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Shape Optimization for Navier-Stokes Equations with Viscosity
Stebel, Jan ; Haslinger, Jaroslav (advisor) ; Feistauer, Miloslav (referee) ; Feireisl, Eduard (referee)
We study the shape optimization problem for the paper machine headbox which distributes a mixture of water and wood fibers in the paper making process. The aim is to find a shape which a priori ensures the given velocity profile on the outlet part. The mathematical formulation leads to the optimal control problem in which the control variable is the shape of the domain representing the header, the state problem is represented by the generalised Navier-Stokes system with nontrivial boundary conditions. The objective is to analyze theoretically this problem (proof of the existence of a solution), its discretization and the numerical realization.
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Non-smooth Newton's method
Balázsová, Monika ; Haslinger, Jaroslav (advisor) ; Ligurský, Tomáš (referee)
In this thesis we generalize classical Newton's method for non-smooth equations. For this purpose we define the Newton approximation of functions. Then we introduce several methods for solving equations with locally Lipschitz and piecewise smooth functions. We prove that their local convergence rate is Q-superlinear or even Q-quadratic. At the end we apply one of the algorithms to the beam problem with the obstacle. Based on the physical model we establish mathematical model and its discretization. Finally we implement the problem in the MATLAB. Results are summarized in tables.
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Shape optimization in contact problems with friction
Pathó, Róbert ; Haslinger, Jaroslav (advisor) ; Knobloch, Petr (referee)
In the present work we formulate a shape optimization problem for the 2D Signorini problem with given friction and a coefficient of friction which depends on the solution. The aim is to find an optimal contact part of an elastic body. A suitable set of admissible domains is given, among which the existence of an optimal one is established for a large class of cost functionals. The shape optimization problem is then approximated. Existence of discrete optimal shapes is proven and convergence analysis is done.
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Approximation and numerical realization of contact problems with given friction and a coefficient of friction depending on the solution in 3D.
Ligurský, Tomáš ; Haslinger, Jaroslav (advisor) ; Knobloch, Petr (referee)
Three-dimensional contact problems with given friction and a coefficient of friction depending on the solution are studied. By means of the fixed-point approach, the existence of at least one solution is proved provided that the coefficient of friction F is represented by a continuous, positive and bounded function. Under an additional assumption, namely the Lipschitz continuity of F with a sufficiently small modulus of the Lipschitz continuity, the uniqueness of the solution is shown. The problem is discretized by the finite element method. The existence and uniqueness of the solution to the discrete problems are investigated in a similar way as it has been done in the continuous setting. Convergence of solutions to the discrete models in an appropriate sense is established. The method of successive approximations is used for finding fixed-points. Each iterative step leads to a contact problem with given friction and a coefficient of friction which does not depend on the solution. We introduce a mixed variational formulation of this problem from which the dual formulation used in computations can be derived. Numerical results of model examples are presented.
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