dynamic steady-state analysis of structures under...

Dynamic Steady-State Analysis of Structures

under Uncertain Harmonic Loads

via Semidefinite Program

Yoshihiro Kanno† Izuru Takewaki‡

†University of Tokyo (Japan)‡Kyoto University (Japan)

July 8, 2009

motivation (static analysis)

Dynamic Steady-State Analysis IUTAM Symposium on the Vibration Analysis of Structures with Uncertainties – 2 / 20

■ 3DOF, 2-bar frame

■ uncertain load & stiffness

(c)

(1)

(2)

(b)

0 x

y

(a)

f∼

x0

y

f f =40000∼

200 kN

200 kN

200 kN

−200 kN

elastic modulus :

180 ≤ Ei ≤ 220 GPa

(24.0 cm2, 72.0 cm4)



■ 3DOF, 2-bar frame [Kanno & Takewaki 06; 08]

■ uncertain load & stiffness → bound of response?

(c)

(1)

(2)

(b)

0 x

y

(a)

f∼

−3 −2.5 −2 −1.5 −1 −0.5

−7.5

−7

−6.5

−6

−5.5

u ( cm )

u y (

cm)

x

distribution of nodal displacement





(c)

(1)

(2)

(b)

0 x

y

(a)

f∼

−3 −2.5 −2 −1.5 −1 −0.5

−7.5

−7

−6.5

−6

−5.5

u ( cm )

u y

(cm

)

x

bounding ellipsoid





(c)

(1)

(2)

(b)

0 x

y

(a)

f∼

−3 −2.5 −2 −1.5 −1 −0.5

−7.5

−7

−6.5

−6

−5.5

u (cm)

uy (

cm)

x

bounding box (interval hull)

uncertainty analysis


■ stochastic model — reliability design etc.

■ non-stochastic model — unknown-but-bounded parameters

◆ convex model [Ben-Haim & Elishakoff 90]

◆ interval analysis [Alefeld & Mayer 00], [Chen et al. 02], etc.

◆ mathematical programming

uncertainty analysis


■ stochastic model — reliability design etc.

■ non-stochastic model — unknown-but-bounded parameters

◆ convex model [Ben-Haim & Elishakoff 90]

■ linear approximation

◆ interval analysis [Alefeld & Mayer 00], [Chen et al. 02], etc.

■ conservative[Neumaier & Pownuk 07] (truss)

◆ mathematical programming

■ SDP & proof of conservativeness ← ♣[Calafiore & El Ghaoui 04] (ULE)

[Kanno & Takewaki 06] (static)

■ MIP & extremal case [Guo et al. 08], [Kanno & Takewaki 07]

objective: uncertainty analysis


K ¨u + C ˙u + Ku = f (eq. of motion)

■ f = feiωt — harmonic excitation

■ consider the uncertainty of f

■ predict the distribution of u

■ do not use the 1st order approximation→ large uncertainties

■ have a proof of conservativeness

mathematical framework


■ robust optimization (RO)

■ semidefinite program (SDP)

■ S-lemma

■ confidence bound detection← optimization / robust optimization

robust optimization


■ nominal (conventional) optimization

minx

{c(x) : g(x) ≥ 0}

e.g.) x: design variables

robust optimization


■ nominal (conventional) optimization

minx

{c(x) : g(x) ≥ 0}

e.g.) x: design variables

■ robust optimization [Ben-Tal & Nemirovski 98; 02]

minx

{c(x) : g(x; ζ) ≥ 0 (∀ζ : α ≥ ‖ζ‖)}

e.g.) ζ: unknown-but-bounded parameters

robust optimization


minx

{cTx : g(x; ζ) ≥ 0 (∀ζ : α ≥ ‖ζ‖)

}

feasible set

optimal solution

c

constraints

(nominal data)

nominal optimization

constraints

(nominal data)

robust optimal

solution

(nominal) optimal

solution

robust

constraints

perturbed

constraints

c

robust optimization

semidefinite program (SDP)


minm∑

i=1

biyi

s.t. C −m∑

i=1

Aiyi � O

variables : y1, . . . , ym

coefficients : b1, . . . , bm,

A1, . . . , Am, C ∈ Sn n× n symmetric matrices

■ P � O (⇔ P is positive semidefinite) ← nonlinear, convex

■ primal-dual interior-point method

■ applications in eigenvalue optimization, etc.

uncertainty model


(−ω2M + iωC + K

)u = f (eq. of motion)

