1 Cesos-Workshop-March-2006 RELIABILITY-BASED STRUCTURAL OPTIMIZATION FOR POSITIONING OF MARINE...

Cesos-Workshop-March-2006 1

RELIABILITY-BASED STRUCTURAL OPTIMIZATION FOR

POSITIONING OF MARINE VESSELS

B J Leira NTNU Trondheim Norway

P I B Berntsen NTNU Trondheim Norway

O M Aamo NTNU Trondheim Norway

Objective

bull To investigate the possibility of implementing structural response and design criteria into the Dynamic Positioning control loop

bull Use a simplified quasistatic response model to derive optimal reliability levels for PID and LQG control schemes in conjunction with two different types of loss functions

bull Implement a control algorithm that is capable of achieving a given target reliability level for a realistic and fully dynamic system

Control of low-frequency response level

Response

time

Possible strategies for control algorithm based on reliability indices

1 Monitoring of reliability indices

2 Weight factors based on reliability indices

3 Derivation of optimal control criteria based on reliability indices

Principle Measure of structural safety is the reliability index β which is related to the failure probability by β = Φ-1(pf)

bull No activation β gt βThreshold

bull Alert interval increasing activation βThreshold gt β gt βCritical

bull Full Activation β lt βCritical

Definition of delta index

(ie due to waves)

is the mean breaking strength of the line

is the standard deviation of the breaking strength

Computation of reliability index

bull Failure probability (pf) is probability that the extreme dynamic response will exceed critical level within a given reference duration

bull Failure probability is estimated for a stationary reference time interval of eg 20 minutes by application of a Gumbel distribution

bull Simplified relationship between delta-index and failure probability is expressed as

pf = (-δ)

Simplified quasistatic loadresponse model is applied for initial rdquooptimization studyrdquo

kTot∙r = FE - FT

where is total linearized stiffness of mooring lines FE is external (low-frequency) excitation and FT is thruster force

Conversely r = (FE ndash FT)kTot

Two types of loss functions are considered

bull Typical LQG type of loss function

L( r ) = KTFT2 + KFr2

(r is response FT is thruster force KT and KF are constants)

bull Loss function based on failure probability

L( r ) = KT FT2 + KPФ(-δ)

Two different types of control schemes are considered

bull PID control scheme

where e here is e= (rTarget ndash rstatic passive) = (rTarget - FEkTot) which (by neglecting second and last term) simplifies into FT = Kp (rTarget ndash rstatic passive) = Kp (rTarget - FEkTot)

bull LQG control scheme FT = -Cr Normalized control factor is xc= CkTot

t

0d)(eiKedKepKTF

First type of loss function versus vessel offset PID type of control scheme (KT∙kTot

2)KF = 10 and FEkTot = 20

Second type of loss function versus vessel offset PID type of control scheme (KT∙kTot

2)KF = 01 (intermediate value) and FEkTot = 20

First type of loss function expressed in terms of normalized control variable - LQG type of control scheme (KT∙kTot

Second type of loss function expressed in terms of normalized control variable -LQG type of control scheme (KT∙kTot

Second type of loss function expressed in terms of normalized control variable - LQG type of control scheme (KT∙kTot

Comparison of optimal offsets for different loss functions

Coefficient ratio (KT∙k Tot

2)KF

and (KT∙k Tot

2)KP

Quadratic loss function

Loss function based on failure probability

100 100m 200m 001 002m 000m

Example Position control of turret moored vessel

Vessel data bull Length of vessel 175mbull Beam 254m bull Draught 95m bull Displaced volume 24 140m3 bull Mooring lines are composed of a mixture of chains and wire

lines bull Representative linearized stiffness of the mooring system is

15∙104 Nm bull Mean value of breaking strength of single line is 1128∙106 Nbull Standard deviation of the breaking strength is 75 of the

mean value

Numerical simulation model

τ

is the mooring force

is the thruster force

wJp

bJτg(ηDννM

2

T

ψ

ψ)

)g(η

η = [pT ψ]T = [x y ψ]T is the position and heading in earth-fixed coordinates

ν = [wT ρ]T = [u v ρ]T is the translational and rotational velocities in body-fixed coordinates

M

is the inertia matrix D is the hydrodynamic damping matrix

b is a slowly varying bias term representing external forces due to wind currents and waves

Feedback control law based on back-stepping technique

bψJηgDνMςτ T

2

(20)

