@ @ Innovative ship design-VariationalMehodand Approximation-

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Innovative Ship Design - Elasticity

@

SDALAdvanced Ship Design Automation Lab.

http://asdal.snu.ac.kr Seoul

National Univ.

N a v a l A rc hi te ct u re & Oc e a n En g in e e ri n g

@ SDAL

Advanced Ship Design Automation Lab.

National Univ.

June 2009

Prof. Kyu-Yeul Lee

Department of Naval Architecture and Ocean Engineering, Seoul National University of College of Engineering

[2009] [15]

Innovative ship design

- Variational Mehod and Approximation-

서울대학교 조선해양공학과 학부4학년 “창의적 선박설계” 강의 교재

(2)

weak form :

‘weighted average’*

Governing Equation

Differential Approach Energy Method

Stress

Strain Displacement

Force Equilibrium

Generalized Hooke’ Law Compatibility

Analytic Solution

Elasticity

Strain Energy , 0

=m =

∑

^F ^{a a} in equilibrium

Virtual Displacement Virtual Work

FEM

Rayleigh-Ritz Method Galerkin Method Deflection Curve of the Beam

Scantling

2

Wang,C.T.,

Applied Elasticity , McGRAW-HILL, 1953 응용탄성학, 이원 역, 숭실대학교 출판부, 1998 Chou,P.C.,

Elasticity (Tensor, Dyadic, and Engineering Approached), D. Van Nostrand, 1967

Gere,J.M.,

Mechanics of Materials, Sixth Edition, Thomson, 2006

Hildebrand,F.B.,

”Methods of Applied Mathematics”, 2^ndedition, Dover, 1965 Becker,E.B.,

“Finite Elements, An Introduction”, Vol.1, Prentice-Hall, 1981 Fletcher,C.A.J.,

“Computational Galerkin Methods”, Springer, 1984

1 2

1

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@

National Univ.

Summary

( ) e 2 0

G G u X

λ+ ^∂x+ ∇ + =

∂

( ) e 2 0

G G v Y

λ+ ^∂y+ ∇ + =

∂

( ) e 2 0

G G w Z

λ+ ^∂z + ∇ + =

∂

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

x x

y y

z z

E E

e

E E

e

E E

e

σ ν ε

ν ν ν

σ ν ε

ν ν ν

σ ν ε

ν ν ν

= +

+ − +

= +

+ − +

= +

+ − +

, e = ε

_x

+ ε

_y

+ ε

_z

, 2( 1)

xy xy

yz yz

zx zx

E E E

τ γ

ν

τ γ

ν

τ γ

ν

= +

6 Relations btw. 6 Strain and 6 Stress 6 Relations btw. Strain and Displacement

, , ,

, ,

x y z

xy yz zx

u v w

x y z

u v v w w u

y x z y x z

ε ε ε

γ γ γ

∂ ∂ ∂

= = =

∂ ∂ ∂

∂ ∂ ∂ ∂ ∂ ∂

= + = + = +

∂ ∂ ∂ ∂ ∂ ∂

6 Equations of force equilibrium 0

0

x yx zx

x

xy y zy

y

xz yz z

z

F X

x y z

F Y

x y z

F Z

x y z

σ τ τ

τ σ τ

τ τ σ

∂ ∂ ∂

= + + + =

∂ ∂ ∂

= + + + =

∂ ∂ ∂

= + + + =

∂ ∂ ∂

∑

0 0 0

x yz zy

y xz zx

z xy yx

M M M

τ τ τ τ τ τ

= − =

∑ ∑

∑

, , , , , , , ,

x yx zx xy y zy xz yz z

σ τ τ τ σ τ τ τ σ

, , u v w

, , , , ,

x y z xy yz zx

ε ε ε γ γ γ

6 Strain 9 Stress 3 Displacement

18 Variables

, , u v w

Given : Body force

X Y Z , ,

Find : Displacement

2 1 3

18 Equations

3 Variables 3 Equations

If we are interested in finding the displacement components in a body, we may reduce the system of equations to three equations with three unknown displacement components.

