Mechanical Engineering in Uncertainties From Classical Approaches to Some Recent Developments
8.1. Introduction 8.2. Mathematical refresher 8.3. Multiobjective optimization and common descent vector 8.4. Descent algorithm for multiobjective optimization and its extension to the stochastic framework 8.5. Illustrations 8.6. References
9 Index
List of Illustrations
1 Chapter 1Figure 1.1. Example of a cumulative distribution function (CDF) and a complement...Figure 1.2. Example of an empirical cumulative distribution function (ECDF, in b...Figure 1.3. Distribution function of a uniform distribution on X between [0,1] (...Figure 1.4. Illustration with a system presenting a resonance of a case where mo...Figure 1.5. Illustration of a probability boxFigure 1.6. Illustration of a trapezoidal membership function in fuzzy set theor...Figure 1.7. Illustrations of (a) a possibility distribution function (PoDF); (b)...Figure 1.8. Comparison between a probability density function f(x) and a possibi...Figure 1.9. Probability density function and possibility distribution function s...Figure 1.10. Distribution function (DF), cumulative possibility function (CPoF) ...Figure 1.11. Example of an uncertainty structure (focal elements Ωi of mass mi) ...Figure 1.12. Illustrations of (a) a BPA function m(x); (b) the corresponding cum...
2 Chapter 2Figure 2.1. “Biloba” concrete core, according to Escoda (2012) (source: EDF). Fo...Figure 2.2. Examples of stochastic models derived from random point processes: (...Figure 2.3. (a) Composite fibrous medium; (b) model (source: Altendorf et al. 20...Figure 2.4. (a) Two-dimensional section of a Johson–Mehl random partition modeli...Figure 2.5. (a and c) SEM image (after segmentation) of fuel cell anode layers; ...Figure 2.6. (a) SEM image of a fuel cell foam; (b) realization of a “Boolean ran...Figure 2.7. Random realizations of three probabilistic models with the same triv...
3 Chapter 3Figure 3.1. Principle of a weakly chained THM+L modeling. For a color version of...Figure 3.2. Containment principle for double-walled nuclear vesselsFigure 3.3. Diagram of the internal wall of the VeRCoRs model, its structural pa...Figure 3.4. Evolution of cracking profile frequencies according to the spatial c...Figure 3.5. Effect of spatial correlation on the distribution of the r.v. of a l...Figure 3.6. Evolution of the relative error on variance εΩ=Gousset according to ...Figure 3.7. Examples of 12-day cracking profiles involved in the ACCD experiment...Figure 3.8. Evolution of the number of cracks per gusset RSV with the tensile st...Figure 3.9. Evaluation of the risk of exceeding the regulatory criterion overall...
4 Chapter 4Figure 4.1. (a) Finite element mesh of the structural part of the wing (skin, sp...Figure 4.2. Displacement field after convergence of the multidisciplinary analys...Figure 4.3. Circulation (a) before and (b) after convergence of the multidiscipl...Figure 4.4. Lift-to-drag ratio YQoI(z) as a function of the angle of attack z, w...Figure 4.5. Twenty random realizations of the lift-to-drag ratio YQoI (z) obtain...Figure 4.6. Histogram of the quantity of interest YQol based on the sample of si...Figure 4.7. Histogram of the quantity of interest YQoI before and after enrichme...Figure 4.8. Histogram of the quantity of interest YQoI before and after enrichme...Figure 4.9. Histogram of the quantity of interest, namely the spin angle YQoI, b...Figure 4.10. Histogram of the new quantity of interest YQoI (wingtip spin angle)...Figure 4.11. Histogram of the new quantity of interest YQoI (wingtip spin angle)...
5 Chapter 5Figure 5.1. Joint PDF fX, limit-state surface |, safe domain | and failure domai...Figure 5.2. Standard normal space. For a color version of this figure, see www.i...Figure 5.3. FORM approximation. For a color version of this figure, see www.iste...Figure 5.4. Successive use of bulges to find multiple MPFPs (Der Kiureghian and ...Figure 5.5. SORM-cf approximation. For a color version of this figure, see www.i...Figure 5.6. Concept of SORM-pf approximation in the (|, |) plane. For a color ve...Figure 5.7. Intermediate failure domains Fu,s for s = 1, …,m (here m = 3). For a...Figure 5.8. Modified Metropolis algorithm: seeds of Markov chains and proposal P...
6 Chapter 6Figure 6.1. Presentation of the AK family of methods (active learning and Krigin...Figure 6.2. Illustration of Kriging prediction with the function y = xsin(x) wit...Figure 6.3. Illustration of an example of the AK-MCS method with a single random...Figure 6.4. Illustration of the AK-MCS method with an example with two random va...Figure 6.5. Illustration of the AK-IS method with an example using two random va...Figure 6.6. Illustration of the AK-SYS method with a serial system using two com...Figure 6.7. Modified oblate spheroid performance function. Influence of interact...
7 Chapter 7Figure 7.1. Illustration of the uncertainty quantification methodology. For a co...Figure 7.2. Flowchart of the proposed methodology to compute | -indices under a ...Figure 7.3. Reference | -indices estimated for Example #1 under a single-level u...Figure 7.4. Reference | -indices estimated for Example #1 under a bi-level uncer...Figure 7.5. A roof trussFigure 7.6. Reference | -indices estimated for Example #2 under a single-level u...Figure 7.7. Reference | -indices estimated for Example #2 under a bi-level uncer...Figure 7.8. Illustration scheme of a launch vehicle first stage fallout phase in...Figure 7.9. | -indices estimated under single-level uncertainty (CMC of Nsim = 1...Figure 7.10. | -indices estimated under bi-level uncertainty (ARA/CMC of Nsim = ...
8 Chapter 8Figure 8.1. Example: calculation of a subdifferential of a convex function. For ...Figure 8.2. Approximation of the subdifferential of a non-convex function. For a...Figure 8.3. Example of dominance according to Pareto for a two-objective problem...Figure 8.4. Pareto fronts and dispersion due to uncertainties. For a color versi...Figure 8.5. Comparison of performance measures. For a color version of this figu...Figure 8.6. Comparison of performance measures. For a color version of this figu...Figure 8.7. Beam under study under compressionFigure 8.8. Pareto front in the objective domain for the reliability problem. Fo...
List of Tables
1 Chapter 3Table 3.1. List of independent random parameters and distributions of the result...Table 3.2. List of independent random parameters and their distributionsTable 3.3. Overall sensitivity analysis, as defined by Sobol’ (2001), applied to...
2 Chapter 4Table 4.1. Estimators of the Sobol’ indices calculated by polynomial chaos, the ...Table 4.2. Wing tip twist. Estimators of the Sobol’ indices calculated by polyno...
3 Chapter 5Table 5.1. Application of the standard Metropolis-Hasting sampler, construction ...Table 5.2. Application of the componentwise Metropolis sampler proposed