*E # ...???? energy 2x\n",
"\n",
" return trace(S*δE')\n",
" end\n",
"\n",
" R = ForwardDiff.gradient(calc_Wint, u[:])\n",
" Kt = ForwardDiff.hessian(calc_Wint, u[:])\n",
" return Kt, -R\n",
" R = ForwardDiff.gradient(W, displacement[:])\n",
" K = ForwardDiff.hessian(W, displacement[:])\n",
" return K, -R\n",
"end\n",
"\n",
"function JuliaFEM.get_lhs(equation::CPS4, ip, time)\n",
@@ -198,14 +171,12 @@
"end\n",
"\n",
"JuliaFEM.has_lhs(equation::CPS4) = true\n",
"JuliaFEM.has_rhs(equation::CPS4) = true\n",
"\n",
"JuliaFEM.integrate_rhs(equation, Inf)"
"JuliaFEM.has_rhs(equation::CPS4) = true"
]
},
{
"cell_type": "code",
"execution_count": 41,
"execution_count": 5,
"metadata": {
"collapsed": false
},
@@ -214,19 +185,90 @@
"name": "stdout",
"output_type": "stream",
"text": [
"testing primary field with point load versus code aster solution\n",
"increment 1, norm = 5.77653, Wint = 0.000, Wext = 81.525, |Wint-Wext| = 81.52516\n",
"increment 2, norm = 0.99988, Wint = 173.895, Wext = 79.037, |Wint-Wext| = 94.85823\n",
"increment 3, norm = 0.28354, Wint = 87.084, Wext = 82.175, |Wint-Wext| = 4.90953\n",
"increment 4, norm = 0.07071, Wint = 82.548, Wext = 83.100, |Wint-Wext| = 0.55141\n",
"increment 5, norm = 0.00082, Wint = 83.116, Wext = 83.109, |Wint-Wext| = 0.00644\n",
"increment 6, norm = 0.00000, Wint = 83.109, Wext = 83.109, |Wint-Wext| = 0.00000\n",
"increment 7, norm = 0.00000, Wint = 83.109, Wext = 83.109, |Wint-Wext| = 0.00000\n",
"increment 8, norm = 0.00000, Wint = 83.109, Wext = 83.109, |Wint-Wext| = 0.00000\n",
"increment 9, norm = 0.00000, Wint = 83.109, Wext = 83.109, |Wint-Wext| = 0.00000\n",
"increment 10, norm = 0.00000, Wint = 83.109, Wext = 83.109, |Wint-Wext| = 0.00000\n",
"elapsed time: 4.257759666 seconds\n",
"1 fact verified.\n"
]
},
{
"data": {
"text/plain": [
"delayed_handler (generic function with 4 methods)"
]
},
"execution_count": 5,
"metadata": {},
"output_type": "execute_result"
}
],
"source": [
"facts(\"testing primary field with point load versus code aster solution\") do\n",
" element = Quad4([1, 2, 3, 4])\n",
" push!(element, FieldSet(\"geometry\", [Field(0.0, Vector[[0.0, 0.0], [10.0, 0.0], [10.0, 1.0], [0.0, 1.0]])]))\n",
" push!(element, FieldSet(\"youngs modulus\", [Field(0.0, 500.0)]))\n",
" push!(element, FieldSet(\"poissons ratio\", [Field(0.0, 0.3)]))\n",
" equation = CPS4(element)\n",
"\n",
" u0 = Field(0.0, Vector[[0.0, 0.0], [0.0, 0.0], [0.0, 0.0], [0.0, 0.0]])\n",
" push!(element[\"displacement\"], u0)\n",
" u = zeros(8)\n",
" du = zeros(8)\n",
" fd = [3, 4, 5, 6]\n",
" f = zeros(8)\n",
" f[6] = -20.0\n",
" tic()\n",
" for i=1:10\n",
" A = JuliaFEM.integrate_lhs(equation, 1.0)\n",
" b = JuliaFEM.integrate_rhs(equation, 1.0)\n",
" du[fd] = A[fd,fd] \\ (b[fd]+f[fd])\n",
" u += du\n",
" new_field = similar(u0, u)\n",
" new_field.time = 1.0\n",
" new_field.increment = i\n",
