Some Questions of Computational Models in Continuum Mechanics

EVELINA PROZOROVA

Mathematical-Mechanical Department

St. Petersburg State University

Av. 28 , Peterhof, 198504

RUSSIA

conditions in Hamiltonian mechanics, especially in frequently used computational solutions to

problems in mechanics. The Runge-Kutta difference scheme for a system of partial differential

equations is analyzed. Examples are given.

Keywords: angular momentum, conservation laws, no symmetrical stress tensor, open systems,

Runge-Kutta method, open system

Received: May 19, 2022. Revised: July 21, 2023. Accepted: August 23, 2023. Published: September 15, 2023.

1. Introduction

The purpose of the work is to establish the

fact that the impossibility of description using

regular theories is practically recognized lies

in the imperfection of mathematical models

and calculation algorithms. The experiment

deals with material objects, i.e. the results are

averaged data over the elementary volumes,

which can be presented in the form of an

integral form. Computational mathematics

always deals with dimensional quantities.

Differential equations are related to the field

mathematically, we are dealing with another

space, i.e., instead of discrete space (for

example, piecewise-continuous), we work in

a narrower-continuous one. These spaces

have different representations when

expanded in a Fourier series. The first can be

represented by a Fourier series only in the

region of smoothness (in difference schemes,

between nodes with stitching at the nodes), in

the second case, in the entire region. This

to the development and use of computational

programs without proper detailed analysis,

the other involves the use of the apparatus of

mathematical physics without checking the

convergence of the obtained asymptotic

formulas and their practical verification by

numbers. However, a mixed approach is

needed. It is impossible to start solving a

problem without an analytical study of the

problem, at least on the simplest reference

comparison with experiment make it possible

to guarantee the reliability of the results.

Most problems are non-linear. This does not

allow proving theorems and writing the

theory in a general form. That is why the role

obtained by previous researchers by

numerical or asymptotic methods. Errors in

solving more "rough" problems can be an

order of magnitude larger or of the same

order as the desired values The question of

building a model is very important. Closed

systems and representations of fields are

currently being investigated. The theory is

based on Hamilton's formalism and

Liouville's formula for a closed system.

zero. It was at this stage that the

phenomenological consideration of the

observed changes in the modules of

quantities for elementary volumes

established the symmetry of the tensor for

equilibrium conditions. Since

mathematicians worked with one-

dimensional and two-dimensional systems,

everyone was satisfied with the idea of the

symmetry of the tensor: at relatively low

appeared that speak about the influence of not

only the physical quantities themselves, but

also their gradients

[4,5]. In mechanics, it is customary to

consider the Lagrange function for non-

interacting and collectively interacting

particles for closed and open systems in the

same way, which is doubtful, especially for

metallic and ionic bonds. It is believed that

the reliability of classical mechanics has been

rely on the kinetic theory, which gives the

same results as continuum mechanics, do not

allow the hypotheses underlying the kinetic

the stress tensor. In all works, including the

works of N.N. Bogolyubov [3], this

hypothesis underlies the theory. The

hypothesis assumes that the values of

macroparameters can be calculated from the

zeroth approximation for the equilibrium

distribution function. Therefore, according to

the hypothesis, the values for the equilibrium

distribution function with the

macroparameters found from the Euler

the classical approach, the law of

conservation of angular momentum is

constructed, but not applied. Theoretically,

this is due to the incomplete taking of the

integral by parts when obtaining conservation

laws and ignoring the out-integral term

present in the Ostrogradsky-Gauss theorem

[6]. The symmetry of the stress tensor leads

to a violation of the “continuity” of the

medium. Mathematically, this circumstance

under the condition of the balance of forces,

we arrive at one or another classical

formulation of continuum mechanics.

The transition to differential equations is

carried out in two ways: using the main

lemma (the theory of elasticity) or taking the

integral by parts. When taking the integral by

parts, as already noted, in addition to the

well-known classical integrals of mechanics,

an external integral term (the Ostrogradsky-

mechanics it was ignored. The possibility of

using the Ostrogradsky-Gauss theorem when

constructing the stress tensor only in the

openness of the system and the exchange of

physical quantities changes the equations.

The dimension of elementary volumes

brings computational and physical models

closer to real objects.

