jn reddy - 1 lecture notes on nonlinear fem meen...

INTRODUCTION AND OVERVIEW OF LINEAR FEM using a 2D model problem

J. N. Reddye-mail: [email protected]

Fall 2016

MEEN 673Nonlinear Finite Element Analysis

Chapter 3

JN Reddy - 1 Lecture Notes on NONLINEAR FEM

3

An Overview of The Finite Element Method

2D Problems involving a single unknown

• Model equation Discretization• Weak form development • Finite element model • Approximation functions• Interpolation functions of

higher-order elements• Post-computation of variables• Numerical examples• Transient analysis of 2-D problems

CONTENTS


MODEL EQUATION

Model Differential Equation

11 22 inu u

a a fx x y y

2-D Problems: 4

coefficients that describe material behaviorsource term

ijaf

11 12 21 22ˆ ˆ, 0x y n

u u u uu u a a n a a n q

x y x y

Type of boundary conditions


EXAMPLES OF MODEL EQUATION(1)− Deflection of a membrane

2-D Problems: 5

11 22 0u u

a a fx x y y

11 22

( , ) transverse deflection of a point in the membrane( , ) applied pressure

( , ), ( , ) tensions in the and directions, respectively

u x y

f x ya x y a x y x

y

=

==

-1

-0.5

0

0.5

1

-1

-0.5

0

0.5

1-1

-0.5

0

0.5

1

Transverse deflection of a square membrane fixed on all its sides and subjected to uniform pressure.

x

y


EXAMPLES OF MODEL EQUATION(2)− Torsion of a cylindrical member

2 2

2 2 0 inGx y

Ω

11 22

( , )

shear modulusangle of twi

warping func

t

tion

s

x ya a GG

0

, , ( , )

( , ) , ( , )

on

xz yz

x yy n x n

u zy v zx w x y

x y G y x y G xy

x

x

y

Γ

L

z

Mz

Mz

y

x

uvzθα =


EXAMPLES OF MODEL EQUATION(3)− 2D Heat transfer

11 22, temperature; ,, thermal conductivities

in the and directions, respectively

internal heat generationheat flux normal to the boundary

x y

x y

n

u T a k a kk k

x y

fq

11 22

11 22

( , ) , ( , )

ˆ on

x y

x y n

u uq x y a q x y ax y

u ua n a n qx y

Γ

11 22 0u u

a a fx x y y

x

y


11 22

, water head (velocity potential), permeabilities in the and directions,

respectively internal infiltrationflow normal to the boundaryn

ua a x y

fq

11 22

11 22

( , ) , ( , )

ˆ on

x y

x y n

v x y a v x y ax y

a n a n qx y

Γ

11 22 0a a f

x x y y

EXAMPLES OF MODEL EQUATION(4) − 2D Inviscid flow

x

y


2-D Problems: 9

WEAK FORM DEVELOPMENT

Approximation ( , ) ( , )hu x y u x y

,e e

e e

x y

F Fdxdy F ds dxdy F dsn n

x y

Green-Gauss Theorem

Consider the identity:

1 11 1 1 1ori i

i i i i

w F F ww F F w w w F F

x x x x x x

11 220

e

h hi

u uw a a f dxdy

x x y y

Weak Form DevelopmentStep 1

1F2F

Step 2: Trade differentiations between and w hu

F


2-D Problems: 10

WEAK FORM DEVELOPMENT continued

e

e

x

Fdxdy n F ds

x

Consider the identity: 11 1

ii i

F ww w F F

x x x

Step 2:

F

1e

e

x i

Fdxdy n w F ds

x

11e

e

hx i

uFdxdy n w a ds

x x

22e

e

hiy

uGdxdy w a s

yd

yn


2-D Problems: 11

Step 1

WEAK FORM DEVELOPMENT (continued)

