FEM – Isoparametric Concept

Universität Stuttgart

Institut für Wasserbau, Lehrstuhl für Hydromechanik und Hydrosystemmodellierung

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Table of contents 1. Interpolation Functions for the Finite Elements 2. Finite Element Types 3. Geometry 4. Lagrange and Hermite Element Family 5. Interpolation Approach Function 6. Cartesian - Natural Coordinates 7. Isoparametric Concept 8. Example to the Isoparametric Concept Universität Stuttgart

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Shape Functions for FEM For the finite elements method, the following is valid: The global function of a sought function consists of a sum of local functions: E Z X e=1

V

ωie dVe

Galerkin method: the interpolation function corresponds to weighted function Ritz method: the global variation principle is constructed from the sum of the local variation principles. Universität Stuttgart

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Shape functions for FEM Critical step in the FEM: Selection of suitable interpolation functions,that are formed through the shape of the finite elements, the approximation order. Choice of Finite elements depends on the geometry of the global area, the desired exactness of the area, the simple integration through the area. Universität Stuttgart

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Finite Element Types In order to be able to formulate special physical problems, several elements are often necessary. They are distinguished by the geometry (1-D, 2-D or 3-D), the selection of the interpolation function (polynomials; Lagrange or Hermite polynomials), the selection of the element coordinates (Cartesian or natural coordinates), the selection of the variables specified at the nodes (Lagrange group or Hermite group of variables).

Universität Stuttgart

Institut für Wasserbau, Lehrstuhl für Hydromechanik und Hydrosystemmodellierung

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Geometry

a) Square elements with straight sides

b) Square elements with curved sidesx

c) Cubic elements Universität Stuttgart

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Interpolation Approach Function Polynomials: One dimensional element approximation u = a 0 + a 1 x + a 2 x2 + a 3 x3 + . . . or i

u = ai x =

X

ai x i

i

with

i = 1 → linear variation i = 2 → quadratic variation i = 3 → cubic variation

1D-element with two nodes:

1

2 1

⇒ For two nodes we need a linear variability. Universität Stuttgart

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Lagrangian polynomials Lagrange interpolation functions save us from inverting the coefficient matrix as is necessary for a standard polynomial. They have the following form u(x) = L1 (x)u1 + L2 (x) + · · · + Ln (x)un

with LN (x) chosen such that

LN (xm ) = δN M . LN (x) takes the form LN (x) = cN (x − x1 )(x − x2 ) . . . (x − xN −1 )(x − xN +1 ) . . . (x − xn ) . Universität Stuttgart

Institut für Wasserbau, Lehrstuhl für Hydromechanik und Hydrosystemmodellierung

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Lagrangian polynomials cN can be found to be 1 . cN = (xN − x1 )(xN − x2 ) . . . (xN − xN −1 )(xN − xN +1 ) . . . (xN − xn ) The polynomials are therefore

(x − x1 )(x − x2 ) . . . (x − xN −1 )(x − xN +1 ) . . . (x − xn ) LN (x) = . (xN − x1 )(xN − x2 ) . . . (xN − xN −1 )(xN − xN +1 ) . . . (xN − xn ) For a quadratic approximation we yield (x − x2 )(x − x3 ) L1 (x) = (x1 − x2 )(x1 − x3 )

(x − x1 )(x − x3 ) L2 (x) = (x2 − x1 )(x2 − x3 ) (x − x1 )(x − x2 ) L3 (x) = . (x3 − x1 )(x3 − x2 )

Universität Stuttgart

Institut für Wasserbau, Lehrstuhl für Hydromechanik und Hydrosystemmodellierung

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Lagrangian polynomials For the following element

