Mathematical Physics - Volume II - Numerical Methods

3.5 Finite element approximations

77

3. Each transformation must be easy to perform, based on element geometry. 4. Functions x ( ξ , η ) and y ( ξ , η ) should be simple in the mathematical sense. These conditions are met by shape functions of finite elements, hence it is natural to use them as the transformation functions:

x = x i ψ i ( ξ , η ) y = y i ψ i ( ξ , η ) ,

(3.127)

where (xi,yi) are the nodal coordinates of point i and M is the number of elements. The following holds:

∂ψ j

∂ψ j

∂ ξ ∂ y

∂ ξ ∂ x ∂η ∂ x

1 | J |

1 | J | 1 | J |

y j

x j

(3.128)

= −

=

∂η ,

∂η ,

∂ψ j

∂ψ j

∂η ∂ y

1 | J |

y j

x j

(3.129)

= −

∂ ξ ,

=

∂ ξ ,

| J | = x i

∂ψ i ∂ ξ

∂ψ j ∂η −

∂ψ i ∂η

∂ψ j ∂ ξ .

y j

x i

y j

(3.130)

As an example, we will illustrate this using a four-node “master” element ˆ Ω and four finite elements, e = 1 , 2 , 3 , 4, obtained by transformation given by (3.127). Shape functions are:

1 4

1 4

ψ 1 ( ξ , η ) =

( 1 − ξ )( 1 − η ) ( 1 + ξ )( 1 + η )

ψ 2 ( ξ , η ) =

( 1 + ξ )( 1 − η ) ( 1 + ξ )( 1 + η ) .

(3.131)

1 4

1 4

ψ 3 ( ξ , η ) =

ψ 4 ( ξ , η ) =

By transforming into element Ω 1 ; fig. 3.18

( x i , y i ) = { ( 3 , 0 ) , ( 3 , 1 ) , ( 0 , 1 ) , ( 0 , 0 ) } ⇒

3 2

x = 3 ψ 1 + 3 ψ 2 =

( 1 − η )

(3.132)

1 2

y = ψ 2 + ψ 3 =

( 1 + ξ ) .

Since | J | > 0, the transformation is invertible. This actually involves simple (linear) expansion and contraction of corresponding element sides, fig. 3.18. By transforming into element number Ω 2 in the same way like before, results in | J | = − 3 4 . It should be noticed that the only difference compared to element Ω 1 is in node numeration – here it has a negative mathematical direction, which should be generally avoided.