I

1

j sj

j jj s j s j s

b j j b j

s

r

T Tr

T2

w

w S

j

jT2 r

ppJ

λ γ and κ are strictly positive constants

rj is the length of the horizontal projection of mooring line number j

Tjrsquois the linearized mooring line tension in line j

pj is the horizontal position of the end-point at the anchor for the same mooring line

σbj is the standard deviation of the breaking strength of line number j

The target value of the reliability index is designated by δs

It can be shown that this controller is global exponentially stable

Notation

Time variation of water current velocity

Time variation of resultant environmental force

Time variation of vessel position in x-direction

Time variation of thruster force

Time variation of delta-index

In order for a delta-index of 44 to be optimal for the present

case study the ratio of (KT∙kTot2)KP needs to be 10-6 ie the

failure cost needs to be very high compared to the rdquounit thruster costrdquo

Summaryconclusions

bull A simplified model is applied in order to study optimal offset values (and corresponding values of the delta-index) when considering both the cost and reliability level

bull Two different loss functions are compared The first type is quadratic in the response while the second is proportional to the failure probability

bull It is demonstrated for a particular example how structural reliability criteria can be incorporated directly into the control algorithm

RELIABILITY-BASED STRUCTURAL OPTIMIZATION FOR POSITIONING OF MARINE VESSELS
Objective
First type of loss function versus vessel offset PID type of control scheme (KT∙kTot2)KF = 10 and FEkTot = 20
Second type of loss function versus vessel offset PID type of control scheme (KT∙kTot2)KF = 10 and FEkTot = 20
Second type of loss function versus vessel offset PID type of control scheme (KT∙kTot2)KF = 01 (intermediate value) and FEkTot = 20
First type of loss function expressed in terms of normalized control variable - LQG type of control scheme (KT∙kTot2)KF = 10 and FEkTot = 20
Second type of loss function expressed in terms of normalized control variable -LQG type of control scheme (KT∙kTot2)KF = 10 and FEkTot = 20
Second type of loss function expressed in terms of normalized control variable - LQG type of control scheme (KT∙kTot2)KF = 001 and FEkTot = 20
Vessel data
Summaryconclusions

Page 2: 1 Cesos-Workshop-March-2006 RELIABILITY-BASED STRUCTURAL OPTIMIZATION FOR POSITIONING OF MARINE VESSELS B. J. Leira, NTNU, Trondheim, Norway P. I. B. Berntsen,

Objective

bull To investigate the possibility of implementing structural response and design criteria into the Dynamic Positioning control loop

bull Use a simplified quasistatic response model to derive optimal reliability levels for PID and LQG control schemes in conjunction with two different types of loss functions

bull Implement a control algorithm that is capable of achieving a given target reliability level for a realistic and fully dynamic system

Response

time

(ie due to waves)

pf = (-δ)

kTot∙r = FE - FT

t

0d)(eiKedKepKTF

2)KF

and (KT∙k Tot

2)KP

100 100m 200m 001 002m 000m

mean value

τ

wJp

bJτg(ηDννM

2

T

ψ

ψ)

)g(η

M

bψJηgDνMςτ T

2

(20)

I

1

j sj

j jj s j s j s

b j j b j

s

r

T Tr

T2

w

w S

j

jT2 r

ppJ

Notation

Summaryconclusions

Objective
Vessel data
Summaryconclusions

Page 3: 1 Cesos-Workshop-March-2006 RELIABILITY-BASED STRUCTURAL OPTIMIZATION FOR POSITIONING OF MARINE VESSELS B. J. Leira, NTNU, Trondheim, Norway P. I. B. Berntsen,

Response

time

(ie due to waves)

pf = (-δ)

kTot∙r = FE - FT

t

0d)(eiKedKepKTF

2)KF

and (KT∙k Tot

2)KP

100 100m 200m 001 002m 000m

mean value

τ

wJp

bJτg(ηDννM

2

T

ψ

ψ)

)g(η

M

bψJηgDνMςτ T

2

(20)

I

1

j sj

j jj s j s j s

b j j b j

s

r

T Tr

T2

w

w S

j

jT2 r

ppJ

Notation

Summaryconclusions

Objective
Vessel data
Summaryconclusions

Page 4: 1 Cesos-Workshop-March-2006 RELIABILITY-BASED STRUCTURAL OPTIMIZATION FOR POSITIONING OF MARINE VESSELS B. J. Leira, NTNU, Trondheim, Norway P. I. B. Berntsen,