−1

0

1

we get

x L1 = (x − 1), 2 L2 = 1 − x 2 , x L3 = (x + 1). 2

L1

−1 Universität Stuttgart

L3

L2

0

1

Institut für Wasserbau, Lehrstuhl für Hydromechanik und Hydrosystemmodellierung

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Hermite polynomials Hermite polynomials do not only assume the continuity of the variable at a common node, but also the derivative at that node. 1d element with two nodes. We need a cubic ansatz.   ∂ uˆ 0 1 u˜(ξ) = Hj (ξ)ˆ uj + Hj (ξ) j = 1, 2 ∂ξ r = 1, 2, 3, 4 u˜(ξ) = Nr wr N1 = H10 = 1 − 3ξ 2 + 2ξ 3 N2 = H20 = 3ξ 2 − 2ξ 3

N3 = H11 = ξ − 2ξ 2 + ξ 3 N4 = H21 = ξ 3 − ξ 2 Universität Stuttgart

Institut für Wasserbau, Lehrstuhl für Hydromechanik und Hydrosystemmodellierung

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Hermite polynomials The four functions have the following shape. 1

0

0

H1

0.8

H2

0.6

0.4

0.2

1

H1

0

H21 -0.2

Universität Stuttgart

0

0.2

0.4

0.6

0.8

1

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Natural coordinates The more general approach is to use the natural coordinates, ξ, with the origin either at the left end of the domain (upper figure) or in the middle (lower figure).

1

2

ξ=0

ξ=1

1

2

ξ=−1

Universität Stuttgart

ξ=0

ξ=1

Institut für Wasserbau, Lehrstuhl für Hydromechanik und Hydrosystemmodellierung

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Natural coordinates The interpolation functions then read for the upper figure ϕ1 = 1 − ξ ϕ2 = ξ,

and for the lower figure 1 ϕ1 = (1 − ξ) 2 1 aϕ2 = (1 + ξ). 2 Universität Stuttgart

Institut für Wasserbau, Lehrstuhl für Hydromechanik und Hydrosystemmodellierung

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Natural coordinates 2D Isoparametric elements Definition: The same parametric function which describes the geometry is used for the interpolation of the variables (shifting, water level, etc.) within an element. 2

1 s r

3

4

Universität Stuttgart

N1 = = N2 N3 N4

1 2

1 r 2



1 2

1 s 2

+ + 1 (1 + r)(1 + s) 4 analogous: = 14 (1 − r)(1 + s) = 14 (1 − r)(1 − s) = 14 (1 + r)(1 − s)



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Cartesian – Natural Coordinates Further examples for interpolation functions: 1. Interpolation function for a 2-D element with a 4 to 9 variable node number s = +1

s

y

2

6

3 r = -1

Universität Stuttgart

Node 1

5 9

8

7

4 s = -1

s=0

r=0

r

r = +1

x

Institut für Wasserbau, Lehrstuhl für Hydromechanik und Hydrosystemmodellierung

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Cartesian – Natural Coordinates

h1 h2 h3 h4 h5 h6 h7 h8 h9

= 14 (1 + r)(1 + s) = 14 (1 − r)(1 + s) = 14 (1 − r)(1 − s) = 14 (1 + r)(1 − s) = 12 (1 − r 2 )(1 + s) = 12 (1 − s2 )(1 − r) = 12 (1 − r 2 )(1 − s) = 12 (1 − s2 )(1 + r) = (1 − r 2 )(1 − s2 )

Universität Stuttgart

i=5 − 12 h5 − 12 h5 ..... ..... ..... ..... ..... .....

i=6 ..... − 12 h6 − 12 h6 ..... ..... ..... ..... .....

i=7 i=8 i=9 ..... − 12 h8 − 14 h9 − 14 h9 − 12 h7 − 14 h9 − 12 h7 − 12 h8 − 14 h9 . . . . . . . . . . − 12 h9 . . . . . . . . . . − 12 h9 . . . . . . . . . . − 12 h9 . . . . . . . . . . − 12 h9

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Isoparametric concept y

7

s 5

Ge

T

ii

+1

i

Re +1

13

r

iii

iv

x 25

We can save a lot of effort, if we construct our shape functions on a reference element and then transfer these functions to the global elements.

