Intro.pptemweb.unl.edu/Negahban/FEMFundamentals/Introduction/Intro.pdf · Title: Microsoft...

ENGM 918

Fundamentals of Finite Elements

Mehrdad Negahbang

W311 Nebraska HallDepartment of Engineering Mechanics

University of Nebraska-Lincoln

Phone: 472-2397E il hb @ l dE-mail: [email protected]

Course description

Objective:

• Solution of partial differential equations of engineering importance usingSolution of partial differential equations of engineering importance using FEM. • Emphasizing the similarity of the development.• Development of common mathematical and numerical tools.• Using consistent and robust methods.• Developing an object oriented programming structure for FEM.• Addressing some advanced topics. 4 Elements

7 7 9 9 N d El t7x7 Element: 16 Load steps, 127 Newton Iterations9x9 Element: 24 Load steps 163 Newton Iterations

AuBv−

7x7 or 9x9 Node Elements 9x9 Element: 24 Load steps, 163 Newton Iterations

F Fw

FA

B

uv

w

Topics

Course

• Object oriented programming in FEMW i h d R id l M h d• Weighted Residual Methods

• Linear Problems (scalar and vector fields, multi physics)• Heat transfer, elasticity, and thermoelasticity

• Constraints• Constraints• Non-Linear Problems

• Nonlinear elasticity (large deformation)• Initial-Value ProblemsInitial Value Problems

• Transient thermal analysis• Vibration of beams

• Bars, Plates and Shells,• Kinematically defined continua

• Numerical Implementation

Peanut Sunflower Eggs

ShellsPeanut Sunflower Eggs

Solanum cinereum germinatingSolanum cinereum germinatingCoconut

ShellsAppleApple

Red blood cell

Cell structure: Procaryote

Coronary artery

Cell structure: Procaryote

Plant cell wall

Shells

Plant cells Plant cell wallPoplar leaf cell

Chloroplast: site of photosynthesis in plant cellsTallow tree

Typical FEM Formulation

Shell Kinematics

Constitutive Characteristics

Finite ElementApproximation

Finite ElementFormulation

Initial and BoundaryConditions

Numerical Solution

Finite Element Formulation

Shell Kinematics

Variational FEM Formulation

Finite ElementApproximation

Formulation FEM Formulation

Constitutive Characteristics

Initial and BoundaryConditionsConditions

Numerical Solution

Shell Kinematics

),(),,( 21321 ξξξξξζ dρ = Q )( ξξ

3ξ Director

ξ

21321 Q ),( 21 ξξdρ

Curvilinear 2ξ

Pt

Coordinate

)( ξξ),,( 321 ξξξx

1ξ

),( 21 ξξsP

Q

“Mid”-surfacetO

Reference Frame

Shell Kinematics

Reissner/Mindlin Shell

⎧ ∂∂∂

+= ),(),,(),(),,( 2132121321

dsdsx

ςξξξξξςξξξξξ jiijg gg o=

⎪

⎪⎪⎨

⎧

∂

=∂∂

+∂∂

+∂∂

=∂∂

=3

2,1

i

iiii

ii

d

ddsxg ς

ξς

ξς

ξξ

),(),,( 21321 ξξξξξζ dρ = Q ),( 21 ξξd

3ξ

ρ

⎪⎪⎩

=∂

∂ 33

ii dξςξ

2ξ

P

ρ

),( 21 ξξd

)( ξξdP

ξ

),( 21 ξξs),,( 321 ξξξx

),( 21 ξξd−

⎧ 1)1( ξξ 1ξ

O⎩⎨⎧

=−=−

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

21

21

ξξςξξς

Making the mappings into material mappings:

Shell Kinematics

1)( −=∇= oJJxF X

dInitial configuration

Current configuration

Making the mappings into material mappings:

D

g

ΩΓ

Γ

)(XJ ξ∇=o s )(xJ ξ∇=oΩ

oΓ

3ξS

ξ

ii egJ ⊗=o

2ξ

e

iio eGJ ⊗=

1ξ2e3e

1eO

Shell Kinematics

1)( −=∇= oJJxF X Current configuration

Deformation Gradient: 1−= oJJFIFH −=Displacement Gradient:

D


)(1 HHε += T )(XJ ξ∇=o s

D

)(xJ ξ∇=

Strains:

