Isoparametric Formulation

MANE 4240 & CIVL 4240Introduction to Finite Elements

Mapped element geometries and shape functions: the isoparametric formulation

Prof. Suvranu De

Reading assignment:

Chapter 10.1-10.3, 10.6 + Lecture notes

Summary:

• Concept of isoparametric mapping• 1D isoparametric mapping• Element matrices and vectors in 1D• 2D isoparametric mapping : rectangular parent elements • 2D isoparametric mapping : triangular parent elements • Element matrices and vectors in 2D

For complex geometries© 2002 Brooks/Cole Publishing / Thomson Learning™

General quadrilateral elements

Elements with curved sides

s

t1 1 12

3 4

1

1

Consider a special 4-noded rectangle in its local coordinate system (s,t)

44332211

44332211

vNvNvNvNv

uNuNuNuNu

Displacement interpolation

)1)(1(4

1),(

)1)(1(4

1),(

)1)(1(4

1),(

)1)(1(4

1),(

4

3

2

1

tstsN

tstsN

tstsN

tstsN

Shape functions in local coord system

Recall that

ttNtNtNtN

ssNsNsNsN

NNNN

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4321 1 Rigid body modes

Constant strain states

Goal is to map this element from local coords to a general quadrilateral element in global coord system

s

t1 1 12

3 4

1

1

Local coordinate system

s

y 1

2

3

4x

s

t

Global coordinate system

),(

),(

tsyy

tsxx

),(

),(

yxtt

yxss

),(1 tsN ),(1 yxN

),()),(),,((),( 111 yxNyxtyxsNtsN

In the mapped coordinates, the shape functions need to satisfy

1. Kronecker delta property

nodesotherallat

inodeatN i 0

1

yyN

xxN

N

ii

i

ii

i

ii

1

Then

2. Polynomial completeness

The relationship

ii

i

ii

i

ytsNy

xtsNx

),(

),(

Provides the required mapping from the local coordinate systemTo the global coordinate system and is known as isoparametric mapping

(s,t): isoparametric coordinates(x,y): global coordinates

s

t1 1

1

1

s

t

y

x

Examples

s

t

1

1

y

x

s

t

1D isoparametric mapping

1 2

x2x1 x1 2

s1 1

Local (isoparametric) coordinates

3 noded (quadratic) element

3

3 x3

23

2

1

1)(

2

1)(

2

1)(

ssN

sssN

sssN

Isoparametric mapping

3

1

)(i

ii xsNx

32

21 12

1

2

1xsx

ssx

ssx

32

21 12

1

2

1xsx

ssx

ssx

NOTES1. Given a point in the isoparametric coordinates, I can obtain the corresponding mapped point in the global coordinates using the isoparametric mapping equation

2

3

1

;1

;0

;1

xxsAt

xxsAt

xxsAt

Question x=? at s=0.5?

2. The shape functions themselves get mappedIn the isoparametric coordinates (s) they are polynomials.In the global coordinates (x) they are in general nonpolynomialsLets consider the following numerical example

4;6;0 321 xxx

1 2

x

3

4 2

Isoparametric mapping x(s)

21 2 3

2

2

1 11

2 21 1

0 6 1 42 2

4 3

s s s sx x x s x

s s s ss

s s

Inverse mapping s(x)

2

4253 xs

Simple polynomial

Complicated function

Now lets compute the shape functions in the global coordinates

)(

4252102

1

2

42531

2

4253

2

1

2

1)(

2

2

xN

xx

xx

sssN

Now lets compute the shape functions in the global coordinates

N2(x) is a complicated function

However, thanks to isoparametric mapping, we always ensure 1. Knonecker delta property2. Rigid body and constant strain states

1 2

s1 1

3

N2(s)

1 2

x

3

4 2 11

N2(x)

N2(s) is a simple polynomial

2

1)(2

sssN

xxxN 425210

2

1)(2

Element matrices and vectors for a mapped 1D bar element

1 2

x1 2

s1 1 3

3

Displacement interpolation dNuNuNuNu 332211

Strain-displacement relation dBudx

dNu

dx

dNu

dx

dN

dx

du 3

32

21

1

Stress dBEE

The strain-displacement matrix

dx

dN

dx

dN

dx

dNB 321

The only difference from before is that the shape functions are in the isoparametric coordinates

23

2

1

1)(

2

1)(

2

1)(

ssN

sssN

sssN

And we will not try to obtain explicitly the inverse map.How to compute the B matrix?

