Assessment of a static test model using Kriging and Sensitivity Analysis

Transcription

1 Assessment of a static test model using Kriging and Sensitivity Analysis Matthias Köchl June 2, 2016 M. Köchl INTALES Workshop 2016 June 2, / 19

2 1 Introduction 2 Kriging prediction 3 Sensitivity analysis 4 Application M. Köchl INTALES Workshop 2016 June 2, / 19

3 1 Introduction 2 Kriging prediction 3 Sensitivity analysis 4 Application M. Köchl INTALES Workshop 2016 June 2, / 19

4 Introduction FE-model of the Winglet with 4.7m degrees of freedom. 60 input parameters: 40 E-moduli, 20 material orientation angles 1170 output parameters: strain on the Winglet. M. Köchl INTALES Workshop 2016 June 2, / 19

5 Introduction Bending test on the Winglet was conducted. Measurement data do not coincide exactly with theoretical strain values. M. Köchl INTALES Workshop 2016 June 2, / 19

6 Introduction Bending test on the Winglet was conducted. Measurement data do not coincide exactly with theoretical strain values. Idea: Extrapolate experimental data onto the whole component and compare with theoretical output values. M. Köchl INTALES Workshop 2016 June 2, / 19

7 Introduction Bending test on the Winglet was conducted. Measurement data do not coincide exactly with theoretical strain values. Idea: Extrapolate experimental data onto the whole component and compare with theoretical output values. Use Kriging M. Köchl INTALES Workshop 2016 June 2, / 19

8 1 Introduction 2 Kriging prediction 3 Sensitivity analysis 4 Application M. Köchl INTALES Workshop 2016 June 2, / 19

9 Kriging Is an extrapolation (interpolation) method, originating in geostatistics, developed by the South African mining engineer Danie G. Krige in M. Köchl INTALES Workshop 2016 June 2, / 19

10 Kriging Is an extrapolation (interpolation) method, originating in geostatistics, developed by the South African mining engineer Danie G. Krige in Advantages: M. Köchl INTALES Workshop 2016 June 2, / 19

11 Kriging Is an extrapolation (interpolation) method, originating in geostatistics, developed by the South African mining engineer Danie G. Krige in Advantages: Minimal variance prediction. M. Köchl INTALES Workshop 2016 June 2, / 19

12 Kriging Is an extrapolation (interpolation) method, originating in geostatistics, developed by the South African mining engineer Danie G. Krige in Advantages: Minimal variance prediction. Exact prediction. (The value at a known site is exact.) M. Köchl INTALES Workshop 2016 June 2, / 19

13 Kriging Is an extrapolation (interpolation) method, originating in geostatistics, developed by the South African mining engineer Danie G. Krige in Advantages: Minimal variance prediction. Exact prediction. (The value at a known site is exact.) Unbiased prediction. M. Köchl INTALES Workshop 2016 June 2, / 19

14 Kriging Is an extrapolation (interpolation) method, originating in geostatistics, developed by the South African mining engineer Danie G. Krige in Advantages: Minimal variance prediction. Exact prediction. (The value at a known site is exact.) Unbiased prediction. Best linear unbiased predictor. M. Köchl INTALES Workshop 2016 June 2, / 19

15 Kriging prediction Goal is to estimate an unknown value T (x) at a site x D R n, where T is a random field on D, using data from surrounding known sample points x 1,..., x k D. M. Köchl INTALES Workshop 2016 June 2, / 19

16 Kriging prediction Goal is to estimate an unknown value T (x) at a site x D R n, where T is a random field on D, using data from surrounding known sample points x 1,..., x k D. M. Köchl INTALES Workshop 2016 June 2, / 19

17 Kriging prediction Goal is to estimate an unknown value T (x) at a site x D R n, where T is a random field on D, using data from surrounding known sample points x 1,..., x k D. Assumptions: E ( T (z) ) = µ for all z D. M. Köchl INTALES Workshop 2016 June 2, / 19

18 Kriging prediction Goal is to estimate an unknown value T (x) at a site x D R n, where T is a random field on D, using data from surrounding known sample points x 1,..., x k D. Assumptions: E ( T (z) ) = µ for all z D. [ (T Semivariogram γ(r) = 1 2 E (xi ) T (x j ) ) ] 2 < and depends only on the distance r = x i x j. M. Köchl INTALES Workshop 2016 June 2, / 19

