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Efficient Re-Analysis Methodology for Probabilistic Vibration of

Large-Scale Structures

Efstratios Nikolaidis, Zissimos Mourelatos

April 14, 2008

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Definition and Significance

m 2m 3m m k k 2k 3k 2k

ttF sin3 tx2

It is very expensive to estimate system reliability of dynamic systems and to optimize them

• Vibratory response varies non-monotonically• Impractical to approximate displacement as a function

of random variables by a metamodel

3

k

m

2max5.18, xmkg

Failure occurs in many disjoint regions

Perform reliability assessment by Monte Carlo simulation and RBDO by gradient-free methods (e.g., GA).

This is too expensive for complex realistic structures

g<0: failureg>0: survival

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Solution1. Deterministic analysis of vibratory response

– Parametric Reduced Order Modeling– Modified Combined Approximations– Reduces cost of FEA by one to two orders of

magnitude

2. Reliability assessment and optimization– Probabilistic reanalysis – Probabilistic sensitivity analysis– Perform many Monte-Carlo simulations at a cost of

a single simulation

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Outline

1. Objectives and Scope2. Efficient Deterministic Re-analysis

– Forced vibration problems by reduced-order modeling– Efficient reanalysis for free vibration

• Parametric Reduced Order Modeling• Modified Combined Approximation Method• Kriging approximation

3. Probabilistic Re-analysis4. Example: Vehicle Model5. Conclusion

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1. Objectives and Scope

• Present and demonstrate methodology that enables designer to; – Assess system reliability of a complex vehicle

model (e.g., 50,000 to 10,000,000 DOF) by Monte Carlo simulation at low cost (e.g., 100,000 sec)

– Minimize mass for given allowable failure probability

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Scope

• Linear eigenvalue analysis, steady-state harmonic response

• Models with 50,000 to 10,000,000 DOF

• System failure probability crisply defined: maximum vibratory response exceeds a level

• Design variables are random; can control their average values

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2. Efficient Deterministic Re-analysis

Problem:

• Know solution for one design (K,M) • Estimate solution for modified design (K+ΔK, M+ΔM)

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2.1 Solving forced vibration analysis by reduced basis modeling

FdMK 2ΦUd Modal Representation:

Modal Basis: nφφφΦ 21

Modal Model: FΦUMΦΦKΦΦ TTT 2

Basis must be recalculated for each new design

Many modes must be retained (e.g. 200)

Calculation of “triple” product expensiveKΦΦT

Φ

Issues:

Reduced Stiffness and Mass Matrices

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Solution

Basis must be recalculated for each new design

Many modes must be retained

Calculation of “triple” product can be expensiveKΦΦT

Φ

Practical Issues:

Re-analysis methods: PROM and CA / MCA

Kriging interpolation

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Efficient re-analysis for free vibrationParametric Reduced Order Modeling (PROM)

Parameter Spacep1

p3p2

Design point

3210 ΦΦΦΦP Reduced Basis

Idea: Approximate modes in basis spanned by modes of representative designs

npnp ΦΦPΘΦ 0 ...0

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PROM (continued)

• Replaces original eigen-problem with reduced size problem

• But requires solution of np+1 eigen-problems for representative designs corresponding to corner points in design space

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Modified Combined Approximation Method (MCA) Reduces cost of solving m eigen-problems

