1
Uncertainties

Taking into account uncertainty in the design process is an innovative approach. This includes dimensioning the structure of the systems, the use of safety coefficients and the most advanced techniques to calculate reliability. The aim is to design a system that statistically achieves the best performance since the system is subject to variations. For a given risk probability, satisfactory system performance can be targeted which has low sensitivity to uncertainties and respects a minimum performance threshold. From a mathematical point of view, an innovative approach to system design can be considered as an optimization problem under constraints. In this chapter, various methods are presented to calculate systems subject to uncertainties.

1.1. Introduction

The methods used to take uncertainties into account are mathematical and statistical tools that make it possible to model and analyze systems whose parameters or use conditions are likely to vary. These methods are used to optimize the design and to balance cost and performance.

These methods are based on:

1. - the development of an approximate mathematical model of the physical system under study;
2. - the identification and characterization of the sources of uncertainty in the model parameters;
3. - the study of the propagation of these uncertainties and their impact on the output signal (response) of the system.

Analysis and estimation of the statistics (moments, distribution parameters, etc.) of the system response are performed in the next step. The methods used to analyze the propagation of uncertainties vary according to the mathematical tools on which they are based. These methods include a reliability based design approach, a probabilistic approach based on design of experiments, and a set based approach.

1.2. The reliability based design approach

The reliability based design approach is based on modeling uncertainties. Depending on the methods used, uncertainties are modeled by random variables, stochastic fields or stochastic processes. These methods make it possible to study and analyze the variability of a system response and to minimize its variability.

The most common methods are the Monte Carlo (MC) method, perturbation method and polynomial chaos method [ELH 13].

1.2.1. The MC method

1.2.1.1. Origin

The first use of this mathematical tool dates back to Fermi's research on the characterization of new molecules in 1930. The MC method has been applied, since 1940, by Von Neumann et al. to perform simulations in the field of atomic physics. The MC method is a powerful and very general mathematical tool. Its field of applications has widened because of the processing power of today's computers.

1.2.1.2. Principle

The MC method is a calculation technique which proceeds by successively solving a determinist system equation in which uncertain parameters are modeled by random variables.

The MC method is used when the problem under study is too complex to solve by using an analytical resolution method. It generates random draws for all uncertain parameters in accordance with their probability distribution laws. The precision of the random generators is very important because for each draw a deterministic calculation is performed using the number of parameters defined by this generator.

1.2.1.3. Advantages and disadvantages

The main advantage of the MC method is that it can be very easily implemented. Potentially, this method can be applied to any system, whatever their dimensions or complexity. The results obtained by this method are exact in a statistical sense, that is their uncertainty decreases as the number of draws increases. This uncertainty of precision for a given confidence level is defined by the Bienaymé-Chebyshev inequality. A reasonable precision requires a large number of draws. This sometimes makes the MC method very costly in terms of calculation time, which is the main disadvantage of this method.

1.2.1.4. Remark

The simplicity of the MC method has made its application popular in the field of engineering sciences. This is a powerful but costly method. Its results are often used to validate new methods that are developed in the framework of fundamental research. It is applied in Chapter 9 in order to characterize carbon nanotubes.

1.2.2. The perturbation method

1.2.2.1. Principle

The perturbation method is another technique used to study the propagation of uncertainties in systems [KLE 92, ELH 13]. It consists of approximating the random variable functions by their Taylor expansion around their mean value. According to the order of the Taylor expansion, the method is described as being the first, second or nth order. The conditions of existence and validity of the Taylor expansion limits the scope of this method to cases where the random variables have a narrow dispersion around their mean value [ELH 13, GUE 15a].

With the perturbation method, the random functions in the expression of the model's response to input parameters are replaced by their Taylor expansions. Terms of the same order are grouped together and, as a result, a system of equations is generated. The resolution is then carried for each order, starting with the zeroth order. The mathematical formalism as well as the general equations for the resolution can be found in the books by El Hami and Radi [ELH 13] and Guerine et al. [GUE 15b].

1.2.2.2. Applications

There are many applications of the perturbation method. This method makes it possible to study the propagation of uncertainties in static and dynamic systems as well as in linear and nonlinear systems. However, it provides precise results only when the uncertain parameters have a low dispersion [ELH 13, GUE 15a].

Guerine et al. [GUE 15b] have used the perturbation method in order to study the aerodynamic properties of elastic structures (stacked flat) subject to several uncertain parameters (structural and geometrical parameters) in the field of modeling and analysis of the vibratory and dynamic behaviors of systems. This work is the first published application of the stochastic finite element method (FEM) combined with the perturbation method for the analysis of aerodynamic stability.

In another study, El Hami and Radi [ELH 13] combine the finite difference method and the perturbation method to model vibration problems in uncertain mechanical structures. This method is used, for example, to determine the probabilistic moments of eigen frequencies and eigen modes of a beam in which the Young modulus varies randomly.

The second order is usually sufficient to determine the first two moments with good precision. In [MUS 99], Muscolino presents a dynamic analysis method for linear systems with uncertain parameters and deterministic excitations. This method improves the first-order perturbation method, which is limited when the dispersion of uncertain parameters is high. The results obtained are compared to the results of the MC method and to the second-order perturbation methods. The results are closely correlated.

1.2.2.3. Remark

The perturbation method consists of expressing all the random variables by their Taylor expansions around their mean values. However, the use of this method is difficult to implement, particularly in the case of systems with many degrees of freedom and in cases where the uncertain parameters have a low dispersion around their mean.

EXAMPLE 1.1.- Application of the perturbation method.

The objective of this example is to demonstrate the advantages of the Muscolino perturbation method to determine the beam response.

A beam which is fixed at its extremities and free to vibrate in the (Oxy) plane is considered (Figure 1.1).

Figure 1.1. Biembedded beam

The mass and stiffness matrices are given by:

The beam has a square section of side b, which is modeled as a Gaussian random variable.

The stiffness matrix [K] can be written as follows:

where [A] is a deterministic matrix.

Likewise, the mass matrix [M] can be written as:

where [B] is a deterministic matrix.

The beam's response to a force F = 600 sin (800t) applied at the beam midpoint is studied. The mean value and standard deviation of the displacement of the beam midpoint are calculated with the secondorder perturbation method and the proposed new method. The results are compared to those obtained with the MC method as the reference using 10,000 draws.

The results (Figures 1.2 and 1.3) show that the two perturbation methods give the same results as the MC method.

Figure 1.2. Mean of the displacement of the beam midpoint

Figure 1.3. Standard deviation of the displacement of the beam midpoint

1.2.3. The polynomial chaos method

1.2.3.1. Origins and principle

The polynomial chaos method is a powerful mathematical tool that was developed by Wiener in the framework of his theory on homogeneous chaos [GUE 15a, GUE 15b]. This method formalizes a separation between the stochastic components and deterministic components of a random function. The polynomial chaos leads to a functional expression of a random response by decomposing its randomness on the basis of orthogonal...