Variables and Equations

, : Lame Elastic constant µ λ

: Shear Moldulus G

: Young's Modulus E

u v w

e x y z

∂ ∂ ∂

= + +

∂ ∂ ∂

, , : bodyforce in x,y, and z direction repectively X Y Z

: Poisson's Ratio ν

2 2 2

2

2 2 2

x z y

∂ ∂ ∂

∇ = + +

∂ ∂ ∂

x y z

σ σ σ

Θ = + +

3/183

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Summary

Given : Body force

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

x x

y y

z z

E E

e

E E

e

E E

e

σ ν ε

ν ν ν

σ ν ε

ν ν ν

σ ν ε

ν ν ν

= +

+ − +

= +

+ − +

= +

+ − +

ε ε ε

= + +

, 2( 1)

xy xy

yz yz

zx zx

E E E

τ γ

ν

τ γ

ν

τ γ

ν

= +

6 Relations btw. 6 Strain and 6 Stress 6 Equations of force equilibrium

0 0

0

x yx zx

x

xy y zy

y

xz yz z

z

F X

x y z

F Y

x y z

F Z

x y z

σ τ τ

τ σ τ

τ τ σ

∂ ∂ ∂

= + + + =

∂ ∂ ∂

= + + + =

∂ ∂ ∂

= + + + =

∂ ∂ ∂

∑

0 0 0

x yz zy

y xz zx

z xy yx

M M M

τ τ τ τ τ τ

= − =

∑ ∑

, , ∑

X Y Z

2

1 3

15 Equations

2 2

1 0

1

1 0

1

1 0

1

xy

yz

zx

Y X

x y x y

Z Y

y z y z

X Z

z x z x

τ ν

∂ ∂  ∂ Θ

+ + ∇ + =

 ∂ ∂  + ∂ ∂

 

∂ ∂  ∂ Θ

+ + ∇ + =

∂ ∂  + ∂ ∂

 

∂ ∂ ∂ Θ

 + + ∇ + =

 ∂ ∂  + ∂ ∂

 

2 2

2

2 2

2

2 2

2

2 1 0

1 1

2 1 0

1 1

2 1 0

1 1

x

y

z

X Y Z X

x y z x x

X Y Z Y

x y z y y

X Y Z Z

x y z z z

ν σ

ν ν

ν σ

ν ν

ν σ

ν ν

∂ ∂ ∂  ∂ ∂ Θ

+ + + + ∇ + =

 

−  ∂ ∂ ∂  ∂ + ∂

∂ +∂ +∂ + ∂ + ∇ + ∂ Θ=

 

−  ∂ ∂ ∂  ∂ + ∂

∂ +∂ +∂ + ∂ + ∇ + ∂ Θ=

 

−  ∂ ∂ ∂  ∂ + ∂

, , , , ,

x y z xy yz zx

σ σ σ τ τ τ

Find : Stress

6 Variables 6 Equations

, , , , , , , ,

x yx zx xy y zy xz yz z

σ τ τ τ σ τ τ τ σ

, , u v w

, , , , ,

x y z xy yz zx

ε ε ε γ γ γ

6 Strain 9 Stress 3 Displacement

15 Variables

6 Relations btw. Strain and Displacement

, , ,

, ,

x y z

xy yz zx

u v w

x y z

u v v w w u

y x z y x z

ε ε ε

γ γ γ

∂ ∂ ∂

= = =

∂ ∂ ∂

∂ ∂ ∂ ∂ ∂ ∂

= + = + = +

∂ ∂ ∂ ∂ ∂ ∂

2

yz xy

x zx

y yz zx xy

yz zx xy

z

y z x x y z

or z x y x y z

x y z x y z

γ γ

ε γ

ε γ γ γ

γ γ γ

ε

 ∂ = ∂ −∂ +∂ +∂ 

 ∂ ∂ ∂  ∂ ∂ ∂ 

 ∂ ∂ ∂ ∂ ∂ 

 = − +

 ∂ ∂ ∂  ∂ ∂ ∂ 

 ∂ ∂ ∂ ∂ ∂ 

 ∂ ∂ =∂  ∂ + ∂ − ∂ 

  