We know the isoparametric mapping

3

1

)(i

ii xsNx

Do I know ?)(

ds

sdN i

Do I know ?dx

ds

I know

3

1

)(i

ii xsNx

Hence )()(3

1

mappingofJacobianJxds

sdN

ds

dx

ii

i

ds

sdN

Jdx

sdN ii )(1)(From (*)

dx

ds

ds

sdN

dx

sdN ii )()(

Using chain rule

(*)

What does the Jacobian do?

dsJdx

Maps a differential element from the isoparametric coordinates to the global coordinates

The strain-displacement matrix

ds

dN

ds

dN

ds

dN

J

dx

dN

dx

dN

dx

dNB

321

321

1

For the 3-noded element

321

3

1

22

12

2

12)(sxx

sx

sx

ds

sdNJ

ii

i

s

ss

JB 2

2

12

2

121

The element stiffness matrix

1

1

2

1

JdsdxJdsBBEA

dxBBEAk

T

x

x

T

NOTES1. The integral on ANY element in the global coordinates in now an integral from -1 to 1 in the local coodinates2. The jacobian is a function of ‘s’ in general and enters the integral. The specific form of ‘J’ is determined by the values of x1, x2 and x3. Gaussian quadrature is used to evaluate the stiffness matrix3. In general B is a vector of rational functions in ‘s’

Isoparametric mapping in 2D: Rectangular parent elements© 2002 Brooks/Cole Publishing / Thomson Learning™

Parent element


Mapped element in global coordinates

ii

i

ii

i

ytsNy

xtsNx

),(

),(

1

2

3

4

1( , ) (1 )(1 )

41

( , ) (1 )(1 )41

( , ) (1 )(1 )41

( , ) (1 )(1 )4

N s t s t

N s t s t

N s t s t

N s t s t


ii

i

ii

i

ytsNy

xtsNx

),(

),(

Shape functions of parent element in isoparametric coordinates

NOTES:1. The isoparametric mapping provides the map (s,t) to (x,y) , i.e.,

if you are given a point (s,t) in isoparametric coordinates, then you can compute the coordinates of the point in the (x,y) coordinate system using the equations

2. The inverse map will never be explicitly computed.

ii

i

ii

i

ytsNy

xtsNx

),(

),(

s

t1 1 34

1 2

1

1

7

8

5

6

8-noded Serendipity element© 2002 Brooks/Cole Publishing / Thomson Learning™

8-noded Serendipity element: element shape functions in isoparametric coordinates

1 5

2 6

3 7

4 8

1 1( , ) (1 )(1 )( 1) ( , ) (1 )(1 )(1 )

4 21 1

( , ) (1 )(1 )( 1) ( , ) (1 )(1 )(1 )4 21 1

( , ) (1 )(1 )( 1) ( , ) (1 )(1 )(1 )4 2

1 1( , ) (1 )(1 )( 1) ( , ) (1 )

4 2

N s t s t s t N s t t s s

N s t s t s t N s t t t s

N s t s t s t N s t t s s

N s t s t s t N s t s

(1 )(1 )t t

NOTES1. Ni(s,t) is a simple polynomial in s and t. But Ni(x,y) is a complex function of x and y.2. The element edges can be curved in the mapped coordinates3. A “midside” node in the parent element may not remain as a midside node in the mapped element. An extreme example

s

t1 1 12

3 4

1

1

5

6

7

8x

y 1

85

7

2,6,3

4. Care must be taken to ensure UNIQUENESS of mapping

s

t1 1 12

3 4

1

1

x

y

1

2

3

4

Isoparametric mapping in 2D: Triangular parent elements

s

t2

3 1

1

1

P(s,t)st

)1(2

12

2

2

1

12

23

31

123

ts

s

t

P

P

P

tsL

tL

sL

P

P

P

1123

123

123

312

123

231

Parent element: a right angled triangles with arms of unit length Key is to link the isoparametric coordinates with the area coordinates

Now replace L1, L2, L3 in the formulas for the shape functions of triangular elements to obtain the shape functions in terms of (s,t)

Example: 3-noded triangle

y

xs

t2

3 1

1

1 (x1,y1)s

t

Parent shape functions

tsN

tN

sN

13

2

1


332211

332211

),(),(),(

),(),(),(

ytsNytsNytsNy

xtsNxtsNxtsNx

2 (x2,y2)

