Ivan Dyyak

Ivan Franko National University of Lviv | ivan franko · Department of Applied Mathematics&Informatics

An algorithm for parallel solution of the dynamic problems of the elasticity theory for axisymmetric objects as a three-dimensional problem of the elasticity theory has been proposed. The semidiscrete approximations reduce the problem to the solution of the Cauchy problem for a system of linear differential equations of the second order. The elemen...

We consider a mathematical model of elastic body with thin inclusion or coating in the form of a thin elastic shell. It is shown that the corresponding Steklov–Poincaré operator of the mathematical model possesses the properties guaranteeing the existence and uniqueness of a weak solution of the boundary-value problem. We propose a method of soluti...

We propose a combined algorithm for the solution of the problems of contact of elastic bodies. This algorithm combines the iterative method of domain decomposition and the h -adaptive scheme based on the comparison of the results of the finite-element and boundary-element methods. The numerical analysis of a test problem shows that the mesh refinem...

We study the domain decomposition methods for the numerical solution of the problems of frictionless unilateral contact of many elastic bodies of finite sizes. By using the finite-element approximations, we solve the problems of the unilateral contact of three elastic bodies compressed by rigid plates and the contact of three fixed bodies one of wh...

In this paper we propose on continuous level a class of domain decomposition methods of Robin-Robin type to solve the problems of unilateral contact between elastic bodies with nonlinear Winkler covers. These methods are based on abstract nonstationary iterative algorithms for nonlinear variational equations in reflexive Banach spaces. We also prov...

In this paper we propose on continuous level several domain decomposition methods to solve unilateral and ideal multibody contact problems of nonlinear elasticity. We also present theorems about convergence of these methods.

The paper is devoted to penalty Robin-Robin domain decomposition methods (DDMs), proposed by us for the solution of unilateral multibody contact problems of elasticity. These DDMs are based on the penalty method for variational inequalities and some stationary and nonstationary iterative methods for nonlinear variational equations. The main result...

Based on the variational formulation and penalty method, we have considered the Neumann parallel scheme of the domain decomposition method for the solution of problems of one-sided contact between three-dimensional elastic bodies. We have shown the existence and uniqueness of a solution of the variational problem with penalty and convergence in the...

A parallel Dirichlet-Dirichlet domain-decomposition algorithm for solving frictionless-contact problems for elastic bodies made of composite materials is proposed and justified. Numerical results that demonstrate the effectiveness of the approach and its software implementation are presented.

The class of parallel Robin (Poincaré) domain decomposition schemes which are based on the penalty method and the simple iteration method for variational equations is proposed for solution of frictionless multibody contact problems of elasticity. The convergence of these schemes is proved. The numerical analysis is made for 2D contact problems usin...

To solve two-dimensional boundary-value problems of elasticity, two iteration algorithms of the domain decomposition method are proposed: parallel Neumann–Neumann and sequential Dirichlet–Neumann. They are based on the hybrid boundary–finite-element approximations. The algorithms are proved to converge. The optimal parameters are selected using the...

An axisymmetric elastic problem is solved by a heterogeneous numerical scheme of the alternating Schwarz method based on the finite- and boundary-element methods. The Schwarz method and the modified scheme are compared. A relaxation parameter is used to accelerate the iterative process

For solving the axisymmetric problem of the theory of elasticity the heterogeneous numerical scheme of alternating Schwarz's method based on the finite and boundary element method is proposed. A comparative numerical analysis of Schwarz's method and the modified scheme is carried out. A choice of the parameter of relaxation is offered for the accel...

This paper presents a common approach to the numerical analysis of elasticity and heat conduction problems in computed structures. It is based on a combination of the linear elasticity theory with Timoshenko's shell theory for elasticity problems, and of the classical heat conduction theory with a dimensionally reduced heat conduction model in thin...

A numerical–analytical approach is proposed to solve a problem on the free vibrations of cylindrical bodies. The approach is based on three-dimensional elastic theory and the semianalytic finite-element method. The free vibrations of isotropic and anisotropic solid cylinders of finite length are examined. It is studied how boundary conditions and m...

A numerical-analytical approach is proposed for solution of the problem on free vibrations of cylindrical bodies. The approach is limited by the framework of the three-dimensional elasticity theory and is based on the semianalytical finite element method. The investigation of free vibrations of isotropic and anisotropic solid finite-long cylinders...

For special problems of the theory of elasticity, coupled boundary and finite element analysis may be a useful alternative to either finite element or boundary element analysis. Herein, we propose a new mode of coupled analysis which is based on combining the linear theory of elasticity with Timoshenko’s shell theory [D(imension)-adaptive model]. F...

The numerical results demonstrate the accuracy and efficiency of the studies performed on the basis of the presented strategy of combining the finite-element method and the boundary-element method. Especially interesting is the use of this approach in solving problems in domains with zones of great gradients of stresses, in particular, problems of...

We propose a method of solving the two-dimensional problem of the theory of elasticity by a direct boundary-element method. We apply Galerkin's method with linear and quadratic approximations of the forces and displacements. We give the numerical results of solving a model problem that shows the effectiveness of the proposed approach.

A numerical approach to solution of dynamics problem based on a model of 3D linear theory of elasticity of anisotropic bodies is proposed. It is based on the use of semi-analytic finite element discretization for space variables and mode superposition for the integration of time coordinate. Efficiency of this approach has been checked by solving so...

The paper presents some aspects of the formulation and numerical implementation of combined mathematical model "elastic body - Timoshenko plate". The variational problem is formulated. The existence of solution of combined model is considered. The numerical investigation of the problem is performed by coupling Direct Boundary Element and Finite Ele...

A variational problem of the theory of elasticity is formulated by combining equations of the elasticity theory and the Timoshenko plate theory. The existence of solution is proved.

Investigation of the free vibration of structural elements or structures often requires the calculation of certain natural frequencies and modes using equations from the three-dimensional theory on elasticity. Other studies have focused on the numerical investigations of problems in such a formulation. Here we propose a new approach to determining...

Free vibrations of axisymmetrical bodies are considered in the framework of spatial theory of elasticity. The mentioned problem is studied by the numerical method which is based on the general Bubnov-Galyorkin's principle and semianalytical method of finite elements. The efficiency of the suggested approach was checked when solving some model probl...

A numeric-experimental method of determining the residual welding stresses in axisymmetric structures is proposed on the basis of analysis of the inverse problem of the theory of elasticity. The problem''s solution is constructed by the self-regulation method using the finite-element method. Isoparametric approximations on curvilinear tetragons of...

A method is proposed for determining thermophysical properties of thin metallic films, using solutions of converse thermal conductivity problems.

This paper presents a numerical approach to elasticity problems for compound structures. It is based on a combination of linear elasticity theory and Timoshenko shell theory for elasticity problems. Subdomains of the compound structure, which are described by different theories, are joined by special interface boundary conditions. Variational state...