1. 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

2. Copyright 2004, Southwest Research Institute, All Rights Reserved. Overview  Sources of error in probabilistic analysis  Proposed strategies MPP search failure detection algorithm Gradient computations for noisy response functions  Example problems  Conclusions

3. Copyright 2004, Southwest Research Institute, All Rights Reserved. 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

4. Copyright 2004, Southwest Research Institute, All Rights Reserved. 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

5. Copyright 2004, Southwest Research Institute, All Rights Reserved. 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

6. Copyright 2004, Southwest Research Institute, All Rights Reserved. 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

7. Copyright 2004, Southwest Research Institute, All Rights Reserved. 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

8. Copyright 2004, Southwest Research Institute, All Rights Reserved. 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

9. Copyright 2004, Southwest Research Institute, All Rights Reserved. MPP Search Failure Detection Algorithm  Many failures of the RF method shows a characteristic cyclic MPP search pattern

10. Copyright 2004, Southwest Research Institute, All Rights Reserved. MPP Search Failure Detection Algorithm  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

11. Copyright 2004, Southwest Research Institute, All Rights Reserved. Gradient Computations for Noisy 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

12. Copyright 2004, Southwest Research Institute, All Rights Reserved. Gradient Computations for Noisy Response Functions  Simulated noisy response z=x 2 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

13. Copyright 2004, Southwest Research Institute, All Rights Reserved. Example Finite Difference Approximation for 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 

14. Copyright 2004, Southwest Research Institute, All Rights Reserved. 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 Information 210/522-6566 nessus@swri.org www.nessus.swri.org

15. Copyright 2004, Southwest Research Institute, All Rights Reserved. MPP Search Failure Detection Algorithm Example  Response function: x 1 =uniform(0,100) x 2 Beta(  =0.5)  Transformation to u-space causes increased non-linearity  Modified RF method does not converge for Z 0 =4.52

16. Copyright 2004, Southwest Research Institute, All Rights Reserved. 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

17. Copyright 2004, Southwest Research Institute, All Rights Reserved. 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

18. Copyright 2004, Southwest Research Institute, All Rights Reserved. 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

19. Copyright 2004, Southwest Research Institute, All Rights Reserved. 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

20. Copyright 2004, Southwest Research Institute, All Rights Reserved. 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

21. Copyright 2004, Southwest Research Institute, All Rights Reserved. AMV+ Solution Using Improved Finite 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

22. Copyright 2004, Southwest Research Institute, All Rights Reserved. Noisy Response Function AMV+ Compared to LHS  AMV+ solution compares well with LHS (2000 samples)  Error may be caused by crude mesh and/or contact surfaces

23. Copyright 2004, Southwest Research Institute, All Rights Reserved. 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