IE-net Managing, handling, and modeling uncertainty in mechanical design Nonintrusive probabilistic quantification of uncertainties with application to the management of manufacturing tolerances Maarten Arnst May 21, 2015 ULg, Liège, Belgium Nonintrusive probabilistic UQ 1 / 33
Motivation Manufacturing tolerances in metal forming Raw materials variability: Product variability: • Material properties. • Final dimensions. • Springback. . . . . . . Process variability: • Blank holder force. • Initial dimensions. • Friction. → → . . . Modeling limitations: Prediction limitations: • Constitutive model. • Numerical noise. • FE discretization. . . . . . . Input variables. Output variables. ULg, Liège, Belgium Nonintrusive probabilistic UQ 2 / 33
Outline Motivation. ■ Outline. ■ Context and current practice. ■ New methods. ■ Example: Metal forming. ■ Conclusion and outlook. ■ References. ■ Contact information. ■ ULg, Liège, Belgium Nonintrusive probabilistic UQ 3 / 33
Selected elements from context and current practice ULg, Liège, Belgium Nonintrusive probabilistic UQ 4 / 33
Context and current practice Example: Bending of a beam p ) = pℓ 3 Young’s modulus y = ( gy, j, p, ℓg u g 3 yj . moment of inertia j ���� ���� � �� � output variable model input variables tip displacement ℓ Let y be uncertain (e.g., imperfect knowledge at design time, imperfect manufacturing when compared to the design,. . . ). Given uncertainty in y , what is the resulting uncertainty in u ? ULg, Liège, Belgium Nonintrusive probabilistic UQ 5 / 33
Context and current practice Example: Bending of a beam (continued) A probabilistic context effects the propagation of uncertainty from y to u as follows: ■ u u u = g ( y ) = pℓ 3 3 yj ρ U y y ρ Y � � � � g − 1 ( u ) ≤ Y P U ≤ u = P because g is a decreasing function . ULg, Liège, Belgium Nonintrusive probabilistic UQ 6 / 33
Context and current practice Example: Bending of a beam (continued) A probabilistic context effects the propagation of uncertainty from y to u as follows: ■ u u u = g ( y ) = pℓ 3 3 yj ρ U y y ρ Y � � � � g − 1 ( u ) ≤ Y P U ≤ u = P because g is a decreasing function . ULg, Liège, Belgium Nonintrusive probabilistic UQ 6 / 33
Context and current practice Example: Bending of a beam (continued) A probabilistic context effects the propagation of uncertainty from y to u as follows: ■ u u u = g ( y ) = pℓ 3 3 yj ρ U y y ρ Y � � � � g − 1 ( u ) ≤ Y P U ≤ u = P because g is a decreasing function . ULg, Liège, Belgium Nonintrusive probabilistic UQ 6 / 33
Context and current practice Example: Bending of a beam (continued) A probabilistic context effects the propagation of uncertainty from y to u as follows: ■ u u u = g ( y ) = pℓ 3 3 yj ρ U y y ρ Y � � � � g − 1 ( u ) ≤ Y P U ≤ u = P because g is a decreasing function . ULg, Liège, Belgium Nonintrusive probabilistic UQ 6 / 33
Context and current practice Example: Bending of a beam (continued) Elaborating this expression by means of the “changes of variables” formula, ■ � + ∞ � u ρ U ( u ) du = ρ Y ( y ) dy 0 g − 1 ( u ) � 0 � dg − 1 � g − 1 ( u ) = du ( u ) du ρ Y u � u �� � dg − 1 � � � g − 1 ( u ) = du ( u ) ρ Y � du, � � � 0 we find the following relationship between the probability density functions of the input and output variables, that is, of the Young’s modulus and the tip displacement: �� � dg − 1 � � � g − 1 ( u ) ρ U ( u ) = ρ Y du ( u ) � . � � � ULg, Liège, Belgium Nonintrusive probabilistic UQ 7 / 33
Context and current practice Example: Bending of a beam (continued) Probability density function, mean, variance, and confidence interval: ■ ρ U ( u ) σ U u u − kσ U u + kσ U u � Mean u = R uρ U ( u ) du , ◆ � Variance σ 2 R ( u − u ) 2 ρ U ( u ) du , U = ◆ � u − kσ U P c -Confidence interval [ u − kσ U , u + kσ U ] such that u − kσ U ρ U ( u ) du ≥ P c . ◆ ULg, Liège, Belgium Nonintrusive probabilistic UQ 8 / 33
Context and current practice Example: Bending of a beam (continued) Using the change-of-variables formula, we can deduce the following expression for the mean u : ■ − 1 � � �� � � dg g − 1 ( u ) g − 1 ( u ) � � � � u = uρ U ( u ) du = uρ Y du � � dy � � R R � dg � − 1 dg � = g ( y ) ρ Y ( y ) dy ( y ) dy ( y ) dy R � = g ( y ) ρ Y ( y ) dy, R and we can deduce the following expression for the variance σ 2 U : − 1 � � �� � � dg σ 2 ( u − u ) 2 ρ U ( u ) du = ( u − u ) 2 ρ U g − 1 ( u ) � g − 1 ( u ) � � � U = du � � dy � � R R � dg � − 1 dg � ( g ( y ) − u ) 2 ρ Y ( y ) = dy ( y ) dy ( y ) dy R � ( g ( y ) − u ) 2 ρ Y ( y ) dy. = R In conclusion, to determine the mean and the variance of the output, knowledge of the probability ■ density function of the input and an integration method are required. ULg, Liège, Belgium Nonintrusive probabilistic UQ 9 / 33
