Mathematical Physics - Volume II - Numerical Methods

3.2 Finite element method – two-dimensional problem

59

k 2

Γ

n

σ (+)

k 1

σ (-)

Figure 3.6: Flux change at the interface.

Let us consider point P i on Γ , Fig. 3.5, i.e. a material strip which contains this point, Fig 3.6. Let us assume that this strip is narrow enough for the flux through its ends, as well as the source (proportional to the surface) can be neglected relative to the net flux through its edges. When the thickness of this layer tends to zero, the conservation law is as follows:

s 2 Z s 1 ( − σ ( − ) · n + σ (+) · n ) d s = 0 ,

Σ =

(3.47)

where s 1 and s 2 are the ends of the strip. Since the integration area is arbitrary, local conservation law at points lying on boundary Γ is reduced to a jump σ · n = σ n with respect to Γ : [ | σ n ( s ) | ] = σ (+) n ( s ) − σ ( − ) n ( s ) = 0 , s ∈ Γ . (3.48)

n

P b

σ ( ) s ^

u s ( ) ^

σ ( ) s

u s ( )

δΩ 2

Figure 3.7: Boundary conditions.

Now let us consider boundary conditions at ∂ Ω 2 , i.e. the area which contains a typical boundary point P b , Fig. 3.7. We will assume that the normal flux ˆ σ ( s ) through the surrounding material in the immediate vicinity of the boundary is proportional to the difference in value of u ( s ) at the boundary and the defined value of ˆ u ( s ) in the external environment: ˆ σ ( s ) ≡ p ( s )[ u ( s ) − ˆ u ( s )] .