(ie due to waves)

pf = (-δ)

kTot∙r = FE - FT

t

0d)(eiKedKepKTF

2)KF

and (KT∙k Tot

2)KP

100 100m 200m 001 002m 000m

mean value

τ

wJp

bJτg(ηDννM

2

T

ψ

ψ)

)g(η

M

bψJηgDνMςτ T

2

(20)

I

1

j sj

j jj s j s j s

b j j b j

s

r

T Tr

T2

w

w S

j

jT2 r

ppJ

Notation

Summaryconclusions

Objective
Vessel data
Summaryconclusions

Page 5: 1 Cesos-Workshop-March-2006 RELIABILITY-BASED STRUCTURAL OPTIMIZATION FOR POSITIONING OF MARINE VESSELS B. J. Leira, NTNU, Trondheim, Norway P. I. B. Berntsen,

(ie due to waves)

pf = (-δ)

kTot∙r = FE - FT

t

0d)(eiKedKepKTF

2)KF

and (KT∙k Tot

2)KP

100 100m 200m 001 002m 000m

mean value

τ

wJp

bJτg(ηDννM

2

T

ψ

ψ)

)g(η

M

bψJηgDνMςτ T

2

(20)

I

1

j sj

j jj s j s j s

b j j b j

s

r

T Tr

T2

w

w S

j

jT2 r

ppJ

Notation

Summaryconclusions

Objective
Vessel data
Summaryconclusions

Page 6: 1 Cesos-Workshop-March-2006 RELIABILITY-BASED STRUCTURAL OPTIMIZATION FOR POSITIONING OF MARINE VESSELS B. J. Leira, NTNU, Trondheim, Norway P. I. B. Berntsen,

(ie due to waves)

pf = (-δ)

kTot∙r = FE - FT

t

0d)(eiKedKepKTF

2)KF

and (KT∙k Tot

2)KP

100 100m 200m 001 002m 000m

mean value

τ

wJp

bJτg(ηDννM

2

T

ψ

ψ)

)g(η

M

bψJηgDνMςτ T

2

(20)

I

1

j sj

j jj s j s j s

b j j b j

s

r

T Tr

T2

w

w S

j

jT2 r

ppJ

Notation

Summaryconclusions

Objective
Vessel data
Summaryconclusions

Page 7: 1 Cesos-Workshop-March-2006 RELIABILITY-BASED STRUCTURAL OPTIMIZATION FOR POSITIONING OF MARINE VESSELS B. J. Leira, NTNU, Trondheim, Norway P. I. B. Berntsen,

pf = (-δ)

kTot∙r = FE - FT

t

0d)(eiKedKepKTF

2)KF

and (KT∙k Tot

2)KP

100 100m 200m 001 002m 000m

mean value

τ

wJp

bJτg(ηDννM

2

T

ψ

ψ)

)g(η

M

bψJηgDνMςτ T

2

(20)

I

1

j sj

j jj s j s j s

b j j b j

s

r

T Tr

T2

w

w S

j

jT2 r

ppJ

Notation

Summaryconclusions

Objective
Vessel data
Summaryconclusions

Page 8: 1 Cesos-Workshop-March-2006 RELIABILITY-BASED STRUCTURAL OPTIMIZATION FOR POSITIONING OF MARINE VESSELS B. J. Leira, NTNU, Trondheim, Norway P. I. B. Berntsen,

kTot∙r = FE - FT

t

0d)(eiKedKepKTF

2)KF

and (KT∙k Tot

2)KP

100 100m 200m 001 002m 000m

mean value

τ

wJp

bJτg(ηDννM

2

T

ψ

ψ)

)g(η

M

bψJηgDνMςτ T

2

(20)

I

1

j sj

j jj s j s j s

b j j b j

s

r

T Tr

T2

w

w S

j

jT2 r

ppJ

Notation

Summaryconclusions

Objective
Vessel data
Summaryconclusions

Page 9: 1 Cesos-Workshop-March-2006 RELIABILITY-BASED STRUCTURAL OPTIMIZATION FOR POSITIONING OF MARINE VESSELS B. J. Leira, NTNU, Trondheim, Norway P. I. B. Berntsen,

t

0d)(eiKedKepKTF

2)KF

and (KT∙k Tot

2)KP

100 100m 200m 001 002m 000m

mean value

τ

wJp

bJτg(ηDννM

2

T

ψ

ψ)

