{"id":262,"date":"2021-11-19T03:30:00","date_gmt":"2021-11-19T02:30:00","guid":{"rendered":"https:\/\/ramyrashad.com\/?p=262"},"modified":"2023-08-11T19:58:23","modified_gmt":"2023-08-11T18:58:23","slug":"fem-of-von-karman-beams","status":"publish","type":"post","link":"https:\/\/ramyrashad.com\/index.php\/2021\/11\/19\/fem-of-von-karman-beams\/","title":{"rendered":"FEM of von Karman Beams"},"content":{"rendered":"\n<p><strong>Research Overview<\/strong><\/p>\n\n\n\n<p>In this contribution, the von Karman beam model is formulated as a port-Hamiltonian system. The selection of energy variables will be such to make the Hamiltonian quadratic in these variables. As a consequence of this choice, the non linearities of the model are included in the interconnection operator, whereas the constitutive relations remain linear.<br>The obtained model can be discretized using mixed finite elements. To this aim, a weak formulation that does not demand for H2 regularity for the vertical displacement (as in classical Galerkin discretization of beams and plates) is obtained. This means that the vertical displacement can be discretized using H1 conforming elements (i.e. Continuous Galerkin elements),<br>rather then using the more computationally demanding H2 conforming elements, like the Hermite polynomials. A numerical test is carried out to evaluate the convergence rate of the discrete solution with respect to an analytical one.<\/p>\n\n\n\n<p>The port-Hamiltonian system representing the von Karman beam model is given by<\/p>\n\n\n<div class=\"wp-block-image\">\n<figure class=\"aligncenter size-large\"><img loading=\"lazy\" decoding=\"async\" width=\"1024\" height=\"224\" src=\"https:\/\/ramyrashad.com\/wp-content\/uploads\/2023\/08\/image-5-1024x224.png\" alt=\"\" class=\"wp-image-445\" srcset=\"https:\/\/ramyrashad.com\/wp-content\/uploads\/2023\/08\/image-5-1024x224.png 1024w, https:\/\/ramyrashad.com\/wp-content\/uploads\/2023\/08\/image-5-300x66.png 300w, https:\/\/ramyrashad.com\/wp-content\/uploads\/2023\/08\/image-5-768x168.png 768w, https:\/\/ramyrashad.com\/wp-content\/uploads\/2023\/08\/image-5.png 1038w\" sizes=\"(max-width: 1024px) 100vw, 1024px\" \/><\/figure><\/div>\n\n\n<p>The numerical solution computed is shown below<\/p>\n\n\n<div class=\"wp-block-image\">\n<figure class=\"aligncenter size-full is-resized\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/ramyrashad.com\/wp-content\/uploads\/2023\/08\/image-6.png\" alt=\"\" class=\"wp-image-448\" width=\"751\" height=\"814\" srcset=\"https:\/\/ramyrashad.com\/wp-content\/uploads\/2023\/08\/image-6.png 827w, https:\/\/ramyrashad.com\/wp-content\/uploads\/2023\/08\/image-6-277x300.png 277w, https:\/\/ramyrashad.com\/wp-content\/uploads\/2023\/08\/image-6-768x832.png 768w\" sizes=\"(max-width: 751px) 100vw, 751px\" \/><\/figure><\/div>\n\n\n<p><strong>Publication<\/strong><\/p>\n\n\n\n<figure class=\"wp-block-table\"><table><tbody><tr><td>Andrea Brugnoli, Ramy Rashad, Federico Califano, Stefano Stramigioli, Denis Matignon&nbsp;(2021)&nbsp;Mixed finite elements for port-Hamiltonian models of von K\u00e1rm\u00e1n beams,&nbsp;IFAC-PapersOnLine&nbsp;54(19),&nbsp;p. 186-191,&nbsp;Elsevier Ltd,&nbsp;<a href=\"https:\/\/doi.org\/10.1016\/j.ifacol.2021.11.076\" target=\"_blank\" rel=\"noreferrer noopener\">doi:10.1016\/j.ifacol.2021.11.076<\/a><\/td><\/tr><\/tbody><\/table><\/figure>\n","protected":false},"excerpt":{"rendered":"<p>Research Overview In this contribution, the von Karman beam model is formulated as a port-Hamiltonian system. The selection of energy variables will be such to make the Hamiltonian [&hellip;]<\/p>\n","protected":false},"author":1,"featured_media":450,"comment_status":"closed","ping_status":"closed","sticky":false,"template":"","format":"standard","meta":{"om_disable_all_campaigns":false,"ub_ctt_via":"","_mi_skip_tracking":false,"footnotes":""},"categories":[11],"tags":[],"aioseo_notices":[],"featured_image_src":"https:\/\/ramyrashad.com\/wp-content\/uploads\/2023\/07\/vonKarman-1.png","author_info":{"display_name":"Ramy","author_link":"https:\/\/ramyrashad.com\/index.php\/author\/ramy-abdelmonemgmail-com\/"},"_links":{"self":[{"href":"https:\/\/ramyrashad.com\/index.php\/wp-json\/wp\/v2\/posts\/262"}],"collection":[{"href":"https:\/\/ramyrashad.com\/index.php\/wp-json\/wp\/v2\/posts"}],"about":[{"href":"https:\/\/ramyrashad.com\/index.php\/wp-json\/wp\/v2\/types\/post"}],"author":[{"embeddable":true,"href":"https:\/\/ramyrashad.com\/index.php\/wp-json\/wp\/v2\/users\/1"}],"replies":[{"embeddable":true,"href":"https:\/\/ramyrashad.com\/index.php\/wp-json\/wp\/v2\/comments?post=262"}],"version-history":[{"count":8,"href":"https:\/\/ramyrashad.com\/index.php\/wp-json\/wp\/v2\/posts\/262\/revisions"}],"predecessor-version":[{"id":464,"href":"https:\/\/ramyrashad.com\/index.php\/wp-json\/wp\/v2\/posts\/262\/revisions\/464"}],"wp:featuredmedia":[{"embeddable":true,"href":"https:\/\/ramyrashad.com\/index.php\/wp-json\/wp\/v2\/media\/450"}],"wp:attachment":[{"href":"https:\/\/ramyrashad.com\/index.php\/wp-json\/wp\/v2\/media?parent=262"}],"wp:term":[{"taxonomy":"category","embeddable":true,"href":"https:\/\/ramyrashad.com\/index.php\/wp-json\/wp\/v2\/categories?post=262"},{"taxonomy":"post_tag","embeddable":true,"href":"https:\/\/ramyrashad.com\/index.php\/wp-json\/wp\/v2\/tags?post=262"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}