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Validation Case: Thick Hollow Thermoelastic Cylinder

This validation case belongs to thermomechanics, with the case of a thick hollow thermoelastic cylinder under temperature gradient conditions. The aim of this test case is to validate the following parameters:

  • Thermomechanical solver

The simulation results of SimScale were compared to the analytical results presented in [HPLA100]\(^1\).


The geometry used for the case is as follows:

geometry thermoelastic cylinder
Figure 1: Geometrical model for the cylinder

It represents a quarter section of a hollow cylinder with an internal radius of 19.5 \(m\) and an external radius of 20.5 \(m\), length of 10.0 \(m\). Face ABCD coincides with the XZ plane, face EFGH with the YZ plane and face CHGD with the XY plane.

Analysis Type and Mesh

Tool Type: Code_Aster

Analysis Type: Thermomechanical, Linear, Steady State.

Mesh and Element Types:

CaseMesh TypeNumber of
Element Type
A, B1st order tetrahedral22105Standard
C, D2nd order tetrahedral39674Standard
Table 1: Mesh details for each case

The meshes were computed using SimScale’s standard mesh algorithm and automatic sizing.

first order mesh thermoelastic cylinder
Figure 2: 1st order tetrahedral mesh, used on cases A and B
second order mesh thermoelastic cylinder
Figure 3: 2nd order tetrahedral mesh, used on cases C and D

Simulation Setup


  • Young’s module \( E\) = 20 \( Pa \)
  • Poisson coefficient \( \nu\) = 0.3
  • Density \( \rho\) = 8000 \( kg/m^3 \)
  • Expansion coefficient \( \alpha\) = 1e-5 \( 1/°C \)
  • Reference temperature \( T_0\) = 0 \(°C\)

Boundary Conditions:

  • Displacements
    • Fixed value DY = DZ = 0 on face ABCD
    • Fixed value DX = DZ = 0 on face EFGH
    • Fixed value DZ = 0 on faces CHGD and AFEB
  • Temperatures (Cases A, C) – Bending deformation
    • Fixed temperature value of \(T\) = -0.5 \(°C\) on the internal face (BEHC)
    • Fixed temperature value of \(T\) = 0.5 \(°C\) on the external face (AGDF)
  • Temperatures (Cases B, D) – Radial expansion
    • Fixed temperature value of \(T\) = 0.1 \(°C\) on the internal face (BEHC)
    • Fixed temperature value of \(T\) = 0.1 \(°C\) on the external face (AGDF)

Reference Solution

The reference solution for the thermoelastic cylinder is of the analytical type, as presented in [HPLA100]\(^1\). It is given in terms of the longitudinal stress onthe internal and external faces:

Cases A and C, internal face: \( \sigma_{ZZ}\) = 1.4321427 \( MPa \)

Cases A and C, external face: \( \sigma_{ZZ}\) = -1.4250001 \( MPa \)

Cases B and D, internal face: \( \sigma_{ZZ}\) = -0.2 \( MPa \)

Cases B and D, external face: \( \sigma_{ZZ} \) = -0.2 \( MPa \)

Result Comparison

Comparison of the computed stresses with the reference solution is presented, at the representative points:

\( \sigma_{ZZ} \) \([Pa]\)
\( \sigma_{ZZ} \) \([Pa]\)
AC1.1593Ee61.4321e619.1 %
D-1.1943e6-1.4250e616.2 %
BC-2.0010e5-2.0e50.0 %
D-1.9993e5-2.0e50.0 %
CC1.4342e61.4321e6-0.1 %
D-1.4228e6-1.4250e60.2 %
DC-2.0000e5-2.0e50.0 %
D-2.0000e5-2.0e50.0 %
Table 2: Results comparison and computed errors for cases A through D

It is found that general results agree with the reference solution. As usual, the first order tetrahedral mesh gives bad accuracy on the bending case, although for the pure expansion it gives good results. It is expected that a finer mesh will improve the accuracy of the solution.

Illustration of the stress distribution for the bending case of the thermoelastic cylinder is shown below:

stress contour thermoelastic cylinder
Figure 4: Longitudinal stress distribution from case C


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Last updated: September 30th, 2020