arXiv subjects

Pavel Krejci

Publications and source records attributed to Pavel Krejci.

16 recordsLinked to original sources

arXiv2022

Phase transitions in porous media

The full quasistatic thermomechanical system of PDEs, describing water diffusion with the possibility of freezing and melting in a visco-elasto-plastic porous solid, is studied in detail under the hypothesis that the pressure-saturation hysteresis relation is given in terms of the Preisach hysteresis operator. The resulting system of balance equations for mass, momentum, and energy coupled with the phase dynamics equation is shown to admit a global solution under general assumptions on the data.

math.AP↗

arXiv2021

Analysis of a tumor model as a multicomponent deformable porous medium

We propose a diffuse interface model to describe tumor as a multicomponent deformable porous medium. We include mechanical effects in the model by coupling the mass balance equations for the tumor species and the nutrient dynamics to a mechanical equilibrium equation with phase-dependent elasticity coefficients. The resulting PDE system couples two Cahn--Hilliard type equations for the tumor phase and the healthy phase with a PDE linking the evolution of the interstitial fluid to the pressure of the system, a reaction-diffusion type equation for the nutrient proportion, and a quasistatic momentum balance. We prove here that the corresponding initial-boundary value problem has a solution in appropriate function spaces.

math.AP↗

arXiv2021

Explicit and implicit non-convex sweeping processes in the space of absolutely continuous functions

We show that sweeping processes with possibly non-convex prox-regular constraints generate a strongly continuous input-output mapping in the space of absolutely continuous functions. Under additional smoothness assumptions on the constraint we prove the local Lipschitz continuity of the input-output mapping. Using the Banach contraction principle, we subsequently prove that also the solution mapping associated with the state-dependent problem is locally Lipschitz continuous.

math.DS↗

arXiv2021

Non-convex sweeping processes in the space of regulated functions

The aim of this paper is to study a wide class of non-convex sweeping processes with moving constraint whose translation and deformation are represented by regulated functions, i.e., functions of not necessarily bounded variation admitting one-sided limits at every point. Assuming that the time-dependent constraint is uniformly prox-regular and has uniformly non-empty interior, we prove existence and uniqueness of solutions, as well as continuous data dependence with respect to the sup-norm.

math.DS↗

arXiv2018

Oscillations of a temperature-dependent piezoelectric rod

Piezoelectricity of some materials has shown to have many applications, in particular in energy harvesting. Due to the inherent hysteresis in the characteristic of such materials, a number of hysteretic models have been proposed minding the fact that hysteresis losses may influence the efficiency of the process. However, hysteresis dissipation is accompanied with heat production, which in turn increases the temperature of the device and may change its physical characteristics. In this paper we propose a phenomenological model for electromechanical coupling in piezoelectric materials where temperature and feedback effects are taken into account. We prove the existence of solution for the resulting PDE system and show that the model is thermodynamically consistent. The main analytical tool is the inverse Preisach operator with temperature-dependent density.

math.AP↗

arXiv2018

Inverse parameter-dependent Preisach operator in thermo-piezoelectricity modeling

Hysteresis is an important issue in modeling piezoelectric materials, for example, in applications to energy harvesting, where hysteresis losses may influence the efficiency of the process. The main problem in numerical simulations is the inversion of the underlying hysteresis operator. Moreover, hysteresis dissipation is accompanied with heat production, which in turn increases the temperature of the device and may change its physical characteristics. More accurate models therefore have to take the temperature dependence into account for a correct energy balance. We prove here that the classical Preisach operator with a fairly general parameter-dependence admits a Lipschitz continuous inverse in the space of right-continuous regulated functions, propose a time-discrete and memory-discrete inversion algorithm, and show that higher regularity of the inputs leads to a higher regularity of the output of the inverse.

math.AP↗

arXiv2017

A continuum of path-dependent equilibrium solutions induced by sticky expectations

We analyze a simple macroeconomic model where rational inflation expectations is replaced by a boundedly rational, and genuinely sticky, response to changes in the actual inflation rate. The stickiness is introduced in a novel way using a mathematical operator that is amenable to rigorous analysis. We prove that, when exogenous noise is absent from the system, the unique equilibrium of the rational expectations model is replaced by an entire line segment of possible equilibria with the one chosen depending, in a deterministic way, upon the previous states of the system. The agents are sufficiently far-removed from the rational expectations paradigm that problems o indeterminacy do not arise. The response to exogenous noise is far more subtle than in a unique equilibrium model. After sufficiently small shocks the system will indeed revert to the same equilibrium but larger ones will move the system to a different one (at the same model parameters). The path to this new equilibrium may be very long with a highly unpredictable endpoint. At certain model parameters exogenously-triggered runaway inflation can occur. Finally, we analyze a variant model in which the same form of sticky response is introduced into the interest rate rule instead.

math.DS↗

arXiv2017

Global stability of a piecewise linear macroeconomic model with a continuum of equilibrium states and sticky expectation

We consider piecewise linear discrete time macroeconomic models, which possess a continuum of equilibrium states. These systems are obtained by replacing rational inflation expectations with a boundedly rational, and genuinely sticky, response of agents to changes in the actual inflation rate in a standard Dynamic Stochastic General Equilibrium model. Both for a low-dimensional variant of the model, with one representative agent, and the multi-agent model, we show that, when exogenous noise is absent from the system, the continuum of equilibrium states is the global attractor. Further, when a uniformly bounded noise is present, or the equilibrium states are destabilized by an imperfect Central Bank policy (or both), we estimate the size of the domain that attracts all the trajectories. The proofs are based on introducing a family of Lyapunov functions and, for the multi-agent model, deriving a formula for the inverse of the Prandtl-Ishlinskii operator acting in the space of discrete time inputs and outputs.

