Aerospace System Analysis and Optimization in Uncertainty

With the increasing complexity of systems such as aerospace vehicles, it has become more and more necessary to adopt a global and integrated approach from the early steps and all along the design process. Tightly coupling aerodynamics, propulsion, structure, guidance and navigation, trajectory, etc. but also taking into account environmental and operational constraints as well as manufacturability, reliability, maintainability is a huge challenge.

The design process of complex systems such as aerospace vehicles involves physics-based and mathematical models. A model is a representation of the reality through a set of simulations and/or experimentations under appropriate assumptions. Due to simplification hypotheses, lack of knowledge, and inherent stochastic quantities, models represent reality with uncertainties. These uncertainties are quite large at the early phases of the design process. The term uncertainty has various definitions and taxonomies depending on the research communities. The concept of uncertainty is related to alternative concepts such as imperfection, ignorance, ambiguity, imprecision, vagueness, incompleteness, etc.

The uncertainty propagation consists in determining the impact of the input uncertainties of a simulation code on the outputs of this model. In the MDO context, the simulation code represents, for instance, a set of coupled disciplines and the uncertainty propagation consists in characterizing the multidisciplinary system outputs considering a given number of uncertain input variables modeled with the mathematical formalisms presented in Chapter 2 (see Chapter 6 for uncertainty propagation on a multidisciplinary system). Before considering a multidisciplinary problem in Chapter 6, a single discipline is considered in this chapter to set the basis of uncertainty propagation. Indeed, due to the presence of uncertainty, the outputs of the discipline are also uncertain variables that need to be characterized in order to be used in a design process (e.g., optimization under uncertainty, see Chapter 5).

Assessing the reliability of a complex system with uncertainty propagation consists in estimating its probability of failure. Common sampling strategies for such tasks are notably based on Monte Carlo sampling. This kind of methods is well suited to characterize events of which associated probabilities are not too low with respect to the simulation budget. However, for critical systems such as aerospace vehicles, the reliability specifications often induce very low probability of failures (said below 10⁻⁴). In this case, Monte Carlo based methods are not efficient inducing unaffordable costs with regard to the available simulation budget. In this chapter, we review the main simulation techniques to estimate low failure probabilities with accuracy.

Optimization under uncertainty is a key problem in order to solve complex system design problem while taking into account inherent physical stochastic phenomena, lack of knowledge, modeling simplifications, etc. Different reviews of optimization techniques in the presence of uncertainty can be found in the literature. The choice of the algorithm is often problem-dependent. The designer has to choose firstly the optimization problem formulation with respect to the system specifications and study but also the optimization algorithm to apply. The objective of this chapter is to present the different existing approaches to solve an optimization problem under uncertainty and to focus specifically on the uncertainty handling mechanisms. The chapter is organized as follows. Firstly, in Section 5.1, different optimization problem formulations are introduced, highlighting the importance of uncertainty measures and the distinctions between robustness-based formulation, reliability-based formulation, and robustness-and-reliability-based formulation. Then, in Section 5.2, different approaches to quantify the uncertainty in optimization are discussed. Finally, in the Section 5.3, an overview of optimization algorithms is presented with a focus on stochastic gradient, population-based algorithms, and surrogate-based approaches. For each type of algorithms the handling of uncertainty is analyzed and discussed.

In Chapter 3, several uncertainty propagation techniques for black-box functions have been introduced. In order to take into account the specific features of multidisciplinary design problems, these methodologies have to be adapted accordingly. This chapter reviews various alternatives described in the literature to efficiently propagate uncertainty in the case of coupled multidisciplinary systems (Figure 6.1).

This chapter is devoted to the description of the MDO formulations in the presence of uncertainty. In Chapter 1, deterministic MDO formulations have been introduced, highlighting the interest of such methodologies to solve complex and multidisciplinary design problems. Uncertainty-based multidisciplinary design optimization (UMDO) deals with the presence of uncertainty in MDO problems. The understanding of the importance of UMDO is spreading among academia and industry quickly. Nevertheless, in comparison with the deterministic MDO approaches, the UMDO methodologies are still in the early stages of development and numerous challenges have still to be solved. In the last decades, important improvements have been made in this field of research and are presented in this chapter. UMDO problems combine the challenges of deterministic MDO (organization of the design process, control of interdisciplinary couplings, etc.) and the difficulties involved by uncertainty propagation for multi-physics problems. Most of the existing UMDO formulations are built on the uncertainty propagation techniques dedicated to multidisciplinary problems presented in Chapter 6. The algorithms used to solve these UMDO problems are not discussed in this chapter, but all the presented optimization techniques in Chapter 5 may be used to solve UMDO problems.

The challenges of handling uncertainties within an MDO process have been discussed in Chapters 6 and 7. Related concepts to multi-fidelity are introduced in this chapter. Indeed, high-fidelity models are used to represent the behavior of a system with an acceptable accuracy. However, these models are computationally intensive and they cannot be repeatedly evaluated, as required in MDO. Low-fidelity models are more suited to the early design phases as they are cheaper to evaluate. But they are often less accurate because of simplifications such as linearization, restrictive physical assumptions, dimensionality reduction, etc. Multi-fidelity models aim at combining models of different fidelities to achieve the desired accuracy at a lower computational cost. In Section 8.2, the connection between MDO, multi-fidelity, and cokriging is made through a review of past works and system representations of code architectures.

In addition to the multi-fidelity aspects in MDO discussed in Chapter 8, two additional topics of interest to solve complex MDO problems are discussed in this chapter: multi-objective MDO and mixed continuous/discrete variable design optimization problems.

In the recent years, aviation is facing with the increasing of fuel price and flights, and the impact is estimated to grow more and more in next years without any action. To reduce its environmental footprint, engineers are trying to design more efficient aircraft, with engines less consuming. However, the classical aircraft’s tube and wing configuration has been developed over half a century, and it still offers small potential gain. A breakthrough in aircraft design is then needed to drastically reduce the problem.

For the coming years, the commercial air transport industry will have to take up the huge challenge of both reducing the costs associated to the aircraft fuel consumption and reduce the global air transport environmental footprint. Reaching these two objectives leads to the reduction of the fuel consumption, the reduction of the pollutant emissions (specifically CO₂ and NO_x emissions), and the reduction of noise emissions.

For many countries (United States of America, Russia, Europe, Japan, etc.), the launch vehicles are cornerstones of an independent access to space. The space agency strategies for Solar system exploration, Earth monitoring and observation, human spaceflight are developed in accordance with their launch vehicle capabilities. Launch vehicle designs are long term projects (around a decade) involving large budgets and requiring efficient organization.