introduction in finite elements modeling
TRANSCRIPT
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
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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
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Slide 3
Scales of Continuum Modeling
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Λ
Rd Ω
l
B d
H
Modeling Scales
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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)
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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 =
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Slide 6
Forces & Conservation Law
1.033/1.57Mechanics ofMaterial Systems
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
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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
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Slide 8
Example: Material Law
1.033/1.57 Mechanics of Material Systems
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
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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 ;
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Slide 10
Convexity of a function
1.033/1.57 Mechanics of Material Systems
x x’
f ( x’ )
f ( x)
Tangent
Secant
∂ f ∂ x x
( x’− x) ≤ f ( x’) − f ( x)
Convexity
of a function
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Slide 11
Convexity of Free Energy
1.033/1.57 Mechanics of Material [email protected]
∂ψ ∂ε ε
(ε’− ε) ≤ ψ (ε’) − ψ (ε)
E 1 [ L]
ε [ L] σ
σ State
Equation
Convexity: Applied to Free Energy
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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
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Slide 13
Elements of 3D Variational
Methods in Linear Elasticity
• Starting Point: Displacement Field u’
• Calculate Potential Energy
• Minimize Potential Energy
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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
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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
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Slide 16
Application II: Water Filling of a
Gravity Dam
α
x
y
H
mH
O A
B
B’
ρd g
ρw g
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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)