The progress achieved by digital and software tools in the past 40 years has allowed scientists to dramatically improve their understanding of the world. The development of mathematical models has allowed us to work on increasingly sophisticated problems in a wide range of fields: predicting the behavior of production tools, transportation, the environment, etc. Managing these complex problems has been facilitated within each discipline separately but also from a cross-disciplinary perspective, allowing more general phenomena to be tackled.

The field of fluid–structure interactions unites the study of all phenomena involving a coupling between the motion of a structure and the motion of a fluid. The range of studied phenomena is very broad, from vibrating cylinders in flows, such as in the nuclear industry, to vibrating structures in turbulent flows and free-surface phenomena in reservoirs. One well-known example of fluid–structure interaction and the complexity of the couplings involved is the collapse of the Tacoma bridge in 1940, which began vibrating to the point of resonance frequency under the effect of violent winds, causing it to be completely destroyed. This shows just how important it is to prepare reliable models in advance of any project so that this kind of behavior may be predicted.

Wind tunnels, such as in aeronautics, allow us to inspect the behavior of the structure on the ground without needing to perform tests in flight. The Euler and Navier–Stokes equations have made it possible to rigorously define a physical framework for characterizing the behavior of the aircraft in terms of a set of different parameters such as the velocity or Mach number. Finite element models have greatly simplified the process of representing an aircraft model and its structure, as well as the way that the aircraft responds to stress.

However, the complexity of the studied phenomena is reflected in the prohibitive computational costs, which motivates us to search for reduced models with more realistic computation times. By a reduced model, we mean a description as a low-dimensional system obtained by analyzing classical numerical formulations. Acheiving this reduction incurs an initial cost, but this cost is largely offset if the reduced model is later found to be applicable for configurations of parameters other than those of the initial formulation.

Thus, just like in other areas of the industry, optimization research is extremely active within the aviation sector. One significant development since the late 1980s has been the introduction of uncertainty parameters into numerical models. Optimization techniques in the presence of uncertainty in aerodynamics have only been studied more recently, beginning in the early 2000s. Their introduction was motivated by the need to account for specific types of situations that make it too difficult to precisely evaluate the aircraft’s behavior. For example, during the aircraft design phase, in order to meet the various different criteria or eliminate certain problems encountered by the model, the model is able to adjust itself to more effectively meet the requirements and needs that it is designed to satisfy. The initial drafts of the model are not fixed, but for safety reasons it is necessary to ensure throughout the development process that the structure is capable of withstanding the stresses that it is likely to encounter in operating conditions. One way of accounting for these potential changes is to introduce uncertainty into the model.

Furthermore, when designing aircraft, manufacturers are naturally interested in maximizing the performance of each vehicle: reducing pollution, noise, drag, increasing the range, maximizing stability, etc. Minimizing the structural mass is an important objective for manufacturers as it allows other optimization criteria such as reducing pollution or extending the range to be satisfied. But less mass will also have negative repercussions on other criteria, including the stability of the aircraft in flight, for example by rendering it susceptible to the phenomenon of “fluttering”.

Manufacturers must therefore perform constrained optimization: minimizing the weight of the wing while ensuring that fluttering cannot affect the airplane within its flight envelope. In such a case, optimization problems have a cross-disciplinary character, since they exhibit behaviors that include both structural and aerodynamic aspects.

The goal is now to integrate the aspect of uncertainty mentioned above into the optimization process. However, to do this, we must first identify the nature of these uncertainties, and decide how we should represent them. Several types of uncertainty have been identified and classified according to their nature.

Accounting for uncertainty has been studied in a number of research areas, but, until recently, in aeronautics research it was not possible to account for or quantify structural uncertainties within the optimization procedures due to the limitations of numerical tools and a lack of theoretical understanding of their impact within reliability studies. Engineers have therefore been forced to implement alternative procedures to simplify the integration of structural uncertainties into model development. The first studies on this topic in the aviation sector were only conducted in the 1990s, at which point this field of research began to produce tangible results.

In the case of optimization problems with probabilistic constraints, reliability-based optimization, which is extremely common in industrial contexts, replaces these probabilistic constraints with another deterministic optimization problem derived by techniques of approximation. The primary difficulty lies in evaluating the reliability of the structure, which is itself the result of another given optimization procedure. Reliability analysis is performed at the optimal point in order to determine the reliability index of the limiting state that is being considered.

This book presents the different aspects of fluid–structure interaction: vibroacoustics and aerodynamics, and the various numerical methods used to simulate them numerically.

One chapter is devoted to the question of model reduction in fluid–structure interaction problems. We begin by presenting dynamic substructuring methods in linear and nonlinear cases. We then give a description of the method of proper orthogonal decomposition for fluid flows. Finally, we present a modal synthesis method for solving large-scale coupled fluid-structure problems. This method couples a dynamic substructuring method of the type proposed by Craig and Bampton with an acoustic subdomain method based on an acoustic formulation of the velocity potential.

To account for uncertainty, one chapter presents concepts associated with reliability and its objectives and benefits in mechanics, methods for calculating the probability of failure, simulation methods such as the Monte Carlo and response surface methods, and approximate methods for analyzing the reliability and calculating the reliability index by the first-order reliability method (FORM) and the second-order reliability method (SORM). We then give a detailed presentation of the implementation of the latter approach in the context of a certain set of reliability-based optimization problems encountered when designing structures that interact with flowing fluids, with the goal of detecting the critical frequency bands that might cause the structure to experience damage or destruction.