{"id":29762,"date":"2020-06-30T09:49:45","date_gmt":"2020-06-30T09:49:45","guid":{"rendered":"https:\/\/www.simscale.com\/?page_id=29762"},"modified":"2020-06-30T13:19:15","modified_gmt":"2020-06-30T13:19:15","slug":"frequency-analysis-ring","status":"publish","type":"page","link":"https:\/\/www.simscale.com\/docs\/validation-cases\/frequency-analysis-ring\/","title":{"rendered":"Validation Case: Frequency Analysis of a Ring"},"content":{"rendered":"\n\n\n\n<p class=\"wp-block-paragraph\">This validation case belongs to solid mechanics. The aim of this test case is to validate the following parameters:<\/p>\n\n\n\n<ul class=\"wp-block-list\"><li>Frequency analysis<\/li><\/ul>\n\n\n\n<p class=\"wp-block-paragraph\">The simulation results of SimScale were compared to the results from [SDLS109]\\(^1\\).<\/p>\n\n\n\n<div class=\"hw-block hw-btnWrapper hw-btnWrapper--alignCenter \">\n    <a href=\"https:\/\/www.simscale.com\/projects\/simscale\/validation_case-_frequency_analysis_of_a_ring\/\" class=\"hw-btn    \" rel=\"noopener \"    data-collateral-name=\"\" data-collateral-type=\"\" data-collateral-campaign=\"\" data-type=\"\" >\n        View Project     <\/a>\n<\/div>\n\n\n\n\n<h2 class=\"wp-block-heading\" id=\"geometry\" >Geometry<\/h2>\n\n\n\n<p class=\"wp-block-paragraph\">The geometry used for the frequency analysis is as follows: <\/p>\n\n\n\n<figure class=\"wp-block-image size-large\"><a href=\"https:\/\/frontend-assets.simscale.com\/media\/2020\/06\/image-15.png\"><img loading=\"lazy\" decoding=\"async\" width=\"1024\" height=\"601\" src=\"https:\/\/frontend-assets.simscale.com\/media\/2020\/06\/image-15-1024x601.png\" alt=\"geometry model parameters for frequency analysis of a ring validation case\" class=\"wp-image-29763\" srcset=\"https:\/\/frontend-assets.simscale.com\/media\/2020\/06\/image-15-1024x601.png 1024w, https:\/\/frontend-assets.simscale.com\/media\/2020\/06\/image-15-300x176.png 300w, https:\/\/frontend-assets.simscale.com\/media\/2020\/06\/image-15-768x451.png 768w, https:\/\/frontend-assets.simscale.com\/media\/2020\/06\/image-15.png 1312w\" sizes=\"auto, (max-width: 1024px) 100vw, 1024px\" \/><\/a><figcaption>Figure 1: Geometry model and parameters<\/figcaption><\/figure>\n\n\n\n<p class=\"wp-block-paragraph\">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\\).<\/p>\n\n\n\n<h2 class=\"wp-block-heading\" id=\"analysis-type-and-mesh\" >Analysis Type and Mesh<\/h2>\n\n\n\n<p class=\"wp-block-paragraph\"><strong>Tool Type<\/strong>: Code_Aster<\/p>\n\n\n\n<p class=\"wp-block-paragraph\"><strong>Analysis Type<\/strong>: Frequency Analysis<\/p>\n\n\n\n<p class=\"wp-block-paragraph\"><strong>Mesh and Element Types<\/strong>:<\/p>\n\n\n\n<p class=\"wp-block-paragraph\">First and second order meshes were computed using the SimScale standard mesh algorithm: <\/p>\n\n\n\n<figure class=\"wp-block-table aligncenter is-style-stripes\"><table><thead><tr><th class=\"has-text-align-center\" data-align=\"center\"><strong>Case<\/strong><\/th><th class=\"has-text-align-center\" data-align=\"center\"><strong>Mesh Type<\/strong><\/th><th class=\"has-text-align-center\" data-align=\"center\"><strong>Number of <\/strong><br><strong>Nodes<\/strong><\/th><th class=\"has-text-align-center\" data-align=\"center\"><strong>Element Type<\/strong><\/th><\/tr><\/thead><tbody><tr><td class=\"has-text-align-center\" data-align=\"center\">A<\/td><td class=\"has-text-align-center\" data-align=\"center\">1st Order Tetrahedral<\/td><td class=\"has-text-align-center\" data-align=\"center\">24755<\/td><td class=\"has-text-align-center\" data-align=\"center\">Standard<\/td><\/tr><tr><td class=\"has-text-align-center\" data-align=\"center\">B<\/td><td class=\"has-text-align-center\" data-align=\"center\">1st Order Tetrahedral<\/td><td class=\"has-text-align-center\" data-align=\"center\">24755<\/td><td class=\"has-text-align-center\" data-align=\"center\">Standard<\/td><\/tr><tr><td class=\"has-text-align-center\" data-align=\"center\">C<\/td><td class=\"has-text-align-center\" data-align=\"center\">1st Order Tetrahedral<\/td><td class=\"has-text-align-center\" data-align=\"center\">24755<\/td><td class=\"has-text-align-center\" data-align=\"center\">Standard<\/td><\/tr><tr><td class=\"has-text-align-center\" data-align=\"center\">D<\/td><td class=\"has-text-align-center\" data-align=\"center\">2nd Order Tetrahedral<\/td><td class=\"has-text-align-center\" data-align=\"center\">172165<\/td><td