copyright 2004, southwest research institute, all rights reserved. nessus capabilities for...

Copyright 2004, Southwest Research Institute, All Rights Reserved.

NESSUS Capabilities for Ill-Behaved Performance Functions

David Riha and Ben Thacker

Southwest Research Institute

Simeon Fitch

Mustard Seed Software

45th AIAA/ASME/ASCE/AHS/ASC Structures,Structural Dynamics & Materials Conference

6th AIAA Non-Deterministic Approaches Forum

Palm Springs, California

April 19-22, 2004


Overview

Sources of error in probabilistic analysis

Proposed strategies• MPP search failure detection

algorithm

• Gradient computations for noisy response functions

Example problems Conclusions

0

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-6 -4 -2 0 2 4 6

x (standard deviations)

z

Exact Function

"Noisy" Function


Source of Error in Probabilistic Analysis

1. Model Approximation• First and second-order approximation

• Calculation of derivatives

• Model simplification

2. Uncertainty Characterization• Insufficient data

• Selection of incorrect distribution

3. Numerical Algorithm• Transformations to standard normal

• Convergence error in locating the MPP

• Algorithm error (wrong or multiple MPP

4. Probability Integration• Insufficient number of samples

• First or second-order approximation

All Forms of Error are Reducible• V&V of the probabilistic analysis

• Increased data collection

• Development of more accurate and robust analysis methods


Algorithm Error

Source of error is inability of the algorithm to locate the correct most probable point (MPP)• local minimum

• multiple minimums

• violations of the assumptions of a smooth and continuous response function

• Highly nonlinear response functions

For robustness, algorithm must be able to locate all MPP’s

Problems can arise after transformation to standard normal space, unbeknownst to the analyst

Failure detection algorithms required for confidence in solutions

0.2750.280.2850.29

ro

1000010200104001060010800

11000 w

- 1500

- 1400

- 1300

z

- 1500

- 1400

- 1300

z

- 4- 2024 u_ro

- 4- 202

4 u_w

20000

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24000

z

20000

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24000

z


Model Approximation

Model approximation can result from a trade-off between computational efficiency and accuracy• Mesh size• Time integration settings

Derivative calculations• Analytical derivatives generally not

available for nonlinear analyses• Source code not available for third party

analysis packages to implement analytical derivatives

• Finite difference approximations are required

Response Surface approach commonly used to avoid derivative computations• May not capture local response• Large number of function evaluations

may be required


Probabilistic Analysis Methods

Fast Probability Integration Methods• Advanced mean value

• First and second-order reliability methods

Sampling Methods• Monte Carlo simulation

• Sphere-based importance sampling

• Latin hypercube simulation

• Adaptive importance sampling

Probabilistic Fault-tree Response Surface Method

u1

u2

f (u)u

Most Probable Point (MPP)


Locating the Most Probable Point (MPP)

Formulation• Minimize:

• Subject to: g(x)=g(u)=0

Standard Optimization Methods• Modified methods of feasible

directions (MMFD)

• Sequential linear programming (SLP)

• Sequential quadratic programming (SQP)

Tailored Methods• Hasofer-Lind

• Rackwitz-Feissler

• Others

uuTD

u1

u2

f (u)u

Most Probable Point (MPP)

Approximate Limit-State

Exact Limit-State

g(x) = 0


MPP Search Failure Detection Algorithm

Literature and experience identifies the Rackwitz-Feissler MPP search algorithm as being more efficient when it converges over other optimization algorithms (MMFD, SQP, etc.)

Approach needed to identify when RF fails• Determine failure is occurring early in the search process to

eliminate potentially expensive function evaluations

• Automatically switch to a more robust yet computationally efficient optimization algorithm to locate the MPP

• Utilize initial search points as a starting point for more robust optimization methods



Many failures of the RF method shows a characteristic cyclic MPP search pattern

-10 -8 -6 -4 -2u1

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u2

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0 20 40 60 80 100

iteration

Be

ta



Algorithm• Monitor the autocorrelation of

during the search process• k is the lag between search points

and a large autocorrelation value indicates similar points in the search

• Initial points in the search are eliminated from the test to avoid the initial search process

• Failure is defined when the autocorrelation exceeds a critical correlation value (e.g., 0.5) for 3 points

• MPP Search algorithm changed upon failure detection

-0.4

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1

0 5 10 15 20 25 30

lag

Au

toc

orr

ela

tio

n

N

ii

kN

ikii

kr

1

1

)(

))((


Gradient Computations forNoisy Response Functions

Gradient information required for efficient probabilistic algorithms• Gradient-based optimization for MPP search

Noisy response• Transient analyses• Impact and blast loading• Sliding friction and contact surfaces• Models using course meshes and time steps

Analytical derivatives not available• Nonlinear solutions• Third party software packages

Finite difference• Forward, backward, central difference• Step size is critical• Capture global response and local sensitivity• Noisy solutions can cause problems


Gradient Computations forNoisy Response Functions

Simulated noisy response• z=x2 x=Normal(5.0,0.5)

Noise term• Sin(100x)

