Isoparametric elements
Piotr Pluciński
e-mail: Piotr.Plucinski@pk.edu.pl
Jerzy Pamin
e-mail: Jerzy.Pamin@pk.edu.pl
Chair for Computational Engineering
Faculty of Civil Engineering, Cracow University of Technology URL: www.CCE.pk.edu.pl
Isoparametric transformation
x y
(xi, yi)
(xj, yj)
(xk, yk) (xl, yl)
i k
j l
e ξ
η
i
k
j l
(-1, -1)
(1, 1)
(1, -1) (-1, 1)
e transformation
actual element ’parent’ element
Isoparametric transformation
x y
(xi, yi)
(xj, yj)
(xk, yk) (xl, yl)
i k
j l
e ξ
η
i
k
j l
(-1, -1)
(1, 1)
(1, -1) (-1, 1)
e transformation
actual element ’parent’ element
Element stiffness matrix and load vector ke=
Z
Ae
BeTDeBehedxdy
pe= Z
Ae
NeTfehedxdy
Ae, he – area and thickness of FE
Isoparametric transformation
x y
uk
vk
i k
j l
e ξ
η
i
k
j l
(-1, -1)
(1, 1)
(1, -1) (-1, 1)
e transformation
actual element ’parent’ element
Interpolation of displacement vector components
u(ξ, η) = N(ξ, η)un v(ξ, η) = N(ξ, η)vn
un = {uiuj uk ul} vn= {vi vj vk vl}
Shape functions
N(ξ, η) = [NiNj NkNl] Ni(ξ, η) =14(1 − ξ)(1 − η) Nj(ξ, η) =14(1 + ξ)(1 − η) Nk(ξ, η) =14(1 + ξ)(1 + η) Nl(ξ, η) =14(1 − ξ)(1 + η)
Isoparametric transformation
x y
(xi, yi)
(xj, yj)
(xk, yk) (xl, yl)
i k
j l
e ξ
η
i
k
j l
(-1, -1)
(1, 1)
(1, -1) (-1, 1)
e transformation
actual element ’parent’ element
Interpolation of actual (model) coordinates
x(ξ, η) = N(ξ, η)xn y(ξ, η) = N(ξ, η)yn
xn = {xi xj xk xl} yn= {yiyj yk yl}
Shape functions
N(ξ, η) = [NiNj NkNl] Ni(ξ, η) =14(1 − ξ)(1 − η) Nj(ξ, η) =14(1 + ξ)(1 − η) Nk(ξ, η) =14(1 + ξ)(1 + η) Nl(ξ, η) =14(1 − ξ)(1 + η)
Isoparametric transformation
x y
(xi, yi)
(xj, yj)
(xk, yk) (xl, yl)
i k
j l
e ξ
η
i
k
j l
(-1, -1)
(1, 1)
(1, -1) (-1, 1)
e transformation
actual element ’parent’ element
Transformation (chain rule) dx = ∂x
∂ξdξ +∂x
∂ηdη dy = ∂y
∂ξdξ + ∂y
∂ηdη
=⇒ dx dy
=
∂x
∂ξ
∂x
∂η
∂y
∂ξ
∂y
∂η
J – Jacobi matrix
dξ dη
Differentiation of multivariable function
Partial derivatives of function f (ξ, η)