)g(η

M

bψJηgDνMςτ T

2

(20)

I

1

j sj

j jj s j s j s

b j j b j

s

r

T Tr

T2

w

w S

j

jT2 r

ppJ

Notation

Summaryconclusions

Objective
Vessel data
Summaryconclusions

Page 10: 1 Cesos-Workshop-March-2006 RELIABILITY-BASED STRUCTURAL OPTIMIZATION FOR POSITIONING OF MARINE VESSELS B. J. Leira, NTNU, Trondheim, Norway P. I. B. Berntsen,

t

0d)(eiKedKepKTF

2)KF

and (KT∙k Tot

2)KP

100 100m 200m 001 002m 000m

mean value

τ

wJp

bJτg(ηDννM

2

T

ψ

ψ)

)g(η

M

bψJηgDνMςτ T

2

(20)

I

1

j sj

j jj s j s j s

b j j b j

s

r

T Tr

T2

w

w S

j

jT2 r

ppJ

Notation

Summaryconclusions

Objective
Vessel data
Summaryconclusions

Page 11: 1 Cesos-Workshop-March-2006 RELIABILITY-BASED STRUCTURAL OPTIMIZATION FOR POSITIONING OF MARINE VESSELS B. J. Leira, NTNU, Trondheim, Norway P. I. B. Berntsen,

2)KF

and (KT∙k Tot

2)KP

100 100m 200m 001 002m 000m

mean value

τ

wJp

bJτg(ηDννM

2

T

ψ

ψ)

)g(η

M

bψJηgDνMςτ T

2

(20)

I

1

j sj

j jj s j s j s

b j j b j

s

r

T Tr

T2

w

w S

j

jT2 r

ppJ

Notation

Summaryconclusions

Objective
Vessel data
Summaryconclusions

Page 12: 1 Cesos-Workshop-March-2006 RELIABILITY-BASED STRUCTURAL OPTIMIZATION FOR POSITIONING OF MARINE VESSELS B. J. Leira, NTNU, Trondheim, Norway P. I. B. Berntsen,

2)KF

and (KT∙k Tot

2)KP

100 100m 200m 001 002m 000m

mean value

τ

wJp

bJτg(ηDννM

2

T

ψ

ψ)

)g(η

M

bψJηgDνMςτ T

2

(20)

I

1

j sj

j jj s j s j s

b j j b j

s

r

T Tr

T2

w

w S

j

jT2 r

ppJ

Notation

Summaryconclusions

Objective
Vessel data
Summaryconclusions

Page 13: 1 Cesos-Workshop-March-2006 RELIABILITY-BASED STRUCTURAL OPTIMIZATION FOR POSITIONING OF MARINE VESSELS B. J. Leira, NTNU, Trondheim, Norway P. I. B. Berntsen,

2)KF

and (KT∙k Tot

2)KP

100 100m 200m 001 002m 000m

mean value

τ

wJp

bJτg(ηDννM

2

T

ψ

ψ)

)g(η

M

bψJηgDνMςτ T

2

(20)

I

1

j sj

j jj s j s j s

b j j b j

s

r

T Tr

T2

w

w S

j

jT2 r

ppJ

Notation

Summaryconclusions

Objective
Vessel data
Summaryconclusions

Page 14: 1 Cesos-Workshop-March-2006 RELIABILITY-BASED STRUCTURAL OPTIMIZATION FOR POSITIONING OF MARINE VESSELS B. J. Leira, NTNU, Trondheim, Norway P. I. B. Berntsen,

2)KF

and (KT∙k Tot

2)KP

100 100m 200m 001 002m 000m

mean value

τ

wJp

bJτg(ηDννM

2

T

ψ

ψ)

)g(η

M

bψJηgDνMςτ T

2

(20)

I

1

j sj

j jj s j s j s

b j j b j

s

r

T Tr

T2

w

w S

j

jT2 r

ppJ

Notation

Summaryconclusions

Objective
Vessel data
Summaryconclusions

Page 15: 1 Cesos-Workshop-March-2006 RELIABILITY-BASED STRUCTURAL OPTIMIZATION FOR POSITIONING OF MARINE VESSELS B. J. Leira, NTNU, Trondheim, Norway P. I. B. Berntsen,

2)KF

and (KT∙k Tot

2)KP

100 100m 200m 001 002m 000m

mean value

τ

wJp

bJτg(ηDννM

2

T

ψ

ψ)