class=\"has-text-align-center\" data-align=\"center\">Standard<\/td><\/tr><tr><td class=\"has-text-align-center\" data-align=\"center\">E<\/td><td class=\"has-text-align-center\" data-align=\"center\">2nd Order Tetrahedral<\/td><td class=\"has-text-align-center\" data-align=\"center\">172165<\/td><td class=\"has-text-align-center\" data-align=\"center\">Standard<\/td><\/tr><tr><td class=\"has-text-align-center\" data-align=\"center\">F<\/td><td class=\"has-text-align-center\" data-align=\"center\">2nd Order Tetrahedral<\/td><td class=\"has-text-align-center\" data-align=\"center\">172165<\/td><td class=\"has-text-align-center\" data-align=\"center\">Standard<\/td><\/tr><\/tbody><\/table><figcaption>Table 1: Mesh refinement per case<\/figcaption><\/figure>\n\n\n\n<p class=\"wp-block-paragraph\">A mesh independence study was also performed for the second order meshes and IRAM \u2013 Sorensen algorithm to ensure the optimal fineness parameter level.<\/p>\n\n\n\n<figure class=\"wp-block-table aligncenter is-style-stripes\"><table><thead><tr><th class=\"has-text-align-center\" data-align=\"center\"><strong>Mesh<\/strong><br>#<\/th><th class=\"has-text-align-center\" data-align=\"center\"><strong>Mesh <\/strong><br><strong>Type<\/strong><\/th><th class=\"has-text-align-center\" data-align=\"center\"><strong>Number of <\/strong><br><strong>Nodes<\/strong><\/th><th class=\"has-text-align-center\" data-align=\"center\">Mesh<br>Fineness<\/th><\/tr><\/thead><tbody><tr><td class=\"has-text-align-center\" data-align=\"center\">1<\/td><td class=\"has-text-align-center\" data-align=\"center\">2nd Order Tetrahedral<\/td><td class=\"has-text-align-center\" data-align=\"center\">677<\/td><td class=\"has-text-align-center\" data-align=\"center\">2<\/td><\/tr><tr><td class=\"has-text-align-center\" data-align=\"center\">2<\/td><td class=\"has-text-align-center\" data-align=\"center\">2nd Order Tetrahedral<\/td><td class=\"has-text-align-center\" data-align=\"center\">1443<\/td><td class=\"has-text-align-center\" data-align=\"center\">4<\/td><\/tr><tr><td class=\"has-text-align-center\" data-align=\"center\">3<\/td><td class=\"has-text-align-center\" data-align=\"center\">2nd Order Tetrahedral<\/td><td class=\"has-text-align-center\" data-align=\"center\">3557<\/td><td class=\"has-text-align-center\" data-align=\"center\">6<\/td><\/tr><tr><td class=\"has-text-align-center\" data-align=\"center\">4<\/td><td class=\"has-text-align-center\" data-align=\"center\">2nd Order Tetrahedral<\/td><td class=\"has-text-align-center\" data-align=\"center\">14004<\/td><td class=\"has-text-align-center\" data-align=\"center\">8<\/td><\/tr><\/tbody><\/table><figcaption>Table 2: Mesh convergence study details<\/figcaption><\/figure>\n\n\n\n<figure class=\"wp-block-image size-large\"><a href=\"https:\/\/frontend-assets.simscale.com\/media\/2020\/06\/mesh-1.png\"><img loading=\"lazy\" decoding=\"async\" width=\"959\" height=\"875\" src=\"https:\/\/frontend-assets.simscale.com\/media\/2020\/06\/mesh-1.png\" alt=\"tetrahedral finite elements mesh for frequency analysis of a ring validation case\" class=\"wp-image-30523\" srcset=\"https:\/\/frontend-assets.simscale.com\/media\/2020\/06\/mesh-1.png 959w, https:\/\/frontend-assets.simscale.com\/media\/2020\/06\/mesh-1-300x274.png 300w, https:\/\/frontend-assets.simscale.com\/media\/2020\/06\/mesh-1-768x701.png 768w\" sizes=\"auto, (max-width: 959px) 100vw, 959px\" \/><\/a><figcaption>Figure 2: Tetrahedral finite element mesh used for all cases<\/figcaption><\/figure>\n\n\n\n<h2 class=\"wp-block-heading\" id=\"simulation-setup\" >Simulation Setup<\/h2>\n\n\n\n<p class=\"wp-block-paragraph\"><strong>Material<\/strong>:<\/p>\n\n\n\n<ul class=\"wp-block-list\"><li>Linear Elastic Isotropic:<ul><li>\\( E = \\) 185 \\(GPa \\)<\/li><li>\\( \\nu&nbsp;= \\) 0.3<\/li><li>\\( \\rho = \\) 7800 \\(kg.m^{-3} \\)<\/li><\/ul><\/li><\/ul>\n\n\n\n<p class=\"wp-block-paragraph\"><strong>Boundary Conditions<\/strong>:<\/p>\n\n\n\n<ul class=\"wp-block-list\"><li>Constraints:<ul><li>Body is free in space.