Piotr Kalita

ANALYSIS OF A CONTACT PROBLEM WITH NORMAL COMPLIANCE, FINITE PENETRATION AND NONMONOTONE SLIP DEPENDENT FRICTION

We consider a mathematical model which describes the frictional contact between a linearly elastic body and an obstacle, the so-called foundation. The process is static and the contact is modeled with normal compliance condition of such a type that the penetration is restricted with unilateral constraint. The friction is modeled with a nonmonotone law in which the friction bound depends both on the tangential displacement and on the value of the penetration. In order to approximate the contact conditions, we consider a regularized problem wherein the contact is modeled by a standard normal compliance condition without finite penetrations. For each problem, we derive a variational formulation and an existence result of the weak solutions of regularized problems is obtained. Next, we prove the convergence of the weak solutions of regularized problems to the weak solution of the initial nonregularized problem. Finally, we provide a numerical validation of this convergence result. To this end we introduce a discrete scheme for the numerical approximation of the frictional contact problems. The solution of the resulting nonsmooth and nonconvex frictional contact problems is found, basing on approximation by a sequence of nonsmooth convex programming problems. Some numerical simulation results are presented in the study of an academic two-dimensional example.

An analytical and numerical approach to a bilateral contact problem with nonmonotone friction

We consider a mathematical model which describes the contact between a linearly elastic body and an obstacle, the so-called foundation. The process is static and the contact is bilateral, i.e., there is no loss of contact. The friction is modeled with a nonmotonone law. The purpose of this work is to provide an error estimate for the Galerkin method as well as to present and compare two numerical methods for solving the resulting nonsmooth and nonconvex frictional contact problem. The first approach is based on the nonconvex proximal bundle method, whereas the second one deals with the approximation of a nonconvex problem by a sequence of nonsmooth convex programming problems. Some numerical experiments are realized to compare the two numerical approaches.

Minimality Properties of Set-Valued Processes and their Pullback Attractors

We discuss the existence of pullback attractors for multivalued dynamical systems on metric spaces. Such attractors are shown to exist without any assumptions in terms of continuity of the solution maps, based only on minimality properties with respect to the notion of pullback attraction. When invariance is required, a very weak closed graph condition on the solving operators is assumed. The presentation is complemented with examples and counterexamples to test the sharpness of the hypotheses involved, including a reaction-diffusion equation, a discontinuous ordinary differential equation, and an irregular form of the heat equation.

Attractors for Navier–Stokes flows with multivalued and nonmonotone subdifferential boundary conditions

Abstract We consider two-dimensional nonstationary Navier–Stokes shear flow with multivalued and nonmonotone boundary conditions on a part of the boundary of the flow domain. We prove the existence of global in time solutions of the considered problem which is governed by a partial differential inclusion with a multivalued term in the form of Clarke subdifferential. Then we prove the existence of a trajectory attractor and a weak global attractor for the associated multivalued semiflow. This research is motivated by control problems for fluid flows in domains with semipermeable walls and membranes.

Compartment model of neuropeptide synaptic transport with impulse control

In this paper a mathematical description of a presynaptic episode of slow synaptic neuropeptide transport is proposed. Two interrelated mathematical models, one based on a system of reaction diffusion partial differential equations and another one, a compartment type, based on a system of ordinary differential equations (ODE) are formulated. Processes of inflow, calcium triggered activation, diffusion and release of neuropeptide from large dense core vesicles (LDCV) as well as inflow and diffusion of ionic calcium are represented. The models assume the space constraints on the motion of inactive LDCVs and free diffusion of activated ones and ions of calcium. Numerical simulations for the ODE model are presented as well. Additionally, an electronic circuit, reflecting the functional properties of the mathematically modelled presynaptic slow transport processes, is introduced.

Numerical simulation for a neurotransmitter transport model in the axon terminal of a presynaptic neuron

Neurotransmitters in the terminal bouton of a presynaptic neuron are stored in vesicles, which diffuse in the cytoplasm and, after a stimulation signal is received, fuse with the membrane and release its contents into the synaptic cleft. It is commonly assumed that vesicles belong to three pools whose content is gradually exploited during the stimulation. This article presents a model that relies on the assumption that the release ability is associated with the vesicle location in the bouton. As a modeling tool, partial differential equations are chosen as they allow one to express the continuous dependence of the unknown vesicle concentration on both the time and space variables. The model represents the synthesis, concentration-gradient-driven diffusion, and accumulation of vesicles as well as the release of neuroactive substances into the cleft. An initial and boundary value problem is numerically solved using the finite element method (FEM) and the simulation results are presented and discussed. Simulations were run for various assumptions concerning the parameters that govern the synthesis and diffusion processes. The obtained results are shown to be consistent with those obtained for a compartment model based on ordinary differential equations. Such studies can be helpful in gaining a deeper understanding of synaptic processes including physiological pathologies. Furthermore, such mathematical models can be useful for estimating the biological parameters that are included in a model and are hard or impossible to measure directly.

Stationary Solutions of the Navier–Stokes Equations

In this chapter we introduce some basic notions from the theory of the Navier–Stokes equations: the function spaces H, V, and V ′, the Stokes operator A with its domain D(A) in H, and the bilinear form B. We apply the Galerkin method and fixed point theorems to prove the existence of solutions of the nonlinear stationary problem, and we consider problems of uniqueness and regularity of solutions.

Stationary Oberbeck–Boussinesq model of generalized Newtonian fluid governed by multivalued partial differential equations

A generalization of the Oberbeck–Boussinesq model consisting of a system of steady state multivalued partial differential equations for incompressible, generalized Newtonian of the p-power type, viscous flow coupled with the nonlinear heat equation is studied in a bounded domain. The existence of a weak solution is proved by combining the surjectivity method for operator inclusions and a fixed point technique.

On Large Time Asymptotics for Two Classes of Contact Problems

We consider two classes of evolution contact problems on two dimensional domains governed by first and second order evolution equations, respectively. The contact is represented by multivalued and nonmonotone boundary conditions that are expressed by means of Clarke subdifferentials of certain locally Lipschitz and semiconvex potentials. For both problems we study the existence and uniqueness of solutions as well as their asymptotic behavior in time. For the first order problem, that is governed by the Navier–Stokes equations with generalized Tresca law, we show the existence of global attractor of finite fractal dimension and existence of exponential attractor. For the second order problem, representing the frictional contact in antiplane viscoelasticity, we show that the global attractor exists, but both the global attractor and the set of stationary states are shown to have infinite fractal dimension.

A Contact Problem with Normal Compliance, Finite Penetration and Nonmonotone Slip Dependent Friction

In this work, we consider a static frictional contact problem between a linearly elastic body and an obstacle, the so-called foundation. This contact is described by a normal compliance condition of such a type that the penetration is restricted with unilateral constraint. The friction is modeled with a nonmonotone law. In order to approximate the contact conditions, we consider a regularized problem wherein the contact is modeled by a standard normal compliance condition without finite penetration. Next, we present a convergence result between the solution of the regularized problem and the original problem. Finally, we provide a numerical validation of this convergence result. To this end we introduce a discrete scheme for the numerical approximation of the frictional contact problems.