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<span lang="EN-US">Hello everyone,</span></p>
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<span lang="EN-US"> </span></p>
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<span lang="EN-US">Based on the Wiki example '2D linear elasticity' I would like to define an axisymmetric problem for a cylinder under external pressure.</span></p>
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<span lang="EN-US">The geometry is simple : a rectangular shape. Pressure is applied on outer surface. Upper and lower edges are clamped along Y direction (Y is axis of symmetry for geometry). I use Jacobian VolAxi and SurAxi. The result is a global displacement
of geometry as if it was just a 2D square and not a symmetry of revolution around Y axis. So my question is rather simple : is VolAxi and SurAxi are suited for axisymmetric linear elasticty problems? If so, could you help me to understand the results.</span></p>
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<span lang="EN-US">Below is the code for .geo and .pro files :</span></p>
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<span lang="EN-US"> </span></p>
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<span lang="EN-US">AxiElasticity.geo file</span></p>
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<span lang="EN-US"> </span></p>
<p class="MsoNormal" style="margin: 0cm 0cm 0.0001pt; font-size: 11pt; font-family: Calibri, sans-serif;">
<span lang="EN-US">r0 = 10e-3;</span></p>
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r1 = 80e-3;</p>
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L = 100e-3;</p>
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lc = 5e-3;</p>
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Point(1) = {r0,0.0,0.0,lc};</p>
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Point(2) = {r1-1e-6,0.0,0.0,lc};</p>
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Point(3) = {r1-1e-6,L,0.0,lc};</p>
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<span lang="EN-US">Point(4) = {r0,L,0.0,lc};</span></p>
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<span lang="EN-US">Line(1) = {1,2};</span></p>
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<span lang="EN-US">Line(2) = {2,3};</span></p>
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<span lang="EN-US">Line(3) = {3,4};</span></p>
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<span lang="EN-US">Line(4) = {4,1};</span></p>
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<span lang="EN-US">Line Loop(5) = {1,2,3,4};</span></p>
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<span lang="EN-US">Plane Surface(6) = {5};</span></p>
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<span lang="EN-US">Physical Surface(100) = {6}; // Cyl</span></p>
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<span lang="EN-US">Physical Line(101) = {2}; // ForceCyl</span></p>
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<span lang="EN-US">Physical Line(102) = {1,3}; // Edges</span></p>
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<span lang="EN-US">Transfinite Line{4,2} = 150;</span></p>
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<span lang="EN-US"> </span></p>
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<span lang="EN-US">AxiElasticity.pro file</span></p>
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<span lang="EN-US"> </span></p>
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<span lang="EN-US">Group {</span></p>
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<span lang="EN-US"> Domain_Disp = Region[100];</span></p>
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<span lang="EN-US"> Domain_Force = Region[101];</span></p>
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<span lang="EN-US"> Edges = Region[102] ; </span></p>
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<span lang="EN-US"> DomainTot = Region[ {Domain_Disp, Domain_Force} ];</span></p>
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<span lang="EN-US"> DefineGroup[ DomainInf ] ;</span></p>
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<span lang="EN-US"> DefineVariable[ Val_Rint, Val_Rext ] ;</span></p>
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}</p>
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</p>
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Function {</p>
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E = 2e11;</p>
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v = 0.3;</p>
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Lambda = E*v/((1+v)*(1-2*v));</p>
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Mu = E/(2*(1+v)); </p>
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Pression = 10e6; </p>
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r0 = 10e-3;</p>
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r1 = 80e-3;</p>
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L = 100e-3;</p>
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</p>
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C_m[ Domain_Disp ] = Tensor[Lambda+2*Mu,Lambda,0,Lambda,Lambda+2*Mu,0,0,0,Mu];</p>
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</p>
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<span lang="EN-US">F[Domain_Force] = Pression*Vector[-1,0,0];</span></p>
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<span lang="EN-US"> </span></p>
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<span lang="EN-US">}</span></p>
