arXiv subjects

Petr N. Vabishchevich

Publications and source records attributed to Petr N. Vabishchevich.

53 records · Page 3Linked to original sources

arXiv2015

Numerical simulation of the stress-strain state of the dental system

We present mathematical models, computational algorithms and software, which can be used for prediction of results of prosthetic treatment. More interest issue is biomechanics of the periodontal complex because any prosthesis is accompanied by a risk of overloading the supporting elements. Such risk can be avoided by the proper load distribution and prediction of stresses that occur during the use of dentures. We developed the mathematical model of the periodontal complex and its software implementation. This model is based on linear elasticity theory and allows to calculate the stress and strain fields in periodontal ligament and jawbone. The input parameters for the developed model can be divided into two groups. The first group of parameters describes the mechanical properties of periodontal ligament, teeth and jawbone (for example, elasticity of periodontal ligament etc.). The second group characterized the geometric properties of objects: the size of the teeth, their spatial coordinates, the size of periodontal ligament etc. The mechanical properties are the same for almost all, but the input of geometrical data is complicated because of their individual characteristics. In this connection, we develop algorithms and software for processing of images obtained by computed tomography (CT) scanner and for constructing individual digital model of the tooth-periodontal ligament-jawbone system of the patient. Integration of models and algorithms described allows to carry out biomechanical analysis on three-dimensional digital model and to select prosthesis design.

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arXiv2015

A splitting scheme to solve an equation for fractional powers of elliptic operators

An equation containing a fractional power of an elliptic operator of second order is studied for Dirichlet boundary conditions. Finite difference approximations in space are employed. The proposed numerical algorithm is based on solving an auxiliary Cauchy problem for a pseudo-parabolic equation. Unconditionally stable vector additive schemes (splitting schemes) are constructed. Numerical results for a model problem in a rectangle calculated using the splitting with respect to spatial variables are presented.

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arXiv2015

Numerical investigation of a space-fractional model of turbulent fluid flow in rectangular ducts

The models that are based of fractional derivatives should be highlighted among promising new models to describe turbulent fluid flows. In the present work, a steady-state flow in a duct is considered under the condition that the turbulent diffusion is governed by a fractional power of the Laplace operator. To study numerically flows in rectangular channels, finite-difference approximations are employed. For approximate solving the corresponding boundary value problem, the iterative method of conjugate gradients is used. At each iteration, the problem with a fractional power of the grid Laplace operator is solved. Predictions of turbulent flows in ducts at different Reynolds numbers are presented via mean velocity fields.

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arXiv2014

Numerical solution of nonstationary problems for a space-fractional diffusion equation

An unsteady problem is considered for a space-fractional diffusion equation in a bounded domain. A first-order evolutionary equation containing a fractional power of an elliptic operator of second order is studied for general boundary conditions of Robin type. Finite element approximation in space is employed. To construct approximation in time, regularized two-level schemes are used. The numerical implementation is based on solving the equation with the fractional power of the elliptic operator using an auxiliary Cauchy problem for a pseudo-parabolic equation. The results of numerical experiments are presented for a model two-dimensional problem.

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arXiv2014

Numerical solving the boundary value problem for fractional powers of elliptic operators

A boundary value problem for a fractional power of the second-order elliptic operator is considered. It is solved numerically using a time-dependent problem for a pseudo-parabolic equation. For the auxiliary Cauchy problem, the standard two-level schemes with weights are applied. Stability conditions are obtained for the fully discrete schemes under the consideration. The numerical results are presented for a model two-dimensional boundary value problem wit a fractional power of an elliptic operator. The dependence of accuracy on grids in time and in space is studied.

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arXiv2013

A priori estimation of a time step for numerically solving parabolic problems

This work deals with the problem of choosing a time step for the numerical solution of boundary value problems for parabolic equations. The problem solution is derived using the fully implicit scheme, whereas a time step is selected via explicit calculations. The selection strategy consists of the following steps. First, using the explicit scheme, we calculate the solution at a new time level. Next, we employ this solution in order to obtain the solution at the previous time level (the implicit scheme, explicit calculations). This solution should be close to the solution of our problem at this time level with a prescribed accuracy. Such an algorithm leads to explicit formulas for the calculation of the time step and takes into account both the dynamics of the problem solution and changes in coefficients of the equation and in its right-hand side. The same formulas for the evaluation of the time step we get using a comparison of two approximate solutions, which are obtained using the explicit scheme with the primary time step and the step that is reduced by half. Numerical results are presented for a model parabolic boundary value problem, which demonstrate the robustness of the developed algorithm for the time step selection.