2 2

2

2 2

2 2 2

2 2

2

2 2

y xy

x

y z yz

x zx

z

y x x y

z y y z

x z z x

ε γ

ε

ε ε γ

ε γ

ε

∂ +∂ =∂

∂ ∂ ∂ ∂

∂ ∂ ∂

 + =

 ∂ ∂ ∂ ∂

∂ +∂ =∂

∂ ∂ ∂ ∂



Compatibility equations 3 independent Equations

18 Variables

18 Equations

If we are interested in finding only the stress components in a body, we may reduce the system of equations to six equations with six unknown stress components

, : Lame Elastic constant µ λ

: Shear Moldulus G

u v w

e x y z

∂ ∂ ∂

= + +

∂ ∂ ∂

, , : bodyforce in x,y, and z direction repectively X Y Z

: Poisson's Ratio ν

2 2 2

2

2 2 2

x z y

∂ ∂ ∂

∇ = + +

∂ ∂ ∂

x y z

σ σ σ

Θ = + +

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Differential Equation (ODE/PDE)

Classification

Integral Equations

Variational formulation

Rayleigh-Ritz Approximate Method

Galerkin Collocation

Least Square

Approximate Method

Volterra

Weak Form²⁾ Approximate Method⁴⁾

Galerkin Collocation Least Square

FEM

( ) ( ) ( ) ^x ( , ) ( )

x y x F x aK x y d

α = +λ∫ ξ ξ ξ

( ) ( ) ( ) ^b ( , ) ( )

x y x F x aK x y d

α = +λ∫ ξ ξ ξ

Fredholm Leibnitz formula¹⁾

( ) ( )

( , )

( , ) [ , ( )] [ , ( )]

B x B x

A x A x

d d F x dB dA

F x d d F x B x F x A x

dx dx x dx dx

ξ ξ= ^∂ ξ ξ+ −

∫ ∫ ∂

1) Jerry, A.j., Introduction to Integral Equations with Applications, Marcel Dekker Inc., 1985, p19~25

2) ‘variational statement of the problem’ -Becker, E.B., et al, Finite Elements An Introduction, Volume 1, Prentice-Hall, 1981, p4

3) Becker, E.B., et al, Finite Elements An Introduction, Volume 1, Prentice-Hall, 1981, p2 . See also Betounes, Partial Differential Equations for Computational Science, Springer, 1988, p408 “…the weak solution is actually a strong (or classical) solution…”

4) some books refer as ‘Method of Weighted Residue’ from the Finite Element Equation point of view and they have different type depending on how to choose the weight functions. See also Fletcher,C.A.J., “Computational Galerkin Methods”, Springer, 1984

5) Jerry, A.j., Introduction to Integral Equations with Applications, Marcel Dekker Inc., 1985, p1 “Problems of a ‘hereditary’ nature fall under the first category, since the state of the system u(t) at any time t depends by the definition on all the previous states u(t-τ) at the previous time t-τ ,which means that we must sum over them, hence involve them under the integral sign in an integral equation.

whenever a smooth ‘classical(strong)’

solution to a (D.E.) problem exists, it is also the solution of the weak

problem³⁾ ¹

0( ) 0

(0 )0, (1) 0 u u x v dx

u u

− + −′′ =

= =

∫

¹

0(−u v′ ′+uv−xv dx) =0

∫

integration by part and demand the test functions

vanish at the endpoints

2 0

d dy

T y p

dx^ dx + ρω + =

Ex.)

