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Generování a optimalizace meshů
Mokriš, Dominik ; Šír, Zbyněk (advisor) ; Hron, Jaroslav (referee)
This thesis is devoted to the problem of finding a suitable geometrical de- scription of the domain for the Finite Element Method (FEM). We present the most important methods used in generation and improvement of unstructured triangular meshes (grids) for two dimensional FEM. Possible measures of mesh quality are discussed with respect to their usage in linear Lagrange FEM. The relationship between mesh geometry (especially angles of particular triangles), discretization error and stiffness matrix condition number is examined. Two methods of mesh improvement, based on Centroidal Voronoi Tessellations (CVT) and Optimal Delaunay Triangulations (ODT), are discussed in detail and some results on convergence of CVT based methods are reviewed. Some aspects of these methods, e.g. the relation between density of boundary points and interior mesh vertices and the treatment of the boundary triangles is reconsidered in a new way. We have implemented these two methods and we discuss possible im- provements and new algorithms. A geometrically very interesting idea of recent alternative to FEM, Isogeometric Analysis (IGA), is outlined and demonstrated on a simple example. Several numerical tests are made in order to the compare the accuracy of solutions of isotropic PDEs obtained by FEM on bad mesh, mesh improved...
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Isogeometric analysis in applications
Bekrová, Martina ; Šír, Zbyněk (advisor) ; Hron, Jaroslav (referee)
Isogeometric analysis (IGA) is a numerical method for solving partial differential equations (PDE). In this master thesis we explain a concept of IGA with special emphasis on problems on closed domains created by a single NURBS patch. For them we show a process how to modify the NURBS basis to ensure the highest possible continuity of the function space. Then we solve the minimal surface problem using two different Newton type methods. The first one is based on the classical approach using PDE, in the second one we use unique advantages of IGA to directly minimize the area functional.
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Generování a optimalizace meshů
Mokriš, Dominik ; Šír, Zbyněk (advisor) ; Hron, Jaroslav (referee)
This thesis is devoted to the problem of finding a suitable geometrical de- scription of the domain for the Finite Element Method (FEM). We present the most important methods used in generation and improvement of unstructured triangular meshes (grids) for two dimensional FEM. Possible measures of mesh quality are discussed with respect to their usage in linear Lagrange FEM. The relationship between mesh geometry (especially angles of particular triangles), discretization error and stiffness matrix condition number is examined. Two methods of mesh improvement, based on Centroidal Voronoi Tessellations (CVT) and Optimal Delaunay Triangulations (ODT), are discussed in detail and some results on convergence of CVT based methods are reviewed. Some aspects of these methods, e.g. the relation between density of boundary points and interior mesh vertices and the treatment of the boundary triangles is reconsidered in a new way. We have implemented these two methods and we discuss possible im- provements and new algorithms. A geometrically very interesting idea of recent alternative to FEM, Isogeometric Analysis (IGA), is outlined and demonstrated on a simple example. Several numerical tests are made in order to the compare the accuracy of solutions of isotropic PDEs obtained by FEM on bad mesh, mesh improved...
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Isogeometric contact analysis: a study of an explicit dynamic contact algorithm
Kopačka, Ján ; Gabriel, Dušan ; Kolman, Radek ; Plešek, Jiří
The isogeometric NURBS based variant of symmetry preserving explicit FE contact-impact algorithm, has been proposed. The algorithm was studied by means of a numerical example, which involves 2d frictionless dynamic Hertz contact problem of two equally shaped cylinders. The attention was paid to the influence of different lumping techniques on the oscillations of contact force and contact pressure. The standard Lagrange finite elements were compared with the NURBS isogeometric elements. Both the first and second order were considered.
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Comparative study of finite element method, isogeometric analysis, and finite volume method in elastic wave propagation of stress discontinuities
Berezovski, A. ; Kolman, Radek ; Blažek, Jiří ; Kopačka, Ján ; Gabriel, Dušan ; Plešek, Jiří
A comparative study of Finite Element Method, Isogeometric Analysis, and Finite Volume Method in numerical simulation of one-dimensional wave propagation problems of stress discontinuities in elastic solids is presented. The special attention is paid to accuracy, convergence, and stability of tested numerical methods and the appearance of spurious oscillations and damping effects occurring close to theoretical sharp wavefronts.
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