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Physics-Mathematical modeling

Лекция

Экономическая теория и математическое моделирование

Using the general laws of the thermodynamics such integrals can be written for any function we are interested in. For example, the equilibrium states can be described by the minimum of the potential energy...

Английский

2014-07-10

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Physics-Mathematical modeling is one of the basic research methods of approach to various phenomenon and processes. A co-ordinate space is associated to an every natural object here and the processes are described by differential equations. Commonly these equations couple a number of physical functions Fк(xi) of the independent variable xi, defined in. As example to an element of a cooler (fig.1а), we can associate co-ordinate space (xi, х=1, 2, 3) (fig.1в). In this space we can define functions of the stress ij(xi), of the temperature T(xi) and of the velocity vi(xi). In the courses on “Mechanics of materials” “Mechanics of fluids” and “Heat and mass transfer” is presented, that differential equations, connecting these functions, can be derived from equilibrium of the infinitesimal area around point in .

а) b)

Фиг.2

If we examine mechanical equilibrium of the volume with infinitesimal edges size dx1, dx2, dx3 (fig.2), we receive:

(1.1) ,

,

,

were Хi are the components of the dead load, is the density and ui are the displacements (the equations will be discussed in the next parts of these materials).

For this volume the next boundary conditions are in force:

33=0 at x3=R - there isn’t any stress on the open surface,

33=p (p - pressure) for x3 = r - the stress on the inner side of the pipe is equal to the pressure.

(1.1а) ij=oij at x1=0 - the stress on the left boundary of the investigated area is equal to the stress on the right side of the next area or is defined by fixing.

ij=lij at x1=l - the stress on the right boundary of the investigated area is equal to the stress on the left side of the next area or is defined by fixing.

In the same way we can derive from the thermal quantity balance the next:

(1.2) ,

were Т is the temperature and а is the conduction coefficient. The boundary conditions are:

Т=Т0 at х1=0 for the temperature,

(1.2а) at х1→0 for thermal flux.

For the fluid stream the equation of the continuity is in force:

(1.3) .

In this case the boundary conditions are:

(1.3а) V1 = 0 за x3 = r; V1 = V1(o) за x1 = 0; V1= V1(l) за x1 = l, V2 = 0, V3 = 0.

So, in existence of sufficiency of boundary conditions 1а-3а, the equations 1-3 can be solved and the functions Fк(xi), which describe the object state, can be determined. But the solution of these equations (by finite differences as example), runs into big difficulties when complex geometries or complex boundary conditions exist. For solving the problem nowadays it’s increasingly used the minimum-principles (variation principles) of Hamilton and Lagrange. According to these principles some definite integrals, representative for the process under investigation, obtain the minimum value if the unknown integral function describes the process in the right way.

Using the general laws of the thermodynamics such integrals can be written for any function we are interested in. For example, the equilibrium states can be described by the minimum of the potential energy, the motion of mechanical systems - by the minimum of the kinetic potential etc. In this way all equations we are interested in can be obtained from one origin – variation principals of the continuum.

Thus the solution of the problem can be obtained by a procedure in which an approximate continuous function is assumed to represent the solution and an above integral formulation is used to create a system of algebraic equations. Finite element method assumes that the investigated area is divided into smaller pieces (elements) and the procedure above mentioned is implemented for every element separately as a first step of the solution (discretization), the results are collated in a global matrix at the second step (assembling) and at the end – a system of algebraic equations for the unknown quantities in the all area is obtained (expanding). A brief description of this three-step procedure is presented in the next part of these materials.