11 22

11 22

11 22

0e

e

e

h hi

i h i hi

h hi x y

u uw a a f dxdy

x x y y

w u w ua a w f dxdy

x x y y

u uw a n a n

x y

11 22

11 22 , flux normal to the boundary

e

e

i h i hi

i n

h hn x y

ds

w u w ua a w f dxdy

x x y y

w q ds

u uq a n a n

x y

Step 2

Step 3


2-D Problems: 12

FINITE ELEMENT APPROXIMATION

x

y

n

q

n qn ˆq

x

y nixˆn

jyˆn

n i jx yˆ ˆˆ n n

ˆn i

ˆn j

ˆn i

n cos i sin jˆ ˆˆ


2-D Problems: 13

Weak form

FINITE ELEMENT APPROXIMATION

11 220e

e

i h i hi

i n

w u w ua a w f dxdy

x x y y

w q ds

1 2 3 4

1

( , ) ( , ) ( terms)

( , )

h

n

j jj

u x y u x y c c x c y c xy n

u x y

Approximation

Finite element model [the ith equation is obtained by replacing the weight function w by ]( 1,2, , )i i n

11 220e

e

i h i hi

i n

u ua a f

x x y y

q ds

dxdy


FINITE ELEMENT MODEL DEVELOPMENT(continued)

11 22

11 221 1

11

0

0

e

e

e

e e

i h i hi

i n

n nj ji i

j jj j

i i n

j

u ua a f dxdy

x x y y

q ds

u a u a dxdyx x y y

f dxdy q ds

u a

221 e

e

e

nj ji i

j

i i n

a dxdyx x y y

f dxdy q ds

2-D Problems: 14


FINITE ELEMENT MODEL DEVELOPMENT(continued)

11 221

1 1

11 22

0

or

,

e

e

e

e

e

e

nj ji i

jj

i i n

n ne e e e e e e e e eij j i i ij j i

j j

j je i iij

ei i i

u a a dxdyx x y y

f dxdy q ds

K u f Q K u F

K a a dxdyx x y y

F f dxdy

K u F

nq ds

2-D Problems: 15


2-D Problems: 16

APPROXIMATION FUNCTIONSLinear Triangular Element

1 2 3

1 1 1 2 1 3 1 1

2 2 1 2 2 3 2 2

3 3 1 2 3 3 3 3

( , )

( , )

( , )

( , )

eh

e eh

e eh

e eh

u x y c c x c y

u x y c c x c y u

u x y c c x c y u

u x y c c x c y u

u A c c A u1 11 1

12 2 2 2

3 33 3

1

1 or

1

e e

e e e e e e e

e e

u cx y

u x y c

x yu c

y

x

1

3

21 1( , )x y2 2( , )x y

3 3( , )x y

1 2

3

·

·

·


2-D Problems: 17

APPROXIMATION FUNCTIONSLinear Triangular Element (continued)

1

1

3

2

ψ1

11

3

2

ψ21

1

3

2

ψ3

1

1 2 33

1 1 2 2 3 31

( , ) 1 1

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

1( , )

2, , ( )

Area of the triangle, 2 Determinant of [ ]

e e eh

e e e e e e e ej j

j

ei i i ie

i j k j k i j k i j ke e

u x y c c x c y x y c x y A u

x y u x y u x y u u x y

x y x y

x y y x y y x x

A


2-D Problems: 18

APPROXIMATION FUNCTIONSLinear Rectangular Element

1 2 3 4

11 1 1

22 1 2 2

3 3 1 2 3 4 3

4 4 1 3 4

( , )

( , )

( , )

( , )

( , )

eh

e eh

e eh

e eh

e eh

u x y c c x c y c x y

u x y c u

u x y c c a u

u x y c c a c b c ab u

u x y c c b u

1 2

3 4

( , ) 1 1 , ( , ) 1 ,

( , ) , ( , ) 1 ,

e e

e e

x y x yx y x y

a b a bx y x y

x y x ya b a b

y

x

1

4 3

2

b

a

y_

x_

( , )ej i i ijx y


2-D Problems: 19

1

1

4

3

2

ψ1

11

4

3

2

ψ2

1

1

4

3

2

ψ3 1

1

43

2

ψ4

APPROXIMATION FUNCTIONSLinear Rectangular Element (continued)


2-D Problems: 20

Interpolation Property of the Approximation Functions

3

1

1, , a fixed value( , ) ; ( , ) 1

0, , a fixed valuee ej i i ij j

j

i jx y x y

i j

x

1 231 2

34

y

x

223

u(x,y)

u1u3

Representation of the domainby 3-node triangles

u2

1

( , )ehu x y

x

1 232

4

yu(x,y)