<\/li><\/ul><\/li><\/ul>\n\n\n\n<p class=\"wp-block-paragraph\"><strong>Computing Algorithm:<\/strong><\/p>\n\n\n\n<p class=\"wp-block-paragraph\">The following available computing algorithms were compared in the different cases:<\/p>\n\n\n\n<figure class=\"wp-block-table aligncenter is-style-stripes\"><table><thead><tr><th class=\"has-text-align-center\" data-align=\"center\"><strong>Case<\/strong><\/th><th class=\"has-text-align-center\" data-align=\"center\">Natural Frequencies<br>Computing <strong>Algorithm<\/strong><\/th><\/tr><\/thead><tbody><tr><td class=\"has-text-align-center\" data-align=\"center\">A<\/td><td class=\"has-text-align-center\" data-align=\"center\">IRAM \u2013 Sorensen<\/td><\/tr><tr><td class=\"has-text-align-center\" data-align=\"center\">B<\/td><td class=\"has-text-align-center\" data-align=\"center\">Lanczos<\/td><\/tr><tr><td class=\"has-text-align-center\" data-align=\"center\">C<\/td><td class=\"has-text-align-center\" data-align=\"center\">Bathe \u2013 Wilson<\/td><\/tr><tr><td class=\"has-text-align-center\" data-align=\"center\">D<\/td><td class=\"has-text-align-center\" data-align=\"center\">IRAM \u2013 Sorensen<\/td><\/tr><tr><td class=\"has-text-align-center\" data-align=\"center\">E<\/td><td class=\"has-text-align-center\" data-align=\"center\">Lanczos<\/td><\/tr><tr><td class=\"has-text-align-center\" data-align=\"center\">F<\/td><td class=\"has-text-align-center\" data-align=\"center\">Bathe \u2013 Wilson<\/td><\/tr><\/tbody><\/table><figcaption>Table 3: Computing algorithms by case<\/figcaption><\/figure>\n\n\n\n<p class=\"wp-block-paragraph\">The main characteristics of each algorithm are summarized below [U4.52.02]\\(^2\\):<\/p>\n\n\n\n<ul class=\"wp-block-list\"><li><strong>IRAM \u2013 Sorensen:<\/strong> Uses a sub-space decomposition method to compute the natural frequencies and modes. Suitable for real and complex, symmetrical or non-symmetrical matrices.<\/li><li><strong>Lanczos:<\/strong> Uses a sub-space decomposition method to compute the natural frequencies and modes. Suitable for real, symmetrical only matrices.<\/li><li><strong>Bathe \u2013 Wilson:<\/strong> Uses a sub-space decomposition method to compute the natural frequencies and modes. Suitable for real, symmetrical only matrices.<\/li><\/ul>\n\n\n\n<p class=\"wp-block-paragraph\">A fourth algorithm is also available in SimScale, called <strong>QZ<\/strong>. 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.<\/p>\n\n\n\n<div class=\"hw-block hw-note hw-note--info hw-note\">\n    <div class=\"hw-note__title\">\n        <p class=\"hw-note__titleText\"><i class=\"fa fa-exclamation-circle\" aria-hidden=\"true\"><\/i>Note<\/p>\n    <\/div>\n    <div class=\"hw-note__body\">\n        <p>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.<\/p>\n    <\/div>\n<\/div>\n\n\n\n<h2 class=\"wp-block-heading\" id=\"frequency-reference-solution\" >Frequency Reference Solution<\/h2>\n\n\n\n<p class=\"wp-block-paragraph\">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.<\/p>\n\n\n\n<p class=\"wp-block-paragraph\">The consulted reference solution is:<\/p>\n\n\n\n<figure class=\"wp-block-table is-style-stripes\"><table><thead><tr><th class=\"has-text-align-center\" data-align=\"center\">Mode<\/th><th class=\"has-text-align-center\" data-align=\"center\">Natural Frequency \\([Hz]\\)<\/th><\/tr><\/thead><tbody><tr><td class=\"has-text-align-center\" data-align=\"center\">Ovalization<\/td><td class=\"has-text-align-center\" data-align=\"center\">210.55<\/td><\/tr><tr><td class=\"has-text-align-center\" data-align=\"center\"><\/td><td class=\"has-text-align-center\" data-align=\"center\">210.55<\/td><\/tr><tr><td class=\"has-text-align-center\" data-align=\"center\">Trifoliate<\/td><td class=\"has-text-align-center\" data-align=\"center\">587.92<\/td><\/tr><tr><td class=\"has-text-align-center\" data-align=\"center\"><\/td><td class=\"has-text-align-center\" data-align=\"center\">587.92<\/td><\/tr><tr><td class=\"has-text-align-center\" data-align=\"center\">Out of Plane<\/td><td class=\"has-text-align-center\" data-align=\"center\">205.89<\/td><\/tr><tr><td class=\"has-text-align-center\" data-align=\"center\"><\/td><td class=\"has-text-align-center\" data-align=\"center\">205.89<\/td><\/tr><tr><td class=\"has-text-align-center\" data-align=\"center\"><\/td><td class=\"has-text-align-center\" data-align=\"center\">588.88<\/td><\/tr><tr><td class=\"has-text-align-center\" data-align=\"center\"><\/td><td class=\"has-text-align-center\" data-align=\"center\">588.88<\/td><\/tr><\/tbody><\/table><figcaption>Table 4: Reference solution for different frequency modes<\/figcaption><\/figure>\n\n\n\n<h2 class=\"wp-block-heading\" id=\"frequency-results-comparison\" >Frequency Results Comparison<\/h2>\n\n\n\n<p class=\"wp-block-paragraph\">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.