Читать книгу An Introduction to the Finite Element Method for Differential Equations - Mohammad Asadzadeh - Страница 26

Let be a fixed spatial domain with boundary . The rate of change of thermal energy with respect to time in is equal to the net flow of energy across the boundary of plus the rate at which heat is generated within .

Let denote the temperature at the position and at time . We assume that the solid is at rest and that it is rigid, so that the only energy present is thermal energy and the density is independent of the time and temperature . Let denote the energy per unit mass. Then the amount of thermal energy in is given by

and the time rate (time derivative) of change of thermal energy in is:

Let denote the heat flux vector and denote the outward unit normal to the boundary , at the point . Then represents the flow of heat per unit cross‐sectional area per unit time crossing a surface element. Thus,

is the amount of heat per unit time flowing into across the boundary . Here, represents the element of surface area. The minus sign reflects the fact that if more heat flows out of the domain than in, the energy in decreases. Finally, in general, the heat production is determined by external sources that are independent of the temperature. In some cases, (such as an air conditioner controlled by a thermostat), it depends on temperature itself, but not on its derivatives. Hence, in the presence of a source (or sink), we denote the corresponding rate at which heat is produced per unit volume by so that the source term becomes

Now, the law of conservation of energy takes the form

(1.5.8)

Applying the Gauss divergence theorem to the integral over , we get

(1.5.9)

where denotes the divergence operator. In the sequel, we shall use the following simple result: