46th SIAA Structures, Dynamics and Materials Conference Austin, TX, April 18-21, 2005 Application of the Generalized Conditional Expectation Method for Enhancing a Probabilistic Design Fatigue Code Faiyazmehadi Momin, Harry Millwater, R. Wes Osborn Department of Mechanical Engineering University of Texas at San Antonio Michael P. Enright Southwest Research Institute

University of Texas at San Antonio Motivation  Probabilistic design codes are specialized for particular application  Highly optimized for particular application  Specific mechanics model  Specific random variables  Specific probabilistic methods  Prominent codes in industry include PROF, DARWIN, VISA  Codes may need to be enhanced  Add more random variables  Source code may not be available

University of Texas at San Antonio Objective  Present the methodology of GCE to enhance a probabilistic design code by considering additional random variables  Compute the sensitivities of the probability-of-failure to ALL random variables  Demonstrate the methodology using a probabilistic fatigue code DARWIN

University of Texas at San Antonio Approach  Basic idea - discrete distribution 40%60% Speed(  ) CPOF 1 CPOF 2 POF TOTAL = CPOF 1 * CPOF 2 * 0.6 = E[CPOF i ] 11 22 CPOF - Conditional Probability of Failure

University of Texas at San Antonio Approach  Generalized Conditional Expectation Multiple runs of the probabilistic design code needed to compute expected value

University of Texas at San Antonio Methodology  Generalized Conditional Expectation (GCE) methodology is implemented without modifying the source code  Random variables are partitioned as “internal” and “external” variables  Internal - random variables already considered in probabilistic design code (called control variables in GCE vernacular)  External - additional random variables to be considered (called conditional variables in GCE vernacular)

University of Texas at San Antonio Variance Reduction Approach  Traditional use for GCE is variance reduction with sampling methods - reduce the sampling variance by eliminating the variance due to the control variables  Conditional (called “external” here)  Control (called “internal” here)  Ayyub, B. M., Haldar, A., “Practical Structural Reliability Techniques,” Journal of Structural Engineering, Vol. 110, No. 8, August 1984, pp  Ayyub, B. M., Chia, “ Generalized Conditional Expectation for Structural Reliability Assessment,” Structural Safety, Vol.11, 1992, pp

University of Texas at San Antonio Generalized Conditional Expectation  The conditional expected value can be approximated by  P fi is the conditional POF of i th realization is conditional variables  The variance and coefficient of variation are given by

University of Texas at San Antonio Implementation 1.Partition random variables into two categories: Internal - random variables within the probabilistic design code External - additional random variables to be considered not within the probabilistic design code 2.Generate a realization of external variables using Monte Carlo sampling 3.Determine the conditional probability-of-fracture (CPOF i ) given this realization of external random variables 4.Compute the expected value (average) of the CPOF results using MC sampling 5.Compute the sensitivities of the POF to the parameters of the internal and external random variables

University of Texas at San Antonio Implementation - Response Surface Option 1.Partition random variables into two categories: Internal - random variables within the probabilistic design code External - additional random variables to be considered not within the probabilistic design code 2.Generate a realization of external variables using response surface designs 3.Determine the conditional probability-of-fracture (CPOF i ) given this realization of external random variables 4.Build a response surface relating the external variables to the CPOF results. 5.Compute the expected value (average) of the CPOF results using MC sampling of the Response Surface 6.Compute the sensitivities of the POF to the parameters of the internal and external random variables

University of Texas at San Antonio Response Surface Option  Build a response surface representing the relationship between the conditional POF and the external random variables  Use classical design of experiments and goodness of fit tests The response surface is not use to approximate the limit state

University of Texas at San Antonio Sensitivities  Methodology developed to compute the sensitivities of the POF to the parameters of the internal and external random variables  Compare the effects of internal and external variables Internal External No additional limit state analyses needed

University of Texas at San Antonio DARWIN ®

University of Texas at San Antonio Implementation with DARWIN 1.Partition random variables into two sets Internal - DARWIN variables (crack size, life scatter, stress scatter) External - non-DARWIN (geometry, loading, structural and thermal material properties, etc.) 2.Generate a realization of external variables using Monte Carlo sampling 3.Run the finite element solver to obtain updated stresses 4.Execute DARWIN given this realization of external random variables and associated stresses to determine CPOF i 5.Compute the expected value (average) of the DARWIN CPOF results using Monte Carlo sampling 6.Compute the sensitivities of the POF to the parameters of the internal and external random variables

University of Texas at San Antonio Implementation - Response Surface Option 1.Partition random variables into two sets Internal - DARWIN variables (crack size, life scatter, stress scatter) External - non-DARWIN (geometry, loading, structural and thermal material properties, etc.) 2.Generate a realization of external variables using response surface design points 3.Run the finite element solver to obtain updated stresses 4.Execute DARWIN given this realization of external random variables and associated stresses to determine CPOF i 5.Build a response surface relating conditional variables to DARWIN CPOF 6.Compute the expected value (average) of the DARWIN CPOF results using Monte Carlo sampling of the response surface 7.Compute the sensitivities of the POF to the parameters of the internal and external random variables

University of Texas at San Antonio FLOW CHART start Parametric Deterministic Model Enter RV Results2NEU.UIF/.UOF DARWINInput file Darwin ResultsDarwin CPOF i = K Expected CPOF Sensitivities NoYes Results file Control Software Finite Element Solver Generate samples, Build RS, Compute Expected CPOF, Sensitivities Design Point Loop

University of Texas at San Antonio Implementation  Ansys probabilistic design system used to control analysis  Ansys FE solver used to compute stresses  ANS2NEU used to extract stresses for DARWIN  DARWIN used to compute the CPOF  Text utility used to extract DARWIN CPOF results and return to Ansys  Sensitivity equations programmed within Ansys  POF determined by computing the expected value of the CPOF using Monte Carlo or Monte Carlo with Response Surface

University of Texas at San Antonio UTSA FLOW CHART start Parametric Deterministic Model Enter RV ANS2NEU.UIF.UOF file DARWINInput file DARWIN ResultsDARWIN POF i = K Expected POF Sensitivities NoYes Results file Ansys PDS Ansys Solver Design Point Loop Generate samples, Build RS, Compute Expected CPOF, Sensitivities

University of Texas at San Antonio Application Example  FA Advisory Circular test case  Internal variables: initial crack size(a)  External variables: rotational speed(RPM), external pressure(Po), inner radius(Ri)  Surface crack on inner bore  Consider POF (assuming a defect is present) at 20,000 cycles  Solve using GCE with Darwin and Ansys  Compare to independent Monte Carlo solution

University of Texas at San Antonio FEM MODEL R2 R1 L  6800 rpm t Speed  Po r x Surface Crack Element type - Plane elements

University of Texas at San Antonio Initial Crack Size a MIN a MAX Exceedance Curve

University of Texas at San Antonio Probabilistic Model Procedure not limited to Normal distributions NoNameType Parameter1 (mean) Parameter2 (COV) 2PressureNormal7250 psi0.1 3SpeedNormal rad/sec0.05 4Inner radiusNormal11.81 inches0.02 No.NameType Parameter 1 a min (mils 2 ) Parameter 2 a max (mils 2 ) 1 Initial crack size Exceedance Curve Internal External

University of Texas at San Antonio Independent Benchmark Solution Developed Method Random variables No. of Samples POF Monte Carlo (Benchmark) Initial crack size (ai) 10, Monte Carlo (DARWIN) Initial crack size (ai) 10,  Approximate analytical fatigue algorithm developed for verification  Uses standard Monte Carlo sampling (no GCE) of all variables

University of Texas at San Antonio Probabilistic Results Using GCE Method Method Random variables No. of Samples POF Monte Carlo (Benchmark) ai Omega Pressure Radius GCE (Ansys (MC) and Darwin) ai Omega Pressure Radius 1000 DARWIN GCE (Ansys (RS) and Darwin) ai Omega Pressure Radius 15 DARWIN &100,000 RS MC-Monte Carlo RS-Response Surface simulations

University of Texas at San Antonio Effects of Response Surface Transformations TransformationNo. of Samples Expected POF Monte Carlo (Comparison) 1000 Darwin Linear 15 Darwin & 100, Quadratic with cross-terms 15 Darwin & 100, Exponential 15 Darwin & 100, Logarithmic 15 Darwin & 100, Power 15 Darwin & 100, Box-Cox 15 Darwin & 100, All quadratic

University of Texas at San Antonio Goodness of Fit Transformation Error sum of Squares (Close to Zero) Coefficient of Determination (R 2 ) (Close to one) Maximum Absolute Residual (Close to Zero) Linear8.634E Non-linear quadratic 2.450E Exponential2.474E Logarithmic7.840E Power1.014E Box-Cox9.196E

University of Texas at San Antonio Response Surface Implementation  Note: Response Surface is only used to compute the expected value of a function  This is completely different from the traditional use of RS in probabilistic analysis, i.e., to approximate the limit state and estimate an often very small probability  Curse-of-Dimensionality is still present if a quadratic model is used; however, only the external random variables enter the equation

University of Texas at San Antonio Sensitivity Results (Mean) Parameter Monte Carlo (Comparison) GCE ANSYS-DARWIN GCE Ansys-Darwin (Finite difference) Rotational speed External pressure Inner radius Sensitivity of POF with respect to mean value Sensitivities have units

University of Texas at San Antonio Sensitivity Results (Mean) Parameter Monte Carlo (Comparison) GCE ANSYS-DARWIN GCE Ansys-Darwin (Finite difference) Rotational speed External pressure Inner radius Sensitivity of POF with respect to mean value

University of Texas at San Antonio Sensitivity Results (Std Dev) Parameter Monte Carlo (Comparison) GCE ANSYS-DARWIN GCE Ansys-Darwin (Finite difference) Rotational speed 0.123E E E-2 External pressure 0.500E E E-4 Inner radius Sensitivity of POF with respect to standard deviations Sensitivities have units

University of Texas at San Antonio Sensitivity Results (Std Dev) Parameter Monte Carlo (Comparison) GCE ANSYS-DARWIN GCE Ansys-Darwin (Finite difference) Rotational speed External pressure Inner radius0.03 Sensitivity of POF with respect to standard deviations

University of Texas at San Antonio Sensitivities of Internal RV Parameter Monte Carlo sampling (1000 Monte Carlo MATLAB runs) GCE ANSYS-DARWIN (1000 DARWIN runs) GCE ANSYS-DARWIN (15 RS and 100,000 MC) Finite Difference ANSYS-DARWIN (15 DARWIN & 100,000 MC) (mils -2 ) (mils -2 ) 6.977E E E-011*** *** The sensitivity of CPOF with respect to a max is too small to be computed using finite difference method Sensitivities have units

University of Texas at San Antonio Sensitivities of Internal RV ParameterMonte Carlo sampling (1000 Monte Carlo MATLAB runs) GCE ANSYS-DARWIN (1000 DARWIN runs) GCE ANSYS-DARWIN (15 RS and 100,000 MC) Finite Difference ANSYS-DARWIN (15 DARWIN & 100,000 MC) (mils -2 ) (mils -2 ) 4E-5 *** Non-dimensionalized sensitivities are significantly smaller than other random variables

Application Problem Zone 14 Zone 13 Zone 12 Zone 11 Zone 10 Zone 9Zone 2 Zone 3 Zone 4 Zone 5 Zone 6 Zone 7 Zone 1 Zone 8

University of Texas at San Antonio POF per Flight results MethodRandom VariablesNo. of Samples Expected POF per Flight Monte Carlo (within DARWIN) Initial crack size 10,000 / Zone1.330E-9 GCE (ANSYS MC and DARWIN) Initial crack size a i Pressure Po Rotational Speed  Inner Radius r i 1000 DARWIN runs 1.917E-9 GCE ANSYS RS and DARWIN (Power Transformation) Initial crack size a i Pressure Po Rotational Speed  Inner Radius r i 15 DARWIN and 100,000 RS 1.935E-9

University of Texas at San Antonio Response Surface Transformations TransformationsNo. of SamplesMean POF Monte-Carlo1000 DARWIN1.957E-9 None - Linear 15 DARWIN & 100,000 MC 2.637E-9 None - Quadratic with Cross Terms 15 DARWIN & 100,000 MC 1.945E-9 Logarithmic 15 DARWIN & 100,000 MC 1.932E-9 Square Root 15 DARWIN & 100,000 MC 1.930E-9 Power (0.45) 15 DARWIN & 100,000 MC 1.935E-9 Box-Cox (0.45) 15 DARWIN & 100,000 MC 1.966E-9 All quadratic

University of Texas at San Antonio Goodness of Fit Measures Transformation Error sum of Squares (Close to Zero) Coefficient of Determination (Close to Zero) Maximum Absolute Residual (Close to Zero) Linear3.077E E-9 Quadratic with Cross Terms 4.322E E-10 Logarithmic3.991E E-9 Square root2.090E E-10 Power (0.45)7.716E E-10

University of Texas at San Antonio Sensitivity Results (Mean) - Parameter GCE ANSYS-DARWIN (1000 DARWIN runs) GCE ANSYS-DARWIN (15 RS and 100,000 MC) Finite Difference ANSYS-DARWIN (15 DARWIN & 100,000 MC) Rotational speed 4.092E E E-11 External pressure 9.974E E E-12 Inner radius 1.609E E E-09 Sensitivities have units

University of Texas at San Antonio Sensitivity Results (Mean) Parameter GCE ANSYS-DARWIN (1000 DARWIN runs) GCE ANSYS-DARWIN (15 RS and 100,000 MC) Finite Difference ANSYS-DARWIN (15 DARWIN & 100,000 MC) Rotational speed External pressure Inner radius

University of Texas at San Antonio Sensitivity Results (Std Dev) Parameter GCE ANSYS-DARWIN (1000 DARWIN runs) GCE ANSYS-DARWIN (15 RS and 100,000 MC) Finite Difference ANSYS-DARWIN (15 DARWIN & 100,000 MC) Rotational speed 2.930E E E-11 External pressure 9.659E E E-13 Inner radius 1.634E E E-10 Sensitivities have units

University of Texas at San Antonio Sensitivity Results (Std Dev) - Parameter GCE ANSYS-DARWIN (1000 DARWIN runs) GCE ANSYS-DARWIN (15 RS and 100,000 MC) Finite Difference ANSYS-DARWIN (15 DARWIN & 100,000 MC) Rotational speed External pressure Inner radius

University of Texas at San Antonio Conclusions  Methodology to consider affects of additional random variables is developed and demonstrated on a probabilistic fatigue analysis  POF from MC sampling and GCE method are in good agreement  Response surface method used to reduce signicantly the computational time with good accuracy  The sensitivities obtained from MC simulations, GCE formulae and finite difference method are in good agreement and indicate the importance of internal and external random variables on the POF

University of Texas at San Antonio Conclusions  Enables the user to consider additional random variables without modifying the source code  Enables the developer to consider the importance of implementing additional random variables