)(21

)(2

IFFG −= TR

3ξS

)(ξ

)(212

IFFG −= TL

2ξ

2e3e

O1ξ2e

1e

ξ

Isoparametric shell

),(),( 2121 ξξξξ ∑=nnpe

pp Nss 2ξ

3ξ

52

Current Configuration),(),,(),(),,( 2132121321 ξξξξξςξξξξξ dsx +=

),(),( 211

21

211

21

ξξξξ ∑

∑

=

=

=nnpe

pp

p

pp

Ndd1d

4

8 1

5

),()(),,( 211

3321 ξξξςξξξς ∑=

=nnpe

pp

p N1ξ 1s

4

3eIsoparametric 2-D

3ξO

2e1e

)1,1(2ξ

12 5

Isoparametric 2 DExtruded 3-D

2ξ

3e4 8 15

21ξ6 89

1ξ 2e1e

ξ

347)1,1( −−

Isoparametric shell

I iti l C fi ti),(),,(),(),,( 2132121321 ξξξξξςξξξξξ DSX o+= Initial Configuration

3ξ3ξ

),(),( 2121 ξξξξ ∑=nnpe

pp NSS

2ξ

1D8 1

52

2ξ

1D8 1

52

),(),( 211

21

211

21

ξξξξ ∑

∑

=

=

=nnpe

pp

p

pp

NDD

Isoparametric 2-D 1ξ 1S

4

1ξ 1S

4),()(),,( 211

3321 ξξξςξξξς ∑=

=nnpe

pp

poo N

3ξ)1,1(2ξ

12 5

Isoparametric 2 DExtruded 3-D

1ξ S

2e3e

1e

1ξ S

2e3e

1e2e

3e

1e

2ξ

3e4 8 15

21ξ6 89 OO

1ξ 2e1e

ξ

347)1,1( −−

)()()( ξξξξξξξξ ρs +

ii

exJ ⊗∂∂

=ξ i

io eXJ ⊗

∂∂

=ξJacobians

),(),,(),(),,( 2132121321 ξξξξξζξξξξξ dsx +=

),,(),(),,( 32121321 ξξξξξξξξ ρsx +=

ppp

ppp

=Δ

−=Δ

DddSss

ρ

),(),()(),(),,( 2121321321 ξξξξξζξξξξξ qq

pp

pp NNN dsx +=

qq

p

ppp NN

Nξξξξ

ξξζ

ξξξ

ξ ∂∂

+∂

∂=

∂∂ ),(),()(),(

2121321 dsx

po

pp ζζζ −=Δ

−=Δ Dddxs

i

qqp

pq

q

i

pp

qpiii

NNN

Nξξξ

ξξξζξξξξξ

ξζ

ξξξξξξξ

∂

∂+

∂

∂+

∂∂∂),(

),()(),(),(

)(

),(),(

2121321

213

2121

dd Thickness unknowns

∑=

+=nif

ii

pi

po

p fa1

333 )()()( ξξζξζ

qq

pi

po

i

pp

i

NN

NNN

ξξξξ

ξξξξξξζ

ξξξ

ξ∂∂

∂∂

+∂

∂=

∂∂

)()(

),(),()(),(2121

321 DSX Thickness basefunctions

i

qqp

poq

q

i

ppo

NNN

Nξξξ

ξξξζξξξξξ

ξζ∂

∂+

∂

∂+

),(),()(),(

),()( 21

2132121

3 DD33)( ξξζ =p

o

3ξNumber of interpolation functions

Thickness interpolation

33)( ξξζ =po

∑+=nif

ii

pi

po

p fa1

333 )()()( ξξζξζ

3ξ1

)( 3ξif

Number of interpolation functions

=i 1

1−Legendre polynomials:

Material surface

⎪

⎪⎨

⎧ −=

+ oddisiPPf

oi

i

)()()(

331

3

ξξξ

⎪⎩ −+ evenisiPPi )()( 3131 ξξ

One term approximation:

