finite elements iarfaria/ae245_05.pdf · finite elements i class notes. instituto tecnológico de...

Instituto Tecnológico de Aeronáutica

AE-245 1

FINITE ELEMENTS I

Class notes


AE-245 2

5. IsoparametricElements


AE-245 3

What are the conditions to ensure that, as the mesh is refined, the Galerkin approximate solution converges to the exact solution?

ISOPARAMETRIC ELEMENTSIntroduction


AE-245 4

Shape functions must be

• Smooth (at least C1) on each element interior.

• Continuous across element boundaries

• Complete



AE-245 5

Conditions 1 and 2 guarantee finite jumps across element interfaces.


n

n

n

Ni,nNi


AE-245 6

Shape functions that satisfy conditions 1 and 2 are of class C0


Euler-Bernoulli beams require higher order of derivatives. Thus, shape functions must be of class C1 (C2 on the element interior and C1 across boundaries).


AE-245 7


Integrands with derivatives of order m: Cm on the element interior and Cm-1 across boundaries.


AE-245 8

ISOPARAMETRIC ELEMENTSCompleteness

Completeness requires that the element interpolation function is capable of exactly representing an arbitrary linear polynomial when the nodal degrees of freedom are assigned values in accordance with it.


AE-245 9


Let

∑=

=m

iii

h dNu1

where di = uh(xi)


AE-245 10


3D situation: shape functions are complete if

iiii zcycxccd 3210 +++=

implies

zcycxccxu h 3210)( +++=


AE-245 11


Argument: as the mesh is refined the exact solution and its derivatives approach constant values over the elements.

In elasticity completeness means that the element can represent all rigid body motions and constant strain states.


AE-245 12

ISOPARAMETRIC ELEMENTSBilinear quadrilateral element

Straight edge four node quadrilateral

x

y

ξ

η

mapping

(−1,−1) (1,−1)

(−1,1) (1,1)

1 2

341

2

34

(x2,y2)

(x1,y1)

(x3,y3)(x4,y4)


AE-245 13


Straight edge four node quadrilateral

∑

∑

=

=

=

=

4

1

4

1

),(),(

),(),(

iii

iii

yNy

xNx

ηξηξ

ηξηξ


AE-245 14


Assume bilinear expansions

ξηβηβξββηξξηαηαξααηξ

3210

3210

),(

),(

+++=+++=

y

x

iii

iii

yy

xx

==

),(

),(

ηξηξ

114

113

112

111

−

−−−

iii ηξ


AE-245 15


Shape functions

)1)(1(4

1),(

)1)(1(4

1),(

)1)(1(4

1),(

)1)(1(4

1),(

4

3

2

1

ηξηξ

ηξηξ

ηξηξ

ηξηξ

+−=

++=

−+=

−−=

N

N

N

N

)1)(1(4

1),( ηηξξηξ iiiN ++=


AE-245 16


Shape functions: properties

Shape functions of class C1 within elements. Ni is always smooth in ξ and η but it may be discontinuous in x and y.

Shape functions of class C0 across element boundaries.


AE-245 17


Shape functions: properties

Completeness:

ycxccycxccN

cyNcxNcN

ycxccNdNu

ii

iii

iii

ii

iiii

iii

h

210210

4

1

2

4

11

4

10

4

1

4

1210

4

1

)(

++=++

=

+

+

=++==

∑

∑∑∑

∑∑

=

===

==


AE-245 18

ISOPARAMETRIC ELEMENTSIsoparametric element

Definition

∑=

=m

iiiN

1

)( xξx ∑=

=m

iii

h dNu1

)(ξ

same shape functions

The bilinear quadrilateral element is isoparametric


AE-245 19

ISOPARAMETRIC ELEMENTSIsoparametric element

If the mapping is one-to-one and if its Jacobian is positive for every ξ then the convergence condition 1 is met.

Isoparametric elements automatically satisfy the three basic convergence conditions.

Isoparametric elements can assume convenient shapes for practical analysis.


AE-245 20

ISOPARAMETRIC ELEMENTSTriangular element

Constant stress/strain element.

Coalescence of two nodes.

η

ξ

η

1 2

34

1

2

3

ξ


AE-245 21


34322111

)()( xxxxξx NNNNNm

iii +++==∑

=

332211

321 )1(2

1)1)(1(

4

1)1)(1(

4

1

xxx

xxxx

NNN ′+′+′=

++−++−−= ηηξηξ


AE-245 22


Exercises

Sketch the shape functions for the triangular element in the xy domain.

