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Minor update. Again working on elasticity solver notebook.

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This commit is contained in:
Jukka Aho
2015-08-17 21:46:58 +03:00
parent 45ea803136
commit 21c9c19f1d
2 changed files with 227 additions and 561 deletions
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notebooks/2015-06-25-elasticity-solver-example.ipynb
+226 -560
View File
@@ -36,7 +36,7 @@
},
{
"cell_type": "code",
"execution_count": 1,
"execution_count": 2,
"metadata": {
"collapsed": false
},
@@ -47,35 +47,14 @@
"Logger(root,DEBUG,Pipe(open, 0 bytes waiting),root)"
]
},
"execution_count": 1,
"execution_count": 2,
"metadata": {},
"output_type": "execute_result"
},
{
"name": "stderr",
"output_type": "stream",
"text": [
" in depwarn at /Applications/Julia-0.4.0-dev-539c818c4e.app/Contents/Resources/julia/lib/julia/sys.dylib\n",
" in int32 at deprecated.jl:49\n",
" in recv at /Users/jukka/.julia/v0.4/ZMQ/src/ZMQ.jl:617\n",
" in recv_ipython at /Users/jukka/.julia/v0.4/IJulia/src/msg.jl:63\n",
" in eventloop at /Users/jukka/.julia/v0.4/IJulia/src/IJulia.jl:120\n",
" in anonymous at task.jl:365\n",
"while loading /Users/jukka/.julia/v0.4/IJulia/src/kernel.jl, in expression starting on line 35\n",
"WARNING: int32(x) is deprecated, use Int32(x) instead.\n",
" in depwarn at /Applications/Julia-0.4.0-dev-539c818c4e.app/Contents/Resources/julia/lib/julia/sys.dylib\n",
" in int32 at deprecated.jl:49\n",
" in recv at /Users/jukka/.julia/v0.4/ZMQ/src/ZMQ.jl:617\n",
" in recv_ipython at /Users/jukka/.julia/v0.4/IJulia/src/msg.jl:63\n",
" in eventloop at /Users/jukka/.julia/v0.4/IJulia/src/IJulia.jl:120\n",
" in anonymous at task.jl:365\n",
"while loading /Users/jukka/.julia/v0.4/IJulia/src/kernel.jl, in expression starting on line 35\n"
]
}
],
"source": [
"using JuliaFEM\n",
"using Logging\n",
"using ForwardDiff\n",
"Logging.configure(level=DEBUG)"
]
},
@@ -89,12 +68,12 @@
"\n",
"*Design principle 5*: we use 4 space indentation like in Python.\n",
"\n",
"First we write some elementary functions to calculate stiffness matrix."
"Our task is: for given $\\mathbf{u}$ calculate $\\mathbf{R}(\\mathbf{u}) = \\mathbf{T}(\\mathbf{u}) - \\mathbf{F}(\\mathbf{u})$ and it's partial derivative with respect to $\\mathbf{u}$, i.e. $\\partial \\mathbf{R}(\\mathbf{u}) / \\partial \\mathbf{u}$."
]
},
{
"cell_type": "code",
"execution_count": 2,
"execution_count": 3,
"metadata": {
"collapsed": false
},
@@ -102,144 +81,56 @@
{
"data": {
"text/plain": [
"calc_local_matrices! (generic function with 1 method)"
"jacobian (generic function with 1 method)"
]
},
"execution_count": 2,
"execution_count": 3,
"metadata": {},
"output_type": "execute_result"
}
],
"source": [
"\"\"\"\n",
"Calculate local tangent stiffness matrix and residual force vector\n",
"R = T - F for elasticity problem.\n",
"\n",
"Parameters\n",
"----------\n",
"X : Element coordinates\n",
"u : Displacement field\n",
"R : Residual force vector\n",
"K : Tangent stiffness matrix\n",
"basis : Basis functions\n",
"dbasis : Derivative of basis functions\n",
"lambda : Material parameter\n",
"mu : Material parameter\n",
"ipoints : integration points\n",
"iweights : integration weights\n",
"\n",
"Returns\n",
"-------\n",
"None\n",
"\n",
"Notes\n",
"-----\n",
"If material parameters are given in list, they are interpolated to gauss\n",
"points using shape functions.\n",
"\n",
"Examples\n",
"--------\n",
"\n",
"\"\"\"\n",
"function calc_local_matrices2!(X, u, R, K, basis, dbasis, lambda_, mu_, ipoints, iweights)\n",
" dim, nnodes = size(X)\n",
" I = eye(dim)\n",
" R[:,:] = 0.0\n",
" K[:,:] = 0.0\n",
"\n",
" dF = zeros(dim, dim)\n",
"\n",
" for m = 1:length(iweights)\n",
" w = iweights[m]\n",
" xi = ipoints[m, :]\n",
" # calculate material parameters\n",
" lambda = typeof(lambda_) == Float64 ? lambda_ : dot(lambda_, basis(xi))\n",
" mu = typeof(mu_) == Float64 ? mu_ : dot(mu_, basis(xi))\n",
" Jt = X*dbasis(xi)\n",
" detJ = det(Jt)\n",
" dbasisdX = dbasis(xi)*inv(Jt)\n",
"function calc_residual_vector_integrand(el::JuliaFEM.Element, xi)\n",
" # Calculate dN/dX\n",
" dbasisdX = JuliaFEM.get_dbasisdX(el, xi)\n",
"\n",
" # Calculate residual force vector R(u) = T(u) - F(u)\n",
" function T(u)\n",
" # kinematics\n",
" gradu = u*dbasisdX\n",
" F = I + gradu # Deformation gradient\n",
" E = 1/2*(gradu' + gradu + gradu'*gradu) # Green-Lagrange strain tensor\n",
" S = lambda*trace(E)*I + 2*mu*E # PK2 stress tensor\n",
" P = F*S # PK1 stress tensor\n",
"\n",
" R[:,:] += w*P*dbasisdX'*detJ\n",
"\n",
" for p = 1:nnodes\n",