2 2 2

0

1 0

2 2

l T dy

y py dx

δ∫ ^ ρω + − ^dx^ ^ =

, 0 1, (0 )0, (1) 0

u u x x

u u

− + =′′ < <

= =

Ex.) Work and Energy Principle

1

( ) ( )

n

k k i i

k

c s x F x

=

∑

=

( ) ( ) ²

1

min

b n

k k

a k

c s x F x dx

=

 − 

 

∑ 

∫

1

( ) ( ) ( ) ( )

n b b

k a i k a i

k

c ψ x s x dx ψ x F x dx

=

∑ ∫ =∫

, ( )ψ x =

∑

ⁿ_k₌1a_{k i}φ( )x

1

( ) ( )

n k k k

c s x F x

=

∑

≈ ^,s x^k^{( )}=φ^k^{( )}x −λ∫a^bK x^{( , ) ( )}ξ yξ ξd

assume:

0 1 1

( ) ( ) ( ) _{n n}( ) y x ≈φ x +cφ x ++cφ x

problem of a “hereditary’ nature⁵⁾ multiply and integration δy

integration by part and B/C

2 0

l d dy

T y p y dx

dx dx ρω δ

   + + 

   

 

∫

multiply and integration v

( ) ⁿ_j 1 _j _j( ) u x ≈

∑

₌ cφ x

( ) ⁿ1 _{i i}( ) v x ≈

∑

i₌ aφ x

- Variation and integration - Integration and variation

( ) ( ) ( ) ( )

{

⁰¹

}

⁰¹ ^{( )}

1 1 1

N N N

i i i j i j i i

i j i

a c φ x φ x φ x φ x dx a xφ x dx

= = =

  ′ ′ +  =

   

 

∑ ∑ ∫ ∑ ∫

1 n

j ij i

i₌ c k =F

∑

kij

Fi

1 2 3 4 x

1( )x φ

( )

2 x φ

3( )x φ

shape function

what is the relationship between ‘week form’ and

‘Variational formulation’?

[ ]

1 1 1

0(−u v dx′′ ) = −u v′ 0+ 0(u v dx′ ′)

∫ ∫

5/183

(6)

Classification

what is the relationship between ‘week form’ and ‘Variational formulation’?

Differential Equation (ODE/PDE)

Integral Equations

Variational formulation

Rayleigh-Ritz Approximate Method

Galerkin Collocation

Least Square Approximate Method

Volterra

Weak Form²⁾ Approximate Method⁴⁾

Galerkin Collocation Least Square

FEM

( ) ( ) ( ) ^x( , ) ( ) a

x y x F x K x y d

α = +λ∫ ξ ξ ξ

( ) ( ) ( ) ^b ( , ) ( ) a

x y x F x K x y d

α = +λ∫ ξ ξ ξ

Fredholm Leibnitz formula¹⁾

( ) ( )

( , )

( , ) [ , ( )] [ , ( )]

B x B x

A x A x

d d F x dB dA

F x d d F x B x F x A x

dx dx x dx dx

ξ ξ= ∂∂ξ ξ+ −

∫ ∫

whenever a smooth ‘classical(strong)’

solution to a (D.E.) problem exists, it is also the solution of the weak

problem³⁾ ¹

0( ) 0

(0 )0, (1) 0 u u x v dx

u u

− + −′′ =

= =

∫ ¹

0(−u v′ ′+uv−xv dx) =0

∫

alternative form (integration by part and B/C)

2 0

d dy

T y p

dx dx + ρω + =

Ex.)

2 22 0

1 0

2 2

l T dy

y py dx

δ∫^ρω + − ^dx^^ = , 0 1,

(0 )0, (1) 0

u u x x

u u

− + =′′ < <

= =

Ex.) Work and Energy Principle

1

( ) ( )

n

k k i i

k

c s x F x

=

∑ =

( ) ( )²

1 min

bn kk a

k

c sx F x dx

=

 − 

 

∑ 

∫

1

( )( ) ( ) ( )

n b b

kai k ai

k

c ψx s x dx ψx F x dx

=

∑ ∫ =∫

, ( ) ⁿ1_{k i}( )

x ka x

ψ =∑₌ φ

1

( ) ( )

n k k k

c s x F x

=

∑ ≈ ^,s x^k^{( )}=φ^k^{( )}x−λ∫a^bK x^{( , ) ( )}ξyξ ξd assume:

