Introduction in Finite Elements Modeling

8/8/2019 Introduction in Finite Elements Modeling

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Slide 1

Introduction to Modeling and Simulation, Spring 2002, MIT

1.992, 2.993, 3.04, 10.94, 18.996, 22.091

Introduction to Finite Element

Modeling in Solid Mechanics

Franz-Josef Ulm



Slide 2

Outline

1. Scales of Continuum Modeling

2. Elements of Continuum Solid Mechanics

3. Variational Principle

4. Application I

5. Application II: Water Filling of a gravity

dam



Slide 3

Scales of Continuum Modeling

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Λ

Rd Ω

l

B d

H

Modeling Scales



Slide 4

Modeling Scales

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10+1 m

Nuclear Waste

Disposal Structure

10−6 m

Material Science

d

d << l << H

10−3 m

10−2 m

Scale of

Continuum Mechanics

l

Modeling Scale (cont’d)



Slide 5

Elements of Continuum

Modeling: 1. Material• 1D-Think Model • Force Application

• Equilibrium

• Material Law E

1 [ L]

u=ε [ L] F d

A F d /=σ σ

ε σ E =



Slide 6

Forces & Conservation Law

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ρ f (x,t )

n T(x,t ,n)

Ωt

d Ωt

∂Ωt

Body & Surface Forces

σ22+

σ21+

σ23+

e1

e2

e3

σ23 _

σ21 _ σ22

_

dx2

dx1 dx3

div σ + ρ (f −γ) = 0 Dynamic Resultant Theorem



Slide 7

Elements of Continuum

Modeling: 2. Field = Structure• Heat Transfer

1. Conservation of Energy

(Field Equation)

2. Heat Flux Boundary

Condition

3. Heat Flux Constitutive

Law (Fourrier - Linear)

• Continuum Mechanics

1. Conservation of

Momentum

2. Stress Vector (Surface

Traction) B. Condition

3. Stress – Strain

Constitutive Law (linear)

0=+ r qdiv 02

2

=

++

dt

ud f div ρ σ

d qnq =⋅d

T n =⋅σ

T k q ∇−= uC s∇= :σ



Slide 8

Example: Material Law

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E 1 [ L]

ε [ L] σ

ϕdt = σ d ε − d ψ = 0

• 1 st + 2nd Law:

Work Stored Energy

= Helmholtz Energy

• 1D-Model:

ψ = 1/2Εε 2 σ =∂ψ ∂ε

• 3D-Model:

ψ = ψ (ε ij) σ ij =∂ψ ∂ε ij

Elasticity Potential



Slide 9

Stationary of Potential Energy

• Potential Energy

• Stationary of Potential

E L

x =ε L F d

A

1-Parameter System

)()( xW E pot Φ−= ε

Internal Energy

AL E W 2

2

1)( ε ε =

External Work

x F x d =Φ )(

0=−= dx F L

dx AL E dE d

pot ε

EA

L F L xdx

d

==∀⇒ ε ;



Slide 10

Convexity of a function

1.033/1.57 Mechanics of Material Systems

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x x’

f ( x’ )

f ( x)

Tangent

Secant

∂ f ∂ x x

( x’− x) ≤ f ( x’) − f ( x)

Convexity

of a function



Slide 11

Convexity of Free Energy

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∂ψ ∂ε ε

(ε’− ε) ≤ ψ (ε’) − ψ (ε)

E 1 [ L]

ε [ L] σ

σ State

Equation

Convexity: Applied to Free Energy



Slide 12

Variational Method (1D)

• Let x be solution of problem • Let x’ be any other approx. solution

External Work = Internal Work

AL x F d σε = AL x F d '' σε =

AL x x F d )'()'( ε ε σ −=−

( ) ( )[ ]')'( ε ψ ε ψ ε ε ε

ψ −≤−

∂

∂ AL

ε

ψ σ

∂

∂=

Convexity

4 434 4214 434 421

ION APPROXIMAT x E

SOLUTION x E pot pot

xW xW

)'()(

)'()'()()( Φ−≤Φ− ε ε Theorem of

Minimum

Potential Energy



Slide 13

Elements of 3D Variational

Methods in Linear Elasticity

• Starting Point: Displacement Field u’

• Calculate Potential Energy

• Minimize Potential Energy



Slide 14

Application to Finite Elements

• Displacement Field:

• Calculate Potential Energy:

• Minimize Potential Energy

∑=

=

= N i

i

jii x N qu1

)(

jiij ji qqk qqW 2

1),( = iii

q F q =Φ )(

)(),( i ji pot qqqW E Φ−=

0:0 =−= i jij pot F qk dE



Slide 15

Application I

• Displacement Field

• Potential Energy

• Minimization with

regard to DOF

E L

q F d

A

1-Parameter System

L

xqu ='

DOF

q F kq E d

pot −=2

1

L

EAk =

00 =−→=∂

∂= d pot

pot F kqdqq

E dE



Slide 16

Application II: Water Filling of a

Gravity Dam

α

x

y

H

mH

O A

B

B’

ρd g

ρw g



Slide 17

From 1D to 3D

• 1D

– Strain-Displacement

– Free Energy (p.vol)

– External Work

• 3D

– Strain-Displacement

– Free Energy (p.vol)

– External Work

( )uu t ∇+∇=2

1ε

( ) ( )( )ε ε µ ε λ ψ ⋅+= tr tr 212

22

1

L

l ∆=ε

2

2

1ε ψ E =

u F d =Φ ∫ ∫ Ω Ω∂

⋅+Ω⋅=Φ dauT d u f d

ρ

(Volume Forces) (Surface Forces)

Introduction in Finite Elements Modeling

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