Validation Case: Design Analysis of a Spherical Pressure Vessel

This design analysis of a spherical pressure vessel validation case belongs to thermomechanics. This test case aims to validate the following parameters:

• Transient thermostructural analysis

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

Geometry

The geometry consists of 1/8th of a sphere, with an inner radius of 0.19 \(m\) and an outer radius of 0.2 \(m\).

The coordinates for the points in the sphere are as tabulated below:

	A	B	C	D	E	F
x	0	0.19	0	0.2	0	0
y	0.19	0	0.2	0	0	0
z	0	0	0	0	0.19	0.2

Analysis Type and Mesh

Tool Type: Code_Aster

Analysis Type: Transient thermomechanical, linear

Mesh and Element Types: The mesh for cases A and B was created with the standard algorithm coupled with a quad-dominant extrusion refinement along the thickness of the vessel. A second order mechanical element and first order thermal element with lumped mass were considered.

The setup from cases A and B is the same, except for the thermal conductivity \(\kappa\).

Case	Mesh Type	Nodes	Thermal Conductivity \(\kappa\)	Element Type
(A)	2nd order Standard	87962	20 \([\frac {W}{m.K}]\)	Hex-dominated
(B)	2nd order Standard	87962	22 \([\frac {W}{m.K}]\)	Hex-dominated

Find below the mesh used for both cases. It’s a standard mesh with second order hexahedral-dominant elements.

Simulation Setup

Material:

• Steel (linear elastic)
  • \(E\) = 190 \(GPa\)
  • \(\nu\) = 0.305
  • \(\rho\) = 7750 \(kg/m³\)
  • \(\kappa\) = 20 \([\frac {W}{m.K}]\) and 22 \([\frac {W}{m.K}]\) for cases A and B, respectively;
  • Expansion coefficient = 9.7e-6 \(1/K\)
  • \(T_0\) Reference temperature = 300 \(K\)
  • Specific heat = 486 \(\frac {J}{kg.K}\)

Initial Conditions

Temperature is 300 \(K\) in the entire pressure vessel.

Boundary Conditions:

• Constraints
  • \(d_x\) = 0 on face ACFE;
  • \(d_y\) = 0 on face BDFE;
  • \(d_z\) = 0 on face ACDB.
• Surface loads
  • Pressure boundary condition on face ABE. The pressure increases linearly from 0 \(MPa\) to 1 \(MPa\) according to formula \(P = (0.2e6).t\), where \(t\) is time from 0 to 5 seconds;
  • Fixed temperature value boundary condition on face ABE. Temperature is increasing linearly, from 300 \(K\) to 500 \(K\) according to formula \(T = 40.t + 300\), where \(t\) is time from 0 to 5 seconds;
  • Convective heat flux boundary condition on face CFD. The heat transfer coefficient is 90 \(\frac {W}{K.m^2}\) and \(T_0\) reference temperature is 300 \(K\).

Reference Solution

The analytical solution is given by the equations presented in [Afkar]\(^1\).

Result Comparison

Since no value for thermal conductivity \(\kappa\) was provided, the values of 20 \(\frac {W}{m.K}\) and 22 \(\frac {W}{m.K}\) were used. For the final time step, the SimScale results for von Mises stress \([MPa]\) and temperature \([K]\) over the edge EF are compared to those from [Afkar]\(^1\).

In Figure 4, we can see how temperature is changing in the sphere’s width, for the last time step:

Last updated: September 26th, 2025