Efficient Multi-Objective Shape Optimization Using Variable Fidelity Performance Data

Optimization problems in many industrial applications are very hard to solve. Many examples of them can be found in the design of aeronautical systems. In this field, the designer is frequently faced with the problem of considering not only a single design objective, but several of them, i.e., the designer needs to solve a Multi-Objective Optimization Problem (MOP). In aeronautical systems design, aerodynamics plays a key role in aircraft design, as well as in the design of propulsion system components, such as turbine engines. Thus, aerodynamic shape optimization is a crucial task, and has been extensively studied and developed. Multi-Objective Evolutionary Algorithms (MOEAs) have gained popularity in recent years as optimization methods in this area, mainly because of their simplicity, their ease of use and their suitability to be coupled to specialized numerical simulation tools. In this chapter, we will review some of the most relevant research on the use of MOEAs to solve multi-objective and/or multi-disciplinary aerodynamic shape optimization problems. In this review, we will highlight some of the benefits and drawbacks of the use of MOEAs, as compared to traditional design optimization methods. In the second part of the chapter, we will present a case study on the application of MOEAs for the solution of a multi-objective aerodynamic shape optimization problem.

Numerical noise is an inevitable by-product of Computational Fluid Dynamics (CFD) simulations which can lead to challenges in finding optimum designs. This article draws attention to the issue, illustrating the difficulties it can cause for road vehicle aerodynamics simulations. Firstly a benchmark problem is used to assess a range of turbulence models and grid types. Large noise amplitudes up to 22% are evident for solutions computed on unstructured tetrahedral grids whereas computations on hexahedral and polyhedral grid structures exhibit substantially less noise. The Spalart Allmaras turbulence model is shown to be far less susceptible to noise levels than two other commonly-used models for this application. Secondly, multi-objective aerodynamic shape optimization is applied to a fairing for a practical road vehicle which is parameterised in terms of three design variables. Moving Least Squares (MLS) metamodels are constructed from 50 high-fidelity CFD solutions for two objective functions. Subsequent optimization is successful for the first objective, however numerical noise levels in excess of 7% give rise to difficulties for the second one. A revision to the problem leads to success and the construction of a small Pareto front. Further analysis underlines the inherent capability of MLS metamodels in dealing with noisy CFD responses.

Surrogate-based method can dramatically reduce the number of expensive function evaluations in real-world multi- objective optimization problems (MOP). When the number of objectives is small, using surrogate models combined with expected hypervolume improvement (EHVI) infill sampling criterion (ISC) has been proved to be efficient to provide a set of solutions with good diversity and good proximity to the Pareto front (PF) in aerodynamic shape optimization. However, traditional hypervolume-based infilling strategies use only one kind of ISC to generate one or multiple sample points, the advantages of various kinds of ISC cannot be comprehensively utilized and the parallelization is not easy to implement. This paper proposes a combined multi-point infilling strategy based on Kriging models and develops an efficient global multi-objective constrained optimization method (EGMOCO) to solve multi- objective aerodynamic shape optimization with complex constraints. Multiple sample points are generated by using four ISC considering hypervolume at each iteration and then evaluated in parallel. Firstly, the performance of EGMOCO is compared with that of single criterion EHVI strategy on six benchmarks within the same computational budget to prove its effectiveness, and then EGMOCO is implemented in an aerodynamic shape optimization problem with complex constraints. The result shows that EGMOCO has good performance in balancing local exploitation and global exploration with faster convergence rate and high robustness, the whole PF can be fully explored in limited evaluations and the constraint handling is effective especially for real-world problems with complex and nonlinear constraints, the comprehensive aerodynamic performance of the airfoil is greatly improved. It can be confirmed that Kriging-based multi-objective optimization method combined with multi-point infilling strategy performs better than single infilling criterion EHVI, since different sample infilling criteria can complement with each other, both local exploitation and global exploration can be considered and well balanced.