" for i = 1:dim\n",
" dF[:,:] = 0.0\n",
" dF[i,:] = dbasisdX[p,:]\n",
" dE = 1/2*(F'*dF + dF'*F)\n",
" dS = lambda*trace(dE)*I + 2*mu*dE\n",
" dP = dF*S + F*dS\n",
" for q = 1:nnodes\n",
" for j = 1:dim\n",
" K[dim*(p-1)+i,dim*(q-1)+j] += w*(dP[j,:]*dbasisdX[q,:]')[1]*detJ\n",
" end\n",
" end\n",
" end\n",
" end\n",
"\n",
" F = I + gradu\n",
" E = 1/2*(gradu' + gradu + gradu'*gradu)\n",
" # constitutive equation\n",
" lambda = JuliaFEM.interpolate(el, \"lambda\", xi)\n",
" mu = JuliaFEM.interpolate(el, \"mu\", xi)\n",
" S = lambda*trace(E)*I + 2*mu*E\n",
" P = F*S\n",
" T = P*dbasisdX'\n",
" return T\n",
" end\n",
"\n",
" u = el.attributes[\"displacement\"]\n",
" R(u) = T(u)\n",
" return R(u) # we skip calculating external force vector for now.\n",
"end\n",
"\n",
"\"\"\"\n",
"Autodiff version.\n",
"\"\"\"\n",
"function calc_local_matrices!(X, u, R, K, basis, dbasis, lambda_, mu_, ipoints, iweights)\n",
" dim, nnodes = size(X)\n",
" I = eye(dim)\n",
" R[:,:] = 0.0\n",
"\n",
" #dF = zeros(dim, dim)\n",
"\n",
" function calc_R!(u, R)\n",
" for m = 1:length(iweights)\n",
" w = iweights[m]\n",
" xi = ipoints[m, :]\n",
" # calculate material parameters\n",
" lambda = typeof(lambda_) == Float64 ? lambda_ : dot(lambda_, basis(xi))\n",
" mu = typeof(mu_) == Float64 ? mu_ : dot(mu_, basis(xi))\n",
" Jt = X*dbasis(xi)\n",
" detJ = det(Jt)\n",
" dbasisdX = dbasis(xi)*inv(Jt)\n",
"\n",
" gradu = u*dbasisdX\n",
" F = I + gradu # Deformation gradient\n",
" E = 1/2*(gradu' + gradu + gradu'*gradu) # Green-Lagrange strain tensor\n",
" S = lambda*trace(E)*I + 2*mu*E # PK2 stress tensor\n",
" P = F*S # PK1 stress tensor\n",
"\n",
" R[:,:] += w*P*dbasisdX'*detJ\n",
" end\n",
" end\n",
"\n",
" # herlper for tangent stiffness matrix\n",
" function R!(u, R)\n",
" R[:] = 0\n",
" calc_R!(reshape(u, dim, nnodes), reshape(R, dim, nnodes))\n",
" #calc_Wext!(reshape(u, 2, 4), reshape(R, 2, 4))\n",
" end\n",
" Jacobian = ForwardDiff.forwarddiff_jacobian(R!, Float64, fadtype=:dual, n=dim*nnodes, m=dim*nnodes)\n",
"\n",
" K[:, :] = Jacobian(reshape(u, dim*nnodes))\n",
" R!(reshape(u, dim*nnodes), reshape(R, dim*nnodes))\n",
"\n",
"end"
"calc_residual_vector = JuliaFEM.integrate(calc_residual_vector_integrand)\n",
"# calculate partial derivatives of R with respect to field \"displacement\"\n",
"calc_tangent_stiffness = JuliaFEM.linearize(calc_residual_vector, \"displacement\")"
]
},
{
"cell_type": "markdown",
"metadata": {},
"source": [
"That was our geometrically nonlinear elasticity solver. Note how we used automatic differentiation to linearize residual vector.\n",
"\n",
"*Design principle 6*: we test our code. We use FactCheck for testing."
]
},
{
"cell_type": "code",
"execution_count": 3,
"execution_count": 4,
"metadata": {
"collapsed": true
},
@@ -250,7 +141,7 @@
},
{
"cell_type": "code",
"execution_count": 4,
"execution_count": 6,
"metadata": {
"collapsed": false,
"scrolled": false
@@ -267,62 +158,23 @@
"name": "stderr",
"output_type": "stream",
"text": [
"13-Aug 00:37:26:DEBUG:root:Converged in 6 iterations.\n",
"13-Aug 00:37:26:DEBUG:root:solution vector: \n",
"17-Aug 18:21:25:DEBUG:root:Converged in 6 iterations.\n",
"17-Aug 18:21:25:DEBUG:root:solution vector: \n",
" [0.0 -0.3991450609547433 -0.07228582695592461 0.0\n",
" 0.0 -2.1779892317073504 -2.222244754401764 0.0]\n",
"13-Aug 00:37:26:DEBUG:root:norm of u: 3.1292483947150047\n",
"13-Aug 00:37:27:DEBUG:root:Converged in 6 iterations.\n",
"13-Aug 00:37:27:DEBUG:root:solution vector: \n",
"17-Aug 18:21:25:DEBUG:root:norm of u: 3.1292483947150047\n",
"17-Aug 18:21:25:DEBUG:root:Converged in 6 iterations.\n",
"17-Aug 18:21:25:DEBUG:root:solution vector: \n",
" [0.0 0.7433248532717796 1.048521014723486 0.0\n",
" 0.0 -2.085766534304891 -1.9606633242166027 0.0]\n",
"13-Aug 00:37:27:DEBUG:root:norm of u: 3.1292483947150056\n",
"13-Aug 00:37:27:DEBUG:root:Iteration 1\n"
"17-Aug 18:21:26:DEBUG:root:norm of u: 3.1292483947150056\n"
]
},
{
"name": "stdout",
"output_type": "stream",
"text": [
"Array(Float64,(12,12)) 12x12 Array{Float64,2}"
]
},
{
"name": "stderr",
"output_type": "stream",
"text": [
"13-Aug 00:37:27:DEBUG:root:Norm of du: 0.5992228342549063\n",
"13-Aug 00:37:27:DEBUG:root:Element displacement: [0.0 -0.02264423092574128 0.022536491965822688 0.0\n",
" 0.0 -0.12688379170176511 -0.12679760053383046 0.0]\n",
"13-Aug 00:37:27:DEBUG:root:Element displacement: [-0.02264423092574128 -0.029998807330416523 0.030242156525002267 0.022536491965822688\n",
" -0.12688379170176511 -0.4041002710283739 -0.40446733271991214 -0.12679760053383046]\n",
"13-Aug 00:37:27:DEBUG:root:solution vector: \n",
" [-0.02264423092574128 -0.029998807330416523 0.022536491965822688 0.030242156525002267 0.0 0.0\n",
" -0.12688379170176511 -0.4041002710283739 -0.12679760053383046 -0.40446733271991214 0.0 0.0]\n",
"13-Aug 00:37:27:DEBUG:root:norm of u: 0.5992228342549063\n"
]
},
{
"name": "stdout",
"output_type": "stream",
"text": [
":\n",
" 132.8 0.0 23.6 -3.0 -113.6 … 23.6 3.0 -33.2 15.0\n",
" 0.0 324.8 3.0 77.6 0.0 -3.0 77.6 15.0 -81.2\n",
" 23.6 3.0 66.4 -15.0 -33.2 0.0 0.0 0.0 0.0\n",
" -3.0 77.6 -15.0 162.4 15.0 0.0 0.0 0.0 0.0\n",
" -113.6 0.0 -33.2 15.0 132.8 -33.2 -15.0 23.6 -3.0\n",
" 0.0 -317.6 15.0 -81.2 0.0 … -15.0 -81.2 3.0 77.6\n",
" -33.2 -15.0 -56.8 3.0 23.6 0.0 0.0 0.0 0.0\n",
" -15.0 -81.2 -3.0 -158.8 3.0 0.0 0.0 0.0 0.0\n",
" 23.6 -3.0 0.0 0.0 -33.2 66.4 15.0 -56.8 3.0\n",
" 3.0 77.6 0.0 0.0 -15.0 15.0 162.4 -3.0 -158.8\n",
" -33.2 15.0 0.0 0.0 23.6 … -56.8 -3.0 66.4 -15.0\n",
" 15.0 -81.2 0.0 0.0 -3.0 3.0 -158.8 -15.0 162.4\n",
"K norm = 708.0378644377365\n",
"du = [-0.02264423092574128 -0.029998807330416523 0.022536491965822688 0.030242156525002267 0.0 0.0\n",
" -0.12688379170176511 -0.4041002710283739 -0.12679760053383046 -0.40446733271991214 0.0 0.0]\n",
"Out of 3 total facts:"
"2 facts verified.\n"
]
},
{
@@ -331,47 +183,52 @@
"delayed_handler (generic function with 4 methods)"
]
},
"execution_count": 4,
"execution_count": 6,
"metadata": {},
"output_type": "execute_result"
}
],
"source": [
"facts(\"test solve one element model\") do\n",
" X = [0.0 0.0; 10.0 0.0; 10.0 1.0; 0.0 1.0]'\n",
" F = [0 0; 0 0; 0 -2; 0 0]'\n",
"\n",
" # Material properties\n",
" E = 90\n",
" nu = 0.25\n",
" mu = E/(2*(1+nu))\n",
" la = E*nu/((1+nu)*(1-2*nu))\n",
" la = 2*la*mu/(la + 2*mu)\n",
"\n",
" u = zeros(2, 4)\n",
" du = zeros(2, 4)\n",
" R = zeros(2, 4)\n",
" K = zeros(8, 8)\n",
"\n",
"function get_test_element()\n",
" # set up one linear quadrangle element\n",
" basis(xi) = [\n",
" (1-xi[1])*(1-xi[2])/4\n",
" (1+xi[1])*(1-xi[2])/4\n",
" (1+xi[1])*(1+xi[2])/4\n",
" (1-xi[1])*(1+xi[2])/4]\n",
"\n",
" dbasis(xi) = [-(1-xi[2])/4.0 -(1-xi[1])/4.0\n",
" (1-xi[2])/4.0 -(1+xi[1])/4.0\n",
" (1+xi[2])/4.0 (1+xi[1])/4.0\n",
" -(1+xi[2])/4.0 (1-xi[1])/4.0]\n",
" ipoints = 1/sqrt(3)*[-1 -1; 1 -1; 1 1; -1 1]'\n",
" iweights = [1.0, 1.0, 1.0, 1.0]\n",
" attributes = Dict()\n",
" e = JuliaFEM.Element(1, [1, 2, 3, 4], basis, dbasis, attributes, ipoints, iweights)\n",
" E = 90.0\n",
" nu = 0.25\n",
" mu = E/(2*(1+nu))\n",
" la = E*nu/((1+nu)*(1-2*nu))\n",
" la = 2*la*mu/(la + 2*mu)\n",
" e.attributes[\"coordinates\"] = [0.0 0.0; 10.0 0.0; 10.0 1.0; 0.0 1.0]'\n",
" e.attributes[\"lambda\"] = la\n",
" e.attributes[\"mu\"] = mu\n",
" e.attributes[\"displacement\"] = [0.0 0.0; 0.0 0.0; 0.0 0.0; 0.0 0.0]'\n",
" return e\n",
"end\n",
"\n",
"facts(\"test solve one element model\") do\n",
"\n",
" e = get_test_element()\n",
" F = [0.0 0.0; 0.0 0.0; 0.0 -2.0; 0.0 0.0]'\n",
"\n",
" du = zeros(2, 4)\n",
"\n",
" ipoints = 1/sqrt(3)*[-1 -1; 1 -1; 1 1; -1 1]\n",
" iweights = [1, 1, 1, 1]\n",
" free_dofs = [3, 4, 5, 6]\n",
"\n",
" for i=1:10\n",
" calc_local_matrices!(X, u, R, K, basis, dbasis, la, mu, ipoints, iweights)\n",
" R = calc_residual_vector(e)\n",
" K = calc_tangent_stiffness(e)\n",
" du[free_dofs] = K[free_dofs, free_dofs] \\ -(R - F)[free_dofs]\n",
" u += du\n",
" e.attributes[\"displacement\"] += du\n",
" if norm(du) < 1.0e-9\n",
" Logging.debug(\"Converged in $i iterations.\")\n",
" break\n",
@@ -379,73 +236,36 @@
" end\n",
"\n",
" # Tested against Elmer solution\n",
" u = e.attributes[\"displacement\"]\n",
" Logging.debug(\"solution vector: \\n $u\")\n",
" @fact u[2, 3] --> roughly(-2.222244754401764)\n",
" norm1 = norm(u)\n",
" Logging.debug(\"norm of u: $(norm(u))\")\n",
"\n",
" # We rotate model a bit and make sure that L2 norm is same\n",
" # We rotate model a bit and make sure that norm remains same\n",
" phi = 30/180*pi\n",
" rmat = [\n",
" cos(phi) -sin(phi)\n",
" sin(phi) cos(phi)]\n",
" X = rmat*X\n",
" e.attributes[\"coordinates\"] = rmat*e.attributes[\"coordinates\"]\n",
" F = rmat*F\n",
" u = zeros(2, 4)\n",
"\n",
" e.attributes[\"displacement\"] = [0.0 0.0; 0.0 0.0; 0.0 0.0; 0.0 0.0]'\n",
" du = zeros(2, 4)\n",
" for i=1:10\n",
" calc_local_matrices!(X, u, R, K, basis, dbasis, la, mu, ipoints, iweights)\n",
" R = calc_residual_vector(e)\n",
" K = calc_tangent_stiffness(e)\n",
" du[free_dofs] = K[free_dofs, free_dofs] \\ -(R - F)[free_dofs]\n",
" u += du\n",
" e.attributes[\"displacement\"] += du\n",
" if norm(du) < 1.0e-9\n",
" Logging.debug(\"Converged in $i iterations.\")\n",
" break\n",
" end\n",
" end\n",