In addition to continuum models, stochastic

methods are often used in mechanics and

physics. Historically, methods for describing

physical processes using distribution

functions (Boltzmann's equations, the

the first case, as a rule, simple basis functions

that are not smooth are used. In the

variational formulation of the problem, the

the integral formulation of the problem

obtained in the experiment, one can obtain

various differential equations. The equations

are identical and transform into each other

with a continuous distribution of quantities,

which is what we have in field theory, but

when passing to a discrete description, they

differ from each other and do not transform

into each other by any transformations,

except for special cases. Sometimes it is

can approach such a scheme [7]. For

example, in the theory of elasticity or

aeromechanics. In this case, additional

requirements for the boundary conditions

arise. It turns out a wide matrix even with the

simplest approximation. They are currently

trying to apply the Rune-Kutta method to

solve a system of partial differential

equations. An analysis of the error at the

"claimed accuracy" of a fourth-order solution

symmetric tensor.

forces is used (Fig. 1) and the equation of

projections on the axes of Cartesian

coordinates [8]



󰇜 , large, but

the zero velocities at the two boundaries of

the wall layer do not allow flow in the

opposite direction. The thickness of the

"resting" liquid at Reynolds numbers is 10-3

in velocity occurs before it vanishes, the

derivative becomes very large for air at one

Kolmogorov). It should be noted that there is

no asymptotic transition from a solution for a

semi-infinite plate to a solution for an infinite

plate. For a semi-infinite plate, friction at

obtain the Prandtl-Karman mixing length.

The classical version corresponds to the

linear profile. Hence it can be concluded that

the viscosity should work in two areas:

directly near the surface and at the outer

boundary. In the stationary case, the amount

component of the velocity and the



Equation with moment

󰇡

󰇢



representation of ln y in a row.

󰇟

󰇛󰇜

of the lower plate.

Friction must also be specified on this

boundary. It can be assumed that since the

velocity near the wall is small, there is an

excess of boundary conditions, then to

determine the constants it will be necessary

to solve a system of equations and one of the

solutions will be the value of friction without

taking into account the moment.

illustrate a method for solving a three-

dimensional problem with an no symmetric

stress tensor [9,10]. Some provisions of the

theory of elasticity are no longer valid for an

no symmetric stress tensor. For example, for

two opposite sides of an elementary volume,

we obtain our own direction of principal

stresses

 

Thus, at each point there is a main direction

of stresses. The common principal axis can

which are based on Kirchhoff's hypotheses

for plates, deformations are considered small

and obey the linear law of viscoelasticity.

Here, the influence of the no symmetric

stress tensor on the acting forces is traced. In

the work

 

oscillations under the action of a load of the

form

for a rectangular plate of finite dimensions, in

which two opposite sides are hinged, and the

other two are arbitrary. We have chosen a

significant. Consider a plate of small

coordinate system, the Oxy plane of which is

aligned with the middle plane of the plate.

The plate is deformed by transverse load

󰇟󰇛󰇜

For projections of the displacement vector,

according to the Kirchhoff hypothesis, we

have

It is assumed that the plate material obeys the

linear viscoelastic law

 



 (󰇜

󰇛󰇜







Finally we get

then the first item disappears.

  󰇛

 󰇜 (19)

We need to write next equation for other

coordinates. If there is a force, the parenthesis

is replaced by that force. An iterative

procedure is performed.

schemes

An important role, in addition to choosing a

method is an integral method. If it comes

from the initial experimental setting, then it

grasps all the components involved in the

change in value physical values in the

volume. However, now the method is being

built on the basis of existing differential

equations. In addition, the method uses the

values of functions on time layers without

passing to intermediate points in time.

Intermediate values are replaced by the

gives them an advantage over implicit ones,

but it is not to get a higher order than

traditional explicitly implicit ones.

First, consider the classic version of the

method

In the work, the implementation of the

method is as follows

The following term is equal to the coefficient

get the Taylor series.

It follows that the circuit has a second order

of accuracy, but its stability increases.

Consider the simplest version of the linear

approximation. Equation of a line passing

through two points

 

By definition of the derivative on the right

and on the left





function satisfies the Dirichlet conditions in

the interval (-π, π) if it is either continuous in

this interval or has a finite number of

discontinuities of the first kind, and if, in

addition, the interval (-π, π) can be divided

into tinite number such intervals, in each of

theory of elasticity

In the theory of elasticity, when passing to a

differential formulation of the problem, the

equations are transformed in order to simplify

them, acting on the resulting equations with

the operators rot, div, which require

differentiation of functions. For example,

when obtaining a biharmonic equation.

Namely, the main task of the theory of

elasticity (written in integral form, is the

In space

allowable displacements, and the

corresponding boundary value problem has

the form

chosen: the Couette problem and

consideration of the influence of the

asymmetric stress tensor in the problem of

Ostrogradsky-Gauss theorem for numerical

Moment in Mathematical Models for Open

Systems.

WSEAS Transactions on Applied and

Theoretical Mechanics, :

10.37394/232011.2021.16.28

methods for studying steady-state oscillations of

viscoelastic rectangular plates and round

1981.1259

Improved Accuracy of High-Order WENO Finite

Volume Methods on Cartesian Grids_

Department of Mathematics, Ruhr-University