)1)(0()()( 23131131 ξζξξζ −Δ==Δ ppp fa

Mid-surface

R id l f FEM

FEM formulationNominal stress TFT 1−= J

∫∫∫∫ΩΩΩΓ

Ω−Ω+Ω∇−Γ= oooooToo

No ddddR aubuTutu X ooo ρρ:)()(

Residual for FEM: Nominal stress TFT = Jo

1)( −=∇= oJJxF X


Current configuration

ΩΩΩΓ oooo

oTo

No NTNTt ==)(

DoΓN)(Not

3ξ

)(XJ ξ∇=o

Ss )(xJ ξ∇=oΩ

⎧ =Δ= jsU q 321

Nodal unknowns:

2ξ⎪⎩

⎪⎨

⎧

===Δ==Δ=

+

nifjaUjdUjsU

qjj

qjjq

jqj

,...,13,2,13,2,1

)6(

)3(

1ξ2e3e

1eO

⎩ + nifjaU jjq ,...,1)6(

Nonlinear elasticity

D f ti di t ( l)

oldnew

UtU

t

Δ∂

+Δ∂

+

Δ+=FFFFF

FFF

&

&

Deformation gradient: (general)11

1−−

−

∂⊗∂

=∂∂

=∂∂

=∂∂

oqj

iio

qjqj

o

qj UUUUJegJJJJF

1⊗∂∂ i JgF ∂x

qjqj

oldqjqj

oldnew UU

tUU

Δ∂

+=Δ∂

+= FFF

Nominal stress: (nonlinear elastic)

1−⊗∂∂

=∂∂

oiqj

i

qj UUJegF

ii ξ∂

∂=

xg

qjqj

ooldooldonewo

oldonewo

UU

t

t

Δ∂∂

∂+=Δ∂+=

Δ+=FTFTFTFTFT

TFTFT

FF :)()(:)()()(

)()(

&

&

From shell kinematics

⎧ =Δ= jsU q 321qj

qjoldonewo U

UΔ

∂∂

+=FFTFT :)()( oE

M t i l ifi

⎪⎩

⎪⎨

⎧

===Δ==Δ=

+

+

nifjaUjdUjsU

qjjq

qjjq

jqj

,...,13,2,13,2,1

)6(

)3(

Material specific

)( oTF∂=oETangent modulus of nominal stress:

Nonlinear elasticity

Two parameter nonlinear-elastic material model:

Nominal stress:

Tangent modulus:g

Material parameters: Shear and bulk modulus matched to linear response

Examples from Nonlinear Elasticity

Cantilevered beam with distribute end load

4 Load steps41 Newton iterations 0.11.00.10

0.40.0102.1 max6

=====×=

bhLFE ν

Cantilevered beam with distribute end load

Ftipw

tipu−

w

u

vw

Examples fromNonlinear Elasticity

Cantilevered beam with distribute end moment

Single (9x3)-node element16 loading steps

Cantilevered beam with distribute end moment

2360oM

M =

M16 loading steps175 Newton iterations

102.1 6×=E2

4360oM

M =

0.11.00.12

0.0

====

bhLν

oMM 36032

=

wTheoretical

oMM 360=tipu−

tipw

vw

u


Cantilevered beam with distribute end momentCantilevered beam with distribute end moment

(a) 2-(4x3)-360o-16LINC-187NI (b) 3-(4x3)-360o-16LINC-147NI (c) 4-(4x3)-360o-16LINC-147NI( ) ( ) ( ) ( ) ( ) ( )

(f) 3-(5x3)-540o-24LINC-224NI(d) 2-(5x3)-360o-16LINC-150NI (e) 3-(5x3)-360o-16LINC-150NI

(g) 3-(6x3)-540o-24LINC-221NI (h) 3-(6x3)-720o-32LINC-309NI (i) 3-(7x3)-720o-32LINC-329NI


Buckling in compression of a columnBuckling in compression of a column

u u

vw

tipu−

301002 11 =×=E ν

u

tipw

0045.0075.05.0

3.0100.2

===

=×=

hbL

E ν

L FL FCritical buckling load


Transverse loading of an annular ringTransverse loading of an annular ring

Aw BwB

8.003.0

0.100.60.01021

max

6

==

===×=

qh

RRE

oi

νA

w

vu

A

B

qA

B

q

32 Load Steps243 Newton steps5 Elements(7x7)-Nodes per element

R

oR

A

B

q

iR


Pinching of hemispherical shell with hole

4 Elements7x7 or 9x9 Node Elements

7x7 Element: 16 Load steps, 127 Newton Iterations9x9 Element: 24 Load steps, 163 Newton Iterations

