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Virtual design analysis has become an indispensable component in most engineering disciplines. Despite the immense developments and availability of computational resources, the relative computational cost of high-fidelity simulations is getting more and more expensive. This opened the chapter of multi-fidelity learning techniques in the field of automated design optimisation. This work presents a novel multi-fidelity surrogate-assisted design optimisation approach for computationally expensive aerospace applications under uncertainty. The proposed optimisation framework overcomes the challenges of probabilistic design optimisation of computationally expensive problems and is capable of finding designs with optimal statistical performance for both single- and multi-objective problems, as well as constrained problems. Our approach performs the design optimisation with a limited computational budget thanks to the integrated multi-fidelity surrogates for design exploration and uncertainty quantification. The design optimisation is realised following the principles of Bayesian optimisation. The acquisition function balances exploration and exploitation of the design space and allocates the available budget efficiently considering the cost and accuracy of the fidelity levels. To validate the proposed optimisation framework, available multi-fidelity test functions were tailored for benchmarking problems under uncertainty. The benchmarks showed that it is profitable to use multi-fidelity surrogates when the computational budget is too limited to allow for the construction of an accurate surrogate with high-fidelity simulations but is large enough to generate a great number of low-fidelity data. The applicability of the proposed optimisation framework for aerospace applications is presented through optimisation studies of a propeller blade airfoil and a 3D propeller blade.

In this work, an optimisation workflow is presented for uncertainty-based design optimisation using a multi-fidelity Gaussian Process Regression. The motivation of the proposed techniques is to solve a propeller design optimisation problem under uncertainty. This problem involves expensive CFD simulations to evaluate the aerodynamic performance which prohibits the application of standard optimisation techniques and the direct calculation of statistical measures. Therefore, a surrogate-based optimisation workflow is presented. The computational budget limits the number of high-fidelity simulations which makes impossible to accurately approximate the design landscape. This motivates the use of cheap low-fidelity simulations to obtain more information about the unexplored locations of the design landscape. The information stemming from the low- and high-fidelity numerical experiments can be fused together with multi-fidelity Gaussian Process Regression to build an accurate surrogate model despite the low number of high-fidelity simulations. Although Gaussian Process can inherently model uncertain processes, here the design and uncertain parameters are treated separately and only the design space is modelled by a Gaussian Process. The probabilistic space is modelled by a Polynomial Chaos Expansion. The combination of the above techniques allows us to efficiently carry out a design optimisation problem which otherwise would be impractical.

In this paper we present a novel approach to the optimisation of complex systems affected by epistemic uncertainty when system and uncertainty evolve dynamically with time; we propose a new modelling approach that uses Evidence Theory to capture epistemic uncertainty A system is considered which is affected by the time during the operational life (failure rate, performance degradation, function degradation, etc.). The goal is to obtain a resilient design: robust with respect to performance variability and reliable against possible partial failures of one or more components. We propose to enhance the Evidence Network Model (ENM) with time-dependent reliability functions and decompose the problem into subproblems of smaller complexity. Through this decomposition uncertainty quantification of complex systems becomes affordable for a range of real-world applications. The method is here applied to a simple resource allocation problem where the goal is to optimally position subsystems within a spacecraft [1]

This paper presents the multi-objective optimisation of the MH114 high-lift airfoil. We seek the set of Pareto optimal solutions that maximise the airfoil lift and minimise the drag. The lift and drag forces are considered uncertain due to geometrical uncertainties. The uncer- tainty quantification of the probabilistic aerodynamic force values re- quires a large number of samples. However, the prediction of the aero- dynamic forces is expensive due to the numerical solution of the Navier- Stokes equations. Therefore, a multi-fidelity surrogate assisted approach is employed to combine expensive RANS simulations with cheap poten- tial flow calculations. The multi-fidelity surrogate-based approach allows us to economically optimise the aerodynamic design of the airfoil under uncertainty.

