Efficient Re-Analysis Methodology for Probabilistic Vibration Analysis

7/28/2019 Efficient Re-Analysis Methodology for Probabilistic Vibration Analysis

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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

2 3 2

ttF sin3 tx2

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

Vibratory response varies non-monotonically

Impractical to approximate displacement as a function

of random variables by a metamodel


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k

m

2

max5.18, 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

g0: survival


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Solution

1. 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 Scope

2. 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 Model

5. 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 failureprobability


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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+K,

M+M)


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2.1 Solving forced vibration analysis by reduced

basis modeling

FdMK 2

Ud Modal Representation:

Modal Basis: n 21

Modal Model: FUMKTTT

2

Basis must be recalculated for each new design

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

Calculation of triple product expensiveKT

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 expensiveKT

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 Space

p1

p3p2

Design point

3210 P Reduced Basis

Idea: Approximate modes in basis spanned by modes of

representative designs

npnp P 0 ...0


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PROM (continued)

Replaces original eigen-problem with

reduced size problem

But requires solution ofnp+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

~~~P

Exact mode shapes for only one design point

Approximate mode shapes forpdesign points using MCA

Cost of original PROM: (p+1) times full analysis

Cost of integrated method: 1 full analysis + npMCA approximations

Full Analysis MCA Approximation


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Basis vectors

siii ,,3,2)()(

)()(

11

0

1

1

TMMKKT

MMKKT

sTTTT 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

npP~~

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:

0


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3. Probabilistic Re-analysis

RBDO problem:

Find average values of random designvariables

To minimize cost functionSo thatpsys pf

all

All design variables are random

PRA analysis: estimate reliabilities of manydesigns at a cost of a single probabilisticanalysis

X

)( Xl


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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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Example: Cost of Deterministic Re-

Analysis.

0 2000 4000 6000 8000 1104

0

5 105

1 106

1.5 106

2 106

2.5 106

NASTRAN

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

Replications

CP

U(

sec)

583

hrs

28 hrs

Deterministic Reanalysis reduces cost to 1/20th of

NASTRAN analysis


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RBDO

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

To minimize mass

Failure probability pfall

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

BedThickness

Mass=2000.167

PF=0.01

CI/PF=0.25

Optimum

Mass 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

Mass(k

g)

Optimum

designsBaseline

Monte Carlo


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5. Conclusion

Presented efficient methodology forRBDO of large-scale structuresconsidering their dynamic response

1. Deterministic re-analysis2. Probabilistic re-analysis

Demonstrated methodology on realistictruck model

Use of methodology enables to performRBDO at a cost of a single simulation.


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Solution: RBDO by

Probabilistic Re-Analysis

Iso-costcurves

Feasible Region

IncreasedPerformance

x2

x1

Optimum

Failure subset

Efficient Re-Analysis Methodology for Probabilistic Vibration Analysis

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