■ f = feiωt — harmonic excitationuncertainty:

f= f j + F 0ζ, ζ ∈ Z

uncertainty model


(−ω2M + iωC + K



f= f j + F 0ζ, ζ ∈ Z

f nominal (best estimate) value

ζ unknown-but-bounded

F 0 coefficients (’magnitudes’ of uncertainties)

Z closed set

uncertainty model


(−ω2M + iωC + K



f= f j + F 0ζ, ζ ∈ Z

f1f1∼

f2∼

f10

f20

f2

Z = {ζ | 1 ≥ ‖T jζ‖2 (∀j)}

f1f1∼

f2∼

f10

f20

f2

Z = {ζ | 1 ≥ ‖ζ‖∞}

■ ‘magnitude & direction’ of amplitude of driving load are uncertain

reduction to real variables


(−ω2M + iωC + K


■ u ∈ Cd (complex vector)

[−ω2M + K −ωC

ωC −ω2M + K

] [v1

v2

]=

[f0

]

reduction to real variables


(−ω2M + iωC + K


■ u ∈ Cd (complex vector)

[−ω2M + K −ωC

ωC −ω2M + K

] [v1

v2

]=

[f0

]

■

[v1

v2

]=

[Re u

Im u

](: real part)(: imag. part)

is a real vector

■ for uq (displacement amplitude):

◆ |uq| (modulus) — r =√

v2

1q + v2

2q

◆ arg uq (argument) — tan θ = v2q/v1q

upper bound of modulus


maximum value of modulus (def.)

rmax = max {‖vq‖ | v solves (♠)} vq = (v1q, v2q)T

eq. of motion (w.r.t. real variables)

Sv =

[f0

], f ∈ F (♠)

■ max{‖vq‖} — nonconvex optimization

■ direct use of nonlinear programming→ no proof of conservativeness





reformulation to robust optimization

rmax = min {r | r ≥ ‖vq‖ (∀v ∈ V)} (RO)

Vq

Re uq

Im uq

rmax

r

(v )1q

(v )2q

conservative method for (RO) −→ find an upper bound of rmax

cf) robust optimization


minx

{cTx : g(x; ζ) ≥ 0 (∀ζ : α ≥ ‖ζ‖)

}

constraints

(nominal data)

robust optimal

solution

(nominal) optimal

solution

robust

constraints

perturbed

constraints

cconservative solution

(nearly optimal)

■ SDP relaxation technique→ find a conservative solution

S-lemma


f(x), g(x) : quadratic functions

(a): f(x) ≥ 0 =⇒ g(x) ≥ 0

m

(b): ∃w ≥ 0, g(x) ≥ wf(x), ∀x

S-lemma


f1(x), . . . , fm(x), g(x) : quadratic functions

(a): f1(x) ≥ 0, . . . , fm(x) ≥ 0 =⇒ g(x) ≥ 0

⇑

(b): ∃w ≥ 0, g(x) ≥m∑

i=1

wifi(x), ∀x

S-lemma



(a): f1(x) ≥ 0, . . . , fm(x) ≥ 0 =⇒ g(x) ≥ 0

⇑

(b): ∃w ≥ 0, g(x) ≥m∑

i=1

wifi(x), ∀x

■ cf.) Farkas’ lemma (f1, . . . , fm, g : linear functions)

(a) ⇔ the system f1(x) ≥ 0, . . . , fm(x) ≥ 0, g(x) < 0

does not have a solution

S-lemma



(a): f1(x) ≥ 0, . . . , fm(x) ≥ 0 =⇒ g(x) ≥ 0

⇑

(b): ∃w ≥ 0, g(x) ≥m∑

i=1

wifi(x), ∀x → p.s.d. constraint of SDP

S-lemma



(a): f1(x) ≥ 0, . . . , fm(x) ≥ 0 =⇒ g(x) ≥ 0

⇑

(b): ∃w ≥ 0, g(x) ≥m∑

i=1

wifi(x), ∀x

■ apply S-lemma to

rmax = min {r | r ≥ ‖vq‖ (∀v ∈ V)} (RO)

■ obtain an SDP providing r∗ (≥ rmax) (conservative bound)

bounds of argument


maximum value of argument (def.)