math.DS↗

arXiv2017

Unsaturated deformable porous media flow with phase transition

In the present paper, a continuum model is introduced for fluid flow in a deformable porous medium, where the fluid may undergo phase transitions. Typically, such problems arise in modeling liquid-solid phase transformations in groundwater flows. The system of equations is derived here from the conservation principles for mass, momentum, and energy and from the Clausius-Duhem inequality for entropy. It couples the evolution of the displacement in the matrix material, of the capillary pressure, of the absolute temperature, and of the phase fraction. Mathematical results are proved under the additional hypothesis that inertia effects and shear stresses can be neglected. For the resulting highly nonlinear system of two PDEs, one ODE and one ordinary differential inclusion with natural initial and boundary conditions, existence of global in time solutions is proved by means of cut-off techniques and suitable Moser-type estimates.

math.AP↗

arXiv2016

Solvability of an unsaturated porous media flow problem with thermomechanical interaction

A PDE system consisting of the momentum balance, mass balance, and energy balance equations for displacement, capillary pressure, and temperature as a model for unsaturated fluid flow in a porous viscoelastoplastic solid is shown to admit a solution under appropriate assumptions on the constitutive behavior. The problem involves two hysteresis operators accounting for plastic and capillary hysteresis.

math.AP↗

arXiv2016

A weighted finite element mass redistribution method for dynamic contact problems

This paper deals with a one-dimensional wave equation being subjected to a unilateral boundary condition. An approximation of this problem combining the finite element and mass redistribution methods is proposed. The mass redistribution method is based on a redistribution of the body mass such that there is no inertia at the contact node and the mass of the contact node is redistributed on the other nodes. The convergence as well as an error estimate in time are proved. The analytical solution associated with a benchmark problem is introduced and it is compared to approximate solutions for different choices of mass redistribution. However some oscillations for the energy associated with approximate solutions obtained for the second order schemes can be observed after the impact. To overcome this difficulty, an new unconditionally stable and a very lightly dissipative scheme is proposed.

math.NA↗

arXiv2014

A thermodynamically consistent phenomenological model for ferroelectric and ferroelastic hysteresis

We propose a hysteretic model for electromechanical coupling in piezoelectric materials, with the strain and the electric field as inputs and the stress and the polarization as outputs. This constitutive law satisfies the thermodynamic principles and exhibits good agreement with experimental measurements. Moreover, when it is coupled with the mechanical and electromagnetic balance equations, the resulting PDE system is well-posed under the hypothesis that hysteretic effects take place only in one preferred direction. We prove the existence and uniqueness of its global weak solutions for each initial data with prescribed regularity. One of the tools is a new Lipschitz continuity theorem for the inverse Preisach operator with time dependent coefficients.

math.AP↗

arXiv2013

A new phase field model for material fatigue in oscillating elastoplastic beam

We pursue the study of fatigue accumulation in an oscillating elastoplastic beam under the additional hypothesis that the material can partially recover by the effect of melting. The full system consists of the momentum and energy balance equations, an evolution equation for the fatigue rate, and a differential inclusion for the phase dynamics. The main result consists in proving the existence and uniqueness of a strong solution.

math.AP↗

arXiv2011

Well-posedness of an extended model for water-ice phase transitions

We propose an improved model explaining the occurrence of high stresses due to the difference in specific volumes during phase transitions between water and ice. The unknowns of the resulting evolution problem are the absolute temperature, the volume increment, and the liquid fraction. The main novelty here consists in including the dependence of the specific heat and of the speed of sound upon the phase. These additional nonlinearities bring new mathematical difficulties which require new estimation techniques based on Moser iteration. We establish the existence of a global solution to the corresponding initial-boundary value problem, as well as lower and upper bounds for the absolute temperature. Assuming constant heat conductivity, we also prove uniqueness and continuous data dependence of the solution.

math.AP↗

arXiv2009

Phase separation in a gravity field

We prove here well-posedness and convergence to equilibria for the solution trajectories associated to a model for solidification of a liquid content of a rigid container in a gravity field. We observe that the gravity effects, which can be neglected without considerable changes of the process on finite time intervals, have a substantial influence on the long time behavior of the evolution system. Without gravity, we find a temperature interval, in which all phase distributions with a prescribed total liquid contents are admissible equilibria, while, under the influence of gravity, the only equilibrium distribution in a connected container consists in two pure phases separated by one plane interface perpendicular to the gravity force.

math.AP↗

arXiv2009

A bottle in a freezer

We propose here a model for solidification of a liquid contents of an elastic bottle in a freezer. The main goal is to explain the occurrence of high stresses inside the bottle. As a by-product, we derive a formula for the undercooling coefficient in terms of the elasticity constants, latent heat, and the phase expansion coefficient. We investigate the well-posedness of the three-dimensional model: we prove the existence and uniqueness of a solution for the corresponding initial-boundary value problem which couples a PDE with an integrodifferential equation and an ordinary differential inclusion ruling the evolution of the phase parameter. Finally, we prove some results on the long time behavior of solutions.

math.AP↗