19 Kriging prediction Kriging: T (x) = with weights ω j R. k ω j T (x j ), j=1 M. Köchl INTALES Workshop 2016 June 2, / 19

20 Kriging prediction Kriging: T (x) = with weights ω j R. k ω j T (x j ), j=1 We choose ω := (ω 1,..., ω k ) such that [ (T σk 2 := E (x) T (x) ) ] 2 minimal, M. Köchl INTALES Workshop 2016 June 2, / 19

21 Kriging prediction Kriging: T (x) = with weights ω j R. k ω j T (x j ), j=1 We choose ω := (ω 1,..., ω k ) such that [ (T σk 2 := E (x) T (x) ) ] 2 minimal, E ( T (x) ) = E ( T (x) ) (unbiasedness). M. Köchl INTALES Workshop 2016 June 2, / 19

22 Kriging prediction Kriging: with weights ω j R. T (x) = k ω j T (x j ), j=1 We choose ω := (ω 1,..., ω k ) such that σ 2 K := E [ (T (x) T (x) ) 2 ] minimal, E ( T (x) ) = E ( T (x) ) (unbiasedness). Leads to a constrained optimization problem, can be solved via Lagrange Multipliers. M. Köchl INTALES Workshop 2016 June 2, / 19

23 1 Introduction 2 Kriging prediction 3 Sensitivity analysis 4 Application M. Köchl INTALES Workshop 2016 June 2, / 19

24 Sensitivity analysis Monte Carlo based sensitivity analysis: M. Köchl INTALES Workshop 2016 June 2, / 19

25 Sensitivity analysis Monte Carlo based sensitivity analysis: 40 E-moduli ±15%, 20 angles ±5% uniformly distributed around the nominal value. M. Köchl INTALES Workshop 2016 June 2, / 19

26 Sensitivity analysis Monte Carlo based sensitivity analysis: 40 E-moduli ±15%, 20 angles ±5% uniformly distributed around the nominal value. Sample size 200. M. Köchl INTALES Workshop 2016 June 2, / 19

27 Sensitivity analysis Monte Carlo based sensitivity analysis: 40 E-moduli ±15%, 20 angles ±5% uniformly distributed around the nominal value. Sample size 200. Get mean values and confidence intervals of the outputs and compare with measurement. M. Köchl INTALES Workshop 2016 June 2, / 19

28 Sensitivity analysis Monte Carlo based sensitivity analysis: 40 E-moduli ±15%, 20 angles ±5% uniformly distributed around the nominal value. Sample size 200. Get mean values and confidence intervals of the outputs and compare with measurement. Compute correlation between inputs and outputs. M. Köchl INTALES Workshop 2016 June 2, / 19

29 Sensitivity analysis Monte Carlo based sensitivity analysis: 40 E-moduli ±15%, 20 angles ±5% uniformly distributed around the nominal value. Sample size 200. Get mean values and confidence intervals of the outputs and compare with measurement. Compute correlation between inputs and outputs. Obtain bootstrapped confidence intervals of the correlation coefficients. M. Köchl INTALES Workshop 2016 June 2, / 19

30 Sensitivity analysis Monte Carlo based sensitivity analysis: 40 E-moduli ±15%, 20 angles ±5% uniformly distributed around the nominal value. Sample size 200. Get mean values and confidence intervals of the outputs and compare with measurement. Compute correlation between inputs and outputs. Obtain bootstrapped confidence intervals of the correlation coefficients. Performed on HPC system MACH, one run-through 11h, 3 parallel computations. M. Köchl INTALES Workshop 2016 June 2, / 19

31 1 Introduction 2 Kriging prediction 3 Sensitivity analysis 4 Application M. Köchl INTALES Workshop 2016 June 2, / 19

32 Application Figure : Illustration with measurement points. M. Köchl INTALES Workshop 2016 June 2, / 19

33 Application Figure : Illustration with Kriging sample points. M. Köchl INTALES Workshop 2016 June 2, / 19

34 Application Figure : Illustration of Kriging. M. Köchl INTALES Workshop 2016 June 2, / 19

35 Application Figure : Comparison of Kriging data with theoretical data. M. Köchl INTALES Workshop 2016 June 2, / 19

36 Application Figure : Correlation coefficients. M. Köchl INTALES Workshop 2016 June 2, / 19

37 Application Figure : Input-Output scatter plots. M. Köchl INTALES Workshop 2016 June 2, / 19

38 Thank you for your attention! M. Köchl INTALES Workshop 2016 June 2, / 19