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210 Bernard Maurin and René ´ Motro
its determinant D = Ni,ξ N Ni,η N (xiyj − xjyi) and its first derivatives:
D,ξ = (Ni,ξξ N Nj,η N + Ni,ξ N Nj,ξη N )(xiyj − xjyi)
D,η = (Ni,ηξ N Nj,η N + Ni,ξ N Nj,ηη N )(xiyj − xjyi)
−1 (D ),ξ = D −1
,ξ = −D−2 D,ξ
(D−1 ),η = D−1 ,η = −D−2 D,η
If we calculate then the components of the inverse jacobian matrix:
(11)
a = ξ,x = D −1 Ni,η N yi ; b = η,x = −D −1 Ni,ξ N yi and
c = ξ,y = −D −1 Ni,η N xi ; d = η,y = D −1 Ni,ξ N xi (12)
The partial first derivatives are defined according to:
Ni,x N = aNi,ξ N + bNi,η N and Ni,y N = cNi,ξ N + dNi,η N (13)
And the partial second derivatives are:
⎧
⎪ Ni,xx ⎪ N = a
⎪
⎨⎪ ⎨
⎪
⎩⎪
2Ni,ξξ N + b2Ni,ηη N +2abNi,ξη N +(aa,ξ + ba,η)Ni,ξ N
+(ab,ξ + bb,η)Ni,η N
Ni,xy N = acNi,ξξ N + bdNi,ηη N + (cb + ad)Ni,ξη N +(ca,ξ + da,η)Ni,ξ N
+(cb,ξ + db,η)Ni,η N
Ni,yy N = c2Ni,ξξ N + d2Ni,ηη N +2cdNi,ξη N +(cc,ξ + dc,η)Ni,ξ N
+(cd,ξ + dd,η)Ni,η N
With the following coefficients:
⎧
⎪
⎨⎪ ⎨
⎪
⎩⎪
⎧
⎪
⎨⎪ ⎨
⎪
⎩⎪
a,ξ = −D −1
,ξ Ni,η
a,η = −D−1 ,η Ni,η
b,ξ = −D −1
η
,ξ Ni,ξ
b,η = −D−1 ,η Ni,ξ
c,ξ = −D −1
,ξ Ni,η
c,η = −D−1 ,η Ni,η
d,ξ = −D −1
,ξ Ni,ξ
d,η = −D−1 ,η Ni,ξ
N yi + D−1Ni,ξη N yi + D−1Ni,ηη N yi − D−1Ni,ξξ N yi − D−1Ni,ξη N yi
N yi
N yi
N yi
N xi − D−1Ni,ξη N xi − D−1Ni,ηη N xi + D−1Ni,ξξ N xi + D−1Ni,ξη N xi
N xi
N xi
N xi
(14)
(15)
Since derivatives N,ξ N ...N,ξη N are dependent on ξ and η values, the calculation
could be achieved at any chosen point within the element, for instance a point
locatedinthemiddleof two nodes.
6.2 Applications
Test
The first application allows the verification of formulations. It deals with an
hyperbolic paraboloid (HP) defined by z = kxy. The gaussian and mean