θmax = max {arctan(v1q/v2q) | v ∈ V}

bounds of argument




aa v =0T

a2

−a

Vq

Re uq

Im uq

θmax1

(v )1q

(v )2q

q

[a1 a2

] [v1q

v2q

]= 0 ⇒

v1q

v2q

= −a1

a2

bounds of argument




reformulation to robust optimization

−1/ tan θmax = min

{a2 |

[1 a2

] [v1q

v2q

]≥ 0 (∀v ∈ V)

}(RO)

aa v =0T

a2

−a

Vq

Re uq

Im uq

θmax1

(v )1q

(v )2q

q

conservative method for (RO) −→ find an upper bound of θmax

ex. (2DOF, 2-bar truss)


■ harmonic driving load — feiωt

■ uncertainty — f = f + ζ α ≥ ‖ζ‖

■ α = 10%

(a)

x

y

(b)

(c)

f∼

(1)

(2) x0

y

f

f =10 kN∼

α kN α kN

α kN

−α kN

■ primal-dual interior-point method: SeDuMi 1.05 [Sturm 99]





■ α = 10% ω = ω0

1: fundamental circular frequency (undamped)

(a)

x

y

(b)

(c)

f∼

(1)

(2)

−0.4 −0.2 0 0.2 0.4 0.6 0.8−1.8

−1.6

−1.4

−1.2

−1

−0.8

−0.6

real (cm)

imag

inar

y (c

m)

displacement amplitude ux(∈ C)





■ α = 10% ω = ω0


(a)

x

y

(b)

(c)

f∼

(1)

(2)

−2 −1 0 1 2

3

4

5

6

7

real (cm)

imag

inar

y (c

m)

displacement amplitude uy(∈ C)





■ α = 10% ω = ω0


(a)

x

y

(b)

(c)

f∼

(1)

(2)

−0.4−0.2 0 0.2 0.4 0.6 0.8−1.8

−1.6

−1.4

−1.2

−1

−0.8

−0.6

real (cm)

imag

inar

y (c

m)

0 100 2000.5

1

1.5

2

sample number

mod

ulus

(cm

)

0 100 200−1.5

−1.45

−1.4

−1.35

sample number

phas

e an

gle

(rad

)random samples (ux)



■ (ω = ω0

1) variations w.r.t. α (‘magnitude’ of uncertainty)

0 0.2 0.4 0.6 0.8 1

0.8

1

1.2

1.4

1.6

1.8

2

α

mod

ulus

(cm

)

|ux|

0 0.2 0.4 0.6 0.8 1−1.5

−1.45

−1.4

−1.35

αph

ase

angl

e (r

ad)

arg ux



■ driving load feiωt (ω = 1.1ω0

1)

−0.15 −0.1 −0.05 0 0.05 0.1−0.25

−0.2

−0.15

−0.1

−0.05

0

real (cm)

imag

inar

y (c

m)

displacement amplitude ux

0.6 0.8 1 1.2 1.4−0.2

−0.1

0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

real (cm)

imag

inar

y (c

m)

displacement amplitude uy



■ (ω = 1.3ω0

1)

0.05 0.06 0.07 0.08 0.09 0.1 0.11 0.12−0.05

−0.04

−0.03

−0.02

−0.01

0

0.01

0.02

0.03

real (cm)

imag

inar

y (c

m)


0.2 0.25 0.3 0.35 0.4 0.45−0.1

−0.05

0

0.05

0.1

0.15

real (cm)

imag

inar

y (c

m)




■ (ω = ω0

2) : 2nd fundamental frequency (undamped)

−0.3 −0.2 −0.1 0 0.1 0.2 0.3

−3.3

−3.2

−3.1

−3

−2.9

−2.8

−2.7

real (cm)

imag

inar

y (c

m)


−0.05 0 0.05 0.1−0.8

−0.78

−0.76

−0.74

−0.72

−0.7

−0.68

−0.66

−0.64

−0.62

real (cm)

imag

inar

y (c

m)




■ distribution of the vector

[|ux||uy|

](ω = ω0

1)

−2 0 2 4 60

1

2

3

4

5

6

7

8

−|ux| (cm)

|uy| (

cm)

−3 −2 −1 0 1

3

4

5

6

7

−|ux| (cm)

|uy| (

cm)




[|ux||uy|

](ω = 1.1ω0

1)

−0.4 −0.2 0 0.2 0.4 0.6 0.80

0.2

0.4

0.6

0.8

1

1.2

−|ux| (cm)

|uy| (

cm)

−0.4 −0.2 0 0.20.5

0.6

0.7

0.8

0.9

1

1.1

1.2

1.3

−|ux| (cm)

|uy| (

cm)




[|ux||uy|

](ω = 1.3ω0

1)

−0.1 0 0.1 0.2 0.3 0.40

0.1

0.2

0.3

0.4

0.5

|ux| (cm)

|uy| (

cm)