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arXiv2013

Explicit schemes for parabolic and hyperbolic equations

Standard explicit schemes for parabolic equations are not very convenient for computing practice due to the fact that they have strong restrictions on a time step. More promising explicit schemes are associated with explicit-implicit splitting of the problem operator (Saul'yev asymmetric schemes, explicit alternating direction (ADE) schemes, group explicit method). These schemes belong to the class of unconditionally stable schemes, but they demonstrate bad approximation properties. These explicit schemes are treated as schemes of the alternating triangle method and can be considered as factorized schemes where the problem operator is splitted into the sum of two operators that are adjoint to each other. Here we propose a multilevel modification of the alternating triangle method, which demonstrates better properties in terms of accuracy. We also consider explicit schemes of the alternating triangle method for the numerical solution of boundary value problems for hyperbolic equations of second order. The study is based on the general theory of stability (well-posedness) for operator-difference schemes.

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arXiv2013

Mathematical modeling of thermal stabilization of vertical wells on high performance computing systems

Temperature stabilization of oil and gas wells is used to ensure stability and prevent deformation of a subgrade estuary zone. In this work, we consider the numerical simulation of thermal stabilization using vertical seasonal freezing columns. A mathematical model of such problems is described by a time-dependent temperature equation with phase transitions from water to ice. The resulting equation is a standard nonlinear parabolic equation. Numerical implementation is based on the finite element method using the package Fenics. After standard purely implicit approximation in time and simple linearization, we obtain a system of linear algebraic equations. Because the size of freezing columns are substantially less than the size of the modeled area, we obtain mesh refinement near columns. Due to this, we get a large system of equations which are solved using high performance computing systems.

cs.CE↗

arXiv2012

Flux-splitting schemes for parabolic problems

To solve numerically boundary value problems for parabolic equations with mixed derivatives, the construction of difference schemes with prescribed quality faces essential difficulties. In parabolic problems, some possibilities are associated with the transition to a new formulation of the problem, where the fluxes (derivatives with respect to a spatial direction) are treated as unknown quantities. In this case, the original problem is rewritten in the form of a boundary value problem for the system of equations in the fluxes. This work deals with studying schemes with weights for parabolic equations written in the flux coordinates. Unconditionally stable flux locally one-dimensional schemes of the first and second order of approximation in time are constructed for parabolic equations without mixed derivatives. A peculiarity of the system of equations written in flux variables for equations with mixed derivatives is that there do exist coupled terms with time derivatives.

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arXiv2011

Additive schemes (splitting schemes) for some systems of evolutionary equations

On the basis of additive schemes (splitting schemes) we construct efficient numerical algorithms to solve approximately the initial-boundary value problems for systems of time-dependent partial differential equations (PDEs). In many applied problems the individual components of the vector of unknowns are coupled together and then splitting schemes are applied in order to get a simple problem for evaluating components at a new time level. Typically, the additive operator-difference schemes for systems of evolutionary equations are constructed for operators coupled in space. In this paper we investigate more general problems where coupling of derivatives in time for components of the solution vector takes place. Splitting schemes are developed using an additive representation for both the primary operator of the problem and the operator at the time derivative. Splitting schemes are based on a triangular two-component representation of the operators.

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arXiv2011

Substructuring domain decomposition scheme for unsteady problems

Domain decomposition methods are used for approximate solving boundary problems for partial differential equations on parallel computing systems. Specific features of unsteady problems are taken into account in the most complete way in iteration-free schemes of domain decomposition. Regionally-additive schemes are based on different classes of splitting schemes. In this paper we highlight a class of domain decomposition schemes which is based on the partition of the initial domain into subdomains with common boundary nodes. Using the partition of unit we have constructed and studied unconditionally stable schemes of domain decomposition based on two-component splitting: the problem within subdomain and the problem at their boundaries. As an example there is considered the Cauchy problem for evolutionary equations of first and second order with non-negative self-adjoint operator in a finite Hilbert space. The theoretical consideration is supplemented with numerical solving a model problem for the two-dimensional parabolic equation.

