Mathematical Physics - Volume II - Numerical Methods

3.2 Finite element method – two-dimensional problem

57

ω

P 0

ω

a)

σ σ ∆σ = + y n y

∆ y

σ σ ∆σ = + n x x

P 0

σ = σ n x

σ σ = n y

∆ x

b)

Figure 3.4: Flux distribution.

With his, we defined the net flux density in a point per unit area, hence the total flux in Ω is given as: Σ = Z Ω ∇ · σ d x d y , (3.41) under the condition that Ω and σ are sufficiently smooth. Based on (3.38), (3.39) and (3.41) we have that: Z Ω ∇ · σ d x d y = Z ∂ Ω σ · n d s , (3.42) which represents the Gauss divergence theorem, which applies to any and all tensor fields. 3.2.3 Two-dimensional elliptical boundary problem In order to formulate a two-dimensional elliptical boundary problem, we will use linear constitutive equations and conservation laws. In other words, the flux in every point is proportional to the state variable gradient, i.e. of the unknown u : σ ( x , y ) = − k ( x , y ) ∇ u ( x , y ) , (3.43) where k ( x , y ) is the material modulus (coefficient or a property) for which we assume that | k ( x , y ) | > k 0 (= const. ) > 0. Conservation (balance) aw suggests that for each part of the domain, the net flux through its boundary is equal to the total flux produced by the internal sources.