" push!(element[\"displacement\"], new_field)\n",
" Wint = (-b'*u)[1]\n",
" Wext = (f'*u)[1]\n",
" W = abs(Wint-Wext)\n",
" @printf(\"increment %2d, norm = %8.5f, Wint = %8.3f, Wext = %8.3f, |Wint-Wext| = %8.5f\\n\", i, norm(du), Wint, Wext, W)\n",
" end\n",
" toc()\n",
" # verified using Code Aster.\n",
" @fact interpolate(element, \"displacement\", [1.0, 1.0], Inf)[2] --> roughly(-4.15546385452579E+00)\n",
"end"
]
},
{
"cell_type": "code",
"execution_count": 6,
"metadata": {
"collapsed": false
},
"outputs": [
{
"name": "stdout",
"output_type": "stream",
"text": [
"testing primary field with volume load versus code aster solution\n",
"increment "
]
},
{
"data": {
"text/plain": [
"Success :: (line:-1) :: fact was true\n",
" Expression: (interpolate(element,\"displacement\",[1.0,1.0],Inf))[2] --> roughly(-4.15546385452579)\n",
" Expected: -4.15546385452579\n",
" Occurred: -4.155463854525783"
"delayed_handler (generic function with 4 methods)"
]
},
"execution_count": 41,
"execution_count": 6,
"metadata": {},
"output_type": "execute_result"
},
@@ -234,47 +276,57 @@
"name": "stdout",
"output_type": "stream",
"text": [
"1, norm = 5.77653, Wint = 0.000, Wext = 81.525, |Wint-Wext| = 81.52516\n",
"increment 2, norm = 0.99988, Wint = 173.895, Wext = 79.037, |Wint-Wext| = 94.85823\n",
"increment 3, norm = 0.28354, Wint = 87.084, Wext = 82.175, |Wint-Wext| = 4.90953\n",
"increment 4, norm = 0.07071, Wint = 82.548, Wext = 83.100, |Wint-Wext| = 0.55141\n",
"increment 5, norm = 0.00082, Wint = 83.116, Wext = 83.109, |Wint-Wext| = 0.00644\n",
"increment 6, norm = 0.00000, Wint = 83.109, Wext = 83.109, |Wint-Wext| = 0.00000\n",
"83.10927709051566\n"
"1, norm = 14.44128, Wint = -509.167, Wext = 0.000, |Wint-Wext| = 509.16667\n",
"increment 2, norm = 4.01742, Wint = 4335.728, Wext = 0.000, |Wint-Wext| = 4335.72801\n",
"increment 3, norm = 1.54645, Wint = 705.913, Wext = 0.000, |Wint-Wext| = 705.91292\n",
"increment 4, norm = 1.12361, Wint = 60.492, Wext = 0.000, |Wint-Wext| = 60.49208\n",
"increment 5, norm = 0.79119, Wint = -1.486, Wext = 0.000, |Wint-Wext| = 1.48555\n",
"increment 6, norm = 0.12733, Wint = 5.743, Wext = 0.000, |Wint-Wext| = 5.74331\n",
"increment 7, norm = 0.00725, Wint = 0.080, Wext = 0.000, |Wint-Wext| = 0.07997\n",
"increment 8, norm = 0.00001, Wint = 0.000, Wext = 0.000, |Wint-Wext| = 0.00045\n",
"increment 9, norm = 0.00000, Wint = 0.000, Wext = 0.000, |Wint-Wext| = 0.00000\n",
"increment 10, norm = 0.00000, Wint = 0.000, Wext = 0.000, |Wint-Wext| = 0.00000\n",
"elapsed time: 0.068575305 seconds\n",
"1 fact verified.\n"
]
}
],
"source": [
"element = Quad4([1, 2, 3, 4])\n",
"push!(element, FieldSet(\"young\", [Field(0.0, 500.0)]))\n",
"push!(element, FieldSet(\"geometry\", [Field(0.0, Vector[[0.0, 0.0], [10.0, 0.0], [10.0, 1.0], [0.0, 1.0]])]))\n",
"push!(element, FieldSet(\"poisson\", [Field(0.0, 0.3)]))\n",
