Grigory Panasenko

METHOD OF ASYMPTOTIC PARTIAL DECOMPOSITION OF DOMAIN

A new method of partial decomposition of a domain is proposed for partial differential equations, depending on a small parameter. It is based on the information about the structure of the asymptotic solution in different parts of the domain. The principal idea of the method is to extract the subdomain of singular behavior of the solution and to simplify the problem in the subdomain of regular behavior of the solution. The special interface conditions are imposed on the common boundary of these partially decomposed subdomains. If, for example, the domain depends on the small parameter and some parts of the domain change their dimension after the passage to the limit, then the proposed method reduces the initial problem to the system of equations posed in the domains of different dimensions with the special interface conditions.

ASYMPTOTIC METHODS FOR MICROPOLAR FLUIDS IN A TUBE STRUCTURE

The steady motion of a micropolar fluid through a wavy tube with the dimensions depending on a small parameter is studied. An asymptotic expansion is proposed and error estimates are proved by using a boundary layer method. We apply the method of partial asymptotic decomposition of domain and we prove that the solution of the partially decomposed problem represents a good approximation for the solution of the considered problem.

Continuous mathematical model of platelet thrombus formation in blood flow

An injury of a blood vessel requires quick repairing of the wound in order to prevent a loss of blood. This is done by the hemostatic system. The key point of its work is the formation of an aggregate from special blood elements, namely, platelets. The construction of a mathematical model of the formation of a thrombocyte aggregate with an adequate representation of its physical, chemical, and biological processes is an extremely complicated problem. A large size of platelets compared to that of molecules, strong inhomogeneity of their distribution across the blood flow, high shear velocities, the moving boundary of the aggregate, the interdependence of its growth and the blood flux hamper the construction of closed mathematical models convenient for biologists. We propose a new PDE-based model of a thrombocyte aggregate formation. In this model, the movement of its boundary due to the adhesion and detachment of platelets is determined by the level set method. The model takes into account the distribution inhomogeneity of erythrocytes and platelets across the blood flow, the invertible adhesion of platelets, their activation, secretion, and aggregation. The calculation results are in accordance with the experimental data concerning the kinetics of the ADP-evoked growth of a thrombus in vivo for different flow velocities.

MEMORY EFFECT IN HOMOGENIZATION OF A VISCOELASTIC KELVIN-VOIGT MODEL WITH TIME-DEPENDENT COEFFICIENTS

This paper is motivated by modeling the procedure of formation of a composite material constituted of solid fibers and of a solidifying matrix. The solidification process for the matrix depends on the temperature and on the reticulation rate which thereby influence the mechanical properties of the matrix. The mechanical properties are described by a viscoelastic medium equation of Kelvin-Voigt type with rapidly oscillating periodic coefficients depending on the temperature and the reticulation rate. That is modeled as an initial boundary value problem with time-dependent elasticity and viscosity tensors to account for the solidification, and the mechanical and/or thermal forcing. First we prove the existence and uniqueness of the solution for the problem and obtain a priori estimates. Then we derive the homogenized problem, characterize its coefficients including explicit memory terms, and prove that it admits a unique solution. Finally, we prove error bounds for the asymptotic solution, and establish some related regularity properties of the homogenized solution.

Asymptotic dimension reduction of a Robin-type elasticity boundary value problem in thin beams

Abstract We consider a linear elasticity boundary value problem in a beam with Robin boundary condition at an end and on a segment of the lateral boundary in the middle of the beam. The Robin parameters are scaled differently in the longitudinal and cross-sectional directions. The dimension of the problem is reduced by a standard asymptotic approach with an additional expansion suggested to fulfil the Robin conditions. The 3D Robin conditions result into 1D Robin boundary conditions for corresponding ODEs. The asymptotic error is estimated and illustrated by a numerical comparison of the 3D and 1D solutions.

Asymptotic analysis of a non-periodic flow in a thin channel with visco-elastic wall

In this paper we continue the study of a fluid-structure interaction problem with the non periodic case. We consider the non stationary flow of a viscous fluid in a thin rectangle with an elastic membrane as the upper part of the boundary. The physical problem which corresponds to non homogeneous boundary conditions is stated. By using a boundary layer method, an asymptotic solution is proposed. The properties of the boundary layer functions are established and an error estimate is obtained.

Asymptotic analysis and domain decomposition for a biharmonic problem in a thin multi-structure

In the paper, we consider the Dirichlet boundary value problem for the biharmonic equation defined in a thin T-like shaped structure. Our goal is to construct an asymptotic expansion of its solution. We provide error estimates and also introduce and justify the asymptotic partial domain decomposition for this problem.

Asymptotic analysis of the nonsteady viscous flow with a given flow rate in a thin pipe

The nonsteady Navier–Stokes equations are considered in a thin infinite pipe with the small diameter ϵ in the case of the Reynolds number of order ϵ. The time-dependent flow rate is a given function. The complete asymptotic expansion is constructed and justified. The error estimate of order O(ϵ J ) for the difference of the exact solution and the J-th asymptotic approximation is proved for any real J.

ASYMPTOTIC ANALYSIS AND NUMERICAL MODELING OF MASS TRANSPORT IN TUBULAR STRUCTURES

In the paper the flow in a thin tubular structure is considered. The velocity of the flow stands for a coefficient in the diffusion-convection equation set in the thin structure. An asymptotic expansion of solution is constructed. This expansion is used further for justification of an asymptotic domain decomposition strategy essentially reducing the memory and the time of the code. A numerical solution obtained by this strategy is compared to the numerical solution obtained by a direct FEM computation.In this paper the flow in a thin tubular structure is considered. The velocity of the flow stands for a coefficient in the convection-diffusion equation set in the thin structure. An asymptotic expansion of solution is constructed. This expansion is used further for justification of an asymptotic domain decomposition strategy essentially reducing the memory and the time of the code. A numerical solution obtained by this strategy is compared to the numerical solution obtained by a direct FEM computation.

THE PARTIAL HOMOGENIZATION: CONTINUOUS AND SEMI-DISCRETIZED VERSIONS

The partial homogenization is a new method for the treatment of the boundary layers in the homogenization theory. It keeps the initial formulation near the boundary, passes to the high order homogenization at some distance from the boundary and prescribes the asymptotically precise junction conditions between the homogenized and the heterogeneous models at the interface. This method is related to the method of asymptotic partial domain decomposition MAPDD (see G. Panasenko, Method of asymptotic partial decomposition of domain, Math. Mod. Meth. Appl. Sci.8 (1998) 139–156). The partial homogenization (as well as the MAPDD) can be interpreted as a multi-scale model coupling the homogenized (macroscopic) description in the internal main part of the domain and the microscopic zoom in the domain of the location of the boundary layers. The semi-discretized partial homogenization uses some high order finite element projection in the homogenized subdomain.