Representation of the domainby 4-node rectangles

1u4u 2u

3u

23

1

4

4


2-D Problems: 21

NUMERICAL EVALUATION OF COEFFICIENT MATRICES

Linear Right-Angled Triangular Element

2 2 2 2

2 2

2 2

[ ] 02

0

1

13

1

e e

ee e

a b b ak

K b bab

a a

ff

b

ax2

3

1

y

2

3

1

Right-angle triangle

2 e ab

11 22

1 1 0 1 0 1

[ ] 1 1 0 0 0 02 2

0 0 0 1 0 1

e e eb aK a a

a b

11 22When , we havee eea a k


2-D Problems: 22

NUMERICAL EVALUATION OF COEFFICIENT MATRICESLinear Rectangular Element

y

x

1

4 3

2

b

a

y_

x_

11 22

2 2 1 1 2 1 1 2 1

2 2 1 1 1 2 2 1 1[ ] ,

1 1 2 2 1 2 2 1 16 6 41 1 2 2 2 1 1 2 1

e e e e ef abb aK a a f

a b


1. Superparametric (m > n): The polynomial degree ofapproximation used for the geometry is of higher order than that used for the dependent variable.

2. Isoparametric (m = n): Equal degree of approximationis used for both geometry and dependent variables.

3. Subparametric (m < n): Higher-order approximation ofthe dependent variable is used.

Geometry:

Solution:

1 1

ˆ ˆ( ), ( )m m

e e e ej j j j

j j

x x , y y ,

1 1

( , ) ( ) ( ( ) ( ))n n

e e e ej j j j

j j

u x y u x,y u x , ,y ,

PARAMETRIC FORMULATIONS (2-D)

Thus, there are two meshes in finite element analysis.

Numer Integration: 23


NUMERICAL EVALUATION OF INTEGRAL COEFFICIENTS

• Transformation of the integrals posed on arbitrary-shaped element to the master element domain so that evaluationof the integrals is made easy.

• The Gauss integration rule that evaluates an integral expression as a linear sum of the integrand evaluated at certain points (Gauss points) and weights (Gauss weights)is used.

x

y

Γ

Γ

Ω e

e

nα

Ω

∧

•

•

•

• eu2i

Degrees of freedom

12

34

eu1

1

( , ) ( , ) ( , )n

e e eh j j

j

u x y u x y u x y

1

( ( , ), ( , ))

( , )

eh

ne ej j

j

u x y

u



ξ

η

ξ = 1

η = 1

x

y

eΩ

1Ω 2Ω 3ΩΩ

ξ

η

eΩ

ηξ ddJdydx =

)1(),1( ηη ,yy,xx ==

)1(),1( ,yy,xx ξξ ==)()(

y,xy,x

ηηξξ

==

)()(

ηξηξ,yy,xx

==

y

x

NUMERICAL INTEGRATION

1

1

ˆ( , ) ( )

ˆ( , ) ( )

me ej j

j

me ej j

j

x x ,

y y ,



( ) ( ) General

( ) ( ) Gauss rule

b

a

NPTx

ij ij ij I Ix

I

NGP

ij ij ij I II

G F x dx F x W

K F d F W

1

1

11

Numerical Integration -26

,

NGPor NGPor NGP

pNGP

p

p

p

12

1 1

2 3 2

4 5 3

Nearest larger integer equal to (p+1)/2

( ) [ ]

( ) ( )

iF Fh

I F d F F

F F W

1

1 21 2

1

1 1

22 2

2 0

•

1

•

( )F 0

1

( )F

NUMERICAL INTEGRATIONJN Reddy - 25 Lecture Notes on NONLINEAR FEM

11 22

ˆ

1 1ˆ

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

ˆ( , ) ( , )

e

e

j je i iij

ij ij

NGP NGP

ij I J ij I JI J

K a a dxdyx x y y

F x y dxdy F x y J d d

F J d d W W F

WI – Weights(xI, yI) – Integration

points

TRANSFORMATION OF THE INTEGRAL

i i i i i i

i ii i i i

x y x yx y x x

x yx yy yx y

ψ ψ ψ ψ ψ ψξ ξ ξ ξ ξ ξ

ψ ψψ ψ ψ ψη ηη η η η

∂ ∂ ∂ ∂ ∂ ∂ ∂ ∂ ∂ ∂ = + ∂ ∂ ∂ ∂ ∂ ∂ ∂ ∂ ∂ ∂ ⇒ = = ∂ ∂ ∂ ∂ ∂ ∂ ∂ ∂∂ ∂ = + ∂ ∂∂ ∂∂ ∂ ∂ ∂ ∂ ∂