Context and current practice Example: Bending of a beam (continued) If the model is linearized, ■ � dg � u = g ( y ) ≈ g ( y ) + dy ( y ) ( y − y ) . then the expression for the mean u can be simplified as follows: � � dg � � � � u = g ( y ) ρ Y ( y ) dy ≈ g ( y ) + dy ( y ) ( y − y ) ρ Y ( y ) dy = g ( y ) , R R and the expression for the variance σ 2 U can be simplified as follows: �� dg � � 2 � dg � 2 � � σ 2 ( g ( y ) − u ) 2 ρ Y ( y ) dy ≈ σ 2 U = dy ( y ) ( y − y ) ρ Y ( y ) dy = dy ( y ) Y . R R In conclusion, linearising the model makes things much simpler!! Now, to approximate the mean ■ and the variance of the output, knowledge of only the mean and variance of the input suffices. ULg, Liège, Belgium Nonintrusive probabilistic UQ 10 / 33
Context and current practice Example: ISO 98 ISO 98: Guide to the expression of uncertainty in measurement. ULg, Liège, Belgium Nonintrusive probabilistic UQ 11 / 33
Context and current practice Example: ISO 98 (continued) ISO 98: Guide to the expression of uncertainty in measurement. ULg, Liège, Belgium Nonintrusive probabilistic UQ 12 / 33
Context and current practice Example: Robust design in aerospace engineering From: A. Karl, B. Farris, L. Brown, and N. Metzger (Rolls-Royce). Robust design and optimization: Key methods and applications. Stanford, 2011. ULg, Liège, Belgium Nonintrusive probabilistic UQ 13 / 33
Context and current practice Some limitations associated with the approaches described so far... Engineering problem ■ ◆ Limited in scope to scalar uncertain quantities. ◆ However, more complex uncertainties can be encountered in engineering problems, such as uncertain geometries, uncertain processes and fields, and uncertain matrices. Characterization of uncertainties ■ ◆ Limited to mean and variance. ◆ No emphasis on constraints that can be imposed by mechanics and physics. Propagation of uncertainties ■ ◆ Approximation entailed by linearization of the model. Sensitivity analysis of uncertainties ■ ◆ Limited to local sensitivity analysis that is also encountered in deterministic problems. ◆ However, global sensitivity analysis can also be of interest; and many new interesting questions can be asked in an uncertainty-quantification-enabled context. ULg, Liège, Belgium Nonintrusive probabilistic UQ 14 / 33
Selected elements from new methods ULg, Liège, Belgium Nonintrusive probabilistic UQ 15 / 33
� � � � New methods Overview �� �� Engineering �� �� problem ② ♣ ☎ ✌ Sensitivity analysis. ✓ �� �� �� �� �� �� �� �� Mechanical modeling. Process optimization. Analysis Characterization UQ of uncertainties Design optimization. of uncertainties Statistics. Model validation. • Gaussian. · • Γ -distribution. . . . �� �� �� �� Propagation of uncertainties Monte Carlo sampling. Stochastic expansion (polynomial chaos). Scientific computing. ULg, Liège, Belgium Nonintrusive probabilistic UQ 16 / 33
New methods Characterization of uncertainties The objective of the characterization of uncertainties is to assign an appropriate probability ■ distribution to the uncertain input variables. An appropriate probability distribution can be obtained by applying methods from mathematical ■ statistics to the available information. In engineering, this available information typically consists not only of observed samples but also of applicable mechanical and physical laws . ◆ Catalogs of probability distributions. ◆ Principles of construction. ◆ Methods for parameter estimation. ◆ Methods for model selection. ◆ . . . If a sufficient amount of data is available, much of this can be automated. ■ Current ressearch allows to consider as uncertain not only scalar input variables but also ■ geometries, fields of mechanical and physical properties, matrix-valued input variables, etc. ULg, Liège, Belgium Nonintrusive probabilistic UQ 17 / 33
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