1.12 The Weak Form of the Energy Balance

Back

We start from the strong form expressed by Equation (1.298)

ρ0

Dε

Dt

= ˙E : S −∇0 ·Q+ ρ0h on V (1.344)

completed by appropriate boundary conditions: we assume that the temperature T and the

flux Q are known on A0T and on A0Q respectively. Furthermore, the flux normal to an

interface Ai is continuous.

T = T on A0T (1.345)

Q = Q on A0Q (1.346)

Q

+ · N

+ +Q

− · N

− =0 onA0i . (1.347)

DISPLACEMENTS, STRAIN, STRESS AND ENERGY 43

To obtain the weak form, we consider again an infinitesimal perturbation of the independent

variable field T , satisfying the “geometric” boundary conditions in Equation (1.345).

Hence,

δT =0 onA0T . (1.348)

Multiplying Equation (1.343) with δT and integrating over V , one obtains

_

V0

ρ0

Dε

Dt

δT dV = _

V0

(˙E : S −∇0 ·Q+ ρ0h)δT dV. (1.349)

The second term on the right can be written as

−_

V0

∇0 ·QδT dV = −_

V0

QK

,KδT dV

= −_

V0

            (QKδT ),K − QKδT,K dV

= −_

A0Q

δTQKNK dA + _

V0

QKδT,K dV. (1.350)

This step is essential to reduce the degree of differentiation of T in the resulting equation

and is similar to Equations (1.318) to (1.320) for the balance of momentum. Indeed, the

constitutive equations in Section 1.13 will show that Q ∼ −∇0T . Consequently, ∇0 ·Q ∼

−∇0 · ∇0T = −∇2

0T and ∇0 ·Q δT is the product of two terms of which the first one is

twice differentiated, the second one not at all. On the other hand, both terms in QKδT,K

are differentiated only once. This implies that the shape functions in the finite element

formulation can have a lesser degree of smoothness and still comply with Equation (1.350).

Substitution of Equation (1.350) into Equation (1.349) yields

−_

V0

Q· δ∇0T dV = _

V0

˙E

: SδT dV − _

A0Q

Q· NδT dA + _

V0

ρ0

_

h − Dε

Dt

_

δT dV.

(1.351)

This equation is the analogue of Equation (1.328). Similar to what was said in Section

1.11.3, the strong form can be derived from the weak form if one allows the temperature

perturbation to be absolutely general provided the “geometric” boundary conditions are

satisfied.

Навигация