" u = e.attributes[\"displacement\"]\n",
" Logging.debug(\"solution vector: \\n $u\")\n",
" Logging.debug(\"norm of u: $(norm(u))\")\n",
" @fact norm(u) --> roughly(norm1) \n",
"\n",
" # test two element model\n",
" X = [0.0 0.0; 5.0 0.0; 5.0 1.0; 0.0 1.0]'\n",
" u = zeros(2, 6)\n",
" du = zeros(2, 6)\n",
" R = zeros(2, 4)\n",
" K = zeros(8, 8)\n",
" ass1 = [9, 10, 1, 2, 5, 6, 11, 12]\n",
" ass2 = [1, 2, 3, 4, 7, 8, 5, 6]\n",
" free_dofs = collect(1:8)\n",
" F = [0 0; 0 0; 0 0; 0 -0.1; 0 0; 0 0]'\n",
"\n",
" A = zeros(12, 12)\n",
" b = zeros(2, 6)\n",
" for i=1:1\n",
" Logging.debug(\"Iteration $i\")\n",
" A[:,:] = 0.0\n",
" b[:] = 0.0\n",
" #Logging.debug(\"Assembling\")\n",
" for ass in (ass1, ass2)\n",
" #Logging.debug(\"ass = $ass, u[ass] = $(u[ass])\")\n",
" calc_local_matrices!(X, u[ass], R, K, basis, dbasis, la, mu, ipoints, iweights)\n",
" A[ass,ass] += K\n",
" b[ass] += R[:]\n",
" end\n",
" dump(round(A, 2))\n",
" println(\"K norm = $(norm(A[free_dofs, free_dofs]))\")\n",
" du[free_dofs] = A[free_dofs, free_dofs] \\ -(b - F)[free_dofs]\n",
" println(\"du = $du\")\n",
" u += du\n",
" Logging.debug(\"Norm of du: $(norm(du))\")\n",
" for ass in (ass1, ass2)\n",
" Logging.debug(\"Element displacement: $(reshape(u[ass], 2, 4))\")\n",
" end\n",
" if norm(du) < 1.0e-9\n",
" Logging.debug(\"Converged in $i iterations.\")\n",
" break\n",
" end\n",
" end\n",
" Logging.debug(\"solution vector: \\n $u\")\n",
" Logging.debug(\"norm of u: $(norm(u))\")\n",
" @pending norm(u) --> :something\n",
"end"
]
},
@@ -456,160 +276,6 @@
"One element solutions are not particularly interesting so next step is to create function that assembles global matrix from local matrices. Some data types:"
]
},
{
"cell_type": "code",
"execution_count": 5,
"metadata": {
"collapsed": false
},
"outputs": [],
"source": [
"#type Node\n",
"# id :: Int\n",
"# #elements :: Array{Int64, 1}\n",
"#end"
]
},
{
"cell_type": "code",
"execution_count": 6,
"metadata": {
"collapsed": false
},
"outputs": [],
"source": [
"type Element\n",
" id :: Int\n",
" node_ids :: Array{Int64, 1}\n",
" coordinates :: Array{Float64, 2}\n",
" attributes :: Dict{ASCIIString, Any}\n",
"end"
]
},
{
"cell_type": "code",
"execution_count": 7,
"metadata": {
"collapsed": false
},
"outputs": [
{
"name": "stdout",
"output_type": "stream",
"text": [
"\n",
" Verified: 2\n",
" Pending: 1\n"
]
}
],
"source": [
"type Assembly\n",
" # LHS\n",
" I :: Array{Int64, 1}\n",
" J :: Array{Int64, 1}\n",
" A :: Array{Float64, 1}\n",
" # RHS\n",
" i :: Array{Int64, 1}\n",
" b :: Array{Float64, 1}\n",
" # global dofs for each element\n",
" gdofs :: Dict{Int64, Array{Int64, 1}}\n",
"end"
]
},
{
"cell_type": "code",
"execution_count": 8,
"metadata": {
"collapsed": false
},
"outputs": [
{
"data": {
"text/plain": [
"get_integration_scheme (generic function with 2 methods)"
]
},
"execution_count": 8,
"metadata": {},
"output_type": "execute_result"
}
],
"source": [
"\"\"\"\n",
"Return shape functions and their derivatives for a element.\n",
"\n",
"Parameters\n",
"----------\n",
"element::Element\n",
"\n",
"Returns\n",
"-------\n",
"tuple (basis, dbasis)\n",
"\"\"\"\n",
"function get_shape_functions(el::Element)\n",
" ndim, nnodes = size(el.coordinates)\n",
" #Logging.debug(\"ndim = $ndim, nnodes=$nnodes\")\n",
" if (nnodes == 4) & (ndim == 2)\n",
" basis(xi) = [\n",
" (1-xi[1])*(1-xi[2])/4\n",
" (1+xi[1])*(1-xi[2])/4\n",
" (1+xi[1])*(1+xi[2])/4\n",
" (1-xi[1])*(1+xi[2])/4]\n",
" dbasis(xi) = [-(1-xi[2])/4.0 -(1-xi[1])/4.0\n",
" (1-xi[2])/4.0 -(1+xi[1])/4.0\n",
" (1+xi[2])/4.0 (1+xi[1])/4.0\n",
" -(1+xi[2])/4.0 (1-xi[1])/4.0]\n",
" return basis, dbasis\n",
" elseif (nnodes == 10) & (ndim == 3)\n",
" basis(xi) = [\n",
" (xi[1] + xi[2] + xi[3] - 1)*(2*xi[1] + 2*xi[2] + 2*xi[3] - 1)\n",
" xi[1]*(2*xi[1] - 1)\n",
" xi[2]*(2*xi[2] - 1)\n",
" xi[3]*(2*xi[3] - 1)\n",
" -4*xi[1]*(xi[1] + xi[2] + xi[3] - 1)\n",
" 4*xi[1]*xi[2]\n",
" -4*xi[2]*(xi[1] + xi[2] + xi[3] - 1)\n",
" -4*xi[3]*(xi[1] + xi[2] + xi[3] - 1)\n",
" 4*xi[1]*xi[3]\n",
" 4*xi[2]*xi[3]\n",
" ]\n",
" dbasis(xi) = [\n",
" 4*xi[1] + 4*xi[2] + 4*xi[3] - 3 4*xi[1] + 4*xi[2] + 4*xi[3] - 3 4*xi[1] + 4*xi[2] + 4*xi[3] - 3\n",
" 4*xi[1] - 1 0 0\n",
" 0 4*xi[2] - 1 0\n",
" 0 0 4*xi[3] - 1\n",
" -4*(2*xi[1] + xi[2] + xi[3] - 1) -4*xi[1] -4*xi[1]\n",
" 4*xi[2] 4*xi[1] 0\n",
" -4*xi[2] -4*(xi[1] + 2*xi[2] + xi[3] - 1) -4*xi[2]\n",
" -4*xi[3] -4*xi[3] -4*(xi[1] + xi[2] + 2*xi[3] - 1)\n",
" 4*xi[3] 0 4*xi[1]\n",
" 0 4*xi[3] 4*xi[2]\n",
" ]\n",
" return basis, dbasis\n",
" end\n",
" throw(\"Unknown function space, ndim=$ndim, nnodes=$nnodes\")\n",