Pinching of hemispherical shell with hole

AuBv−

ric

Symm

etric

Hole

z

F F

FA

B

uv

wSym

met

r c

F

F

R = 10h 0 04

A

B y

FuF h = 0.04E = 6.825x107

ν = 0.3Hole = 18o

x


Pinching of cylindrical shell with free endsPinching of cylindrical shell with free ends

v

40 Load Steps174 Newton steps4 Elements(9x9) Nodes per element

FF

Bv−Cv−Aw(9x9)-Nodes per element

A

Cv−

Bv−

FFww

R

2L

B

C

400003125010510 6 ==×= FE νvu

vu 094.0953.435.10

400003125.0105.10 max

=====×=

hRLFE ν


Point load on a hinged cylindrical roof

Hinged

FreeF

Hinged

FreeF

3.075.31021.02542540

=====

νθ

EradLR

Point load on a hinged cylindrical roof

Hinged

H

Free

R

LHinged

H

Free

R

L

Rθ

Rθ

Four parameter elastic model

⎤⎡ ⎞⎛ 2

τ ⎟⎠⎞

⎜⎝⎛ −⎥⎥⎥⎥⎤

⎢⎢⎢⎢⎡

+⎟⎟⎠

⎞⎜⎜⎝

⎛−

+−= ∞

∗

IBBIT )(31

)3(tanh

)1( 3/52

12

2

3/5 trJG

G

IG

JGJ

o

o

oo

τκ

G

⎠⎝⎥⎥

⎦⎢⎢

⎣−∗

3)3( 12

JIGJ

o

o

τ

∞G1 )det(F=J

)(tr Coτ

( )GG κτ ,,, FFC T

32

*1

)(

J

trI C=

oGγ1

( )ooo GG κτ ,,, ∞ FFC T=

γ1

Large deformation elastic-plastic model

E

⎟⎟⎠

⎞⎜⎜⎝

⎛−+−= IBBIT

3)(1)1(

35

ee

e

e tr

JGJκ )21(3 ν

κ−

=E

EG

Cauchy stress:

⎠⎝3J)1(2 ν+

=G

be1e TTFFΔT −= − 0Tb =Over-stress:

22f ΔSΔS IΔTΔTΔS )(tr

Yield function:

2

3: yf σ−= ΔSΔS IΔTΔS

3)(

−=

Power-law hardening rule:

nyoy a )1( ξσσ +=

pLΔT :Tξ& pLΔT :=ξ

Two models in tension

3500

4000 Four Parameter Model

Large Deformation Elastic-Plastic Model

2500

3000

4 parameter Elastic-plastic

2000

2500

Stre

ss

000,45000,90

==

Go

oκ000,45000,90

==

Gκ

4-parameter Elastic plastic

1000

1500

700520,1

,

==

∞Go

o

τ6.0

700,1

=

=

ayoσ

500115.0=n

00 0.05 0.1 0.15 0.2 0.25 0.3 0.35

Strain

Beam in compression

12000

Four Parameter Brick

8000

10000

on)

Four Parameter Brick

Four Parameter Shell

Large Deformation Elastic-Plastic Brick

6000

8000

Com

pres

sio

u

4000Loa

d (C

2000L=10, b=1, h=1

0-7 -6 -5 -4 -3 -2 -1 0

Displacement

Beam in tension

3000

3500

2500

3000

1500

2000

Loa

d

u

1000

1500Four Parameter Brick

Four Parameter Shell

500Large Deformation Elastic-Plastic Brick

L=10, b=1, h=1

00.0 0.4 0.8 1.2 1.6

Displacement

Cantilevered beam

1200Four Parameter BrickFour parameter brickLarge Deformation Elastic-Plastic BrickL D f ti El ti Pl ti B i k

uv

800

1000Large Deformation Elastic-Plastic BrickFour Parameter ShellFour Parameter Shell

v

600

Loa

d

L=10, b=1, h=1

uv

400

0

200

0-12 -10 -8 -6 -4 -2 0

Displacement

Multi-scale analysis

Each integration point can have an RVE

NaBRO

Questions?Q

Intro.pptemweb.unl.edu/Negahban/FEMFundamentals/Introduction/Intro.pdf · Title: Microsoft...

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