0 1 1

( ) ( ) ( ) n n( )

y x≈φ x+cφx++cφ x

problem of a “hereditary’ nature⁵⁾ multiply and integration δy

integration by part and B/C 2

0

l d dy

T y p y dx

dx dx ρω δ

   + + 

   

 

∫

multiply and integration v

( ) ⁿj1jj( ) u x≈∑₌cφ x ( ) ⁿ1_{i i}( ) v x≈∑i₌aφx

- Variation and integration - Integration and variation

1 n

j ij i

i₌c k =F

∑

1 2 34

shape function

d dy 2

T y p

dx^ dx + ρω = −

Ex: Rotating String)

2 2 2

0

1 0

2 2

l T dy

y py dx

δ

∫

^ ρω + − ^dx^ ^ =

differential equation

2 2 2

0 0

1 0

2 2

l l

T dy y y T dy

py dx d

x

ρ dx δ

δ ^ ω + − ^ ^ ^ + =

 

 

 

 

∫

2

0^l d dy 0

T y p dx

dx d y

x ρω δ

   + +  =

   

 

∫

multiply and integration δy

integration by part and B/C

Variational formulation:

physical meaning : “Minimize the difference of kinetic energy and

potential energy”

2

0 0

l d dy l

T y dx p dx

dx dx ρω ν ν

   +  = −

   

 

∫ ∫

multiply and ‘week form’

ν

Weighted Residual

if is has a meaning of minimizing the weighted residual, we may rewrite is as

virtual displacement (it has a physical meaning)

self-adjointform Natural B/C

2

0^l d dy 0

T y p dx

dx dx ρω ν

   + +  =

   

 

∫

Ritz method ( ) ⁿ 1 _j _j( ) y x ≈

∑

j₌ cφ x

( ) ⁿ_j 1 _j _j( ) y x ≈

∑

₌ cφ x

( ) ⁿ1 _{i i}( ) v x ≈

∑

i₌ aφ x

2

0^l d dy 0

T y p dx

dx dx

δ

∫

^ ^  + ρω + ^ν =

in case of using Galerkin method, weight function can be regarded as a kind of y since they have same basis functions

2 0

l d dy

T y p ydx

δ

∫

^ ^  + ρω + ^ =

a similarity

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Classification

1) Becker E.B., Finite Elements An Introduction, Volume1, Prentice-Hall, 1981

Our reference to certain weak forms of boundary-value problems as

“variational” statements arises from the fact that, whenever the

operators involved possess a certain symmetry, a weak form of the problem can be obtained which is precisely that arising in standard problems in the calculus of variations.

7/183

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Classification

Remark 14.1¹⁾In the case of positive definite operators, the Galerkin method brings nothing new in comparison with the Ritz method ; the two methods lead to the solution of identical systems of linear equations and to identical sequences of approximate solution. However, the possibility of the application of the Galerkin method is much broader that that of the Ritz method.

(Au_n − f,ϕ_k)=0, k=1,...,n

The Galerkin method, which is characterized by the condition does not impose beforehand any essential restrictive conditions on operator

It is in no way necessary that the operator A be positive definite, it need not even be symmetric, above all it need not ne linear.

Formally speaking, the Galerkin method can thus applied even in the case of very general operators

A

Remark 14.2^.Although both the Ritz and the Galerkin methods lead to the same results in the case of linear positive operators, the basic ideas of these methods are entirely different.

ex: deflection of beam

[

^EIu^{′′ =}

]

^′′ ^{q x}^{( )} with the B/C u(0) =u′(0) =0, ( )u l =u l′( )=0,

2

0 0

1 ( )

2

l l

EI u′′ dx− q u dx

∫ ∫

or minimize the function of energy

Definition 8.15²⁾ An operator is called positive in its domain if it is symmetric and if for all , the relations A D_A u∈D_A (Au u, )≥0 and (Au u, )=0 ⇒ =u 0 hold.

