Mathematical Physics - Volume II - Numerical Methods

3.5 Finite element approximations

75

x x = ( 1) ξ, y y = ( 1) ξ,

η

ξ

x x = (1 ) , η y y = (1 ) , η

y

e Ω

x

x x = ( ) ξ, η y y = ( ) ξ, η

ξ ξ , = ( ) x y η η , = ( ) x y

-1 T : e

T : e

η

η =1

(1,1)

ξ

Ω ^

ξ =1

(-1,-1)

Figure 3.16: Finite element mapping.

Local to global transformations

Let us introduce the “master” element Ω , of simple form, whose coordinates are defined as − 1 ≤ ξ ≤ 1 and − 1 ≤ η ≤ 1. In this case, coordinate transformation can be written as:

x = x ( ξ , η ) y = y ( ξ , η ) .

(3.120)

The basic idea behind introducing the “master” element is the ability to interpret is a finite element mesh generation achieved by a series of transformations, [ T e ] , 3.120), shown in fig. 3.17. To this end, we will assume that functions x and y are continuously differentiable with respect to ξ and η , hence we have that:

∂ x ∂ ξ d

∂ x ∂η d

∂ y ∂ ξ d

∂ y ∂η d

ξ +

η d y =

ξ +

η

d x =

(3.121)

i.e. in matrix form

d y =    ∂ x ∂ ξ ∂ y ∂ ξ

   d ξ

∂ x ∂η ∂ y ∂η

d x

d η

(3.122)