<\/p>\n\n\n\n<figure class=\"wp-block-image size-large\"><a href=\"https:\/\/frontend-assets.simscale.com\/media\/2020\/06\/image-29.png\"><img loading=\"lazy\" decoding=\"async\" width=\"1024\" height=\"705\" src=\"https:\/\/frontend-assets.simscale.com\/media\/2020\/06\/image-29-1024x705.png\" alt=\"mesh convergence analysis plot for frequency analysis of a ring validation case\" class=\"wp-image-29882\" srcset=\"https:\/\/frontend-assets.simscale.com\/media\/2020\/06\/image-29-1024x705.png 1024w, https:\/\/frontend-assets.simscale.com\/media\/2020\/06\/image-29-300x207.png 300w, https:\/\/frontend-assets.simscale.com\/media\/2020\/06\/image-29-768x529.png 768w, https:\/\/frontend-assets.simscale.com\/media\/2020\/06\/image-29.png 1038w\" sizes=\"auto, (max-width: 1024px) 100vw, 1024px\" \/><\/a><figcaption>Figure 3: Mesh convergence analysis results for second order mesh. <br>Here, F1 means first frequency, F2 second frequency, and so on.<\/figcaption><\/figure>\n\n\n\n<p class=\"wp-block-paragraph\">Comparison of computed natural frequencies with the reference solution for each case can be seen below:<\/p>\n\n\n\n<figure class=\"wp-block-table is-style-stripes\"><table><thead><tr><th class=\"has-text-align-center\" data-align=\"center\">Mode<\/th><th class=\"has-text-align-center\" data-align=\"center\">Reference Solution<\/th><th class=\"has-text-align-center\" data-align=\"center\">SimScale Solution<\/th><th class=\"has-text-align-center\" data-align=\"center\">Error<\/th><\/tr><\/thead><tbody><tr><td class=\"has-text-align-center\" data-align=\"center\">Ovalization<\/td><td class=\"has-text-align-center\" data-align=\"center\">210.55<\/td><td class=\"has-text-align-center\" data-align=\"center\">213.332<\/td><td class=\"has-text-align-center\" data-align=\"center\">1.32 %<\/td><\/tr><tr><td class=\"has-text-align-center\" data-align=\"center\">\ufeff<\/td><td class=\"has-text-align-center\" data-align=\"center\">210.55<\/td><td class=\"has-text-align-center\" data-align=\"center\">213.966<\/td><td class=\"has-text-align-center\" data-align=\"center\">1.62 %<\/td><\/tr><tr><td class=\"has-text-align-center\" data-align=\"center\">Trifoliate<\/td><td class=\"has-text-align-center\" data-align=\"center\">587.92<\/td><td class=\"has-text-align-center\" data-align=\"center\">596.343<\/td><td class=\"has-text-align-center\" data-align=\"center\">1.43 %<\/td><\/tr><tr><td class=\"has-text-align-center\" data-align=\"center\">\ufeff<\/td><td class=\"has-text-align-center\" data-align=\"center\">587.92<\/td><td class=\"has-text-align-center\" data-align=\"center\">596.698<\/td><td class=\"has-text-align-center\" data-align=\"center\">1.49 %<\/td><\/tr><tr><td class=\"has-text-align-center\" data-align=\"center\">Out of Plane<\/td><td class=\"has-text-align-center\" data-align=\"center\">205.89<\/td><td class=\"has-text-align-center\" data-align=\"center\">210.017<\/td><td class=\"has-text-align-center\" data-align=\"center\">2.00 %<\/td><\/tr><tr><td class=\"has-text-align-center\" data-align=\"center\">\ufeff<\/td><td class=\"has-text-align-center\" data-align=\"center\">205.89<\/td><td class=\"has-text-align-center\" data-align=\"center\">210.244<\/td><td class=\"has-text-align-center\" data-align=\"center\">2.11 %<\/td><\/tr><tr><td class=\"has-text-align-center\" data-align=\"center\">\ufeff<\/td><td class=\"has-text-align-center\" data-align=\"center\">588.88<\/td><td class=\"has-text-align-center\" data-align=\"center\">599.512<\/td><td class=\"has-text-align-center\" data-align=\"center\">1.81 %<\/td><\/tr><tr><td class=\"has-text-align-center\" data-align=\"center\">\ufeff<\/td><td class=\"has-text-align-center\" data-align=\"center\">588.88<\/td><td class=\"has-text-align-center\" data-align=\"center\">599.521<\/td><td class=\"has-text-align-center\" data-align=\"center\">1.81 %<\/td><\/tr><\/tbody><\/table><figcaption>Table 5: Case A results<\/figcaption><\/figure>\n\n\n\n<figure class=\"wp-block-table is-style-stripes\"><table><thead><tr><th class=\"has-text-align-center\" data-align=\"center\">Mode<\/th><th class=\"has-text-align-center\" data-align=\"center\">Reference Solution<\/th><th class=\"has-text-align-center\" data-align=\"center\">SimScale Solution<\/th><th class=\"has-text-align-center\" data-align=\"center\">Error<\/th><\/tr><\/thead><tbody><tr><td class=\"has-text-align-center\" data-align=\"center\">Ovalization<\/td><td class=\"has-text-align-center\" data-align=\"center\">210.55<\/td><td class=\"has-text-align-center\" data-align=\"center\">215.181<\/td><td