0 0.05 0.1 0.15 0.2 0.25

0.2

0.25

0.3

0.35

0.4

0.45

|ux| (cm)

|uy| (

cm)




[|ux||uy|

](ω = ω0

2)

0 1 2 30

0.5

1

1.5

2

2.5

3

3.5

4

|ux| (cm)

|uy| (

cm)

2.8 3 3.20.4

0.5

0.6

0.7

0.8

0.9

1

1.1

|ux| (cm)

|uy| (

cm)



(b) (a) x

y

f∼f

∼

■ nominal driving loadf : 8 kN, 12 kN

■ uncertainty nodal loads

◆ at all the nodes

◆ perturb independently

■ complex damping:ωC = 2βK (β = 0.02)



■ displacement amplitude (bottom-right node) (ω = ω0

1)

−1 −0.5 0 0.5 1

1.5

2

2.5

3

real (cm)

imag

inar

y (c

m)

ux

−8 −6 −4 −2 0 2 4 6

10

12

14

16

18

20

22

real (cm)

imag

inar

y (c

m)

uy



■ displacement amplitude (bottom-right node)variations w.r.t. α (‘magnitude’ of uncertainty)

0 0.2 0.4 0.6 0.8 11

1.5

2

2.5

3

3.5

α

mod

ulus

(cm

)

|ux|

0 0.2 0.4 0.6 0.8 11.54

1.56

1.58

1.6

1.62

1.64

1.66

1.68

α

phas

e an

gle

(rad

)

arg ux



■ displacement amplitude (bottom-right node)variations w.r.t. α (‘magnitude’ of uncertainty)

0 0.2 0.4 0.6 0.8 18

10

12

14

16

18

20

22

α

mod

ulus

(cm

)

|uy|

0 0.2 0.4 0.6 0.8 11.575

1.58

1.585

1.59

1.595

1.6

αph

ase

angl

e (r

ad)

arg uy



■ displacement amplitude (bottom-right node)variations w.r.t. ω (circular frequency)

1.3 1.4 1.5 1.6 1.7 1.8 1.9 2−0.5

0

0.5

1

1.5

2

2.5

3

3.5

ω (rad/sec)

mod

ulus

(cm

)

|ux|

1.3 1.4 1.5 1.6 1.7 1.8 1.9 20

5

10

15

20

25

ω (rad/sec)m

odul

us (

cm)

|uy|




[|ux||uy|

](ω = ω0

1)

0 10 20−5

0

5

10

15

20

|ux| (cm)

|uy| (

cm)

−5 0 58

10

12

14

16

18

20

22

|ux| (cm)

|uy| (

cm)




[|ux||uy|

](ω = 1.2ω0

1)

0 0.5 1 1.5

0

0.5

1

1.5

|ux| (cm)

|uy| (

cm)

−0.4 −0.2 0 0.2 0.4 0.6 0.80.6

0.8

1

1.2

1.4

1.6

1.8

|ux| (cm)

|uy| (

cm)




[|ux||uy|

](ω = 1.3ω0

1)

0 0.5 1

−0.2

0

0.2

0.4

0.6

0.8

1

|ux| (cm)

|uy| (

cm)

−0.2 0 0.2 0.4 0.60.3

0.4

0.5

0.6

0.7

0.8

0.9

1

1.1

|ux| (cm)

|uy| (

cm)




[|ux||uy|

](ω = ω0

2)

−1 −0.5 0 0.5

−0.5

0

0.5

1

−|ux| (cm)

|uy| (

cm)

−1 −0.8 −0.6 −0.4 −0.2 00

0.2

0.4

0.6

0.8

1

1.2

−|ux| (cm)

|uy| (

cm)




[|ux||uy|

]

variation w.r.t. α (‘magnitude’ of uncertainty)

0 0.2 0.4 0.6 0.8 18

10

12

14

16

18

20

22

α

mod

ulus

(cm

)




[|ux||uy|

]

variation w.r.t. ω (circular frequency)

1.3 1.4 1.5 1.6 1.7 1.8 1.9 20

5

10

15

20

25

ω (rad/sec)

mod

ulus

(cm

)

conclusions


■ uncertainty of dynamic load

◆ uncertainty of harmonic exciting load

◆ steady state of a damped structure

■ bounds of response

◆ modulus & argument of displacement amplitude

◆ RO formulation

◆ SDP approximation

◆ conservativeness← an approximate optimal solution of (RO)

dynamic steady-state analysis of structures under...

Documents