3 (x3,y3)1

Element matrices and vectors for a mapped 2D element

Stress approximation

Recall: For each element

Element stiffness matrix

Element nodal load vector

dNu

dBD

dBε

eV

k dVBDBT

S

eT

b

e

f

S ST

f

V

T dSTdVXf NN

Displacement approximation

Strain approximation

STe

e

In isoparametric formulation1. Shape functions first expressed in (s,t) coordinate systemi.e., Ni(s,t)2. The isoparamtric mapping relates the (s,t) coordinates with the

global coordinates (x,y)

3. It is laborious to find the inverse map s(x,y) and t(x,y) and we do not do that. Instead we compute the integrals in the domain of the parent element.

ii

i

ii

i

ytsNy

xtsNx

),(

),(

NOTE1. Ni(s,t) s are already available as simple polynomial functions2. The first task is to find and

x

N i

y

N i

Use chain rule

t

y

y

N

t

x

x

N

t

yxN

s

y

y

N

s

x

x

N

s

yxN

iii

iii

),(

),(

In matrix form

y

Nx

N

t

y

t

xs

y

s

x

t

Ns

N

i

i

i

i

Can be computed

We want to compute these for the B matrix

This is known as the Jacobian matrix (J) for the mapping(s,t) → (x,y)

y

Nx

N

J

t

Ns

N

i

i

i

i

t

Ns

N

J

y

Nx

N

i

i

i

i

1

How to compute the Jacobian matrix?Start from

ii

i

ii

i

ytsNy

xtsNx

),(

),(

ii

i

ii

i

ii

i

ii

i

yt

tsN

t

yy

s

tsN

s

y

xt

tsN

t

xx

s

tsN

s

x

),(;

),(

),(;

),(

ii

i

ii

i

ii

i

ii

i

yt

tsNx

t

tsN

ys

tsNx

s

tsN

t

y

t

xs

y

s

x

J

),(),(

),(),(

Need to ensure that det(J) > 0 for one-to-one mapping

3. Now we need to transform the integrals from (x,y) to (s,t)

Case 1. Volume integrals

1

1

1

1

)det(),(),(),( dsdtJhtsfdAhtsfdVtsfee AV

h=thickness of element

This depends on the key result

dsdtJdxdydA )det(

s

t1 1 12

3 4

1

1

Problem: Consider the following isoparamteric map

y

12

3 4(3,1) (6,1)

(6,6)(3,6)

xISOPARAMETRIC COORDINATES

y

GLOBAL COORDINATES

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44332211

vNvNvNvNv

uNuNuNuNu

Displacement interpolation

)1)(1(4

1),(

)1)(1(4

1),(

)1)(1(4

1),(

)1)(1(4

1),(

4

3

2

1

tstsN

tstsN

tstsN

tstsN

Shape functions in isoparametric coord system

1 1 2 2 3 3 4 4

1 1 2 2 3 3 4 4

( , ) ( , ) ( , ) ( , )

( , ) ( , ) ( , ) ( , )

x N s t x N s t x N s t x N s t x

y N s t y N s t y N s t y N s t y

The isoparamtric map

(1 )(1 ) (1 )(1 ) (1 )(1 ) (1 )(1 )6 3 3 6

4 4 4 43(1 )

2(1 )(1 ) (1 )(1 ) (1 )(1 ) (1 )(1 )

6 6 1 14 4 4 4

7 5

2

s t s t s t s tx

s

s t s t s t s ty

t

In this case, we may compute the inverse map, but we will NOT do that!

The Jacobian matrix

NOTE: The diagonal terms are due to stretching of the sides along the x-and y-directions. The off-diagonal terms are zero because the element does not shear.

30

25

02

x y

s sJx y

t t

3(1 )

27 5

2

sx

ty

since

1 2 / 3 0 15and det( )

0 2 / 5 4J J

Hence, if I were to compute the first column of the B matrix along the positive x-direction

I would use

1

1

1

0

N

x

N

y

B

1 1

1

1 1

112 / 3 0 64

0 2 / 5 1 1

4 10

N N ttx sJN N s sy t

1

1

1

1

60 0

1

10

N t

x

N s

y

B

Hence

The element stiffness matrix

1 1T T

1 1B D B dV B D B det( ) dsdt

eVk J h

Isoparametric Formulation

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