Compute the Jacobiandeterminant at ξ =η = 0 for the element shown.

y

x

(0,1)

(0,0) (x1,0)1

2

3


AE-245 23

ISOPARAMETRIC ELEMENTSTrilinear hexahedral element

ξ η

ζ

12

34

56

7 8

1

23

4

5

6 7

8


AE-245 24


ξηζαηζαξζαξηαζαηαξααζηξ 76543210),,( +++++++=x

=

−−−−

−−−−−−−−

−−−−−−−−

−−−−−−−−

8

7

6

5

4

3

2

1

7

6

5

4

3

2

1

0

11111111

11111111

11111111

11111111

11111111

11111111

11111111

11111111

x

x

x

x

x

x

x

x

αααααααα


AE-245 25


)1)(1)(1(8

1),,( ζζηηξξζηξ iiiiN +++=

1118

1117

1116

1115

1114

1113

1112

1111

−

−−−

−−−−−−−−

iiii ζηξ


AE-245 26


Degenerated wedge

1

23 ≡ 4

5

6 7 ≡ 8

1

23

4

5

6 7

8

coalescence


AE-245 27


Degenerated tetrahedral

1

23 ≡ 4

5 ≡ 6 ≡ 7 ≡ 8

1

23 ≡ 4

5

6 7 ≡ 8

coalescence


AE-245 28

ISOPARAMETRIC ELEMENTSHigher order elements

• Shape functions of order higher than linear

• More accurate representations

• Boundaries may be curved

• Computationally more expensive


AE-245 29


Lagrange polynomials

)()( )())((

)()( )())((

)(

)(

)(1121

1121

1

1

miiiiiii

miim

ijj

ji

m

ijj

j

il ξξξξξξξξξξξξξξξξξξξξ

ξξ

ξξ

ξ−−−−−

−−−−−=

−

−

=+−

+−

≠=

≠=

∏

∏LL

LL

ith term omitted


AE-245 30


Lagrange polynomials

ijjil δξ =)(

ii lN =m-noded element in one dimension:


AE-245 31


Examples

2-noded, one dimensional element:

3-noded, one dimensional element:

1 2

31 2

−1

−1

1

10


AE-245 32


Lagrange elements: higher order two - and three- dimensional elements can be obtained by taking products of Lagrange polynomials.

ξ

1 2

34

η

5

6

7

8 9


AE-245 33

ISOPARAMETRIC ELEMENTSVariable number of nodes

2D quadrilateral element with 5 to 9 nodes

)1)(1(4

1),( ηηξξηξ iiiN ++=

Add fifth node to side 1-2: ξ

1 2

34

5

η


AE-245 34


)1)(1(2

1)()(),( 211

225 ηξηξηξ −−== llN

Now, shape functions N1 and N2 must be modified to be zero at node 5.

522 2

1NNN −←511 2

1NNN −←


AE-245 35


)(2

1

)(2

1

)(2

1

)(2

1

8744

7633

6522

8511

NNNN

NNNN

NNNN

NNNN

+−←

+−←

+−←

+−←

)1)(1(2

1),(

)1)(1(2

1),(

)1)(1(2

1),(

)1)(1(2

1),(

28

27

26

25

ξηηξ

ηξηξ

ξηηξ

ηξηξ

−−=

+−=

+−=

−−=

N

N

N

N


AE-245 36


Bubble function: N9(ξ,η) = (1−ξ 2)(1−η 2)