"equation = CPS4(element)\n",
"facts(\"testing primary field with volume load versus code aster solution\") do\n",
" element = Quad4([1, 2, 3, 4])\n",
" push!(element, FieldSet(\"geometry\", [Field(0.0, Vector[[0.0, 0.0], [10.0, 0.0], [10.0, 1.0], [0.0, 1.0]])]))\n",
" push!(element, FieldSet(\"youngs modulus\", [Field(0.0, 500.0)]))\n",
" push!(element, FieldSet(\"poissons ratio\", [Field(0.0, 0.3)]))\n",
" push!(element, FieldSet(\"displacement volume load\",\n",
" [Field(0.0, Vector[[0.0, -10.0], [0.0, -10.0], [0.0, -10.0], [0.0, -10.0]])]))\n",
" equation = CPS4(element)\n",
"\n",
"u0 = Field(0.0, Vector[[0.0, 0.0], [0.0, 0.0], [0.0, 0.0], [0.0, 0.0]])\n",
"push!(element[\"displacement\"], u0)\n",
"u = zeros(8)\n",
"du = zeros(8)\n",
"fd = [3, 4, 5, 6]\n",
"f = zeros(8)\n",
"f[6] = -20.0\n",
"for i=1:6\n",
" A = JuliaFEM.integrate_lhs(equation, 1.0)\n",
" b = JuliaFEM.integrate_rhs(equation, 1.0)\n",
" du[fd] = A[fd,fd] \\ (b[fd]+f[fd])\n",
" u += du\n",
" new_field = similar(u0, u)\n",
" new_field.time = 1.0\n",
" new_field.increment = i\n",
" push!(element[\"displacement\"], new_field)\n",
" Wint = (-b'*u)[1]\n",
" Wext = (f'*u)[1]\n",
" W = abs(Wint-Wext)\n",
" @printf(\"increment %2d, norm = %8.5f, Wint = %8.3f, Wext = %8.3f, |Wint-Wext| = %8.5f\\n\", i, norm(du), Wint, Wext, W)\n",
"end\n",
"println((f'*u)[1])\n",
"@fact interpolate(element, \"displacement\", [1.0, 1.0], Inf)[2] --> roughly(-4.15546385452579E+00)\n",
"#interpolate(element, \"displacement\", [1.0, 1.0], Inf)[2], (f'*u)[1]"
" u0 = Field(0.0, Vector[[0.0, 0.0], [0.0, 0.0], [0.0, 0.0], [0.0, 0.0]])\n",
" push!(element[\"displacement\"], u0)\n",
" u = zeros(8)\n",
" du = zeros(8)\n",
" fd = [3, 4, 5, 6]\n",
" f = zeros(8)\n",
" f[6] = -20.0*0\n",
" tic()\n",
" for i=1:10\n",
" A = JuliaFEM.integrate_lhs(equation, 1.0)\n",
" b = JuliaFEM.integrate_rhs(equation, 1.0)\n",
" du[fd] = A[fd,fd] \\ (b[fd]+f[fd])\n",
" u += du\n",
" new_field = similar(u0, u)\n",
" new_field.time = 1.0\n",
" new_field.increment = i\n",
" push!(element[\"displacement\"], new_field)\n",
" Wint = (-b'*u)[1]\n",
" Wext = (f'*u)[1]\n",
" W = abs(Wint-Wext)\n",
" @printf(\"increment %2d, norm = %8.5f, Wint = %8.3f, Wext = %8.3f, |Wint-Wext| = %8.5f\\n\", i, norm(du), Wint, Wext, W)\n",
" end\n",
" toc()\n",
" # verified using Code Aster.\n",
" @fact interpolate(element, \"displacement\", [1.0, 1.0], Inf)[2] --> roughly(-8.77303119819776E+00)\n",
"end"
]
},
{
src/elements.jl
+5
View File
@@ -186,6 +186,11 @@ function dinterpolate(element::Element, field_name, ip::IntegrationPoint, time::
dinterpolate(element, field_name, ip.xi, time)
end
""" Check does fieldset exist. """
function Base.haskey(element::Element, what)
haskey(element.fields, symbol(what))
end
"""
Get jacobian of element evaluated at point ξ on element in reference configuration.