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arXiv2011

Domain decomposition schemes for the Stokes equation

Numerical algorithms for solving problems of mathematical physics on modern parallel computers employ various domain decomposition techniques. Domain decomposition schemes are developed here to solve numerically initial/boundary value problems for the Stokes system of equations in the primitive variables pressure-velocity. Unconditionally stable schemes of domain decomposition are based on the partition of unit for a computational domain and the corresponding Hilbert spaces of grid functions.

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arXiv2011

Domain decomposition schemes for evolutionary equations of first order with not self-adjoint operators

Domain decomposition methods are essential in solving applied problems on parallel computer systems. For boundary value problems for evolutionary equations the implicit schemes are in common use to solve problems at a new time level employing iterative methods of domain decomposition. An alternative approach is based on constructing iteration-free methods based on special schemes of splitting into subdomains. Such regionally-additive schemes are constructed using the general theory of additive operator-difference schemes. There are employed the analogues of classical schemes of alternating direction method, locally one-dimensional schemes, factorization methods, vector and regularized additive schemes. The main results were obtained here for time-dependent problems with self-adjoint elliptic operators of second order. The paper discusses the Cauchy problem for the first order evolutionary equations with a nonnegative not self-adjoint operator in a finite-dimensional Hilbert space. Based on the partition of unit, we have constructed the operators of decomposition which preserve nonnegativity for the individual operator terms of splitting. Unconditionally stable additive schemes of domain decomposition were constructed using the regularization principle for operator-difference schemes. Vector additive schemes were considered, too. The results of our work are illustrated by a model problem for the two-dimensional parabolic equation.

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arXiv2010

Splitting schemes for hyperbolic heat conduction equation

Rapid processes of heat transfer are not described by the standard heat conduction equation. To take into account a finite velocity of heat transfer, we use the hyperbolic model of heat conduction, which is connected with the relaxation of heat fluxes. In this case, the mathematical model is based on a hyperbolic equation of second order or a system of equations for the temperature and heat fluxes. In this paper we construct for the hyperbolic heat conduction equation the additive schemes of splitting with respect to directions. Unconditional stability of locally one-dimensional splitting schemes is established. New splitting schemes are proposed and studied for a system of equations written in terms of the temperature and heat fluxes.

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arXiv2010

SM stability for time-dependent problems

Various classes of stable finite difference schemes can be constructed to obtain a numerical solution. It is important to select among all stable schemes such a scheme that is optimal in terms of certain additional criteria. In this study, we use a simple boundary value problem for a one-dimensional parabolic equation to discuss the selection of an approximation with respect to time. We consider the pure diffusion equation, the pure convective transport equation and combined convection-diffusion phenomena. Requirements for the unconditionally stable finite difference schemes are formulated that are related to retaining the main features of the differential problem. The concept of SM stable finite difference scheme is introduced. The starting point are difference schemes constructed on the basis of the various Pad$\acute{e}$ approximations.

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arXiv2010

On a new class of additive (splitting) operator-difference schemes

Many applied time-dependent problems are characterized by an additive representation of the problem operator. Additive schemes are constructed using such a splitting and associated with the transition to a new time level on the basis of the solution of more simple problems for the individual operators in the additive decomposition. We consider a new class of additive schemes for problems with additive representation of the operator at the time derivative. In this paper we construct and study the vector operator-difference schemes, which are characterized by a transition from one initial the evolution equation to a system of such equations.

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arXiv2010

VAGO method for the solution of elliptic second-order boundary value problems

Mathematical physics problems are often formulated using differential oprators of vector analysis - invariant operators of first order, namely, divergence, gradient and rotor operators. In approximate solution of such problems it is natural to employ similar operator formulations for grid problems, too. The VAGO (Vector Analysis Grid Operators) method is based on such a methodology. In this paper the vector analysis difference operators are constructed using the Delaunay triangulation and the Voronoi diagrams. Further the VAGO method is used to solve approximately boundary value problems for the general elliptic equation of second order. In the convection-diffusion-reaction equation the diffusion coefficient is a symmetric tensor of second order.

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