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Documentation

Validation Case: Frequency Analysis of a Ring

This validation case belongs to solid mechanics. The aim of this test case is to validate the following parameters:

  • Frequency analysis

The simulation results of SimScale were compared to the results from [SDLS109]\(^1\).

Geometry

The geometry used for the frequency analysis is as follows:

geometry model parameters for frequency analysis of a ring validation case
Figure 1: Geometry model and parameters

The ring has a length \(L\) of 0.05 \(m\), a thickness \(t\) of 0.048 \(m\), and a medium radius \(Rm\) of 0.369 \(m\).

Analysis Type and Mesh

Tool Type: Code_Aster

Analysis Type: Frequency Analysis

Mesh and Element Types:

First and second order meshes were computed using the SimScale standard mesh algorithm:

CaseMesh TypeNumber of
Nodes
Element Type
A1st Order Tetrahedral24755Standard
B1st Order Tetrahedral24755Standard
C1st Order Tetrahedral24755Standard
D2nd Order Tetrahedral172165Standard
E2nd Order Tetrahedral172165Standard
F2nd Order Tetrahedral172165Standard
Table 1: Mesh refinement per case

A mesh independence study was also performed for the second order meshes and IRAM – Sorensen algorithm to ensure the optimal fineness parameter level.

Mesh
#
Mesh
Type
Number of
Nodes
Mesh
Fineness
12nd Order Tetrahedral6772
22nd Order Tetrahedral14434
32nd Order Tetrahedral35576
42nd Order Tetrahedral140048
Table 2: Mesh convergence study details
tetrahedral finite elements mesh for frequency analysis of a ring validation case
Figure 2: Tetrahedral finite element mesh used for all cases

Simulation Setup

Material:

  • Linear Elastic Isotropic:
    • \( E = \) 185 \(GPa \)
    • \( \nu = \) 0.3
    • \( \rho = \) 7800 \(kg.m^{-3} \)

Boundary Conditions:

  • Constraints:
    • Body is free in space.

Computing Algorithm:

The following available computing algorithms were compared in the different cases:

CaseNatural Frequencies
Computing Algorithm
AIRAM – Sorensen
BLanczos
CBathe – Wilson
DIRAM – Sorensen
ELanczos
FBathe – Wilson
Table 3: Computing algorithms by case

The main characteristics of each algorithm are summarized below [U4.52.02]\(^2\):

  • IRAM – Sorensen: Uses a sub-space decomposition method to compute the natural frequencies and modes. Suitable for real and complex, symmetrical or non-symmetrical matrices.
  • Lanczos: Uses a sub-space decomposition method to compute the natural frequencies and modes. Suitable for real, symmetrical only matrices.
  • Bathe – Wilson: Uses a sub-space decomposition method to compute the natural frequencies and modes. Suitable for real, symmetrical only matrices.

A fourth algorithm is also available in SimScale, called QZ. This algorithm suffers from high memory consumption, which limits its application to cases with less than 1000 degrees of freedom. Therefore, it is not suitable for the current validation case.

Note

Complex matrices appear in the frequency analysis of materials with frequency damping. As this model is not available in SimScale, the difference between the algorithms comes down to robustness and speed.

Frequency Reference Solution

The reference solution is of numerical type, as developed in [SDLS109]\(^1\). The solution is presented in terms of all the natural frequencies and their corresponding shapes in the frequency range [200, 800] \(Hz\). This solution was achieved by a convergence analysis using hexahedral elements, and as reported in the reference, a precision of 5% of the computed frequencies is estimated.

The consulted reference solution is:

ModeNatural Frequency \([Hz]\)
Ovalization210.55
210.55
Trifoliate587.92
587.92
Out of Plane205.89
205.89
588.88
588.88
Table 4: Reference solution for different frequency modes

Frequency Results Comparison

Below can be found the results of the mesh independence study. For each natural frequency (F1 through F8), the variation of the result (in percent) with respect to the previous solution is plotted against the number of nodes in the mesh. At the final, finer mesh, the solution precision is 0.6% or lower.

mesh convergence analysis plot for frequency analysis of a ring validation case
Figure 3: Mesh convergence analysis results for second order mesh.
Here, F1 means first frequency, F2 second frequency, and so on.

Comparison of computed natural frequencies with the reference solution for each case can be seen below:

ModeReference SolutionSimScale SolutionError
Ovalization210.55213.3321.32 %
210.55213.9661.62 %
Trifoliate587.92596.3431.43 %
587.92596.6981.49 %
Out of Plane205.89210.0172.00 %
205.89210.2442.11 %
588.88599.5121.81 %
588.88599.5211.81 %
Table 5: Case A results
ModeReference SolutionSimScale SolutionError
Ovalization210.55215.1812.20 %
210.55216.092.63 %
Trifoliate587.92601.832.37 %
587.92602.3162.45 %
Out of Plane205.89212.8223.37 %
205.89213.0983.50 %
588.88606.6853.02 %
588.88606.7943.04 %
Table 6: Case B results
ModeReference SolutionSimScale SolutionError
Ovalization210.55215.1812.20 %
210.55216.092.63 %
Trifoliate587.92601.832.37 %
587.92602.3162.45 %
Out of Plane205.89212.8223.37 %
205.89213.0983.50 %
588.88606.6853.02 %
588.88606.7943.04 %
Table 7: Case C results
ModeReference SolutionSimScale SolutionError
Ovalization210.55209.998-0.26 %
210.55209.998-0.26 %
Trifoliate587.92586.293-0.28 %
587.92586.293-0.28 %
Out of Plane205.89205.143-0.36 %
205.89205.144-0.36 %
588.88586.975-0.32 %
588.88586.975-0.32 %
Table 8: Case D results
ModeReference SolutionSimScale SolutionError
Ovalization210.55209.998-0.26 %
210.55209.998-0.26 %
Trifoliate587.92586.293-0.28 %
587.92586.293-0.28 %
Out of Plane205.89205.143-0.36 %
205.89205.144-0.36 %
588.88586.975-0.32 %
588.88586.975-0.32 %
Table 9: Case E results
ModeReference SolutionSimScale SolutionError
Ovalization210.55209.998-0.26 %
210.55209.998-0.26 %
Trifoliate587.92586.293-0.28 %
587.92586.293-0.28 %
Out of Plane205.89205.143-0.36 %
205.89205.144-0.36 %
588.88586.975-0.32 %
588.88586.975-0.32 %
Table 10: Case F results

Results are mesh dependent instead of algorithm dependent because all algorithms produce similar results. The difference between algorithms can be seen when looking at the running times. Cases using IRAM – Sorensen and Lanczos algorithm are much faster than Bathe – Wilson. The recommendation is then to stay with the default algorithm (IRAM – Sorensen), because of its known robustness.

AlgorithmRuntime
1st Order Mesh
Runtime
2nd Order Mesh
IRAM – Sorensen2 min13 min
Lanczos2 min13 min
Bathe – Wilson9 min139 min
Table 11: Comparison of runtime for each algorithm

Following are the referenced natural mode shapes as seen on the online post-processor:

natural vibration shapes plot for frequency analysis of a ring validation case
Figure 4: Natural vibration shapes for each natural frequency taken from case D.

References

  • SDLS109 – Fréquences propres d’un anneau cylindrique épais – Code_Aster validation case
  • [U4.52.02] – Opérateur CALC_MODES – Code_Aster utilization manual

Note

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