"end\n",
"\n",
"\"\"\"\n",
"\"\"\"\n",
"function get_integration_scheme(el::Element, order=2)\n",
" ndim, nnodes = size(el.coordinates)\n",
" if (nnodes == 4) & (order == 2) & (ndim == 2)\n",
" ipoints = 1/sqrt(3)*[-1 -1; 1 -1; 1 1; -1 1]\n",
" iweights = [1, 1, 1, 1]\n",
" return ipoints, iweights\n",
" elseif (nnodes == 10) & (order == 2) & (ndim == 3) # c3d10\n",
" # from code aster documentation\n",
" a = 1/20*(5-sqrt(5))\n",
" b = 1/20*(5+3*sqrt(5))\n",
" ipoints = [a a a; a a b; a b a; b a a]\n",
" iweights = 1/24*[1 1 1 1]\n",
" return ipoints, iweights\n",
" end\n",
"end"
]
},
{
"cell_type": "code",
"execution_count": 9,
@@ -629,7 +295,7 @@
}
],
"source": [
"function assemble_element!(ass::Assembly, el::Element, io=2)\n",
"function assemble_element!(ass::Assembly, el::Element)\n",
"\n",
" # Material properties\n",
" E = el.attributes[\"Young\"]\n",
@@ -699,28 +365,28 @@
"name": "stderr",
"output_type": "stream",
"text": [
"13-Aug 00:37:30:DEBUG:root:Adding nodes to array\n",
"13-Aug 00:37:30:DEBUG:root:Creating elements\n",
"13-Aug 00:37:30:DEBUG:root:Starting iteration 1\n",
"13-Aug 00:37:30:DEBUG:root:Assembling\n",
"13-Aug 00:37:30:DEBUG:root:Solution norm = 3.090022136728999\n",
"13-Aug 00:37:30:DEBUG:root:Starting iteration 2\n",
"13-Aug 00:37:30:DEBUG:root:Assembling\n",
"13-Aug 00:37:30:DEBUG:root:Solution norm = 0.32121316021535135\n",
"13-Aug 00:37:30:DEBUG:root:Starting iteration 3\n",
"13-Aug 00:37:30:DEBUG:root:Assembling\n",
"13-Aug 00:37:30:DEBUG:root:Solution norm = 0.040431781939994194\n",
"13-Aug 00:37:30:DEBUG:root:Starting iteration 4\n",
"13-Aug 00:37:30:DEBUG:root:Assembling\n",
"13-Aug 00:37:30:DEBUG:root:Solution norm = 0.0009291101052124042\n",
"13-Aug 00:37:30:DEBUG:root:Starting iteration 5\n",
"13-Aug 00:37:30:DEBUG:root:Assembling\n",
"13-Aug 00:37:30:DEBUG:root:Solution norm = 1.5638899022213175e-7\n",
"13-Aug 00:37:30:DEBUG:root:Starting iteration 6\n",
"13-Aug 00:37:30:DEBUG:root:Assembling\n",
"13-Aug 00:37:30:DEBUG:root:Solution norm = 1.0118539067290854e-14\n",
"13-Aug 00:37:30:DEBUG:root:Converged in 6 iterations.\n",
"13-Aug 00:37:30:DEBUG:root:Displacement of element = \n",
"15-Aug 17:03:55:DEBUG:root:Adding nodes to array\n",
"15-Aug 17:03:55:DEBUG:root:Creating elements\n",
"15-Aug 17:03:55:DEBUG:root:Starting iteration 1\n",
"15-Aug 17:03:55:DEBUG:root:Assembling\n",
"15-Aug 17:03:57:DEBUG:root:Solution norm = 3.090022136728999\n",
"15-Aug 17:03:57:DEBUG:root:Starting iteration 2\n",
"15-Aug 17:03:57:DEBUG:root:Assembling\n",
"15-Aug 17:03:57:DEBUG:root:Solution norm = 0.32121316021535135\n",
"15-Aug 17:03:57:DEBUG:root:Starting iteration 3\n",
"15-Aug 17:03:57:DEBUG:root:Assembling\n",
"15-Aug 17:03:57:DEBUG:root:Solution norm = 0.040431781939994194\n",
"15-Aug 17:03:57:DEBUG:root:Starting iteration 4\n",
"15-Aug 17:03:57:DEBUG:root:Assembling\n",
"15-Aug 17:03:57:DEBUG:root:Solution norm = 0.0009291101052124042\n",
"15-Aug 17:03:57:DEBUG:root:Starting iteration 5\n",
"15-Aug 17:03:57:DEBUG:root:Assembling\n",
"15-Aug 17:03:57:DEBUG:root:Solution norm = 1.5638899022213175e-7\n",
"15-Aug 17:03:58:DEBUG:root:Starting iteration 6\n",
"15-Aug 17:03:58:DEBUG:root:Assembling\n",
"15-Aug 17:03:58:DEBUG:root:Solution norm = 1.0118539067290854e-14\n",
"15-Aug 17:03:58:DEBUG:root:Converged in 6 iterations.\n",
"15-Aug 17:03:58:DEBUG:root:Displacement of element = \n",
"[-0.39914506095474334 -0.0722858269559246 0.0 0.0\n",
" -2.1779892317073504 -2.2222447544017645 0.0 0.0]\n"
]
@@ -844,48 +510,48 @@
"name": "stderr",
"output_type": "stream",
"text": [
"13-Aug 00:37:31:DEBUG:root:Creating elements\n",
"13-Aug 00:37:31:INFO:root:create_ldof2gdofmap: dofs per node: 2\n",
"13-Aug 00:37:31:DEBUG:root:Dict(4=>[7,8],2=>[3,4],3=>[5,6],5=>[9,10],6=>[11,12],1=>[1,2])\n",
"13-Aug 00:37:32:INFO:root:solve!: dofs per node: 2\n",
"13-Aug 00:37:32:DEBUG:root:Problem size = 12\n",
"13-Aug 00:37:32:DEBUG:root:Starting iteration 1\n",
"13-Aug 00:37:32:DEBUG:root:Assembling\n",
"13-Aug 00:37:32:DEBUG:root:Adding Dirichlet boundary conditions using Lagrange multipliers\n",
"13-Aug 00:37:32:DEBUG:root:Added 5 Lagrange multipliers to model\n",
"13-Aug 00:37:32:DEBUG:root:Adding Neumann boundary conditions\n",
"13-Aug 00:37:32:DEBUG:root:Solving system of equations. Total size = 16\n",
"13-Aug 00:37:32:DEBUG:root:Solution norm du = 0.6015838690633517\n",
"13-Aug 00:37:32:DEBUG:root:Starting iteration 2\n",
"13-Aug 00:37:32:DEBUG:root:Assembling\n",
"13-Aug 00:37:32:DEBUG:root:Adding Dirichlet boundary conditions using Lagrange multipliers\n",
"13-Aug 00:37:32:DEBUG:root:Added 5 Lagrange multipliers to model\n",