Response Surface Methods (RSMs) are statistical and numerical models that approximate the relationship between multiple input variables and an output variable. This chapter introduces the methodology and its importance for engineer- ing design optimisation. The basic steps to build RSMs and validate the model accuracy are explained. An overview of three classical methods (Least Squares, Radial Basis Functions, and Kriging) is provided. A simple wing structure design optimisation problem is used to illustrate the different phases of the response surface methodology and its application to design optimisation. This example also includes the case of noisy data.

Uncertainty-based optimisation techniques provide optimal airfoil de- signs that are less vulnerable to the presence of uncertainty in the operational conditions (i.e., Mach number, angle-of-attack, etc.) at which an airfoil is func- tioning. These uncertainty-based techniques typically require numerous function evaluations to accurately calculate the statistical measure of the quantity of inter- est. To render the computational burden down, the design optimisation of the air- foil is performed by a multi-fidelity surrogate-based technique. The high-fidelity aerodynamic performance is calculated with a compressible RANS solver using a fine grid. At the low-fidelity level a coarser grid is used. To obtain accurate drag predictions despite the lower grid resolution the so-called far-field drag approxi- mation is employed.

Ducted propellers are promising candidates for electrical aircrafts where the propulsion efficiency must be highly optimised. The aerodynamic performance and efficiency of ducted propellers can be obtained through models and experiments of various fidelity, ranging from cheap analytic formulas to expensive Computational Fluid Dynamics simulations. The ducted propeller design problem also involves several uncertainties arising from environmental conditions or manufacturing imperfections. To alleviate the computational burden of the uncertainty-based design optimisation of ducted propellers, we propose a new approach that combines multi-fidelity information fusion and dimensionality reduction by using different surrogate models for the design and probabilistic spaces. The method is validated on a simplified propeller model.

This paper introduces the concept of Resilience Engineering in the context of space systems design and a model of Global System Reliability and Robustness that accounts for epistemic uncertainty and imprecision. In particular, Dempster-Shafer Theory of evidence is used to model uncertainty in both system and environmental parameters. A resilience model is developed to account for the transition from functional to degraded states, and back, during the operational life and the dependency of these transitions on system level design choices and uncertainties. The resilience model is embedded in a network representation of a complex space system. This network representation, called Evidence Network Model (ENM), allows for a fast quantification of the global robustness and reliability of system. A computational optimisation algorithm is then proposed to derive design solutions that provide an optimal compromise between resilience and performance. The result is a set of design solutions that maximise the probability of a system to recover functionalities in the case of a complete or partial failure and at the same time maximises the belief in the desired target value of the performance index.

The paper introduces the concept of design for resilience in the context of space systems engineering and proposes a method to account for imprecision and epistemic uncertainty. Resilience can be seen as the ability of a system to adjust its functioning prior to, during, or following changes and disturbances, so that it can sustain required operations under both expected and unexpected conditions. Mathematically speaking this translates into the attribute of a dynamical system (or time dependent system) to be simultaneously robust and reliable. However, the quantification of robustness and reliability in the early stage of the design of a space systems is generally affected by uncertainty that is epistemic in nature. As the design evolves from Phase A down to phase E, the level of epistemic uncertainty is expected to decrease but still a level of variability can exist in the expected operational conditions and system requirements. The paper proposes a representation of a complex space system using the so called Evidence Network Models (ENM): a non-directed (unlike Bayesian network models) network of interconnected nodes where each node represents a subsystem with associated epistemic uncertainty on system performance and failure probability. Once the reliability and uncertainty on the performance of the spacecraft are quantified, a design optimisation process is applied to improve resilience and performance. The method is finally applied to an example of preliminary design of a small satellite in Low Earth Orbit (LEO). The spacecraft is divided in 5 subsystems, AOCS, TTC, OBDH, Power and Payload. The payload is a simple camera acquiring images at scheduled times. The assumption is that each component has multiple functionalities and both the performance of the component and the reliability associated to each functionality are affected by a level of imprecision. The overall performance indicator is the sum of the performance indicators of all the components.