class=\"has-text-align-center\" data-align=\"center\">2.20 %<\/td><\/tr><tr><td class=\"has-text-align-center\" data-align=\"center\">\ufeff<\/td><td class=\"has-text-align-center\" data-align=\"center\">210.55<\/td><td class=\"has-text-align-center\" data-align=\"center\">216.09<\/td><td class=\"has-text-align-center\" data-align=\"center\">2.63 %<\/td><\/tr><tr><td class=\"has-text-align-center\" data-align=\"center\">Trifoliate<\/td><td class=\"has-text-align-center\" data-align=\"center\">587.92<\/td><td class=\"has-text-align-center\" data-align=\"center\">601.83<\/td><td class=\"has-text-align-center\" data-align=\"center\">2.37 %<\/td><\/tr><tr><td class=\"has-text-align-center\" data-align=\"center\">\ufeff<\/td><td class=\"has-text-align-center\" data-align=\"center\">587.92<\/td><td class=\"has-text-align-center\" data-align=\"center\">602.316<\/td><td class=\"has-text-align-center\" data-align=\"center\">2.45 %<\/td><\/tr><tr><td class=\"has-text-align-center\" data-align=\"center\">Out of Plane<\/td><td class=\"has-text-align-center\" data-align=\"center\">205.89<\/td><td class=\"has-text-align-center\" data-align=\"center\">212.822<\/td><td class=\"has-text-align-center\" data-align=\"center\">3.37 %<\/td><\/tr><tr><td class=\"has-text-align-center\" data-align=\"center\">\ufeff<\/td><td class=\"has-text-align-center\" data-align=\"center\">205.89<\/td><td class=\"has-text-align-center\" data-align=\"center\">213.098<\/td><td class=\"has-text-align-center\" data-align=\"center\">3.50 %<\/td><\/tr><tr><td class=\"has-text-align-center\" data-align=\"center\">\ufeff<\/td><td class=\"has-text-align-center\" data-align=\"center\">588.88<\/td><td class=\"has-text-align-center\" data-align=\"center\">606.685<\/td><td class=\"has-text-align-center\" data-align=\"center\">3.02 %<\/td><\/tr><tr><td class=\"has-text-align-center\" data-align=\"center\">\ufeff<\/td><td class=\"has-text-align-center\" data-align=\"center\">588.88<\/td><td class=\"has-text-align-center\" data-align=\"center\">606.794<\/td><td class=\"has-text-align-center\" data-align=\"center\">3.04 %<\/td><\/tr><\/tbody><\/table><figcaption>Table 6: Case B results<\/figcaption><\/figure>\n\n\n\n<figure class=\"wp-block-table is-style-stripes\"><table><thead><tr><th class=\"has-text-align-center\" data-align=\"center\">Mode<\/th><th class=\"has-text-align-center\" data-align=\"center\">Reference Solution<\/th><th class=\"has-text-align-center\" data-align=\"center\">SimScale Solution<\/th><th class=\"has-text-align-center\" data-align=\"center\">Error<\/th><\/tr><\/thead><tbody><tr><td class=\"has-text-align-center\" data-align=\"center\">Ovalization<\/td><td class=\"has-text-align-center\" data-align=\"center\">210.55<\/td><td class=\"has-text-align-center\" data-align=\"center\">215.181<\/td><td class=\"has-text-align-center\" data-align=\"center\">2.20 %<\/td><\/tr><tr><td class=\"has-text-align-center\" data-align=\"center\">\ufeff<\/td><td class=\"has-text-align-center\" data-align=\"center\">210.55<\/td><td class=\"has-text-align-center\" data-align=\"center\">216.09<\/td><td class=\"has-text-align-center\" data-align=\"center\">2.63 %<\/td><\/tr><tr><td class=\"has-text-align-center\" data-align=\"center\">Trifoliate<\/td><td class=\"has-text-align-center\" data-align=\"center\">587.92<\/td><td class=\"has-text-align-center\" data-align=\"center\">601.83<\/td><td class=\"has-text-align-center\" data-align=\"center\">2.37 %<\/td><\/tr><tr><td class=\"has-text-align-center\" data-align=\"center\">\ufeff<\/td><td class=\"has-text-align-center\" data-align=\"center\">587.92<\/td><td class=\"has-text-align-center\" data-align=\"center\">602.316<\/td><td class=\"has-text-align-center\" data-align=\"center\">2.45 %<\/td><\/tr><tr><td class=\"has-text-align-center\" data-align=\"center\">Out of Plane<\/td><td class=\"has-text-align-center\" data-align=\"center\">205.89<\/td><td class=\"has-text-align-center\" data-align=\"center\">212.822<\/td><td class=\"has-text-align-center\" data-align=\"center\">3.37 %<\/td><\/tr><tr><td class=\"has-text-align-center\" data-align=\"center\">\ufeff<\/td><td class=\"has-text-align-center\" data-align=\"center\">205.89<\/td><td class=\"has-text-align-center\" data-align=\"center\">213.098<\/td><td class=\"has-text-align-center\" data-align=\"center\">3.50 %<\/td><\/tr><tr><td class=\"has-text-align-center\" data-align=\"center\">\ufeff<\/td><td class=\"has-text-align-center\" data-align=\"center\">588.88<\/td><td