Metody polygonálních konečných prvků
| Thesis title in Czech: | Metody polygonálních konečných prvků |
|---|---|
| Thesis title in English: | Polygonal finite element methods |
| Key words: | Diferenciální rovnice|metoda konečných prvků|Voronoiovy buňky|polygonální prvky|numerické metody|konečná elasticita |
| English key words: | Differential equations|finite element method|Voronoi cells|polygonal elements|numerical methods|finite elasticity |
| Academic year of topic announcement: | 2025/2026 |
| Thesis type: | diploma thesis |
| Thesis language: | |
| Department: | Department of Numerical Mathematics (32-KNM) |
| Supervisor: | Scott Congreve, Ph.D. |
| Author: |
| Guidelines |
| Traditionally the finite element method [1, 2] uses, in two dimensions, triangular or square elements for
the solution of partial differential equations. In recent years, there has been extensive research into the use of finite element methods that use general n-sided polygons as elements [3, 4, 5]. In this thesis, we will study the analysis for some of these methods for a simple partial differential equation, implement them in MATLAB (or another programming language), and perform numerical simulations. Furthermore, we will compare them to standard finite element methods, and potentially study their advantage in solving problems in finite elasticity [6]. |
| References |
| [1] S. C. Brenner and L. R. Scott. The Mathematical Theory of Finite Element Methods. SpringerVerlag, 2008.
[2] V. Dolejší, P. Knobloch, V. Kučera and M. Vlasák. Finite element methods: Theory, applications and implementations. Matfyzpress, Praha, 2013. [3] S. Ghosh and R. L. Mallett. Voronoi cell finite elements. Computers & Structures, 50(1):33-46, 1994. url: https://doi.org/10.1016/0045-7949(94)90435-9. [4] N. Sukumar and A. Tabarraei. Conforming polygonal finite elements. International Journal for Numerical Methods in Engineering, 61(12):2045-2066, 2004. url: https://doi.org/10.1002/nme.1141 [5] K. Dai, G. Liu and T. Nguyen. An n-sided polygonal smoothed finite element method (nSFEM) for solid mechanics. Finite Elements in Analysis and Design, 43(11):847-860, 2007. url: https://doi.org/10.1016/j.finel.2007.05.009 [6] H. Chi, C. Talischi, O. Lopez-Pamies and G. H. Paulino. Polygonal finite elements for finite elasticity. International Journal for Numerical Methods in Engineering, 101(4):305-328, 2015. url: https://doi.org/10.1002/nme.4802 |
| Preliminary scope of work |
| Cílem této práce je studovat různé metody konečných prvků, které na rozdíl od standardních
čtvercových nebo trojúhelníkových prvků používají pro řešení parciálních diferenciálních rovnic obecné polygonální prvky. |
| Preliminary scope of work in English |
| The goal of this thesis is to study various finite element methods that use general polygonal
elements, rather than standard square or triangular elements, for the solution of partial differential equations. |