Mathematical Physics - Volume II - Numerical Methods

Chapter 3. Comparison of finite element method and finite difference method

76

e Ω

y

T e

x

η

(1,1)

(-1,1)

ξ

Ω ^

(1,-1)

(-1,-1)

Figure 3.17: Local to global transformation.

Partial derivative matrix of the 2 × 2 order is called the transformation Jacobian and is denoted by J . In order for the inverse transformation to hold, it is obvious that | J |̸ = 0. In this case, we have that: d ξ d η = J − 1 d x d y = 1 | J |    ∂ y ∂η ∂ x ∂η − ∂ y ∂ ξ ∂ x ∂ ξ    d x d y , (3.123)

thus

ξ = ξ ( x , y ) η = η ( x , y )

(3.124)

Defines the inverse transformation. Considering the below expression, analogous to (3.122)

d η =    ∂ ξ ∂ x ∂η ∂ x

  

∂ ξ ∂ y ∂η ∂ y

d ξ

d x d y

(3.125)

it is clear that the following holds: ∂ ξ ∂ x = 1 | J | ∂ y ∂η ,

∂ ξ ∂ y

∂ x ∂η , ∂ x ∂ ξ .

1 | J | 1 | J |

= −

(3.126)

∂η ∂ x

∂ y ∂ ξ ,

∂η ∂ y

1 | J |

= −

=

Now we can define the conditions necessary for a function in order to be used as a transformation function: 1. Within each element, functions ξ = ξ ( x , y ) and η = η ( x , y ) must be invertible and continu ously differentiable in order for the relation (3.126) to be applicable to them. 2. Mesh generated using a series of transformations [ T e ] must not contain overlapping elements and empty spaces between them.