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# Validation Case: NAFEMS LE11 “Solid cylinder/taper/sphere-temperature” benchmark

## Problem statement

This thermomechanical simulation is part of the NAFEMS benchmark cases. It consists of a quarter of a solid cylindrical object with an imposed temperature field. The predicted stress due to thermomechanical expansion from the simulation results is compared against the reference solution given by the NAFEMS benchmark book [1].

## Geometry

The geometry is a thick hollow cylinder as shown in Fig. 1. All planar surfaces have symmetry boundary conditions such that only a quarter of the cylinder is represented in the simulation.

The CAD file for this geometry is available as an Onshape document. The dimensions of the base sketch are shown in Fig. 2.

## Boundary Conditions

Symmetry boundary conditions are imposed on all planar surfaces. The temperature field is specified with an algebraic expression and imposed on the entire domain.

\begin{align*} T(x,y,z) = \sqrt{x^2+y^2} + z\end{align*}

The reference temperature is 0 ºC.

## Material Properties

Homogeneous isotropic linear elastic properties are used, as specified in the original problem formulation [1].

Figure 1. NAFEMS LE11 3D CAD

Figure 2. NAFEMS LE11 dimensions

PropertySymbolValue
Young modulusE210 GPa
Poisson’s ratio\nu0.3
Thermal expansion coefficient \alpha 2.3 \times 10^{-4}~\text{ºC}^{-1}

## Mesh

OnScale Solve generated a second-order tetrahedral mesh.

Five simulations are run with the mesh density ranging from very coarse to very fine. Convergence toward the reference solution is shown in the results.

Mesh density# of elements# of dofs
very coarse7655,912
coarse2,20015,380
medium5,59936,936
fine14,31890,368
very fine82,198494,148

## Results

The z component of stress \sigma_z at point A from the simulation is compared against the reference solution \sigma_z = -105 MPa [1].

In this case we are interested in the stress at a single point which requires a relatively refined mesh in that region. OnScale computes the stress precisely at the requested point coordinates. Often if only deflections are of interest, a less refined mesh can be used.

Mesh density# of dofsRef. \sigma_z MPaSim. \sigma_z MPaDiff. %
very coarse5,912-105-97.007.6
coarse15,380-105-101.383.4
medium39,936-105-102.362.5
fine90,368-105-102.991.9
very fine494,148-105-104.650.3

## Simulation Definition

The complete simulation definition is given below. The metadata tags for parts and faces correspond to the version of the CAD file that was imported from Onshape.

"""
Auto-generated simulation code.
"""
import  onscale  as  on
with  on.Simulation('Simulation' , 'Generated in SOLVE', version = '0.8.2')  as  sim:
# General simulation settings
on.settings.AmbientTemperature(0)
on.settings.EnabledPhysics(["mechanical", "thermal"])
on.settings.Mesher(use_feature_detection=True, mesh_index=4)
coord = on.GlobalCoordinates()
# Define geometry
geometry = on.CadFile('Assembly_1.x_t', unit="m")
point = on.Point(1, 0, 0)
# Define material database and materials 
materials  =on.CloudMaterials('onscale')
structural_steel = materials['Structural steel']
structural_steel.alias = 'Customer Structural steel 1'
structural_steel >> geometry.parts.search('Mvy45FHalSAFz5PDD', field='partIDTag')
structural_steel.set('youngs_modulus', 210000000000.0)
structural_steel.set('thermal_expansion', 0.00023)
# Define and apply loads 
symmetry = on.loads.Symmetry(alias='Symmetry 3')
symmetry_2 = on.loads.Symmetry(alias='Symmetry 4')
symmetry_3 = on.loads.Symmetry(alias='Symmetry 1')
symmetry_3 >> geometry.parts.search('Mvy45FHalSAFz5PDD', field='partIDTag').faces.search('JFW', field='faceIDTag')
symmetry_4 = on.loads.Symmetry(alias='Symmetry 2')
symmetry_4 >> geometry.parts.search('Mvy45FHalSAFz5PDD', field='partIDTag').faces.search('JFG', field='faceIDTag')
symmetry >> geometry.parts.search('Mvy45FHalSAFz5PDD', field='partIDTag').faces.search('JFK', field='faceIDTag')
symmetry_2 >> geometry.parts.search('Mvy45FHalSAFz5PDD', field='partIDTag').faces.search('JFS', field= 'faceIDTag')
temperature = on.loads.Temperature(f'sqrt({coord.x}**2+{coord.y}**2)+{coord.z}', alias='Temperature 1')
temperature >> geometry.parts.search('Mvy45FHalSAFz5PDD', field='partIDTag')
# Define output variables
probe_sensor = on.sensors.ProbeSensor(data="Stress")
probe_sensor >> point
field_sensor = on.sensors.FieldSensor(data=["Displacement",  "VonMises", "Stress", "Strain", "PrincipalStress", "PrincipalStrain", "StrainEnergyDensity", "EigenVector", "Temperature"], alias='Global Sensor')
field_sensor >> geometry
reaction_sensor = on.sensors.ReactionSensor()
reaction_sensor >> symmetry
reaction_sensor_2 = on.sensors.ReactionSensor()
reaction_sensor_2 >>  symmetry_2
reaction_sensor_3 = on.sensors.ReactionSensor()
reaction_sensor_3 >> symmetry_3
reaction_sensor_4 = on.sensors.ReactionSensor()
reaction_sensor_4 >> symmetry_4

• ## References

[1]       National Agency for Finite Element Methods & Standards,
The standard NAFEMS benchmarks.
Glasgow, U.K.: East Kilbride, 1990.