class=\"has-text-align-center\" data-align=\"center\">606.685<\/td><td class=\"has-text-align-center\" data-align=\"center\">3.02 %<\/td><\/tr><tr><td class=\"has-text-align-center\" data-align=\"center\">\ufeff<\/td><td class=\"has-text-align-center\" data-align=\"center\">588.88<\/td><td class=\"has-text-align-center\" data-align=\"center\">606.794<\/td><td class=\"has-text-align-center\" data-align=\"center\">3.04 %<\/td><\/tr><\/tbody><\/table><figcaption>Table 7: Case C results<\/figcaption><\/figure>\n\n\n\n<figure class=\"wp-block-table is-style-stripes\"><table><thead><tr><th class=\"has-text-align-center\" data-align=\"center\">Mode<\/th><th class=\"has-text-align-center\" data-align=\"center\">Reference Solution<\/th><th class=\"has-text-align-center\" data-align=\"center\">SimScale Solution<\/th><th class=\"has-text-align-center\" data-align=\"center\">Error<\/th><\/tr><\/thead><tbody><tr><td class=\"has-text-align-center\" data-align=\"center\">Ovalization<\/td><td class=\"has-text-align-center\" data-align=\"center\">210.55<\/td><td class=\"has-text-align-center\" data-align=\"center\">209.998<\/td><td class=\"has-text-align-center\" data-align=\"center\">-0.26 %<\/td><\/tr><tr><td class=\"has-text-align-center\" data-align=\"center\">\ufeff<\/td><td class=\"has-text-align-center\" data-align=\"center\">210.55<\/td><td class=\"has-text-align-center\" data-align=\"center\">209.998<\/td><td class=\"has-text-align-center\" data-align=\"center\">-0.26 %<\/td><\/tr><tr><td class=\"has-text-align-center\" data-align=\"center\">Trifoliate<\/td><td class=\"has-text-align-center\" data-align=\"center\">587.92<\/td><td class=\"has-text-align-center\" data-align=\"center\">586.293<\/td><td class=\"has-text-align-center\" data-align=\"center\">-0.28 %<\/td><\/tr><tr><td class=\"has-text-align-center\" data-align=\"center\">\ufeff<\/td><td class=\"has-text-align-center\" data-align=\"center\">587.92<\/td><td class=\"has-text-align-center\" data-align=\"center\">586.293<\/td><td class=\"has-text-align-center\" data-align=\"center\">-0.28 %<\/td><\/tr><tr><td class=\"has-text-align-center\" data-align=\"center\">Out of Plane<\/td><td class=\"has-text-align-center\" data-align=\"center\">205.89<\/td><td class=\"has-text-align-center\" data-align=\"center\">205.143<\/td><td class=\"has-text-align-center\" data-align=\"center\">-0.36 %<\/td><\/tr><tr><td class=\"has-text-align-center\" data-align=\"center\">\ufeff<\/td><td class=\"has-text-align-center\" data-align=\"center\">205.89<\/td><td class=\"has-text-align-center\" data-align=\"center\">205.144<\/td><td class=\"has-text-align-center\" data-align=\"center\">-0.36 %<\/td><\/tr><tr><td class=\"has-text-align-center\" data-align=\"center\">\ufeff<\/td><td class=\"has-text-align-center\" data-align=\"center\">588.88<\/td><td class=\"has-text-align-center\" data-align=\"center\">586.975<\/td><td class=\"has-text-align-center\" data-align=\"center\">-0.32 %<\/td><\/tr><tr><td class=\"has-text-align-center\" data-align=\"center\">\ufeff<\/td><td class=\"has-text-align-center\" data-align=\"center\">588.88<\/td><td class=\"has-text-align-center\" data-align=\"center\">586.975<\/td><td class=\"has-text-align-center\" data-align=\"center\">-0.32 %<\/td><\/tr><\/tbody><\/table><figcaption>Table 8: Case D results<\/figcaption><\/figure>\n\n\n\n<figure class=\"wp-block-table is-style-stripes\"><table><thead><tr><th class=\"has-text-align-center\" data-align=\"center\">Mode<\/th><th class=\"has-text-align-center\" data-align=\"center\">Reference Solution<\/th><th class=\"has-text-align-center\" data-align=\"center\">SimScale Solution<\/th><th class=\"has-text-align-center\" data-align=\"center\">Error<\/th><\/tr><\/thead><tbody><tr><td class=\"has-text-align-center\" data-align=\"center\">Ovalization<\/td><td class=\"has-text-align-center\" data-align=\"center\">210.55<\/td><td class=\"has-text-align-center\" data-align=\"center\">209.998<\/td><td class=\"has-text-align-center\" data-align=\"center\">-0.26 %<\/td><\/tr><tr><td class=\"has-text-align-center\" data-align=\"center\">\ufeff<\/td><td class=\"has-text-align-center\" data-align=\"center\">210.55<\/td><td class=\"has-text-align-center\" data-align=\"center\">209.998<\/td><td class=\"has-text-align-center\" data-align=\"center\">-0.26 %<\/td><\/tr><tr><td class=\"has-text-align-center\" data-align=\"center\">Trifoliate<\/td><td class=\"has-text-align-center\" data-align=\"center\">587.92<\/td><td