Node 9 present

N1-N8 do not vanish at node 9. Hence,

94

1NNN ii −← 92

1NNN jj −←

i = 1, 2, 3, 4 j = 5, 6, 7, 8


AE-245 37

ISOPARAMETRIC ELEMENTSElement families

Serendipity family of quadrilateral elements


AE-245 38


Serendipity quadrilaterals

mm

mm

ξηηξηξ

ξηηξηξηηξξ

ηξηηξξηξηξ

ηξ

ON

ON44

4334

3223

22

1


AE-245 39


Lagrange family of quadrilateral elements


AE-245 40


Lagrange quadrilaterals

mm

mm

mm

mm

mm

mm

ηξ

ηξηξηξηξ

ηξηξηξξηηξηξηξ

ηηξηξηξξξηηξηξηξ

ηξηηξηξξηξηηξξ

ηξηξηξ

NO

ON

44

33

2442

4334

423324

432234

432234

3223

22

1


AE-245 41


Standard triangular elements


AE-245 42


Triangles

mmmmmm ηξηηξηξηξξ

ξηηξηξηξηξηηξηξξ

ηξηηξξηξηξ

ηξ

122221

43234

432234

3223

22

1

−−−−L

OONN

ON


AE-245 43

ISOPARAMETRIC ELEMENTSNumerical integration

Integration of

∫Ω

Ωdf )(x

In one dimension

∫b

a

dxxf )(


AE-245 44

ISOPARAMETRIC ELEMENTSNumerical integration

Numerical integration methods

• Trapezoidal rule (2nd order accurate)

• Simpson ’s rule (4th order accurate)

• Newton -Cotes formulas

• Gaussian quadrature (fewer points)


AE-245 45

ISOPARAMETRIC ELEMENTSGaussian quadrature

nnn

b

a

Rxfxfxfdxxf ++++=∫ )(...)()()( 2211 ααα

αi = weights

xi = sampling points


AE-245 46


...))(()()( 2210 ++++= xxxpxxf βββψ

where ))...()(()( 21 nxxxxxxxp −−−=

∑∑==

==n

iii

n

iii xlfxlxfx

11

)()()()(ψ

)()( )())((

)()( )())(()(

1121

1121

niiiiiii

niii xxxxxxxxxx

xxxxxxxxxxxl

−−−−−−−−−−

=+−

+−

LL

LL


AE-245 47


At the sampling points

iiiiiii fxxxxpxxf ==++++= )(...))(()()(2

210 ψβββψ

Integration of f(x) yields

∑ ∫∑ ∫∫∞

==

+

=

01

)()()(i

b

a

ii

n

i

b

a

ii

b

a

dxxpxdxxlfdxxf β


AE-245 48


Require that: 1,...,0for 0)( −==∫ nidxxpxb

a

i

p(x) is of order n. Hence, integrals of polynomials of order n to 2n−1 are zero.

Sampling weights: ∫=b

a

ii dxxl )(α


AE-245 49


Change of variables

∫∫+

−

=1

1

,)()( ξξ ξ dxfdxxfb

a

Advantage: standardize results


AE-245 50


αirin

0.467913934572691±0.238619186083197

0.360761573048139±0.661209386466265

0.171324492379170±0.9324695142031526

0.5688888888888890.0

0.478628670499366±0.538469310105683

0.236926885056189±0.9061798459386645

0.652145154862546±0.339981043584856

0.347854845137454±0.8611363115940534

0.8888888888888890.0

0.555555555555556±0.7745966692414833

1.0±0.5773502691896262

2.00.01

2=∑ iα

ri symmetrically positioned


AE-245 51


Two and three dimensions:

∑∫∑∫∫ ==ji

jijii

ii srfdssrfdsdrsrf,

),(),(),( ααα


AE-245 52

ISOPARAMETRIC ELEMENTSDerivatives of shape functions

• Construct stiffness matrices

• Recover strains and stresses

• Compute N,x and N,y

• Chain rule

yyy

xxx

NNN

NNN

,,,,,

,,,,,

ηξηξ

ηξ

ηξ

+=

+=


AE-245 53


Chain rule

yyy

xxx

NNN

NNN

,,,,,

,,,,,

ηξηξ

ηξ

ηξ

+=

+=

=

η

ξ

ηξηξ

,

,

,,

,,

,

,

N

N

N

N

yy

xx

y

x

The Jacobian matrix must be computed first!