( , ) ^b 2

Au u =

∫

a Au dx′

If, moreover, there exists a constant such that for all the relation holds, then the operator is called positive definite in

0

C > u∈D_A (Au u, )≥C u² ²

A D_A

DA

in

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Classification

1) Rectorys, K. Variational Methods in Mathematics, Science and Engineering, Second edition, D.Reidel Publishing, 1980, p163

Remark 14.2^.Although both the Ritz and the Galerkin methods lead to the same results in the case of linear positive operators, the basic ideas of these methods are entirely different.

ex: deflection of beam

[

^EIu^{′′ =}

]

^′′ ^{q x}^{( )} with the B/C u(0) =u′(0) =0, ( )u l =u l′( )=0, ₂

0 0

1 ( )

2

l l

EI u′′ dx− q u dx

∫ ∫

or minimize the function of energy

1 ,

n

n k i i

u =

∑

₌ aϕ

approximate solution where satisfy the B/Cϕ_i

Galerkin method

( ) ( ) ( )

1 0 1 1 2 0 2 1 0 1 0 1

1 0 1 2 2 0 2 2 0 2 0 2

1 0 1 2 0 2 0 0

l l l l

n n

l l l l

n n

l l l l

n n n n n n

a EI dx a EI dx a EI dx q dx

ϕ ϕ ϕ ϕ ϕ ϕ ϕ

′′ ′′ ′′

′′ + ′′ + + ′′ =

′′ ′′ ′′

′′ + ′′ + + ′′ =

′′ ′′ ′′

′′ + ′′ + + ′′ =

∫ ∫ ∫ ∫







: multiply and integration

ϕ

Ritz method

( )

2

1 0 1 2 0 1 2 0 1 0 1

2

1 1 2 2 2 2 2

0 0 0 0

2

1 0 1 2 0 2 0 0

l l l l

n n

l l l l

n n

l l l l

n n n n n

a EI dx a EI dx a EI dx q dx

ϕ ϕ ϕ ϕ ϕ ϕ

′′ + ′′ ′′ + + ′′ ′′ =

′′ ′′ + ′′ + + ′′ ′′ =

′′ ′′ + ′′ ′′ + + ′′ =

∫ ∫ ∫ ∫







different?

The Galerkin method starting with the differential equation of the problem and the Ritz method with the respective functional

( ) ( ) ( ) ( ) ( )

( )

0 0 0

0

l l l l

i k i k i k i k i k

l

i k

EI dx EI EI dx EI EI dx

EI dx

ϕ ϕ ϕ ϕ ϕ ϕ ϕ ϕ ϕ ϕ

ϕ ϕ

 

′′ ′ ′

′′ = ′′  − ′′ ′ = − ′′ ′+ ′′ ′′

′′ ′′

=

∫ ∫ ∫

∫

identical

9/183

@ @ Innovative ship design-VariationalMehodand Approximation-

@

N a v a l A rc hi te ct u re & Oc e a n En g in e e ri n g

@ SDAL

June 2009

Prof. Kyu-Yeul Lee

Department of Naval Architecture and Ocean Engineering, Seoul National University of College of Engineering

Innovative ship design

- Variational Mehod and Approximation-

Contents

Governing Equation

Elasticity

∑

@

Summary

, e = ε

+ ε

+ ε

∑ ∑

∑

, , u v w

X Y Z , ,

Variables and Equations

Summary

∑ ∑

, , ∑

X Y Z

, , , , ,

σ σ σ τ τ τ

@

Classification

∫

∫

∑

∑

∑

∫

∑

∑

{

}

∑

Classification

∫

∫

∫

∫ ∫

ν

∫

∑

∑

∑

∫

∫

@

Classification

Our reference to certain weak forms of boundary-value problems as

“variational” statements arises from the fact that, whenever the

operators involved possess a certain symmetry, a weak form of the problem can be obtained which is precisely that arising in standard problems in the calculus of variations.

Classification

[

]

∫ ∫

∫

@

Classification

[

]

∫ ∫

∑

∫ ∫ ∫ ∫

∫ ∫ ∫ ∫

∫ ∫ ∫ ∫

ϕ

∫ ∫ ∫ ∫

∫ ∫ ∫ ∫

∫ ∫ ∫ ∫

∫ ∫ ∫

∫