Topology optimization of a 2D cantilever beam using body-fitted meshes#
The goal of this tutorial is to run a topology optimization of a bidimensional cantilever beam using using body-fitted meshes.
Note
For running this tutorial Code_Aster 2 needs to be installed and available from the command line.
Test case#
Consider the following specifications
domain of size \(2\) m \(\times 1\) m
elastic material with Young modulus equal to \(210.e9\) Pa and Poisson ratio equal to \(0.3\)
beam clamped on the plane \(x = 0\) m and submitted to a vertical load g = \((0, -10000)\) N on the point \((2, 0.5)\)
minimization of the volume under a compliance constraint
Data setting : header#
We start by editing a Cantilever2D.xml file.
The data file starts with the following header
<data dim="2" >
Data setting : implicit zones#
Define implicit zones
<Zones>
<Plane id="1" point="0.0001 0.25 0.5" normal="1. 0. 0." desc="x0 plane "/>
<Plane id="2" point="0.05 0.25 0.5" normal="1. 0. 0." desc="dirichlet"/>
<Sphere id="4" center="2. 0.5 0.0" radius="0.0009" desc="nodal force"/>
<Cylinder id="5" center1="1.9999 0.5 0." center2="2.1 0.5 0." radius="0.01" desc="force"/>
<Holes id="6" n="6 4 2" r="0.2" boundingMin="0. 0. 0." boundingMax="2.0 1.0 1.0" type="Original" desc="Holes"/>
<Union id="7" z="2 4 6" desc="initialization"/>
<Union id="8" z="2 5" desc="onzone" />
</Zones>
Data setting : grids#
Define the background mesh which is a uniform grid
<Grids>
<Grid id="1" n="61 41" origin="0. 0." length="2.0 1.0" ofSimplex="True"/>
</Grids>
Data setting : levelsets#
Define the level set together with a set of options. The option conform=”True” means that the level set is conformal. A set of meshing options are defined to drive the remeshing process.
<LevelSets>
<LevelSet id="1" support="1" conform="True" gradientMethod="gradPhi">
<MesherInfo hmax=".03" hmin="0.008" iso="0.0" rmc="1e-5"
nr="True" computeDistanceWith="meshdist"/>
</LevelSet>
</LevelSets>
Data setting : physical problems#
Define the physical problem, which is a linear elastic static problem.
We define material properties, one dirichlet condition and one neumann condition.
By default we use linear elements.
<PhysicalProblems>
<GeneralAster type="static_elastic" id="1" >
<Material young="210.e9" poisson="0.3" />
<Dirichlet eTag="eTag1" dofs="0 1" value="0.0"/>
<Force nTag="nTag4" value="0. -1000" />
</GeneralAster>
</PhysicalProblems>
Data setting : outputs#
Define the desired output
<Outputs>
<Output id="1" name="Cantilever2DHistory.xdmf" inTempDirectory="True" />
<Output id="2" name="Cantilever2DHistory.csv" inTempDirectory="True" />
<Output id="3" name="PROXY" outputs="1 2" />
</Outputs>
Data setting : optimization problems#
Then, we define the optimization problem by entering the objective, the constraints, the initial levelset, the implicit zones, the output and a set of options.
Note that we have to define an upper bound value for the constraint.
Note that, since the constraint is a physical criterion, we precise which physical problem should be used with the keyword useProblem=”1”.
<OptimProblems>
<OptimProblem type="TopoGeneric" id="1"
OnZone="8"
useLevelset="1"
printMeshQualityInfo = "True"
output="3"
outputEveryLevelsetUpdate = "False">
<Objective type="Volume" name="Volume"/>
<Constraint type="Compliance" name="Compliance" UpperBound="2*1.90489800e-04" useProblem="1" />
</OptimProblem>
</OptimProblems>
Data setting : optimization algorithms#
Then, we define the optimization algorithm which is linked to the optimization problem
<TopologicalOptimizations>
<OptimAlgoNullSpace id="1"
useOptimProblem="1"/>
</TopologicalOptimizations>
Actions setting#
Here we declare actions which must be runned. At first the level set “1” is initialized with the implicit zone “7”
<Actions>
<Action type="InitLevelset" ls="1" zone="7"/>
We create a nodal tag “nTag4” from implicit zone “4” for the nodal force and we set it as required to preserve it during the remeshing
<Action type="CreateTagsFromZones" ls="1" nodalTags="4" nodalprefix="nTag"/>
<Action type= "SetAsRequired" ls="1" nTags="nTag4" />
We perform a remeshing to generate a conformal mesh
<Action type="Remesh" ls="1" hmax=".03" hmin="0.008" iso="0.0" rmc="1e-5"
nr="False" computeDistanceWith="meshdist" />
We create an element tag to support the Dirichlet boundary condition
<Action type="CreateTagsFromZones" ls="1" elementTags="1" dim="1" elemenprefix="eTag"/>
Then, we run the topology optimization
<Action type="RunTopoOp" TopoOp="1"/>
Eventually mesh quality indicators are printed
<Action type="LsQualityInfo" ls="1" />
</Actions>
</data>
Fig. 11 Visualisation of the initial material distribution in Paraview#
Run#
To run the script using the command line application execute in a terminal
OpenPiscoCL Cantilever2D.xml
To run the script using the GUI application execute in a terminal
OpenPisco Cantilever2D.xml
Output#
After running the tutorial script the following output files have been created in the working directory
Cantilever2D.xml.log : run summary
Cantilever2DHistory.xdmf : (heavy) file containing the mesh and fields at each iteration in xdmf format
Cantilever2DHistory.csv : convergence history in csv format
Fig. 12 Visualisation of the optimal design in Paraview#
Fig. 13 Visualisation of the displacement field in Paraview#
The demo xml file created is available in the module OpenPisco.Demos() of the OpenPisco Python library 1 .