"13-Aug 00:37:32:DEBUG:root:Adding Neumann boundary conditions\n",
"13-Aug 00:37:32:DEBUG:root:Solving system of equations. Total size = 16\n",
"13-Aug 00:37:32:DEBUG:root:Solution norm du = 0.013420417380980414\n",
"13-Aug 00:37:33:DEBUG:root:Starting iteration 3\n",
"13-Aug 00:37:33:DEBUG:root:Assembling\n",
"13-Aug 00:37:33:DEBUG:root:Adding Dirichlet boundary conditions using Lagrange multipliers\n",
"13-Aug 00:37:33:DEBUG:root:Added 5 Lagrange multipliers to model\n",
"13-Aug 00:37:33:DEBUG:root:Adding Neumann boundary conditions\n",
"13-Aug 00:37:33:DEBUG:root:Solving system of equations. Total size = 16\n",
"13-Aug 00:37:33:DEBUG:root:Solution norm du = 0.00032202957936854873\n",
"13-Aug 00:37:33:DEBUG:root:Starting iteration 4\n",
"13-Aug 00:37:33:DEBUG:root:Assembling\n",
"13-Aug 00:37:33:DEBUG:root:Adding Dirichlet boundary conditions using Lagrange multipliers\n",
"13-Aug 00:37:33:DEBUG:root:Added 5 Lagrange multipliers to model\n",
"13-Aug 00:37:33:DEBUG:root:Adding Neumann boundary conditions\n",
"13-Aug 00:37:33:DEBUG:root:Solving system of equations. Total size = 16\n",
"13-Aug 00:37:33:DEBUG:root:Solution norm du = 9.906677094801476e-8\n",
"13-Aug 00:37:33:DEBUG:root:Starting iteration 5\n",
"13-Aug 00:37:33:DEBUG:root:Assembling\n",
"13-Aug 00:37:33:DEBUG:root:Adding Dirichlet boundary conditions using Lagrange multipliers\n",
"13-Aug 00:37:33:DEBUG:root:Added 5 Lagrange multipliers to model\n",
"13-Aug 00:37:33:DEBUG:root:Adding Neumann boundary conditions\n",
"13-Aug 00:37:33:DEBUG:root:Solving system of equations. Total size = 16\n",
"13-Aug 00:37:33:DEBUG:root:Solution norm du = 5.027624635820698e-15\n",
"13-Aug 00:37:33:DEBUG:root:Converged in 5 iterations.\n",
"13-Aug 00:37:33:DEBUG:root:Displacement of element = \n",
"15-Aug 17:03:59:DEBUG:root:Creating elements\n",
"15-Aug 17:03:59:INFO:root:create_ldof2gdofmap: dofs per node: 2\n",
"15-Aug 17:04:00:DEBUG:root:Dict(4=>[7,8],2=>[3,4],3=>[5,6],5=>[9,10],6=>[11,12],1=>[1,2])\n",
"15-Aug 17:04:04:INFO:root:solve!: dofs per node: 2\n",
"15-Aug 17:04:04:DEBUG:root:Problem size = 12\n",
"15-Aug 17:04:04:DEBUG:root:Starting iteration 1\n",
"15-Aug 17:04:04:DEBUG:root:Assembling\n",
"15-Aug 17:04:05:DEBUG:root:Adding Dirichlet boundary conditions using Lagrange multipliers\n",
"15-Aug 17:04:05:DEBUG:root:Added 5 Lagrange multipliers to model\n",
"15-Aug 17:04:05:DEBUG:root:Adding Neumann boundary conditions\n",
"15-Aug 17:04:05:DEBUG:root:Solving system of equations. Total size = 16\n",
"15-Aug 17:04:05:DEBUG:root:Solution norm du = 0.6015838690633517\n",
"15-Aug 17:04:05:DEBUG:root:Starting iteration 2\n",
"15-Aug 17:04:05:DEBUG:root:Assembling\n",
"15-Aug 17:04:05:DEBUG:root:Adding Dirichlet boundary conditions using Lagrange multipliers\n",
"15-Aug 17:04:05:DEBUG:root:Added 5 Lagrange multipliers to model\n",
"15-Aug 17:04:05:DEBUG:root:Adding Neumann boundary conditions\n",
"15-Aug 17:04:05:DEBUG:root:Solving system of equations. Total size = 16\n",
"15-Aug 17:04:05:DEBUG:root:Solution norm du = 0.013420417380980414\n",
"15-Aug 17:04:05:DEBUG:root:Starting iteration 3\n",
"15-Aug 17:04:05:DEBUG:root:Assembling\n",
"15-Aug 17:04:05:DEBUG:root:Adding Dirichlet boundary conditions using Lagrange multipliers\n",
"15-Aug 17:04:05:DEBUG:root:Added 5 Lagrange multipliers to model\n",
"15-Aug 17:04:05:DEBUG:root:Adding Neumann boundary conditions\n",
"15-Aug 17:04:06:DEBUG:root:Solving system of equations. Total size = 16\n",
"15-Aug 17:04:06:DEBUG:root:Solution norm du = 0.00032202957936854873\n",
"15-Aug 17:04:06:DEBUG:root:Starting iteration 4\n",
"15-Aug 17:04:06:DEBUG:root:Assembling\n",
"15-Aug 17:04:06:DEBUG:root:Adding Dirichlet boundary conditions using Lagrange multipliers\n",
"15-Aug 17:04:06:DEBUG:root:Added 5 Lagrange multipliers to model\n",
"15-Aug 17:04:06:DEBUG:root:Adding Neumann boundary conditions\n",
"15-Aug 17:04:06:DEBUG:root:Solving system of equations. Total size = 16\n",
"15-Aug 17:04:06:DEBUG:root:Solution norm du = 9.906677094801476e-8\n",
"15-Aug 17:04:06:DEBUG:root:Starting iteration 5\n",
"15-Aug 17:04:06:DEBUG:root:Assembling\n",
"15-Aug 17:04:06:DEBUG:root:Adding Dirichlet boundary conditions using Lagrange multipliers\n",
"15-Aug 17:04:06:DEBUG:root:Added 5 Lagrange multipliers to model\n",
"15-Aug 17:04:06:DEBUG:root:Adding Neumann boundary conditions\n",
"15-Aug 17:04:06:DEBUG:root:Solving system of equations. Total size = 16\n",
"15-Aug 17:04:06:DEBUG:root:Solution norm du = 5.027624635820698e-15\n",
"15-Aug 17:04:06:DEBUG:root:Converged in 5 iterations.\n",
