Martin Kružík - Academia.edu (original) (raw)
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Papers by Martin Kružík
Springer Proceedings in Mathematics & Statistics, 2013
We propose a phenomenological model describing stress and temperature induced transformations in ... more We propose a phenomenological model describing stress and temperature induced transformations in polycrystalline shape memory alloys. Polycrystallinity is mimicked on the level of a finite element discretization: Each element is treated as a single grain of a randomly chosen orientation. This heterogeneity destroys the undesirable effect of instantaneous transformation under spatially homogeneous loading or heating. We present various computational experiments on the hysteretic effects.
Abstract Evolution strategies are inspired in biology and form part of a larger research field kn... more Abstract Evolution strategies are inspired in biology and form part of a larger research field known as evolutionary algorithms. Those strategies perform a random search in the space of admissible functions, aiming to optimize some given objective function. We show that simple evolution strategies are a useful tool in optimal control, permitting one to obtain, in an efficient way, good approximations to the solutions of some recent and challenging optimal control problems.
Looking at severe plastic deformation experiments, it seems that crystalline materials at yield b... more Looking at severe plastic deformation experiments, it seems that crystalline materials at yield behave as a special kind of anisotropic, highly viscous fluids flowing through an adjustable crystal lattice space. High viscosity provides a possibility to describe the flow as a quasi-static process, where inertial and other body forces can be neglected. The flow through the lattice space is restricted to preferred crystallographic planes and directions causing anisotropy. In the deformation process the lattice is strained and rotated. The proposed model is based on the rate form of the decomposition rule: the velocity gradient consists of the lattice velocity gradient and the sum of the velocity gradients corresponding to the slip rates of individual slip systems. The proposed crystal plasticity model allowing for large deformations is treated as the flow-adjusted boundary value problem. As a test example we analyze a plastic flow of an single crystal compressed in a channel die. We propose three step algorithm of finite element discretization for a numerical solution in the Arbitrary Lagrangian Eulerian (ALE) configuration.
A continuum evolutionary model for micromagnetics is presented that, beside the standard magnetic... more A continuum evolutionary model for micromagnetics is presented that, beside the standard magnetic balance laws, includes thermo-magnetic coupling. To allow conceptually efficient computer implementation, inspired by relaxation method of static minimization problems, our model is mesoscopic in the sense that possible fine spatial oscillations of the magnetization are modeled by means of Young measures. Existence of weak solutions is proved by backward Euler time discretization. AMS Subj. Classification: 35K85, 35Q60, 49S05, 78A30, 78M30, 80A17.
The review is focused on two methods of formulation and solution of the subgrain formation proble... more The review is focused on two methods of formulation and solution of the subgrain formation problem: an energetic approach and a model of incremental deformations. Both methods are based on a reduced single slip version of crystal plasticity. The mathematical analysis of the energetic approach is done for a single slip model only; in the incremental approach the deformation are assumed small, hence, multi slip can be treated as a sum of single slips. The energetic approach has been employed in analysis of the crystal plasticity model of shear and kink bands. The incremental higher strain gradient model provides an insight into an initial stage of the subgrain formation and the mechanism controlling the subgrain size.
We propose a new approach to the numerical treatment of non(quasi)convex rateindependent evolutio... more We propose a new approach to the numerical treatment of non(quasi)convex rateindependent evolutionary problems. The main idea is to replace the original microscopic energy density with its polyconvexification. For this problem, first-order optimality conditions are derived and used in finding a discrete solution. The effectiveness of the method is illustrated with some numerical experiments.
We derive necessary and su cient optimality conditions for a relaxed (in terms of Young measures)... more We derive necessary and su cient optimality conditions for a relaxed (in terms of Young measures) variational problem governing steady states of ferromagnetic materials. Such conditions here stated in the form of a generalized Weierstrass maximum principle enable us to establish uniqueness of a solution in some speci c situations and can also be used in e cient n umerical algorithms solving the relaxed problems, for instance.