∂f
∂ξ =∂f
∂x
∂x
∂ξ +∂f
∂y
∂y
∂ξ
∂f
∂η =∂f
∂x
∂x
∂η +∂f
∂y
∂y
∂η
=⇒
∂f
∂ξ
∂f
∂η
=
∂x
∂ξ
∂y
∂ξ
∂x
∂η
∂y
∂η
JT
∂f
∂x
∂f
∂y
∂f
∂x
∂f
∂y
= J−T
∂f
∂ξ
∂f
∂η
Integration
Z
A
f (x, y)dxdy = Z 1
-1
Z 1 -1
f (x(ξ, η), y(ξ, η)) det Jdξdη
Differentiation of multivariable function
Partial derivatives of function f (ξ, η)
∂f
∂ξ =∂f
∂x
∂x
∂ξ +∂f
∂y
∂y
∂ξ
∂f
∂η =∂f
∂x
∂x
∂η +∂f
∂y
∂y
∂η
=⇒
∂f
∂ξ
∂f
∂η
=
∂x
∂ξ
∂y
∂ξ
∂x
∂η
∂y
∂η
JT
∂f
∂x
∂f
∂y
∂f
∂x
∂f
∂y
= J−T
∂f
∂ξ
∂f
∂η
Integration
Z
A
f (x, y)dxdy = Z 1
-1
Z 1 -1
f (x(ξ, η), y(ξ, η)) det Jdξdη
Differentiation of multivariable function
Partial derivatives of function f (ξ, η)
∂f
∂ξ =∂f
∂x
∂x
∂ξ +∂f
∂y
∂y
∂ξ
∂f
∂η =∂f
∂x
∂x
∂η +∂f
∂y
∂y
∂η
=⇒
∂f
∂ξ
∂f
∂η
=
∂x
∂ξ
∂y
∂ξ
∂x
∂η
∂y
∂η
JT
∂f
∂x
∂f
∂y
∂f
∂x
∂f
∂y
= J−T
∂f
∂ξ
∂f
∂η
Integration
Z
A
f (x, y)dxdy = Z 1
-1
Z 1 -1
f (x(ξ, η), y(ξ, η)) det Jdξdη
Example - determination of conductivity matrix
x y
(-2, 1)
(0, -2)
(3, 1) (-1, 3)
i k
j l
e ξ
η
i
k
j l
(-1, -1)
(1, 1)
(1, -1) (-1, 1)
e transformation
actual element ’parent’ element
Interpolation of actual (model) coordinates
x(ξ, η) = −2·1
4(1−ξ)(1−η) + 0·1
4(1+ξ)(1−η) + 3·1
4(1+ξ)(1+η) − 1·1
4(1−ξ)(1+η)
=3
2ξ + η +1 2ξη y(ξ, η) = 1·1
4(1−ξ)(1−η) − 2·1
4(1+ξ)(1−η) + 1·1
4(1+ξ)(1+η) + 3·1
4(1−ξ)(1+η)
=3 4−5
4ξ +5 4η +1
4ξη
Example - determination of conductivity matrix
Jacobian
J =
∂x
∂ξ
∂x
∂η
∂y
∂ξ
∂y
∂η
=
" 3
2+12η 1 +12ξ
−54 +14η 54+14ξ
#
det J = 25
8 + ξ + 3 8η
Conductivity matrix k = 100, h = 0.1
k = Z
A
BTkhBdxdy, B=
∂N
∂x
∂N
∂y
= J−T
∂N
∂ξ
∂N
∂η
x y
(-2, 1)
(0, -2)
(3, 1) (-1, 3)
i k
j l
e
ξ η
i k
j l
(-1, -1)
(1, 1)
(1, -1) (-1, 1)
e
x(ξ, η) =3 2ξ +η +1
2ξη
y(ξ, η) =3 4
−5 4
ξ +5 4
η +1 4 ξη
Matrix of shape function derivatives
B = 1
25 + 8ξ + 3η
−5 + 2ξ + 3η −2ξ − 2η 5 + 3ξ + 2η −3ξ − 3η
−1 + 4ξ − 3η −5 − 4ξ + η 1 + 2ξ − η 5 − 2ξ + 3η
Example - determination of conductivity matrix
Jacobian
J =
∂x
∂ξ
∂x
∂η
∂y
∂ξ
∂y
∂η
=
" 3
