Implicit direct time integration of the heat conduction problem in the Method of Matched Sections
The paper is devoted to further elaboration of the Method of Matched Sections as a new branch of finite element method in application to the transient 2D temperature problem. The main distinction of MMS from conventional FEM consist in that the conjugation is provided between the adjacent sections r...
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| Format: | Article |
| Language: | English |
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National Technical University of Ukraine "Igor Sikorsky Kyiv Polytechnic Institute"
2024-03-01
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| Series: | Mechanics and Advanced Technologies |
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| Online Access: | https://journal.mmi.kpi.ua/article/view/299059 |
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| author | Ігор Ориняк Антон Цибульник Кірілл Даниленко Андрій Ориняк |
| author_facet | Ігор Ориняк Антон Цибульник Кірілл Даниленко Андрій Ориняк |
| author_sort | Ігор Ориняк |
| collection | DOAJ |
| description | The paper is devoted to further elaboration of the Method of Matched Sections as a new branch of finite element method in application to the transient 2D temperature problem. The main distinction of MMS from conventional FEM consist in that the conjugation is provided between the adjacent sections rather than in the nodes of the elements. Important feature is that method is based on approximate strong form solution of the governing differential equations called here as the Connection equations. It is assumed that for each small rectangular element the 2D problem can be considered as the combination of two 1D problems – one is x-dependent, and another is y-dependent. Each problem is characterized by two functions – the temperature, , and heat flux . In practical realization for rectangular finite elements the method is reduced to determination of eight unknowns for each element – two unknowns on each side, which are related by the Connection equations, and requirement of the temperature continuity at the center of element. Another salient feature of the paper is an implementation of the original implicit time integration scheme, where the time step became the parameter of shape function within the element, i.e. it determines the behavior of the Connection equations. This method was early proposed by first author for number of 1D problem, and here in first time it is applied for 2D problems. The number of tests for rectangular plate exhibits the remarkable properties of this “embedded” time integration scheme with respect to stability, accuracy, and absence of any restrictions as to increasing of the time step. |
| format | Article |
| id | doaj-art-974bf88c618240fcab00de05018d3476 |
| institution | DOAJ |
| issn | 2521-1943 2522-4255 |
| language | English |
| publishDate | 2024-03-01 |
| publisher | National Technical University of Ukraine "Igor Sikorsky Kyiv Polytechnic Institute" |
| record_format | Article |
| series | Mechanics and Advanced Technologies |
| spelling | doaj-art-974bf88c618240fcab00de05018d34762025-08-20T02:52:24ZengNational Technical University of Ukraine "Igor Sikorsky Kyiv Polytechnic Institute"Mechanics and Advanced Technologies2521-19432522-42552024-03-0181(100)879710.20535/2521-1943.2024.8.1(100).299059337411Implicit direct time integration of the heat conduction problem in the Method of Matched SectionsІгор Ориняк0https://orcid.org/0000-0003-4529-0235Антон Цибульник1https://orcid.org/0009-0004-4418-4064Кірілл Даниленко2https://orcid.org/0009-0007-6101-6582Андрій Ориняк3https://orcid.org/0000-0003-4856-9960Igor Sikorsky Kyiv Polytechnic InstituteIgor Sikorsky Kyiv Polytechnic Institute Igor Sikorsky Kyiv Polytechnic InstitutePublic Joint Stock Company “Ukrtransnafta”The paper is devoted to further elaboration of the Method of Matched Sections as a new branch of finite element method in application to the transient 2D temperature problem. The main distinction of MMS from conventional FEM consist in that the conjugation is provided between the adjacent sections rather than in the nodes of the elements. Important feature is that method is based on approximate strong form solution of the governing differential equations called here as the Connection equations. It is assumed that for each small rectangular element the 2D problem can be considered as the combination of two 1D problems – one is x-dependent, and another is y-dependent. Each problem is characterized by two functions – the temperature, , and heat flux . In practical realization for rectangular finite elements the method is reduced to determination of eight unknowns for each element – two unknowns on each side, which are related by the Connection equations, and requirement of the temperature continuity at the center of element. Another salient feature of the paper is an implementation of the original implicit time integration scheme, where the time step became the parameter of shape function within the element, i.e. it determines the behavior of the Connection equations. This method was early proposed by first author for number of 1D problem, and here in first time it is applied for 2D problems. The number of tests for rectangular plate exhibits the remarkable properties of this “embedded” time integration scheme with respect to stability, accuracy, and absence of any restrictions as to increasing of the time step.https://journal.mmi.kpi.ua/article/view/299059method of matched sectionsimplicit time integrationtime step dependent shape functionsrectangular platetransient temperature |
| spellingShingle | Ігор Ориняк Антон Цибульник Кірілл Даниленко Андрій Ориняк Implicit direct time integration of the heat conduction problem in the Method of Matched Sections Mechanics and Advanced Technologies method of matched sections implicit time integration time step dependent shape functions rectangular plate transient temperature |
| title | Implicit direct time integration of the heat conduction problem in the Method of Matched Sections |
| title_full | Implicit direct time integration of the heat conduction problem in the Method of Matched Sections |
| title_fullStr | Implicit direct time integration of the heat conduction problem in the Method of Matched Sections |
| title_full_unstemmed | Implicit direct time integration of the heat conduction problem in the Method of Matched Sections |
| title_short | Implicit direct time integration of the heat conduction problem in the Method of Matched Sections |
| title_sort | implicit direct time integration of the heat conduction problem in the method of matched sections |
| topic | method of matched sections implicit time integration time step dependent shape functions rectangular plate transient temperature |
| url | https://journal.mmi.kpi.ua/article/view/299059 |
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