)g(η

M

bψJηgDνMςτ T

2

(20)

I

1

j sj

j jj s j s j s

b j j b j

s

r

T Tr

T2

w

w S

j

jT2 r

ppJ

Notation

Summaryconclusions

Objective
Vessel data
Summaryconclusions

Page 16: 1 Cesos-Workshop-March-2006 RELIABILITY-BASED STRUCTURAL OPTIMIZATION FOR POSITIONING OF MARINE VESSELS B. J. Leira, NTNU, Trondheim, Norway P. I. B. Berntsen,

2)KF

and (KT∙k Tot

2)KP

100 100m 200m 001 002m 000m

mean value

τ

wJp

bJτg(ηDννM

2

T

ψ

ψ)

)g(η

M

bψJηgDνMςτ T

2

(20)

I

1

j sj

j jj s j s j s

b j j b j

s

r

T Tr

T2

w

w S

j

jT2 r

ppJ

Notation

Summaryconclusions

Objective
Vessel data
Summaryconclusions

Page 17: 1 Cesos-Workshop-March-2006 RELIABILITY-BASED STRUCTURAL OPTIMIZATION FOR POSITIONING OF MARINE VESSELS B. J. Leira, NTNU, Trondheim, Norway P. I. B. Berntsen,

2)KF

and (KT∙k Tot

2)KP

100 100m 200m 001 002m 000m

mean value

τ

wJp

bJτg(ηDννM

2

T

ψ

ψ)

)g(η

M

bψJηgDνMςτ T

2

(20)

I

1

j sj

j jj s j s j s

b j j b j

s

r

T Tr

T2

w

w S

j

jT2 r

ppJ

Notation

Summaryconclusions

Objective
Vessel data
Summaryconclusions

Page 18: 1 Cesos-Workshop-March-2006 RELIABILITY-BASED STRUCTURAL OPTIMIZATION FOR POSITIONING OF MARINE VESSELS B. J. Leira, NTNU, Trondheim, Norway P. I. B. Berntsen,

2)KF

and (KT∙k Tot

2)KP

100 100m 200m 001 002m 000m

mean value

τ

wJp

bJτg(ηDννM

2

T

ψ

ψ)

)g(η

M

bψJηgDνMςτ T

2

(20)

I

1

j sj

j jj s j s j s

b j j b j

s

r

T Tr

T2

w

w S

j

jT2 r

ppJ

Notation

Summaryconclusions

Objective
Vessel data
Summaryconclusions

Page 19: 1 Cesos-Workshop-March-2006 RELIABILITY-BASED STRUCTURAL OPTIMIZATION FOR POSITIONING OF MARINE VESSELS B. J. Leira, NTNU, Trondheim, Norway P. I. B. Berntsen,

2)KF

and (KT∙k Tot

2)KP

100 100m 200m 001 002m 000m

mean value

τ

wJp

bJτg(ηDννM

2

T

ψ

ψ)

)g(η

M

bψJηgDνMςτ T

2

(20)

I

1

j sj

j jj s j s j s

b j j b j

s

r

T Tr

T2

w

w S

j

jT2 r

ppJ

Notation

Summaryconclusions

Objective
Vessel data
Summaryconclusions

Page 20: 1 Cesos-Workshop-March-2006 RELIABILITY-BASED STRUCTURAL OPTIMIZATION FOR POSITIONING OF MARINE VESSELS B. J. Leira, NTNU, Trondheim, Norway P. I. B. Berntsen,

2)KF

and (KT∙k Tot

2)KP

100 100m 200m 001 002m 000m

mean value

τ

wJp

bJτg(ηDννM

2

T

ψ

ψ)

)g(η

M

bψJηgDνMςτ T

2

(20)

I

1

j sj

j jj s j s j s

b j j b j

s

r

T Tr

T2

w

w S

j

jT2 r

ppJ

Notation

Summaryconclusions

Objective
Vessel data
Summaryconclusions

Page 21: 1 Cesos-Workshop-March-2006 RELIABILITY-BASED STRUCTURAL OPTIMIZATION FOR POSITIONING OF MARINE VESSELS B. J. Leira, NTNU, Trondheim, Norway P. I. B. Berntsen,

mean value

τ

wJp

bJτg(ηDννM

2

T

ψ

ψ)

)g(η

M

bψJηgDνMςτ T

2

(20)