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<span lang="EN-US"> </span></p>
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<span lang="EN-US">Constraint {</span></p>
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<span lang="EN-US"> { Name DeplacementX ; Type Assign ;</span></p>
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<span lang="EN-US"> Case {</span></p>
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<span lang="EN-US"> }</span></p>
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<span lang="EN-US"> }</span></p>
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<span lang="EN-US"> { Name DeplacementY ; Type Assign ;</span></p>
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<span lang="EN-US"> Case {</span></p>
<p class="MsoNormal" style="margin: 0cm 0cm 0.0001pt; font-size: 11pt; font-family: Calibri, sans-serif;">
<span lang="EN-US"> { Region Edges; Value 0.0 ; }</span></p>
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<span lang="EN-US"> }</span></p>
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<span lang="EN-US"> }</span></p>
<p class="MsoNormal" style="margin: 0cm 0cm 0.0001pt; font-size: 11pt; font-family: Calibri, sans-serif;">
<span lang="EN-US"> { Name DeplacementZ ; Type Assign ;</span></p>
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<span lang="EN-US"> Case {</span></p>
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<span lang="EN-US"> </span></p>
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<span lang="EN-US"> }</span></p>
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<span lang="EN-US"> }</span></p>
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<span lang="EN-US">}</span></p>
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<span lang="EN-US"> </span></p>
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<span lang="EN-US">Jacobian {</span></p>
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<span lang="EN-US"> { Name MyJac ;</span></p>
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<span lang="EN-US"> Case {</span></p>
<p class="MsoNormal" style="margin: 0cm 0cm 0.0001pt; font-size: 11pt; font-family: Calibri, sans-serif;">
<span lang="EN-US"> { Region DomainInf ; Jacobian VolAxiSphShell {Val_Rint, Val_Rext} ; }</span></p>
<p class="MsoNormal" style="margin: 0cm 0cm 0.0001pt; font-size: 11pt; font-family: Calibri, sans-serif;">
<span lang="EN-US"> { Region Domain_Force ; Jacobian SurAxi ; }</span></p>
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<span lang="EN-US"> { Region All ; Jacobian VolAxi ; }</span></p>
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<span lang="EN-US"> }</span></p>
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<span lang="EN-US"> }</span></p>
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<span lang="EN-US">}</span></p>
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<span lang="EN-US"> </span></p>
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<span lang="EN-US">Integration {</span></p>
<p class="MsoNormal" style="margin: 0cm 0cm 0.0001pt; font-size: 11pt; font-family: Calibri, sans-serif;">
<span lang="EN-US"> { Name GradGrad ;</span></p>
<p class="MsoNormal" style="margin: 0cm 0cm 0.0001pt; font-size: 11pt; font-family: Calibri, sans-serif;">
<span lang="EN-US"> Case { </span></p>
<p class="MsoNormal" style="margin: 0cm 0cm 0.0001pt; font-size: 11pt; font-family: Calibri, sans-serif;">
<span lang="EN-US"> { Type Gauss ;</span></p>
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<span class="apple-tab-span"><span lang="EN-US"> </span></span><span lang="EN-US">Case { </span></p>
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<span class="apple-tab-span"><span lang="EN-US"> </span></span><span lang="EN-US"> { GeoElement Point ; NumberOfPoints 1 ; }</span></p>
<p class="MsoNormal" style="margin: 0cm 0cm 0.0001pt; font-size: 11pt; font-family: Calibri, sans-serif;">
<span class="apple-tab-span"><span lang="EN-US"> </span></span><span lang="EN-US"> { GeoElement Line ; NumberOfPoints 2 ; } </span></p>
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<span class="apple-tab-span"><span lang="EN-US"> </span></span><span lang="EN-US"> { GeoElement Triangle ; NumberOfPoints 4 ; } </span></p>
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<span class="apple-tab-span"><span lang="EN-US"> </span></span><span lang="EN-US"> { GeoElement Quadrangle ; NumberOfPoints 4 ; }</span></p>
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<span class="apple-tab-span"><span lang="EN-US"> </span></span><span lang="EN-US">}</span></p>
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<span lang="EN-US"> }</span></p>
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<span lang="EN-US"> }</span></p>
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<span lang="EN-US"> }</span></p>
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<span lang="EN-US">}</span></p>
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<span lang="EN-US"> </span></p>
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<span lang="EN-US">/* --------------------------------------------------------------------------*/</span></p>
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<span lang="EN-US"> </span></p>
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<span lang="EN-US">/* Mechanics - Displacement ux, uy, uz formulations */</span></p>
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<span lang="EN-US"> </span></p>