Universität Stuttgart

Institut für Wasserbau, Lehrstuhl für Hydromechanik und Hydrosystemmodellierung

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Isoparametric concept ˆ terms In technical applications (e.g. ground water flow ∇ W h), have to be often differentiated or integrated according to the Cartesian coordinates. Since we formulate our shape functions in isoparametric coordinates, one looks for a transformation relation between the two coordinate systems. With chain rule:        ∂ ∂ ∂r ∂ ∂s ∂s ∂r ∂ = ∂r ∂x + ∂s ∂x   ∂x  ∂x ∂x    ∂r  ·      =  ∂ ∂ ∂r ∂ ∂s ∂s ∂r ∂  = + ∂y ∂r ∂y ∂s ∂y ∂y ∂y ∂s Universität Stuttgart

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Isoparametric concept ∂r The calculation of ∂x , is not easily possible. Therefore, the opposite transformation is calculated first:

  ∂ ∂r



    = ∂ ∂s

∂x ∂r

∂y ∂r

∂x ∂s

∂y ∂s



 ·

"

∂ ∂x ∂ ∂y

#



 =J ·

∂ ∂x ∂ ∂y

  

ˆ Jacobi–Matrix, can be calculated more easily. J=

Universität Stuttgart

Institut für Wasserbau, Lehrstuhl für Hydromechanik und Hydrosystemmodellierung

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Isoparametric concept In order to calculate the Jacobian, we use the property of our shape functions. We can express the variable x with the shape functions and the nodal values. (linear interpolation of the coordinates x= Ni x i between the nodes, i=1 ne = number of nodes per element)     x1 y1 x  y   2  2 x = [N1 , N2 , N3 , N4 ]   y = [N1 , N2 , N3 , N4 ]    x3   y3  x4 y4 ne X

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Isoparametric concept Now we differentiate this expression. As the nodal values are no functions of x and y, we can take them out of the derivative and differentiate the local shape functions only. The product of these two matrices is the Jacobian matrix.       x 1 y1   ∂ ∂N2 ∂N3 ∂N4 ∂N1 ∂ x y  ∂x ∂r ∂r ∂r ∂r ∂r      2 2   = ·       x 3 y3  ∂ ∂N2 ∂N3 ∂N4 ∂N1 ∂ ∂y ∂s ∂s ∂s ∂s ∂s x 4 y4 {z } | J

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Isoparametric concept – Example We want to calculate the Jacobian of the element shown on the left. We can do this with the help of the previously introduced local shape functions. y

1 cm

1

P1 = (1|1, 25)

1 cm

P2 = (−1|0, 25)

2

x

1 cm

0.75 cm 3

Universität Stuttgart

2 cm

P3 = (−1| − 0, 75)

P4 = (1| − 0, 75)

4

Institut für Wasserbau, Lehrstuhl für Hydromechanik und Hydrosystemmodellierung

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Isoparametric concept – Example The global coordinates can be transformed into local coordinates as follows. x=

n X

Ni x i =

1

+ (1 − r)(1 − s)(−1) + (1 + r)(1 − s)(1)}

=r y=

n X

1 {(1 + r)(1 + s)(1) + (1 − r)(1 + s)(−1) 4

Ni y i =

1

3 1 1 = r + s + rs 4 4 4

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1 4



1 5 (1 + r)(1 + s) + (1 − r)(1 + s) 4 4  −3 −3 + (1 − r)(1 − s) + (1 + r)(1 − s) 4 4

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Isoparametric concept – Example These terms have to be differentiated   ∂y  J =

This yields 

 J =

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∂x ∂r

∂r

∂x ∂s

∂y ∂s

 .

∂ (r) ∂r

∂ 1 ( r ∂r 4

+ 34 s + 14 rs)

∂ (r) ∂s

∂ 1 ( r ∂s 4

+ 34 s + 14 rs)



" # 1 ( 14 + 14 s)  . = 3 1 0 ( 4 + 4 r)

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