I

1

j sj

j jj s j s j s

b j j b j

s

r

T Tr

T2

w

w S

j

jT2 r

ppJ

Notation

Summaryconclusions

Objective
Vessel data
Summaryconclusions

Page 22: 1 Cesos-Workshop-March-2006 RELIABILITY-BASED STRUCTURAL OPTIMIZATION FOR POSITIONING OF MARINE VESSELS B. J. Leira, NTNU, Trondheim, Norway P. I. B. Berntsen,

mean value

τ

wJp

bJτg(ηDννM

2

T

ψ

ψ)

)g(η

M

bψJηgDνMςτ T

2

(20)

I

1

j sj

j jj s j s j s

b j j b j

s

r

T Tr

T2

w

w S

j

jT2 r

ppJ

Notation

Summaryconclusions

Objective
Vessel data
Summaryconclusions

Page 23: 1 Cesos-Workshop-March-2006 RELIABILITY-BASED STRUCTURAL OPTIMIZATION FOR POSITIONING OF MARINE VESSELS B. J. Leira, NTNU, Trondheim, Norway P. I. B. Berntsen,

τ

wJp

bJτg(ηDννM

2

T

ψ

ψ)

)g(η

M

bψJηgDνMςτ T

2

(20)

I

1

j sj

j jj s j s j s

b j j b j

s

r

T Tr

T2

w

w S

j

jT2 r

ppJ

Notation

Summaryconclusions

Objective
Vessel data
Summaryconclusions

Page 24: 1 Cesos-Workshop-March-2006 RELIABILITY-BASED STRUCTURAL OPTIMIZATION FOR POSITIONING OF MARINE VESSELS B. J. Leira, NTNU, Trondheim, Norway P. I. B. Berntsen,

bψJηgDνMςτ T

2

(20)

I

1

j sj

j jj s j s j s

b j j b j

s

r

T Tr

T2

w

w S

j

jT2 r

ppJ

Notation

Summaryconclusions

Objective
Vessel data
Summaryconclusions

Page 25: 1 Cesos-Workshop-March-2006 RELIABILITY-BASED STRUCTURAL OPTIMIZATION FOR POSITIONING OF MARINE VESSELS B. J. Leira, NTNU, Trondheim, Norway P. I. B. Berntsen,

Notation

Summaryconclusions

Objective
Vessel data
Summaryconclusions

Page 26: 1 Cesos-Workshop-March-2006 RELIABILITY-BASED STRUCTURAL OPTIMIZATION FOR POSITIONING OF MARINE VESSELS B. J. Leira, NTNU, Trondheim, Norway P. I. B. Berntsen,

Summaryconclusions

Objective
Vessel data
Summaryconclusions

Page 27: 1 Cesos-Workshop-March-2006 RELIABILITY-BASED STRUCTURAL OPTIMIZATION FOR POSITIONING OF MARINE VESSELS B. J. Leira, NTNU, Trondheim, Norway P. I. B. Berntsen,

Summaryconclusions

Objective
Vessel data
Summaryconclusions

Page 28: 1 Cesos-Workshop-March-2006 RELIABILITY-BASED STRUCTURAL OPTIMIZATION FOR POSITIONING OF MARINE VESSELS B. J. Leira, NTNU, Trondheim, Norway P. I. B. Berntsen,

Summaryconclusions

Objective
Vessel data
Summaryconclusions

Page 29: 1 Cesos-Workshop-March-2006 RELIABILITY-BASED STRUCTURAL OPTIMIZATION FOR POSITIONING OF MARINE VESSELS B. J. Leira, NTNU, Trondheim, Norway P. I. B. Berntsen,

Summaryconclusions

Objective
Vessel data
Summaryconclusions

Page 30: 1 Cesos-Workshop-March-2006 RELIABILITY-BASED STRUCTURAL OPTIMIZATION FOR POSITIONING OF MARINE VESSELS B. J. Leira, NTNU, Trondheim, Norway P. I. B. Berntsen,

Summaryconclusions

Objective
Vessel data
Summaryconclusions

Page 31: 1 Cesos-Workshop-March-2006 RELIABILITY-BASED STRUCTURAL OPTIMIZATION FOR POSITIONING OF MARINE VESSELS B. J. Leira, NTNU, Trondheim, Norway P. I. B. Berntsen,

Summaryconclusions

Objective
Vessel data
Summaryconclusions

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