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<span lang="EN-US">FunctionSpace {</span></p>
<p class="MsoNormal" style="margin: 0cm 0cm 0.0001pt; font-size: 11pt; font-family: Calibri, sans-serif;">
<span lang="EN-US"> { Name H_u_Mec2D ; Type Vector ;</span></p>
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<span lang="EN-US"> BasisFunction {</span></p>
<p class="MsoNormal" style="margin: 0cm 0cm 0.0001pt; font-size: 11pt; font-family: Calibri, sans-serif;">
<span lang="EN-US"> { Name sxn ; NameOfCoef uxn ; Function BF_NodeX ;</span></p>
<p class="MsoNormal" style="margin: 0cm 0cm 0.0001pt; font-size: 11pt; font-family: Calibri, sans-serif;">
<span class="apple-tab-span"><span lang="EN-US"> </span></span><span lang="EN-US">dFunction {BF_NodeX_D12, BF_Zero} ;</span></p>
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<span lang="EN-US"> Support DomainTot ; Entity NodesOf[ All ] ; }</span></p>
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<span lang="EN-US"> { Name syn ; NameOfCoef uyn ; Function BF_NodeY ;</span></p>
<p class="MsoNormal" style="margin: 0cm 0cm 0.0001pt; font-size: 11pt; font-family: Calibri, sans-serif;">
<span class="apple-tab-span"><span lang="EN-US"> </span></span><span lang="EN-US">dFunction {BF_NodeY_D12, BF_Zero} ;</span></p>
<p class="MsoNormal" style="margin: 0cm 0cm 0.0001pt; font-size: 11pt; font-family: Calibri, sans-serif;">
<span lang="EN-US"> Support DomainTot ; Entity NodesOf[ All ] ; }</span></p>
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<span lang="EN-US"> }</span></p>
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<span lang="EN-US"> Constraint {</span></p>
<p class="MsoNormal" style="margin: 0cm 0cm 0.0001pt; font-size: 11pt; font-family: Calibri, sans-serif;">
<span lang="EN-US"> { NameOfCoef uxn ;</span></p>
<p class="MsoNormal" style="margin: 0cm 0cm 0.0001pt; font-size: 11pt; font-family: Calibri, sans-serif;">
<span lang="EN-US"> EntityType NodesOf ; NameOfConstraint DeplacementX ; }</span></p>
<p class="MsoNormal" style="margin: 0cm 0cm 0.0001pt; font-size: 11pt; font-family: Calibri, sans-serif;">
<span lang="EN-US"> { NameOfCoef uyn ;</span></p>
<p class="MsoNormal" style="margin: 0cm 0cm 0.0001pt; font-size: 11pt; font-family: Calibri, sans-serif;">
<span lang="EN-US"> EntityType NodesOf ; NameOfConstraint DeplacementY ; }</span></p>
<p class="MsoNormal" style="margin: 0cm 0cm 0.0001pt; font-size: 11pt; font-family: Calibri, sans-serif;">
<span lang="EN-US"> }</span></p>
<p class="MsoNormal" style="margin: 0cm 0cm 0.0001pt; font-size: 11pt; font-family: Calibri, sans-serif;">
<span lang="EN-US"> }</span></p>
<p class="MsoNormal" style="margin: 0cm 0cm 0.0001pt; font-size: 11pt; font-family: Calibri, sans-serif;">
<span lang="EN-US">}</span></p>
<p class="MsoNormal" style="margin: 0cm 0cm 0.0001pt; font-size: 11pt; font-family: Calibri, sans-serif;">
<span lang="EN-US"> </span></p>
<p class="MsoNormal" style="margin: 0cm 0cm 0.0001pt; font-size: 11pt; font-family: Calibri, sans-serif;">
<span lang="EN-US">/*------------------------------------------------------------------------*/</span></p>
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<span lang="EN-US"> </span></p>
<p class="MsoNormal" style="margin: 0cm 0cm 0.0001pt; font-size: 11pt; font-family: Calibri, sans-serif;">
<span lang="EN-US">Formulation {</span></p>
<p class="MsoNormal" style="margin: 0cm 0cm 0.0001pt; font-size: 11pt; font-family: Calibri, sans-serif;">
<span lang="EN-US"> { Name Mec2D_u ; Type FemEquation ;</span></p>
<p class="MsoNormal" style="margin: 0cm 0cm 0.0001pt; font-size: 11pt; font-family: Calibri, sans-serif;">
<span lang="EN-US"> Quantity {</span></p>
<p class="MsoNormal" style="margin: 0cm 0cm 0.0001pt; font-size: 11pt; font-family: Calibri, sans-serif;">
<span lang="EN-US"> { Name u ; Type Local ; NameOfSpace H_u_Mec2D ; }</span></p>
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<span lang="EN-US"> }</span></p>
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<span lang="EN-US"> Equation {</span></p>
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<span lang="EN-US"> Galerkin { [ C_m[] * Dof{D1 u}, {D1 u} ] ;</span></p>
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<span lang="EN-US"> In Domain_Disp ; Jacobian MyJac ; Integration GradGrad ; } </span></p>
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<span lang="EN-US"> Galerkin { [ -F[] , {u} ] ; </span></p>
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<span lang="EN-US"> In Domain_Force; Jacobian MyJac; Integration GradGrad; } </span></p>
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<span lang="EN-US"> }</span></p>
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<span lang="EN-US"> }</span></p>
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<span lang="EN-US">}</span></p>
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<span lang="EN-US"> </span></p>
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<span lang="EN-US">/*------------------------------------------------------------------------*/</span></p>
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<span lang="EN-US"> </span></p>
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<span lang="EN-US">Resolution {</span></p>
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<span lang="EN-US"> { Name Mec2D_u ;</span></p>
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<span lang="EN-US"> System {</span></p>
