Isoparametric elements

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Isoparametric elements

Piotr Pluciński e-mail: pplucin@L5.pk.edu.pl

Jerzy Pamin

e-mail: jpamin@L5.pk.edu.pl

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Transformation

x y

(ˆ x i , ˆ y i )

(ˆ x j , ˆ y j )

(ˆ x k , ˆ y k ) (ˆ x l , ˆ y l )

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

(3)

Transformation

x y

(ˆ x i , ˆ y i )

(ˆ x j , ˆ y j )

(ˆ x k , ˆ y k ) (ˆ x l , ˆ y l )

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

∂ξ dξ + ∂x

∂η dη dy = ∂y

∂ξ dξ + ∂y

∂η dη

=⇒

dx dy

=







∂x

∂ξ

∂x

∂η

∂y

∂ξ

∂y

∂η







J – Jacobi matrix

dξ dη

(4)

Transformation

x y

(ˆ x i , ˆ y i )

(ˆ x j , ˆ y j )

(ˆ x k , ˆ y k ) (ˆ x l , ˆ y l )

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)ˆ x y(ξ, η) = N(ξ, η)ˆ y ˆ

x = {ˆ x _i x ˆ _j x ˆ _k x ˆ _l } ˆ

y = {ˆ y i y ˆ j y ˆ k y ˆ l }

Shape functions

N(ξ, η) = [N i N j N k N l ]

N i (ξ, η) = ¹ ₄ (1 − ξ)(1 − η)

N j (ξ, η) = ¹ ₄ (1 + ξ)(1 − η)

N k (ξ, η) = ¹ ₄ (1 + ξ)(1 + η)

N l (ξ, η) = ¹ ₄ (1 − ξ)(1 + η)

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Differentiation of multivariable function

Partial derivatives of function f (ξ, η)

∂f

∂ξ = ∂f

∂x

∂ξ + ∂f

∂y

∂ξ

∂f

∂η = ∂f

∂x

∂η + ∂f

∂y

∂η

=⇒







∂f

∂ξ

∂f

∂η







=







∂x

∂ξ

∂y

∂ξ

∂x

∂η

∂y

∂η







J ^T







∂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

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Differentiation of multivariable function

Partial derivatives of function f (ξ, η)

∂f

∂ξ = ∂f

∂x

∂ξ + ∂f

∂y

∂ξ

∂f

∂η = ∂f

∂x

∂η + ∂f

∂y

∂η

=⇒







∂f

∂ξ

∂f

∂η







=







∂x

∂ξ

∂y

∂ξ

∂x

∂η

∂y

∂η







J ^T







∂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

(7)

Differentiation of multivariable function

Partial derivatives of function f (ξ, η)

∂f

∂ξ = ∂f

∂x

∂ξ + ∂f

∂y

∂ξ

∂f

∂η = ∂f

∂x

∂η + ∂f

∂y

∂η

=⇒







∂f

∂ξ

∂f

∂η







=







∂x

∂ξ

∂y

∂ξ

∂x

∂η

∂y

∂η







J ^T







∂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

(8)

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

2 ξ + η + 1 2 ξη 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

4 ξ + 5 4 η + 1

4 ξη

(9)

Example - determination of conductivity matrix

Jacobian

J =







∂x

∂ξ

∂x

∂η

∂y

∂ξ

∂y

∂η







=





3 2 + ¹ ₂ η 1 + ¹ ₂ ξ

− ⁵ ₄ + ¹ ₄ η ⁵ ₄ + ¹ ₄ ξ





det J = 25

8 + ξ + 3 8 η

Conductivity matrix k = 100, h = 0.1

k = Z

A

B ^T khBdxdy, 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 2 ξ +η + 1

2 ξη

y(ξ, η) = 3 4

− 5 4

ξ + 5 4

η + 1 4 ξη

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

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Example - determination of conductivity matrix

Jacobian

J =







∂x

∂ξ

∂x

∂η

∂y

∂ξ

∂y

∂η







=





3 2 + ¹ ₂ η 1 + ¹ ₂ ξ

− ⁵ ₄ + ¹ ₄ η ⁵ ₄ + ¹ ₄ ξ





det J = 25

8 + ξ + 3 8 η

Conductivity matrix k = 100, h = 0.1

k = Z

A

B ^T khBdxdy, 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 2 ξ +η + 1

2 ξη

y(ξ, η) = 3 4

− 5 4

ξ + 5 4

η + 1 4 ξη

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

(11)

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

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

(12)

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

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

(13)

Example - determination of conductivity matrix

Conductivity matrix k = 100, h = 0.1

k = Z

A

B(x, y) ^T khB(x, y)dxdy

= Z 1

-1

Z 1 -1

B(x(ξ, η), y(ξ, η)) ^T khB(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

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

Isoparametric elements