Mathematical Physics - Volume II - Numerical Methods

3.5 Finite element approximations

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which is not greater than zero for ∀ ξ , η ∈ ( − 1 , 1 ) - for example for 3 ξ = 5 − 4 η | J | = 0, beneath the greater-than-zero line, and above the less-than-zero line! The area of line 3 ξ = 5 − 4 η is transformed outside of Ω 4 , hence this element is obviously unacceptable. In addition, there is a problem in the node 3 corner, where it is greater than π . It can be shown that the transformation function (3.127) is integrable if the angles of the rectangle are < π . For square shape functions (e.g. the eight-node finite elements), constraints, apart from those previously mentioned, are related to nodes which are not in element corners (fig. 3.19). Namely, in order to preserve the invertibility of transformation functions, it is necessary for these nodes to be located at midpoints of sides, i.e. elements (in the case internal nodes exist).

e +1 Ω

η

e Ω

(0,1)

y

x

Ω ^

ξ

(1,0)

(0,0)

η

(1,1)

e +1 Ω

e Ω

y

ξ

(0,0)

Ω ^

x

(-1,-1)

Figure 3.19: Quadratic finite elements.

3.5.2 Calculating of finite element matrices

We can now calculate all of the necessary finite element matrices and vectors:

i j = Z Ω e i = Z Ω e i = Z Ω e

k e

∇ ψ e

e j d x d y ,

i · ∇ ψ

(3.134)

f e

f ψ e

i d x d y ,

(3.135)

s e

ˆ σψ e

i d x d y .

(3.136)