class=\"has-text-align-center\" data-align=\"center\">586.293<\/td><td class=\"has-text-align-center\" data-align=\"center\">-0.28 %<\/td><\/tr><tr><td class=\"has-text-align-center\" data-align=\"center\">\ufeff<\/td><td class=\"has-text-align-center\" data-align=\"center\">587.92<\/td><td class=\"has-text-align-center\" data-align=\"center\">586.293<\/td><td class=\"has-text-align-center\" data-align=\"center\">-0.28 %<\/td><\/tr><tr><td class=\"has-text-align-center\" data-align=\"center\">Out of Plane<\/td><td class=\"has-text-align-center\" data-align=\"center\">205.89<\/td><td class=\"has-text-align-center\" data-align=\"center\">205.143<\/td><td class=\"has-text-align-center\" data-align=\"center\">-0.36 %<\/td><\/tr><tr><td class=\"has-text-align-center\" data-align=\"center\">\ufeff<\/td><td class=\"has-text-align-center\" data-align=\"center\">205.89<\/td><td class=\"has-text-align-center\" data-align=\"center\">205.144<\/td><td class=\"has-text-align-center\" data-align=\"center\">-0.36 %<\/td><\/tr><tr><td class=\"has-text-align-center\" data-align=\"center\">\ufeff<\/td><td class=\"has-text-align-center\" data-align=\"center\">588.88<\/td><td class=\"has-text-align-center\" data-align=\"center\">586.975<\/td><td class=\"has-text-align-center\" data-align=\"center\">-0.32 %<\/td><\/tr><tr><td class=\"has-text-align-center\" data-align=\"center\">\ufeff<\/td><td class=\"has-text-align-center\" data-align=\"center\">588.88<\/td><td class=\"has-text-align-center\" data-align=\"center\">586.975<\/td><td class=\"has-text-align-center\" data-align=\"center\">-0.32 %<\/td><\/tr><\/tbody><\/table><figcaption>Table 9: Case E results<\/figcaption><\/figure>\n\n\n\n<figure class=\"wp-block-table is-style-stripes\"><table><thead><tr><th class=\"has-text-align-center\" data-align=\"center\">Mode<\/th><th class=\"has-text-align-center\" data-align=\"center\">Reference Solution<\/th><th class=\"has-text-align-center\" data-align=\"center\">SimScale Solution<\/th><th class=\"has-text-align-center\" data-align=\"center\">Error<\/th><\/tr><\/thead><tbody><tr><td class=\"has-text-align-center\" data-align=\"center\">Ovalization<\/td><td class=\"has-text-align-center\" data-align=\"center\">210.55<\/td><td class=\"has-text-align-center\" data-align=\"center\">209.998<\/td><td class=\"has-text-align-center\" data-align=\"center\">-0.26 %<\/td><\/tr><tr><td class=\"has-text-align-center\" data-align=\"center\">\ufeff<\/td><td class=\"has-text-align-center\" data-align=\"center\">210.55<\/td><td class=\"has-text-align-center\" data-align=\"center\">209.998<\/td><td class=\"has-text-align-center\" data-align=\"center\">-0.26 %<\/td><\/tr><tr><td class=\"has-text-align-center\" data-align=\"center\">Trifoliate<\/td><td class=\"has-text-align-center\" data-align=\"center\">587.92<\/td><td class=\"has-text-align-center\" data-align=\"center\">586.293<\/td><td class=\"has-text-align-center\" data-align=\"center\">-0.28 %<\/td><\/tr><tr><td class=\"has-text-align-center\" data-align=\"center\">\ufeff<\/td><td class=\"has-text-align-center\" data-align=\"center\">587.92<\/td><td class=\"has-text-align-center\" data-align=\"center\">586.293<\/td><td class=\"has-text-align-center\" data-align=\"center\">-0.28 %<\/td><\/tr><tr><td class=\"has-text-align-center\" data-align=\"center\">Out of Plane<\/td><td class=\"has-text-align-center\" data-align=\"center\">205.89<\/td><td class=\"has-text-align-center\" data-align=\"center\">205.143<\/td><td class=\"has-text-align-center\" data-align=\"center\">-0.36 %<\/td><\/tr><tr><td class=\"has-text-align-center\" data-align=\"center\">\ufeff<\/td><td class=\"has-text-align-center\" data-align=\"center\">205.89<\/td><td class=\"has-text-align-center\" data-align=\"center\">205.144<\/td><td class=\"has-text-align-center\" data-align=\"center\">-0.36 %<\/td><\/tr><tr><td class=\"has-text-align-center\" data-align=\"center\">\ufeff<\/td><td class=\"has-text-align-center\" data-align=\"center\">588.88<\/td><td class=\"has-text-align-center\" data-align=\"center\">586.975<\/td><td class=\"has-text-align-center\" data-align=\"center\">-0.32 %<\/td><\/tr><tr><td class=\"has-text-align-center\" data-align=\"center\">\ufeff<\/td><td class=\"has-text-align-center\" data-align=\"center\">588.88<\/td><td class=\"has-text-align-center\" data-align=\"center\">586.975<\/td><td class=\"has-text-align-center\" data-align=\"center\">-0.32 %<\/td><\/tr><\/tbody><\/table><figcaption>Table 10: Case F results<\/figcaption><\/figure>\n\n\n\n<p class=\"wp-block-paragraph\">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 \u2013 Sorensen and Lanczos algorithm are much faster than Bathe \u2013 Wilson. The recommendation is then to stay with the default algorithm (IRAM \u2013 Sorensen), because of its known robustness.