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:

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:

Case	Mesh Type	Number of Nodes	Element Type
A	1st Order Tetrahedral	24755	Standard
B	1st Order Tetrahedral	24755	Standard
C	1st Order Tetrahedral	24755	Standard
D	2nd Order Tetrahedral	172165	Standard
E	2nd Order Tetrahedral	172165	Standard
F	2nd Order Tetrahedral	172165	Standard

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
1	2nd Order Tetrahedral	677	2
2	2nd Order Tetrahedral	1443	4
3	2nd Order Tetrahedral	3557	6
4	2nd Order Tetrahedral	14004	8

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:

Case	Natural Frequencies Computing Algorithm
A	IRAM – Sorensen
B	Lanczos
C	Bathe – Wilson
D	IRAM – Sorensen
E	Lanczos
F	Bathe – Wilson

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.

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:

Mode	Natural Frequency \([Hz]\)
Ovalization	210.55
	210.55
Trifoliate	587.92
	587.92
Out of Plane	205.89
	205.89
	588.88
	588.88

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.

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

Mode	Reference Solution	SimScale Solution	Error
Ovalization	210.55	213.332	1.32 %
	210.55	213.966	1.62 %
Trifoliate	587.92	596.343	1.43 %
	587.92	596.698	1.49 %
Out of Plane	205.89	210.017	2.00 %
	205.89	210.244	2.11 %
	588.88	599.512	1.81 %
	588.88	599.521	1.81 %

Mode	Reference Solution	SimScale Solution	Error
Ovalization	210.55	215.181	2.20 %
	210.55	216.09	2.63 %
Trifoliate	587.92	601.83	2.37 %
	587.92	602.316	2.45 %
Out of Plane	205.89	212.822	3.37 %
	205.89	213.098	3.50 %
	588.88	606.685	3.02 %
	588.88	606.794	3.04 %

Mode	Reference Solution	SimScale Solution	Error
Ovalization	210.55	215.181	2.20 %
	210.55	216.09	2.63 %
Trifoliate	587.92	601.83	2.37 %
	587.92	602.316	2.45 %
Out of Plane	205.89	212.822	3.37 %
	205.89	213.098	3.50 %
	588.88	606.685	3.02 %
	588.88	606.794	3.04 %

Mode	Reference Solution	SimScale Solution	Error
Ovalization	210.55	209.998	-0.26 %
	210.55	209.998	-0.26 %
Trifoliate	587.92	586.293	-0.28 %
	587.92	586.293	-0.28 %
Out of Plane	205.89	205.143	-0.36 %
	205.89	205.144	-0.36 %
	588.88	586.975	-0.32 %
	588.88	586.975	-0.32 %

Mode	Reference Solution	SimScale Solution	Error
Ovalization	210.55	209.998	-0.26 %
	210.55	209.998	-0.26 %
Trifoliate	587.92	586.293	-0.28 %
	587.92	586.293	-0.28 %
Out of Plane	205.89	205.143	-0.36 %
	205.89	205.144	-0.36 %
	588.88	586.975	-0.32 %
	588.88	586.975	-0.32 %

Mode	Reference Solution	SimScale Solution	Error
Ovalization	210.55	209.998	-0.26 %
	210.55	209.998	-0.26 %
Trifoliate	587.92	586.293	-0.28 %
	587.92	586.293	-0.28 %
Out of Plane	205.89	205.143	-0.36 %
	205.89	205.144	-0.36 %
	588.88	586.975	-0.32 %
	588.88	586.975	-0.32 %

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.

Algorithm	Runtime 1st Order Mesh	Runtime 2nd Order Mesh
IRAM – Sorensen	2 min	13 min
Lanczos	2 min	13 min
Bathe – Wilson	9 min	139 min

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

Last updated: June 30th, 2020