<p class="MsoNormal" style="margin: 0cm 0cm 0.0001pt; font-size: 11pt; font-family: Calibri, sans-serif;">
<span lang="EN-US"> { Name Sys_Mec ; NameOfFormulation Mec2D_u ; }</span></p>
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<span lang="EN-US"> }</span></p>
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<span lang="EN-US"> Operation { Generate Sys_Mec ; Solve Sys_Mec ; SaveSolution Sys_Mec ; }</span></p>
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<span lang="EN-US"> }</span></p>
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<span lang="EN-US">}</span></p>
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<span lang="EN-US"> </span></p>
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<span lang="EN-US">/*------------------------------------------------------------------------*/</span></p>
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<span lang="EN-US"> </span></p>
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<span lang="EN-US">PostProcessing {</span></p>
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<span lang="EN-US"> { Name Mec2D_u ; NameOfFormulation Mec2D_u ; </span></p>
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<span lang="EN-US"> PostQuantity {</span></p>
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<span lang="EN-US"> { Name u ; Value { Term { [ {u} ] ; In Domain_Disp ; Jacobian MyJac ; Integration GradGrad ;} } }</span></p>
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<span lang="EN-US"> { Name um ; Value { Term { [Norm[{u}]] ; In Domain_Disp ; Jacobian MyJac ; Integration GradGrad ;} } }</span></p>
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<span lang="EN-US"> { Name Sx ; Value { Term { [CompX[C_m[]*{D1 u}]] ; In Domain_Disp ; Jacobian MyJac ; Integration GradGrad ;} } }</span></p>
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<span lang="EN-US"> { Name Sy ; Value { Term { [CompY[C_m[]*{D1 u}]] ; In Domain_Disp ; Jacobian MyJac ; Integration GradGrad ;} } }</span></p>
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<span lang="EN-US"> { Name Sz ; Value { Term { [CompZ[C_m[]*{D1 u}]] ; In Domain_Disp ; Jacobian MyJac ; Integration GradGrad ;} } }</span></p>
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<span lang="EN-US"> { Name F ; Value { Term { [F[]] ; In Domain_Force ; Jacobian MyJac ; Integration GradGrad ;} } }</span></p>
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<span lang="EN-US"> { Name Vm ; Value { Term { </span></p>
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<span class="apple-tab-span"><span lang="EN-US"> </span></span><span lang="EN-US"> [Sqrt[ CompX[C_m[]*{D1 u}]#1^2 + CompY[C_m[]*{D1 u}]#2^2 - #1*#2 </span></p>
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<span lang="EN-US"> + 3*(CompZ[C_m[]*{D1 u}])^2 ] ] ; In Domain_Disp ; Jacobian MyJac ; Integration GradGrad ;} } } </span></p>
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<span lang="EN-US"> } </span></p>
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<span lang="EN-US"> }</span></p>
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<span lang="EN-US">}</span></p>
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<span lang="EN-US"> </span></p>
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<span lang="EN-US">PostOperation Map_u UsingPost Mec2D_u { </span></p>
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<span lang="EN-US"> Print [ u, OnElementsOf Domain_Disp, File "u.pos"] ;</span></p>
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<span lang="EN-US"> Print [ um, OnElementsOf Domain_Disp, File "um.pos"] ;</span></p>
<p class="MsoNormal" style="margin: 0cm 0cm 0.0001pt; font-size: 11pt; font-family: Calibri, sans-serif;">
<span lang="EN-US"> Print [ Sx, OnElementsOf Domain_Disp, File "Sx.pos"] ;</span></p>
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<span lang="EN-US"> Print [ Sy, OnElementsOf Domain_Disp, File "Sy.pos"] ;</span></p>
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<span lang="EN-US"> Print [ Sz, OnElementsOf Domain_Disp, File "Sz.pos"] ;</span></p>
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<span lang="EN-US"> Print [ F, OnElementsOf Domain_Force, File "F.pos"] ;</span></p>
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<span lang="EN-US"> Print [ Vm , OnElementsOf Domain_Disp, File "Vm.pos"] ;</span></p>
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<span lang="EN-US">}</span></p>
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<span lang="EN-US"> </span></p>
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<span lang="EN-US"> </span></p>
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<span lang="EN-US">Best regards,</span></p>
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<span lang="EN-US"> </span></p>
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<span lang="EN-US"> </span></p>
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<b><span lang="EN-US">Yves DUPUIS</span></b></p>
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<span lang="EN-US">Doctor / Engineer in Mechatronics</span></p>
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2, rue des champs</p>
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F-27950 St-Marcel – France</p>
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Tél : +33 2 76 48 51 25</p>
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<a href="mailto:Yves.dupuis@s2m.fr" target="_blank" style="color: purple;">Yves.dupuis@s2m.fr</a></p>
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