J



NUMERICAL INTEGRATION

1 1

1 1

1 11 2

2 2

1 2

ˆ ˆ

Jˆ ˆ

ˆˆ ˆ

ˆˆ ˆ

m mi ii ii i

m mi ii ii i

m

m

m m

x y x y

x yx y

x yx y

x y

1 *

J J

i ii

i i i

ξ ξx

y η η

−

∂ ∂∂ ∂ ∂∂ = = ∂ ∂ ∂

∂ ∂ ∂

Jacobian matrix

Global derivatives in terms of the local derivatives



ELEMENT CALCULATIONS

11 22

* * * *11 11 12 11 12

ˆ

* * * *22 21 22 21 22

( , )

( , )

e

j je i iij

j ji i

j ji i

K a a dxdyx x y y

a J J J J

a J J J J

1 1

ˆ 1 1

1 1

ˆ ˆ( , ) ( , )

ˆ ( , )

e eij ij

NGPNGPeij I J I J

I J

J d d

F d d F d d

F WW


,dxdy J dξ d Jη= = J


ξ

η

1 2

34

ξ

η

51 2

346

7

89

ξ

η

1

3

2

ξ

η

21

3

4

56

ξ

η

1

2

34

ξ

η

1

3

2

ξ

η

1

3

24

56

ξ

η

1

2

34

5

6

78

9

Master element Actual element



Computational Issues: - 31

η

ξ

ξ = − 13 ξ = 1

3

η = 13

η = − 13

η= 0

η

ξ

ξ = − 35 ξ = 3

5

η = 35

η = − 35

ξ= 0

η= 0.861...

ξ= − 0.861...

ξ

ηξ= 0.861...

η= 0.339...

η= − 0.339...η= − 0.861...

GAUSS QUADRATURE

Domain of the masterelement

Domain of the physical element

Gauss points Gauss weights

ˆ

, 1

ˆ,

ˆ ,

( ) ( )

( )

e e

ij ij

N

I J ij I JI J

F x y dxdy F ξ,η dξ dη

W W F ξ η


2-D Problems: 32

ASSEMBLY OF ELEMENTS/EQUATIONS(1 DoF per node)

•

•

•

• 1 3

2

4

•

•

••

•

1

2

43

5

1

1

1

11

2

3

4

5

6

7

9

8

a relationship between certain property of global

node and global node .0, if node and do not

belong to the same elemen .t

IJK

I JI J

(1) (1) (1)11 11 13 14 15 13 14

(1) (2) (1) (2)22 22 11 25 23 14 26

(1) (2) (3) (4)55 33 44 22 11

(3) (4) (1) (1) (2)56 23 13 1 1 2 2 1

(1) (2) (3) (4)5 3 4 2 1

, , , 0,

, , 0,

,

, , ,

K K K K K K K

K K K K K K K

K K K K K

K K K F F F F F

F F F F F

1 2 3 4

1 2 5 3

2 4 7 5

3 5 6[ ]

5 7 6

7 8 9 6

B

Connectivity matrix

Element local node numbers


2-D Problems: 33

POST-COMPUTATION OF VARIABLES

( ) ( )

1

( ) ( )( ) ( )

1 1

( , ) ( , ), ( , )

, , ( , )

ne e e eh j j

j

e en ne ej je e eh h

j jj j

u x y u x y x y

u uu u x y

x x y y

6

•

•

•

• 1 3

2

4

•

•

••

•

1

2

43

5

1

1

1

11

2

3

4

5

7

9

8


2-D Problems: 34

Derivatives of the Solution

( )( )

1 1

( )( )

1 1

1, ( , )

2

1, ( , )

2

en neje e e eh

j j jej j

en neje e e eh

j j jej j

uu u x y

x x

uu u x y

y y

Linear triangular element

Linear Rectangular element( )

( )

1 1

( )( )