AE-245 54


Jacobian matrix

∑=

=m

iii xNx

1

),( ηξ ∑=

=m

iii yNy

1

),( ηξ

∑=

=m

iii xNx

1,, ξξ

∑=

=m

iii xNx

1,, ηη

∑=

=m

iii yNy

1,, ξξ

∑=

=m

iii yNy

1,, ηη


AE-245 55


Chain rule

−−

−=

−−

=

ξξ

ηη

ξηηξξξ

ηη

ηξηξ

,,

,,

,,,,,,

,,

,,

,, 1

)det(

1xy

xy

yxyxxy

xy

Jyy

xx


AE-245 56

ISOPARAMETRIC ELEMENTSTriangular and tetrahedral elements

• Elements that model complex geometries

• Facilitate mesh transitions

• Avoid distorted quadrilaterals/hexahedra

• Area/volume coordinates


AE-245 57

ISOPARAMETRIC ELEMENTSTriangular elements

Area coordinates

0 1

1

r

s

srtsrN

ssrN

rsrN

−−====

1),(

),(

),(

3

2

1

0

t constant


AE-245 58


Area coordinates

1 2

3

rs

t


AE-245 59


Example: 6 node triangle

1 2

3

4

5 6

stN

rtN

rsN

ttN

ssN

rrN

4

4

4

)12(

)12(

)12(

6

5

4

3

2

1

===

−=−=−=


AE-245 60

ISOPARAMETRIC ELEMENTSTetrahedral elements

Volume coordinates

r

s

t

1

1

1

tsrutsrN

ttsrN

stsrN

rtsrN

−−−=====

1),,(

),,(

),,(

),,(

4

3

2

1


AE-245 61

ISOPARAMETRIC ELEMENTSTriangular and tetrahedral elements

Numerical Integration

∑∫=Ω

=Ωm

iiii Jffd

1

)det(α

0.1116979480.1081030180.4459484910.445948491

0.1116979480.4459484910.1081030180.445948491

0.1116979480.445948491 0.445948491 0.108103018

0.0549758720.81684757300.0915762140.091576214

0.0549758720.0915762140.81684757300.091576214

0.0549758720.091576214 0.09157621350.81684757306

25/9611/152/152/15

25/962/1511/152/15

25/962/152/1511/15

−27/961/31/31/34

1/62/31/61/6

1/61/62/31/6

1/61/61/62/33

0.51/31/31/31

αtsrn


AE-245 62

ISOPARAMETRIC ELEMENTSSpecial shape functions

• Elements with special properties

• Infinite elements

• Modeling of singularities (cracks)


AE-245 63


One dimensional example:

0 La

ξ = 0 ξ = 1/2 ξ = 1 )12()()1(4)(

)1)(21()(

3

2

1

−=−=

−−=

ξξξξξξ

ξξξ

N

N

N

332211 uNuNuNu ++=

32332211 LNaNxNxNxNx +=++=


AE-245 64


Obtain strain:

0 La

ξ = 0 ξ = 1/2 ξ = 1

ξξξξξξ )4()42()12()1(4 2 LaaLLax −+−=−+−=

If x is a purely quadratic function of ξ then a = L/42ξLx =


AE-245 65


Obtain strain:

0 LL/4

ξ = 0 ξ = 1/2 ξ = 1

LxLLddx

ddu

dx

du

222/

/ 0101 ααξαξα

ξξε +=+===

∞=→

)(lim0

xx

ε Singularity modeling


AE-245 66

Membrane FE code


AE-245 67

Membrane FE codeMain routines

• input_data

• mechanical

• renum

• crout_dec

• crout_sol


AE-245 68

Membrane FE codeRoutine input_data

Purpose: generate mesh, input material, allocate memory, find band width

Features: user defined, geometry dependant


AE-245 69

Membrane FE codeRoutine mechanical

Purpose: generate element stiffness matrix and global stiffness matrix

Features: Gaussian numerical integration, biquadratic and 6-node triangular elements


AE-245 70

Membrane FE codeRoutine renum

Purpose: Apply essential boundary conditions

Features: Zeroes rows and columns


AE-245 71

Membrane FE codeRoutine crout_dec

Purpose: Decompose banded matrix

Features: In -place decomposition, symmetric matrix

ULK =upper triangular matrix

lower triangular matrix


AE-245 72

Membrane FE codeRoutine crout_sol

Purpose: Solve system Kq = f

Features: Two step procedure

fqUL =


AE-245 73

Membrane FE codeResults: Circular plate

radius = 2.0 m

thickness = 1.0 mm

E = 208 MPa

ν = 0.3


AE-245 74


x

y

-2 -1 0 1 2-2

-1.5

-1

-0.5

0

0.5

1

1.5

2

u6.13E-015.25E-014.38E-013.50E-012.63E-011.75E-018.76E-020.00E+00

-8.76E-02-1.75E-01-2.63E-01-3.50E-01-4.38E-01-5.25E-01-6.13E-01

x

y

-2 -1 0 1 2-2

-1.5

-1

-0.5

0

0.5

1

1.5

2

v6.13E-015.25E-014.38E-013.50E-012.63E-011.75E-018.76E-020.00E+00

-8.76E-02-1.75E-01-2.63E-01-3.50E-01-4.38E-01-5.25E-01-6.13E-01


AE-245 75


x

y

-2 -1 0 1 2-2

-1.5

-1

-0.5

0

0.5

1

1.5

2

u6.64E-016.20E-015.76E-015.32E-014.87E-014.43E-013.99E-013.54E-013.10E-012.66E-012.21E-011.77E-011.33E-018.86E-024.43E-02

