Mathematical Physics - Volume II - Numerical Methods

3.5 Finite element approximations

81

!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!! Sada cˇlanovi matrice krutosti možemo da napišemo u obliku

j 1 j 2 #

e i 2 ]

e 12 e 22 "

i j = Z Ω e

ψ e ψ e

g e

k J

− g g

k e

[ ψ e

i 1 ψ

d ξ d η .

11

(3.149)

− g e

21

The basic problem with the integration of the above expression is the occurrence of a relatively complex function – the Jacobian in the denominator, and due to this, numerical squaring is typically used. In a general case, the expression for numerical integration is defined by determining of coordi nates ( ξ t , η t ) of a specific number of N t integration points within the domain for which the integral is being calculated, along with the quantity w t which is referred to as the integration weight function ( t = 1 , 2 , . . . , N t ). Thus, if expression G ( ξ , η ) needs to be integrated along the area Ω e , we will use the following relation: Z Ω e G ( ξ , η ) d ξ d η = N t ∑ t = 1 ˆ G ( ξ , η ) w t + E , where E represents the integration error. If the integrand G ( ξ , η ) is a polynomial, the integration order should be sufficiently high for the integration to be accurate. For example if we use the Gaussian square method to integrate a linear function in the canonical triangle, this integration will be exact when using the single point rule ( ξ 1 , η 1 ) , in the centre, with w 1 = 1 2 . As for expression (3.135), it is obvious that it is easier to calculate than (3.134), hence we will leave it for the practice examples. Expression (3.136) is a boundary integral and its calculation is slightly more complicated. We calculate it for contours of elements ∂ Ω e 2 h of domain Ω e , along which the natural boundary conditions are defined. We will demonstrate this calculation for the edge where ξ = const . , for the canonical element, which needs to be projected onto ∂ Ω e 2 h . In the observed case g αβ is reduced to g 22 : g 22 = ∂ z 1 ∂ ξ 2 2 + ∂ z 2 ∂ ξ 2 2 , which, at the same time, represents the “determinant” for expression g αβ . As can be seen, the Jacobian is equal to the square root of the matrix: J = " ∂ x ∂η 2 + ∂ y ∂η 2 # 1 / 2 , Hence the integral is: s e i = Z ∂ Ω e 2 h ˆ σψ e i " ∂ x ∂η 2 + ∂ y ∂η 2 # 1 / 2 d η . It is common practice to calculate these integrals using numerical integration.