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Reliable computation and local mesh adaptivity in limit analysis
Sysala, Stanislav ; Haslinger, Jaroslav ; Repin, S.
The contribution is devoted to computations of the limit load for a perfectly plastic model with the von Mises yield criterion. The limit factor of a prescribed load is defined by a specific variational problem, the so-called limit analysis problem. This problem is solved in terms of deformation fields by a penalization, the finite element and the semismooth Newton methods. From the numerical solution, we derive a guaranteed upper bound of the limit factor. To achieve more accurate results, a local mesh adaptivity is used.
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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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Approximation, numerical realization and qualitative analysis of contact problems with friction
Ligurský, Tomáš ; Haslinger, Jaroslav (advisor) ; Segeth, Karel (referee) ; Rohan, Eduard (referee)
Title: Approximation, numerical realization and qualitative analysis of contact problems with friction Author: Tomáš Ligurský Department: Department of Numerical Mathematics Supervisor: prof. RNDr. Jaroslav Haslinger, DrSc., Department of Numerical Mathe- matics Abstract: This thesis deals with theoretical analysis and numerical realization of dis- cretized contact problems with Coulomb friction. First, discretized 3D static contact prob- lems with isotropic and orthotropic Coulomb friction and solution-dependent coefficients of friction are analyzed by means of the fixed-point approach. Existence of at least one solution is established for coefficients of friction represented by positive, bounded and con- tinuous functions. If these functions are in addition Lipschitz continuous and upper bounds of their values together with their Lipschitz moduli are sufficiently small, uniqueness of the solution is guaranteed. Second, properties of solutions parametrized by the coefficient of friction or the load vector are studied in the case of discrete 2D static contact problems with isotropic Coulomb friction and coefficient independent of the solution. Conditions under which there exists a local Lipschitz continuous branch of solutions around a given reference point are established due to two variants of the...
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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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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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