"15-Aug 17:04:06:DEBUG:root:Displacement of element = \n",
"[-0.02442313597467864 -0.039356000063335075 0.021097993207232584 0.020877031423993653\n",
" -0.12673626841485705 -0.40433021969759375 -0.40656320177872923 -0.1275776940913048]\n"
]
@@ -1136,7 +802,7 @@
"\n",
"WARNING: deprecated syntax \"{a=>b, ...}\" at /Users/jukka/.julia/v0.4/JuliaFEM/src/abaqus_reader.jl:32.\n",
"Use \"Dict{Any,Any}(a=>b, ...)\" instead.\n",
"13-Aug 00:37:37:INFO:root:Registered handlers: Any[\"ELEMENT\",\"NODE\",\"NSET\"]\n",
"15-Aug 17:04:20:INFO:root:Registered handlers: Any[\"ELEMENT\",\"NODE\",\"NSET\"]\n",
"WARNING: beginswith is deprecated, use startswith instead.\n",
" in depwarn at /Applications/Julia-0.4.0-dev-539c818c4e.app/Contents/Resources/julia/lib/julia/sys.dylib\n",
" in beginswith at deprecated.jl:30\n",
@@ -1155,8 +821,8 @@
" in eventloop at /Users/jukka/.julia/v0.4/IJulia/src/IJulia.jl:123\n",
" in anonymous at task.jl:365\n",
"while loading In[13], in expression starting on line 3\n",
"13-Aug 00:37:38:DEBUG:root:Found NODE section\n",
"13-Aug 00:37:38:DEBUG:root:Found ELEMENT section\n",
"15-Aug 17:04:21:DEBUG:root:Found NODE section\n",
"15-Aug 17:04:22:DEBUG:root:Found ELEMENT section\n",
"WARNING: integer(s::AbstractString) is deprecated, use parse(Int,s) instead.\n",
" in depwarn at /Applications/Julia-0.4.0-dev-539c818c4e.app/Contents/Resources/julia/lib/julia/sys.dylib\n",
" in integer at deprecated.jl:49\n",
@@ -1169,14 +835,14 @@
" in eventloop at /Users/jukka/.julia/v0.4/IJulia/src/IJulia.jl:123\n",
" in anonymous at task.jl:365\n",
"while loading In[13], in expression starting on line 3\n",
"13-Aug 00:37:39:DEBUG:root:120 elements found\n",
"13-Aug 00:37:40:INFO:root:Creating ELSET Body1\n",
"13-Aug 00:37:40:DEBUG:root:Found NSET section\n",
"13-Aug 00:37:40:DEBUG:root:Creating node set SUPPORT\n",
"13-Aug 00:37:40:DEBUG:root:Found NSET section\n",
"13-Aug 00:37:40:DEBUG:root:Creating node set LOAD\n",
"13-Aug 00:37:40:DEBUG:root:Found NSET section\n",
"13-Aug 00:37:40:DEBUG:root:Creating node set TOP\n"
"15-Aug 17:04:27:DEBUG:root:120 elements found\n",
"15-Aug 17:04:27:INFO:root:Creating ELSET Body1\n",
"15-Aug 17:04:27:DEBUG:root:Found NSET section\n",
"15-Aug 17:04:28:DEBUG:root:Creating node set SUPPORT\n",
"15-Aug 17:04:28:DEBUG:root:Found NSET section\n",
"15-Aug 17:04:28:DEBUG:root:Creating node set LOAD\n",
"15-Aug 17:04:28:DEBUG:root:Found NSET section\n",
"15-Aug 17:04:28:DEBUG:root:Creating node set TOP\n"
]
},
{
@@ -1214,47 +880,47 @@
"name": "stderr",
"output_type": "stream",
"text": [
"13-Aug 00:37:41:DEBUG:root:Creating elements\n",
"13-Aug 00:37:41:INFO:root:create_ldof2gdofmap: dofs per node: 3\n",
"13-Aug 00:37:41:INFO:root:solve!: dofs per node: 3\n",
"13-Aug 00:37:41:DEBUG:root:Problem size = 894\n",
"13-Aug 00:37:41:DEBUG:root:Starting iteration 1\n",
"13-Aug 00:37:41:DEBUG:root:Assembling\n",
"13-Aug 00:37:43:DEBUG:root:Adding Dirichlet boundary conditions using Lagrange multipliers\n",
"13-Aug 00:37:43:DEBUG:root:Added 28 Lagrange multipliers to model\n",
"13-Aug 00:37:43:DEBUG:root:Adding Neumann boundary conditions\n",
"13-Aug 00:37:43:DEBUG:root:Solving system of equations. Total size = 921\n",
"13-Aug 00:37:43:DEBUG:root:Solution norm du = 54.19642700242575\n",
"13-Aug 00:37:43:DEBUG:root:Starting iteration 2\n",
"13-Aug 00:37:43:DEBUG:root:Assembling\n",
"13-Aug 00:37:45:DEBUG:root:Adding Dirichlet boundary conditions using Lagrange multipliers\n",
"13-Aug 00:37:45:DEBUG:root:Added 28 Lagrange multipliers to model\n",
"13-Aug 00:37:45:DEBUG:root:Adding Neumann boundary conditions\n",
"13-Aug 00:37:45:DEBUG:root:Solving system of equations. Total size = 921\n",
"13-Aug 00:37:45:DEBUG:root:Solution norm du = 1.68729144400063\n",
"13-Aug 00:37:45:DEBUG:root:Starting iteration 3\n",
"13-Aug 00:37:45:DEBUG:root:Assembling\n",
"13-Aug 00:37:47:DEBUG:root:Adding Dirichlet boundary conditions using Lagrange multipliers\n",
"13-Aug 00:37:47:DEBUG:root:Added 28 Lagrange multipliers to model\n",
"13-Aug 00:37:47:DEBUG:root:Adding Neumann boundary conditions\n",
"13-Aug 00:37:47:DEBUG:root:Solving system of equations. Total size = 921\n",
"13-Aug 00:37:47:DEBUG:root:Solution norm du = 0.04175080278208098\n",
"13-Aug 00:37:47:DEBUG:root:Starting iteration 4\n",
"13-Aug 00:37:47:DEBUG:root:Assembling\n",
"13-Aug 00:37:48:DEBUG:root:Adding Dirichlet boundary conditions using Lagrange multipliers\n",
"13-Aug 00:37:48:DEBUG:root:Added 28 Lagrange multipliers to model\n",
"13-Aug 00:37:48:DEBUG:root:Adding Neumann boundary conditions\n",
"13-Aug 00:37:48:DEBUG:root:Solving system of equations. Total size = 921\n",