Gradient Overall response seems well

behaved Small fluctuations in the

response can cause large errors in the gradient computation using finite difference

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z

Exact Function

"Noisy" Function

xdx

dz2


Example Finite Difference Approximationfor a Noisy Response function

Small step size causes large error in the gradient (0.1) Inspecting the response variation provides a step size that

captures the local sensitivity but eliminates the majority of the noise (0.5

-20

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40

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dz/

dx

Analytical Derivative

Finite Difference (0.1*sigma)0

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-6 -4 -2 0 2 4 6


dz/

dx

Analytical DerivativeFinite Difference (0.5*sigma)


Inputs-Java-based graphical user interface-Free format keyword interface-Ten probability density functions-Correlated random variables-Users/Theory/Examples manual

Outputs-Cumulative distribution function-Prob. of failure given performance-Performance given prob. of failure-Probabilistic sensitivity factors wrt and -Confidence Bounds-Empirical CDF and histogram

Results Visualization-XY, bar, pie charts-Comparison of multiple solutions-3D model visualization

Deterministic Analysis-Parameter variation analysis

Probabilistic Analysis Methods-First-order reliability method (FORM)-Second-order reliability method (SORM)-Fast probability integration (FPI)-Advanced mean value (AMV+)-Response surface method (RSM)-Automatic Monte Carlo simulation (MC)-Importance sampling (ISAM)-Latin hypercube simulation (LHS)-Adaptive importance sampling (AIS)-Probabilistic fault-tree (PFTA)

Applications-Component/system reliability-Reliability-based optimization-Reliability test planning-Inspection scheduling-Design certification-Risk-based cost analysis-MVFO probability contouring

Performance Functions-Analytical (Fortran)-Analytical (direct)-Numerical (FEM, CFD, other)-Failure Models (Fortran, ext.

models)-Hierarchical failure models

Interfaces-ABAQUS/Standard/Explicit-MSC.NASTRAN-ANSYS-NASA/GRC-FEM-PRONTO-DYNA/PARADYN-LS-DYNA-MADYMO-NASA analysis modules-User-defined

Other-Automated restart-Batch processing

Hardware-PC (NT4, W2000, XP)-Unix workstations (HP, Sun, SGI)-Other systems

NESSUS 8.1 Capabilities

Further Information210/522-6566 [email protected] www.nessus.swri.org


MPP Search Failure Detection Algorithm Example

Response function:• x1=uniform(0,100)

• x2 Beta(=0.5)

Transformation to u-space causes increased non-linearity Modified RF method does not converge for Z0=4.52

50100 25.3

1 xxz

-4 -2 2 4u1

-4

-2

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4

u2

-4 -2 2 4u1

-4

-2

2

4

u2


MPP Search Failure Detection Algorithm Example

Autocorrelation computed for incrementally increasing iteration numbers• used to eliminate initial search

points

Failure detected after 40 steps• 0.5 correlation cutoff

• 3 successive points

• detected after 2 cycles

SQP method used after failure is detected and identifies the correct MPP

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Starting Iteration

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toco

rrel

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n 30

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iteration

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ta


Noisy Response Function Example

Stainless steel float crush between two platens

Response is the total energy to crush the float

Anticipate a “noisy” solution because of the contact surfaces and coarseness of the model

r=4.4735

r

t

Platen location is a functionof the thickness and radius


Sphere Crush Problem Statement in NESSUS

The response is the total energy Z

Functional relationships define how the random variables change the numerical model input

The function “fe” is assigned to the LS-DYNA explicit finite element code


MV and AMV Solution Using 0.1 Step Size

Default step size used AMV solution is

required to be tangent to the MV solution at the median value

Behavior typical of inaccurate sensitivities


Parameter Variation Analysis

Parameter variation analysis provides a tool to understand the deterministic response

Select step sizes for finite difference

Verify model for parameter values away from the mean

A step size of 1.0 is selected to capture the gradient while maintaining the local sensitivity


AMV+ Solution Using ImprovedFinite Difference Step Size

Step size selection critical for noisy response functions

AMV and AMV+ solutions tangent to MV solution at the median value

AMV+ converges within a 5% tolerance on Z


Noisy Response FunctionAMV+ Compared to LHS

AMV+ solution compares well with LHS (2000 samples)

Error may be caused by crude mesh and/or contact surfaces


Conclusions

MPP search failure detection algorithm developed• Able to identify characteristic cyclic behavior of the RF MPP search

algorithm early in the solution• Once failure is detected, an alternative solution strategy can be

automatically employed• Research continues to identify critical correlation values to identify failure

and the use of function solutions for starting the next solution strategy NESSUS visualization capabilities provide a practical tool for

performing probabilistic analysis with noisy response functions where analytical derivatives are not available• Parameter variation analysis aides in understanding the deterministic

problem and selecting step sizes for finite difference approximations to the gradients

• Overlay of different steps in the AMV+ solution indicates the potential of inaccurate gradient solutions

Continued robustness improvements in probabilistic methods and tools is required for acceptance of the probabilistic design approach

copyright 2004, southwest research institute, all rights reserved. nessus capabilities for...

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