2+12η 1 +12ξ
−54 +14η 54+14ξ
#
det J = 25
8 + ξ + 3 8η
Conductivity matrix k = 100, h = 0.1
k = Z
A
BTkhBdxdy, B=
∂N
∂x
∂N
∂y
= J−T
∂N
∂ξ
∂N
∂η
x y
(-2, 1)
(0, -2)
(3, 1) (-1, 3)
i k
j l
e
ξ η
i k
j l
(-1, -1)
(1, 1)
(1, -1) (-1, 1)
e
x(ξ, η) =3 2ξ +η +1
2ξη
y(ξ, η) =3 4
−5 4
ξ +5 4
η +1 4 ξη
Matrix of shape function derivatives
B = 1
25 + 8ξ + 3η
−5 + 2ξ + 3η −2ξ − 2η 5 + 3ξ + 2η −3ξ − 3η
−1 + 4ξ − 3η −5 − 4ξ + η 1 + 2ξ − η 5 − 2ξ + 3η
Example - determination of conductivity matrix
Shape function derivatives
J-1= 1 25 + 8ξ + 3η
10 + 2ξ −8 − 4ξ 10 − 2η 12 + 4η
∂N
∂ξ
∂N
∂η
= 1 4
−1 + η 1 − η 1 + η −1 − η
−1 + ξ −1 − ξ 1 + ξ 1 − ξ
x y
(-2, 1)
(0, -2)
(3, 1) (-1, 3)
i k
j l
e
ξ η
i k
j l
(-1, -1)
(1, 1)
(1, -1) (-1, 1)
e
x(ξ, η) =3 2
ξ +η +1 2
ξη
y(ξ, η) =3 4
−5 4
ξ +5 4
η +1 4 ξη
Matrix of shape function derivatives
B = 1
25 + 8ξ + 3η
−5 + 2ξ + 3η −2ξ − 2η 5 + 3ξ + 2η −3ξ − 3η
−1 + 4ξ − 3η −5 − 4ξ + η 1 + 2ξ − η 5 − 2ξ + 3η
Example - determination of conductivity matrix
Shape function derivatives
J-1= 1 25 + 8ξ + 3η
10 + 2ξ −8 − 4ξ 10 − 2η 12 + 4η
∂N
∂ξ
∂N
∂η
=1 4
−1 + η 1 − η 1 + η −1 − η
−1 + ξ −1 − ξ 1 + ξ 1 − ξ
x y
(-2, 1)
(0, -2)
(3, 1) (-1, 3)
i k
j l
e
ξ η
i k
j l
(-1, -1)
(1, 1)
(1, -1) (-1, 1)
e
Matrix of shape function derivatives
B = 1
25 + 8ξ + 3η
−5 + 2ξ + 3η −2ξ − 2η 5 + 3ξ + 2η −3ξ − 3η
−1 + 4ξ − 3η −5 − 4ξ + η 1 + 2ξ − η 5 − 2ξ + 3η
Example - determination of conductivity matrix
Conductivity matrix k = 100, h = 0.1
k = Z
A
B(x, y)TkhB(x, y)dxdy
= Z 1
-1
Z 1 -1
B(x(ξ, η), y(ξ, η))TkhB(x(ξ, η), y(ξ, η)) det Jdξdη
k =
8.9488 −1.0827 −3.7421 −4.1240
−1.0827 6.1570 −1.8099 −3.2644
−3.7421 −1.8099 5.7670 −0.2149
−4.1240 −3.2644 −0.2149 7.6034
x y
(-2, 1)
(0, -2)
(3, 1) (-1, 3)
i k
j l
e
ξ η
i k
j l
(-1, -1)
(1, 1)
(1, -1) (-1, 1)
e
Matrix of shape function derivatives
B = 1
25 + 8ξ + 3η
−5 + 2ξ + 3η −2ξ − 2η 5 + 3ξ + 2η −3ξ − 3η
−1 + 4ξ − 3η −5 − 4ξ + η 1 + 2ξ − η 5 − 2ξ + 3η