"13-Aug 00:37:49:DEBUG:root:Solution norm du = 2.4176157836844963e-5\n",
"13-Aug 00:37:49:DEBUG:root:Starting iteration 5\n",
"13-Aug 00:37:49:DEBUG:root:Assembling\n",
"13-Aug 00:37:50:DEBUG:root:Adding Dirichlet boundary conditions using Lagrange multipliers\n",
"13-Aug 00:37:50:DEBUG:root:Added 28 Lagrange multipliers to model\n",
"13-Aug 00:37:50:DEBUG:root:Adding Neumann boundary conditions\n",
"13-Aug 00:37:50:DEBUG:root:Solving system of equations. Total size = 921\n",
"13-Aug 00:37:50:DEBUG:root:Solution norm du = 1.4448231751500831e-11\n",
"13-Aug 00:37:50:DEBUG:root:Converged in 5 iterations.\n",
"13-Aug 00:37:50:INFO:root:Maximum absolute displacement in y direction: 5.245400568184194\n"
"15-Aug 17:04:30:DEBUG:root:Creating elements\n",
"15-Aug 17:04:31:INFO:root:create_ldof2gdofmap: dofs per node: 3\n",
"15-Aug 17:04:31:INFO:root:solve!: dofs per node: 3\n",
"15-Aug 17:04:31:DEBUG:root:Problem size = 894\n",
"15-Aug 17:04:31:DEBUG:root:Starting iteration 1\n",
"15-Aug 17:04:31:DEBUG:root:Assembling\n",
"15-Aug 17:04:36:DEBUG:root:Adding Dirichlet boundary conditions using Lagrange multipliers\n",
"15-Aug 17:04:36:DEBUG:root:Added 28 Lagrange multipliers to model\n",
"15-Aug 17:04:36:DEBUG:root:Adding Neumann boundary conditions\n",
"15-Aug 17:04:36:DEBUG:root:Solving system of equations. Total size = 921\n",
"15-Aug 17:04:36:DEBUG:root:Solution norm du = 72.87727091053921\n",
"15-Aug 17:04:36:DEBUG:root:Starting iteration 2\n",
"15-Aug 17:04:36:DEBUG:root:Assembling\n",
"15-Aug 17:04:40:DEBUG:root:Adding Dirichlet boundary conditions using Lagrange multipliers\n",
"15-Aug 17:04:40:DEBUG:root:Added 28 Lagrange multipliers to model\n",
"15-Aug 17:04:40:DEBUG:root:Adding Neumann boundary conditions\n",
"15-Aug 17:04:40:DEBUG:root:Solving system of equations. Total size = 921\n",
"15-Aug 17:04:40:DEBUG:root:Solution norm du = 2.737285770100854\n",
"15-Aug 17:04:40:DEBUG:root:Starting iteration 3\n",
"15-Aug 17:04:40:DEBUG:root:Assembling\n",
"15-Aug 17:04:44:DEBUG:root:Adding Dirichlet boundary conditions using Lagrange multipliers\n",
"15-Aug 17:04:44:DEBUG:root:Added 28 Lagrange multipliers to model\n",
"15-Aug 17:04:44:DEBUG:root:Adding Neumann boundary conditions\n",
"15-Aug 17:04:44:DEBUG:root:Solving system of equations. Total size = 921\n",
"15-Aug 17:04:44:DEBUG:root:Solution norm du = 0.07997112801214978\n",
"15-Aug 17:04:44:DEBUG:root:Starting iteration 4\n",
"15-Aug 17:04:44:DEBUG:root:Assembling\n",
"15-Aug 17:04:48:DEBUG:root:Adding Dirichlet boundary conditions using Lagrange multipliers\n",
"15-Aug 17:04:48:DEBUG:root:Added 28 Lagrange multipliers to model\n",
"15-Aug 17:04:48:DEBUG:root:Adding Neumann boundary conditions\n",
"15-Aug 17:04:48:DEBUG:root:Solving system of equations. Total size = 921\n",
"15-Aug 17:04:48:DEBUG:root:Solution norm du = 6.406557430235748e-5\n",
"15-Aug 17:04:48:DEBUG:root:Starting iteration 5\n",
"15-Aug 17:04:48:DEBUG:root:Assembling\n",
"15-Aug 17:04:52:DEBUG:root:Adding Dirichlet boundary conditions using Lagrange multipliers\n",
"15-Aug 17:04:52:DEBUG:root:Added 28 Lagrange multipliers to model\n",
"15-Aug 17:04:52:DEBUG:root:Adding Neumann boundary conditions\n",
"15-Aug 17:04:52:DEBUG:root:Solving system of equations. Total size = 921\n",
"15-Aug 17:04:52:DEBUG:root:Solution norm du = 6.870072007793185e-11\n",
"15-Aug 17:04:52:DEBUG:root:Converged in 5 iterations.\n",
"15-Aug 17:04:52:INFO:root:Maximum absolute displacement in y direction: 6.9925206227884695\n"
]
}
],
@@ -1367,8 +1033,8 @@
"name": "stderr",
"output_type": "stream",
"text": [
"13-Aug 00:37:51:INFO:root:Number of nodes in model: 298\n",
"13-Aug 00:37:51:INFO:root:Number of elements in model: 120\n"
"15-Aug 17:04:54:INFO:root:Number of nodes in model: 298\n",
"15-Aug 17:04:54:INFO:root:Number of elements in model: 120\n"
]
}
],
@@ -1498,31 +1164,31 @@
"data": {
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" ⋮ => ⋮"
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},
@@ -1552,9 +1218,9 @@
"data": {
"text/plain": [
"3x298 Array{Float64,2}:\n",
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]
},
"execution_count": 20,
@@ -1602,7 +1268,7 @@
{
"data": {
"text/plain": [
"28215"
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]
},
"execution_count": 22,
src/math.jl
+1 -1
View File
@@ -75,7 +75,7 @@ Return partial derivatives of shape functions w.r.t X using chain rule.
"""
function get_dbasisdX(el::Element, xi)
J = interpolate(el, "coordinates", xi; derivative=true)
dbasisdX = el.dbasis(xi)*inv(J)
dbasisdX = el.dbasis(xi)*inv(J')
return dbasisdX
end