<\/p>\n\n\n\n<figure class=\"wp-block-table is-style-stripes\"><table><thead><tr><th class=\"has-text-align-center\" data-align=\"center\">Algorithm<\/th><th class=\"has-text-align-center\" data-align=\"center\">Runtime<br>1st Order Mesh<\/th><th class=\"has-text-align-center\" data-align=\"center\">Runtime<br>2nd Order Mesh<\/th><\/tr><\/thead><tbody><tr><td class=\"has-text-align-center\" data-align=\"center\">IRAM \u2013 Sorensen<\/td><td class=\"has-text-align-center\" data-align=\"center\">2 min<\/td><td class=\"has-text-align-center\" data-align=\"center\">13 min<\/td><\/tr><tr><td class=\"has-text-align-center\" data-align=\"center\">Lanczos<\/td><td class=\"has-text-align-center\" data-align=\"center\">2 min<\/td><td class=\"has-text-align-center\" data-align=\"center\">13 min<\/td><\/tr><tr><td class=\"has-text-align-center\" data-align=\"center\">Bathe \u2013 Wilson<\/td><td class=\"has-text-align-center\" data-align=\"center\">9 min<\/td><td class=\"has-text-align-center\" data-align=\"center\">139 min<\/td><\/tr><\/tbody><\/table><figcaption>Table 11: Comparison of runtime for each algorithm<\/figcaption><\/figure>\n\n\n\n<p class=\"wp-block-paragraph\">Following are the referenced natural mode shapes as seen on the online post-processor:<\/p>\n\n\n\n<div class=\"wp-block-image\"><figure class=\"aligncenter size-large\"><a href=\"https:\/\/frontend-assets.simscale.com\/media\/2020\/06\/image-41.png\"><img decoding=\"async\" src=\"https:\/\/frontend-assets.simscale.com\/media\/2020\/06\/image-41.png\" alt=\"natural vibration shapes plot for frequency analysis of a ring validation case\" class=\"wp-image-30178\"\/><\/a><figcaption>Figure 4: Natural vibration shapes for each natural frequency taken from case D.<\/figcaption><\/figure><\/div>\n\n\n\n\n<div class='hw-block hw-references hw-references'>\n    <p class='hw-references__title'>References<\/p>\n    <ul class='hw-references__list'>\n\n        <li><cite>SDLS109 \u2013 Fr\u00e9quences propres d\u2019un anneau cylindrique \u00e9pais \u2013 Code_Aster validation case<\/cite><\/li><li><cite>[U4.52.02] \u2013 Op\u00e9rateur CALC_MODES \u2013 Code_Aster utilization manual<\/cite><\/li>\n    <\/ul>\n<\/div>\n\n\n\n<div class=\"hw-block hw-note hw-note--info hw-note\">\n    <div class=\"hw-note__title\">\n        <p class=\"hw-note__titleText\"><i class=\"fa fa-exclamation-circle\" aria-hidden=\"true\"><\/i>Note<\/p>\n    <\/div>\n    <div class=\"hw-note__body\">\n        <p>If you still encounter problems validating you simulation, then please post the issue on our <a href=\"https:\/\/www.simscale.com\/forum\/\">forum<\/a> or <a href=\"mailto:support@simscale.com\">contact us<\/a>.<\/p>\n    <\/div>\n<\/div>\n","protected":false},"excerpt":{"rendered":"<p>This validation case belongs to solid mechanics. The aim of this test case is to validate the following parameters:...","protected":false},"author":115,"featured_media":29764,"parent":17191,"menu_order":0,"comment_status":"closed","ping_status":"closed","template":"templates\/template-documentation.php","meta":{"_acf_changed":false,"_crdt_document":"","inline_featured_image":false,"footnotes":""},"class_list":["post-29762","page","type-page","status-publish","has-post-thumbnail","hentry"],"acf":[],"_links":{"self":[{"href":"https:\/\/www.simscale.com\/wp-json\/wp\/v2\/pages\/29762","targetHints":{"allow":["GET"]}}],"collection":[{"href":"https:\/\/www.simscale.com\/wp-json\/wp\/v2\/pages"}],"about":[{"href":"https:\/\/www.simscale.com\/wp-json\/wp\/v2\/types\/page"}],"author":[{"embeddable":true,"href":"https:\/\/www.simscale.com\/wp-json\/wp\/v2\/users\/115"}],"replies":[{"embeddable":true,"href":"https:\/\/www.simscale.com\/wp-json\/wp\/v2\/comments?post=29762"}],"version-history":[{"count":0,"href":"https:\/\/www.simscale.com\/wp-json\/wp\/v2\/pages\/29762\/revisions"}],"up":[{"embeddable":true,"href":"https:\/\/www.simscale.com\/wp-json\/wp\/v2\/pages\/17191"}],"wp:featuredmedia":[{"embeddable":true,"href":"https:\/\/www.simscale.com\/wp-json\/wp\/v2\/media\/29764"}],"wp:attachment":[{"href":"https:\/\/www.simscale.com\/wp-json\/wp\/v2\/media?parent=29762"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}