1 1

1( ), ( , )

2

1( ), ( , )

2

en neje e e e eh

j j j jej j

en neje e e e eh

j j j jej j

uu u y x y

x x

uu u x x y

y y


1

1 2 3

6

9

1 1

1

1 2

3 4

x

y

a = 1

a = 1

u = 0

0=∂∂

−=∂∂

=yu

nuqn

(a)

4

0=∂∂

−xu

u = 0

7 8

5

52

6 10

1 3 4

15

25

1 11 x

ya = 1

u = 0

0=∂∂

−=∂∂

=yu

nuqn

(b)

16

1

20

11

21

u = 0

0=∂∂

−xu

a = 1

9

1 2 3 4

8

12

5

1613

13

8

18

Example 8.3.1 from the book2×2 mesh 4×4 mesh

11 1 12 2 14 4 15 5 1

1 1 1 1 1 1 11 2 4 511 12 14 13 1 1 1

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

For example, equation for node 1 is:

orK U K U K U K U F

K U K U K U K U f Q f

12 =∇− u


Coordinate, x

Solu

tion,

u(x

,0)

• Ritz solution∆ FEM (mesh T2)

Analytical

FEM (mesh T1)

x

y

0=∂∂

xu

0=∂∂

yu

0=u

0=u

12 =∇− u

Coordinate, x

Solu

tion,

qx(x

,0)

• FEM (mesh T2)FEM (mesh T1)Analytical

x

y

0=∂∂

xu

0=∂∂

yu

0=u

0=u

12 =∇− u

•

•

•

Finite Element, Ritz, and Exact Solutions


DISCUSSION (distributed boundary source and convection type boundary condition)

0

0

( ) ( )

( ) 1

h

i n i

n

Q q s s ds

sq s q

h

8 6

2 1,

3 2 3 2n nq h q h

Q Q

For linear elements

2-D Problems: 37

1

1

1

11 2

3 4

5

6

7 8

x

y

1.0

1.0

1 2

3 41

(0, ) 4 (1 )

0 1

u y y y

y

3 0nq u

( , 0) 4 (1 ), 0 1u x x x x

0 5q 0nq

s

h


DISCUSSION(computation of convection type BC)

00 0

010

01 0 0

( ) ( ) ( ) ( )

( ) ( )

h h

i n i i

h n

j j ij

h hn

j i j ij

Q q s s ds u u s ds

u u s ds

u ds u ds

0

General form of the BC:

( ) 0nq u u

1

1

1

11 2

3 4

5

6

7 8

x

y

1.0

1.0

1 2

3 41

(0, ) 4 (1 )

0 1

u y y y

y

03 0 ( 3, 0)nq u u

( , 0) 4 (1 ), 0 1u x x x x

5nq

0nq

2-D Problems: 38


Transient Analysis of 2-D Problems

( )2

1 2 11 222, ,

u u u uc c a a f x y t

t t x x y y

( )

2

1 2 11 222

2

1 2 11 222

0

, ,e

e

e

h h h hi

h h i h i hi i

i i

u u u uw c c a a

t t x x y yf x y t dxdy

u u w u w uw c w c a a

t t x x y y

w f dxdy qw ds

Weak formulation

Model Governing Differential Equation

2-D Problems: 39


1

( ) ( ) ( ) ( ), , , , ,n

e e eh j j

j

u x y t u x y t u t x y

Approximation

Finite element model

1 2

11 22

Cu Mu Ku F

,e e

e

e e

e eij i j ij i j

j je i iij

ei i i

C c dxdy M c dxdy

K a a dxdyx x y y

F f dxdy q ds

+ + =

= =

∂ ∂ ∂ ∂= + ∂ ∂ ∂ ∂

= +

∫ ∫

∫

∫ ∫

SPATIAL DISCRETIZATION:Finite Element Model

2-D Problems: 40


TIME APPROXIMATIONS (Parabolic)

K u F , K K C

F K C u F F

1 1 1 1 1 1

1 1 1 1 1

ˆ ˆ ˆ ,

ˆ (1 ) (1 )

s s s s s s

s s s s s s s

t

t t

u u u u

Cu Cu Cu Cu1 1 1

1 1 1

(1 )