x

y

-2 -1 0 1 2-2

-1.5

-1

-0.5

0

0.5

1

1.5

2

v-5.18E-15-1.07E-14-1.61E-14-2.16E-14-2.71E-14-3.26E-14-3.81E-14-4.35E-14-4.90E-14-5.45E-14-6.00E-14-6.54E-14-7.09E-14-7.64E-14-8.19E-14


AE-245 76


Radius = 2.0 m

Thickness = 1.0 mm

E = 208 MPa

ν = 0.3


AE-245 77


x

y

-2 -1 0 1 2-2

-1.5

-1

-0.5

0

0.5

1

1.5

2

u6.95E+005.96E+004.97E+003.97E+002.98E+001.99E+009.93E-010.00E+00

-9.93E-01-1.99E+00-2.98E+00-3.97E+00-4.97E+00-5.96E+00-6.95E+00

x

y

-2 -1 0 1 2-2

-1.5

-1

-0.5

0

0.5

1

1.5

2

v1.06E+009.10E-017.58E-016.07E-014.55E-013.03E-011.52E-010.00E+00

-1.52E-01-3.03E-01-4.55E-01-6.07E-01-7.58E-01-9.10E-01-1.06E+00


AE-245 78


x

y

-2 -1 0 1 2-2

-1.5

-1

-0.5

0

0.5

1

1.5

2

u8.52E+007.30E+006.09E+004.87E+003.65E+002.43E+001.22E+000.00E+00

-1.22E+00-2.43E+00-3.65E+00-4.87E+00-6.09E+00-7.30E+00-8.52E+00

x

y

-2 -1 0 1 2-2

-1.5

-1

-0.5

0

0.5

1

1.5

2

v1.06E+009.10E-017.58E-016.07E-014.55E-013.03E-011.52E-010.00E+00

-1.52E-01-3.03E-01-4.55E-01-6.07E-01-7.58E-01-9.10E-01-1.06E+00


AE-245 79


x

y

-2 -1 0 1 2-2

-1.5

-1

-0.5

0

0.5

1

1.5

2

u1.01E+018.64E+007.20E+005.76E+004.32E+002.88E+001.44E+000.00E+00

-1.44E+00-2.88E+00-4.32E+00-5.76E+00-7.20E+00-8.64E+00-1.01E+01

x

y

-2 -1 0 1 2-2

-1.5

-1

-0.5

0

0.5

1

1.5

2

v1.18E+001.01E+008.40E-016.72E-015.04E-013.36E-011.68E-010.00E+00

-1.68E-01-3.36E-01-5.04E-01-6.72E-01-8.40E-01-1.01E+00-1.18E+00


AE-245 80

Membrane FE codeResults: L shaped plate

2 m

2 m

P

thickness = 1.0 mm

E = 208 MPa

ν = 0.31 m


AE-245 81


x

y

0 0.5 1 1.5 20

0.25

0.5

0.75

1

1.25

1.5

1.75

2

u59.2142.3425.468.59

-8.29-25.16-42.04-58.92-75.79-92.67

-109.54-126.42-143.29-160.17-177.04

x

y

0 0.5 1 1.5 20

0.25

0.5

0.75

1

1.25

1.5

1.75

2

v286.51264.02241.53219.04196.55174.07151.58129.09106.60

84.1161.6239.1316.64-5.85

-28.34


AE-245 82


x

y

0 0.5 1 1.5 20

0.25

0.5

0.75

1

1.25

1.5

1.75

2

u61.6344.1426.65

9.16-8.32

-25.81-43.30-60.79-78.27-95.76

-113.25-130.74-148.23-165.71-183.20

x

y

0 0.5 1 1.5 20

0.25

0.5

0.75

1

1.25

1.5

1.75

2

v298.78275.43252.09228.74205.39182.04158.69135.35112.00

88.6565.3041.9618.61-4.74

-28.09

finite elements iarfaria/ae245_05.pdf · finite elements i class notes. instituto tecnológico de...

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