Springer Proceedings in Mathematics & Statistics, 2013
We propose a phenomenological model describing stress and temperature induced transformations in ... more We propose a phenomenological model describing stress and temperature induced transformations in polycrystalline shape memory alloys. Polycrystallinity is mimicked on the level of a finite element discretization: Each element is treated as a single grain of a randomly chosen orientation. This heterogeneity destroys the undesirable effect of instantaneous transformation under spatially homogeneous loading or heating. We present various computational experiments on the hysteretic effects.
Abstract Evolution strategies are inspired in biology and form part of a larger research field kn... more Abstract Evolution strategies are inspired in biology and form part of a larger research field known as evolutionary algorithms. Those strategies perform a random search in the space of admissible functions, aiming to optimize some given objective function. We show that simple evolution strategies are a useful tool in optimal control, permitting one to obtain, in an efficient way, good approximations to the solutions of some recent and challenging optimal control problems.
Looking at severe plastic deformation experiments, it seems that crystalline materials at yield b... more Looking at severe plastic deformation experiments, it seems that crystalline materials at yield behave as a special kind of anisotropic, highly viscous fluids flowing through an adjustable crystal lattice space. High viscosity provides a possibility to describe the flow as a quasi-static process, where inertial and other body forces can be neglected. The flow through the lattice space is restricted to preferred crystallographic planes and directions causing anisotropy. In the deformation process the lattice is strained and rotated. The proposed model is based on the rate form of the decomposition rule: the velocity gradient consists of the lattice velocity gradient and the sum of the velocity gradients corresponding to the slip rates of individual slip systems. The proposed crystal plasticity model allowing for large deformations is treated as the flow-adjusted boundary value problem. As a test example we analyze a plastic flow of an single crystal compressed in a channel die. We propose three step algorithm of finite element discretization for a numerical solution in the Arbitrary Lagrangian Eulerian (ALE) configuration.
A continuum evolutionary model for micromagnetics is presented that, beside the standard magnetic... more A continuum evolutionary model for micromagnetics is presented that, beside the standard magnetic balance laws, includes thermo-magnetic coupling. To allow conceptually efficient computer implementation, inspired by relaxation method of static minimization problems, our model is mesoscopic in the sense that possible fine spatial oscillations of the magnetization are modeled by means of Young measures. Existence of weak solutions is proved by backward Euler time discretization. AMS Subj. Classification: 35K85, 35Q60, 49S05, 78A30, 78M30, 80A17.
The review is focused on two methods of formulation and solution of the subgrain formation proble... more The review is focused on two methods of formulation and solution of the subgrain formation problem: an energetic approach and a model of incremental deformations. Both methods are based on a reduced single slip version of crystal plasticity. The mathematical analysis of the energetic approach is done for a single slip model only; in the incremental approach the deformation are assumed small, hence, multi slip can be treated as a sum of single slips. The energetic approach has been employed in analysis of the crystal plasticity model of shear and kink bands. The incremental higher strain gradient model provides an insight into an initial stage of the subgrain formation and the mechanism controlling the subgrain size.
We propose a new approach to the numerical treatment of non(quasi)convex rateindependent evolutio... more We propose a new approach to the numerical treatment of non(quasi)convex rateindependent evolutionary problems. The main idea is to replace the original microscopic energy density with its polyconvexification. For this problem, first-order optimality conditions are derived and used in finding a discrete solution. The effectiveness of the method is illustrated with some numerical experiments.
We derive necessary and su cient optimality conditions for a relaxed (in terms of Young measures)... more We derive necessary and su cient optimality conditions for a relaxed (in terms of Young measures) variational problem governing steady states of ferromagnetic materials. Such conditions here stated in the form of a generalized Weierstrass maximum principle enable us to establish uniqueness of a solution in some speci c situations and can also be used in e cient n umerical algorithms solving the relaxed problems, for instance.