(1 )

s s s s s

s s s s s

t

t

Alfa-family of approximation

Cu F K u1 1 1 1s s s s

C K u C K u

F F1 1 1

1 1

(1 )

(1 )

s s s s s s

s s s

t t

t

Cu F K us s s s

Cu Ku F, 0 t T

Semidiscrete FE model

Fully discretized model

2-D Problems: 41

NOTE: From here, the procedure is the same as in 1-D


TIME APPROXIMATIONS (Hyperbolic)

C u M u K u Fe e e e e e e

u u u uu u u u u u

2121 ,

1 , , 1, (1 )

s s s s

s s s s s s

t t

t

Semidiscrete FE model

Newmark scheme (second-order equations)

K u F , K K M C

F F M u u u C u u u

1 1 1 1 1 3 1 5 1

1 1 1 3 4 5 1 5 6 7

ˆ ˆ ˆ

ˆ

s s s s s s s

s s s s s s s s s s

a a

a a a a a a

Fully discretized model

2-D Problems: 42


An Example: Transient Heat Conduction Problem

1 ( , , )T T Tc k k f x y tt x x y y

52

6 10

1 3 4

15

25

x

y

16 20

11

21

12

34

78

910

2625 31

32

x

y

1.01.0

T = 0

)(insulated0=∂∂

−yT

T = 0

)(insulated

0=∂∂

−xT

0 0 0

0 1 1

x yT Tk n n x yx y

T y x

on and

on and0 1( , , )T x y

Governing equation

Boundary conditions

Initial condition

2-D Problems: 43


Stability Characteristics

0.00 0.02 0.04 0.06 0.08 0.10 0.12Time, t

-0.15

-0.10

-0.05

0.00

0.05

0.10

0.15Te

mpe

ratu

re,

00518.0triangles linear of mesh44

cri =∆×t

01050 .,. =∆= tα005000 .,. =∆= tα01000 .,. =∆= tα

T(0,

0,t)

2-D Problems: 44


0.00 0.20 0.40 0.60 0.80 1.00Distance, x

0.00

0.05

0.10

0.15

0.20

0.25

0.30

0.35

Tem

pera

ture

,T(x

,0,t f

ixed

)

500050 .,. ==∆ αtt = 1.0 (steady state)

t = 0.005

t = 0.1

t = 0.2

t = 0.5

0.0 0.4 0.8 1.2 1.6 2.0Time, t

0.00

0.05

0.10

0.15

0.20

0.25

0.30

0.35

Tem

pera

ture

, 500050 .,. ==∆ αt

T(0,

0,t)

A NUMERICAL EXAMPLE (Parabolic)

•

2-D Problems: 45


y

2ft4ft

x

x

y

52

6 10

u = 0

0=∂∂

−yu

1 3 4

15

2516

0=∂∂

−xu

2011

21u = 0

y

x1 2 45 8

121613

9

3

14 15

TRANSIENT RESPONSE OF A MEMBRANE

0.0 0.5 1.0 1.5 2.0 2.5Time, t

-0.50

-0.25

0.00

0.25

0.50

Defle

ction

,

u(2,

1, t

)

1058.0domain original the of quadrant a in

elements triangular linear of mesh44

cri =∆

×

t

• 5.0,5.0,125.0 ===∆ γαt

3/1,5.0,125.0 ===∆ γαt3/1,5.0,1.0 ===∆ γαt

0.0 0.5 1.0 1.5 2.0 2.5 3.0 3.5Time, t

-8

-6

-4

-2

0

2

4

6

8

Defle

ction

,

♦

2-D Problems: 46


2-D Problems: 47

SUMMARY 1. Starting point is a model differential equation in u2. Construct an integral statement – weak form,

which has three steps. Integration by parts that (a) relaxes (“weakens” differentiability on the variable u, and (b) brings in secondary variable into the integral form.

3. Substitute suitable approximation for u and obtain the finite element model (i.e., a set of algebraic relations between nodal values of u and Q).

4. Numerical evaluation of coefficients 5. Assemble equations, impose boundary conditions, and

solve the assembled equations. 6. Post-computation of variables follows same procedure as

in 1-D FEM.7. Reviewed time-dependent problems.

andij iK f


jn reddy - 1 lecture notes on nonlinear fem meen...

Documents