p1

p3p2

Parameter Space

3210

~~~ΦΦΦΦP

Exact mode shapes for only one design point Approximate mode shapes for p design points using MCA Cost of original PROM: (p+1) times full analysis Cost of integrated method: 1 full analysis + np MCA approximations

Full Analysis MCA Approximation

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Basis vectors

siii ,,3,2)()(

)()(

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01

1

TΔMMΔKKT

ΦΔMMΔKKT

sTTTΦT 210

Idea: Approximate modes of representative designs in subspace T

•Recursive equation converges to modes of modified design.•High quality basis, only 1-3 basis vectors are usually needed. •Original eigen-problem (size nxn) reduces to eigen-problem of size (sxs, s=1 to 3)

MCA method

Approximate reduced mass and stiffness matrices of a new design by using Kriging

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Deterministic Re-Analysis Algorithm

p1

p3 p2 2. Calculate np approximate mode shapes by MCA

4. Generate reduced matrices at a specific number of sample design points

5. Establish Kriging model for predicting reduced matrices

npΦΦΦP~~

10 3. Form basis

1. Calculate exact mode shape by FEA

6. Obtain reduced matrices by Kriging interpolation

7. Perform eigen-analysis of reduced matrices

8. Obtain approximate mode shapes of new design

9. Find forced vibratory response using approximate modes

Repeat steps 6-9 for each new design:

0Φ

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3. Probabilistic Re-analysis

• RBDO problem:Find average values of random design variables To minimize cost function

So that psys ≤ pfall

• All design variables are random• PRA analysis: estimate reliabilities of many

designs at a cost of a single probabilistic analysis

Xμ

)( Xμl

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4. Example: RBDO of Truck

Model:Pickup truck with 65,000 DOF

Excitation: Unit harmonic force applied at engine mount points in X, Y and Z directions

Response:Displacement at 5 selected points on the right door

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Design variables

1.860 mm2.480 mm1.240 mmLeft and right doors5

3.555 mm4.740 mm2.370 mmBed4

3.750 mm5.000 mm2.500 mmCabin3

5.417 mm7.222 mm3.611 mmChassis cross link2

4.706 mm6.274 mm3.137 mmChassis1

BaselineValues

UpperBound

LowerBound

Description(thickness of)

Param.#

1.860 mm2.480 mm1.240 mmLeft and right doors5

3.555 mm4.740 mm2.370 mmBed4

3.750 mm5.000 mm2.500 mmCabin3

5.417 mm7.222 mm3.611 mmChassis cross link2

4.706 mm6.274 mm3.137 mmChassis1

BaselineValues

UpperBound

LowerBound

Description(thickness of)

Param.#

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Example: Cost of Deterministic Re-Analysis.

0 2000 4000 6000 8000 1 104

0

5 105

1 106

1.5 106

2 106

2.5 106

NASTRANMCA+PROM (Section 3.3)MCA+PROM+Kriging (Section 3.4)

Replications

CPU

(sec

)

583 hrs

28 hrs

Deterministic Reanalysis reduces cost to 1/20th of NASTRAN analysis

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Re-analysis: Failure probability and its sensitivity to cabin thickness

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RBDO

• Find average thickness of chassis, cross link, cabin, bed and doors

• To minimize mass• Failure probability pf

all

• Half width of 95% confidence interval 0.25 pfall

• Plate thicknesses normal• Failure: max door displacement>0.225 mm• Repeat optimization for pf

all : 0.005-0.015• Conjugate gradient method for optimization

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Optimum in space of design variables

Optimum Truck Design

3.52

3.53

3.54

3.55

3.56

3.57

3.58

3.59

3.6

3.63 3.64 3.65 3.66 3.67 3.68 3.69 3.7 3.71

Cabin Thickness

Bed

Th

ickn

ess

Mass=2000.167

PF=0.01

CI/PF=0.25

OptimumMass decreases

Baseline: mass=2027, PF=0.011

Feasible Region

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Mass of optimum designs vs. allowable failure probability(gray dashed curves show 95% confidence bounds of Monte-Carlo

simulation results)

1980

1990

2000

2010

2020

2030

0 0.005 0.01 0.015 0.02 0.025

Failure probability

Mas

s (k

g)

OptimumdesignsBaseline

Monte Carlo

MC Conf.Bound=

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5. Conclusion

• Presented efficient methodology for RBDO of large-scale structures considering their dynamic response

1. Deterministic re-analysis 2. Probabilistic re-analysis

• Demonstrated methodology on realistic truck model

• Use of methodology enables to perform RBDO at a cost of a single simulation.

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Solution: RBDO by Probabilistic Re-Analysis

Iso-cost curves

Feasible Region

Increased Performance

x2

x1

Optimum

Failure subset