In this research, a proper orthogonal decomposition (POD)-based re-parameterization approach is utilized to realize efficient multi-objective aerodynamic shape optimization (MOASO) by reducing the number of design variables, which results in reducing the number of computational fluid dynamics evaluations. The approach developed is examined in a three-dimensional wing-shape optimization problem of a supersonic transport configuration. As for the multi-objective design optimization problem, aerodynamic/sonic boom evaluations are performed to minimize the drag coefficient and sonic boom strength. By introducing a variable fidelity concept in the POD approach, the optimization problem can be solved with much smaller computational cost. Important design insights can be extracted and discussed from the most dominant (first) POD mode.

Many real-world multi-objective optimization problems have hundreds or even thousands of decision variables, which contrast with the current practice of multi-objective metaheuristics whose performance is typically assessed using benchmark problems with a relatively low number of decision variables (normally, no more than 30). In this paper, we propose a cooperative coevolution framework that is capable of optimizing large scale (in decision variable space) multi-objective optimization problems. We adopt a benchmark that is scalable in the number of decision variables (the ZDT test suite) and compare our proposed algorithm with respect to two state-of-the-art multi-objective evolutionary algorithms (GDE3 and NSGA-II) when using a large number of decision variables (from 200 up to 5000). The results clearly indicate that our proposed approach is effective as well as efficient for solving large scale multi-objective optimization problems.

A proper orthogonal decomposition (POD) method is used to interpolate the flow around an airfoil for various Mach numbers and angles of attack in the transonic regime. POD uses a few numerical simulations, called snapshots, to create eigenfunctions. These eigenfunctions are combined using weighting coefficients to create a new solution for different values of the input parameters. Since POD methods are linear, their interpolation capabilities are quite limited when dealing with flow presenting nonlinearities, such as shocks. In order to improve their performance for cases involving shocks, a new method is proposed using variable fidelity. The main idea is to use POD to interpolate the difference between the CFD solution obtained on two different grids, a coarse one and a fine one. Then, for any new input parameter value, a coarse grid solution is computed using CFD and the POD interpolated difference is added to predict the fine grid solution. This allows some nonlinearities associated with the flow to be introduced. Results for various Mach numbers and angles of attack are compared to full CFD results. The variable fidelity-based POD method shows good improvement over the classical approach.

The effect of strain rate on mechanical properties and microstructure of a metastable FeMnCoCr high entropy alloy

In this paper the use of Genetic Algorithms for multi objective optimisation in aerodynamic optimisation is outlined. After a review of existing GA methodologies the operators considered at present the most promising one are described. A simple mathematical test is used for preliminary algorithmic perfomance while in more applicative cases the pressure reconstruction problem of two conflicting aerodynamic profiles is used as benchmark. A full potential transonic solver is at first used showing the performances of the optimisation algorithm employed while final results are obtained using a commercial Navier-Stokes solver with k-e turbulence modelling to reconstruct the geometry of two airfoils working at Mach=0.2 Re=5E6 and Mach=0.77 Re=19.6E6. Even thogh the test case presented might not have a practical application, it shows that direct multi objective optimisation with Navier Stokes solver can be faced with GA.

Expedited constrained multi-objective aerodynamic shape optimization by means of physics-based surrogates

In this paper, a numerical optimization of a radial turbine impeller is developed, which combines popular techniques such as artificial neural networks, genetic algorithm, database and CFD analysis tools. Multi-objective aerodynamic optimization is performed by using software FINETM/Design 3D. The performance parameters in the terms of total-to-static efficiency, expansion ratio are defined as optimized objectives. The performance of the new impeller is compared to the original one at the design point as well as at off-design points with a significant performance improvement. The design method provided very good predictions leading to an efficiency increase of 8.01 percent and an expansion ratio increase of 2.76 percent. An analysis of the CFD results is also discussed within the original and the new impeller. The whole results indicate that the multi-objective aerodynamic optimization process can be used in practical terms.

The traditional multi-objective efficient global optimization (EGO) algorithms have been hybridized and adapted to solving the expensive aerodynamic shape optimization problems based on high-fidelity numerical simulations. Although the traditional EGO algorithms are highly efficient in solving some of the optimization problems with very complex landscape, it is not preferred to solve most of the aerodynamic shape optimization problems with relatively low-degree multi-modal design spaces. A new infill criterion encouraging more local exploitation has been proposed by hybridizing two traditional multi-objective expected improvements (EIs), namely, statistical multi-objective EI and expected hypervolume improvement, in order to improve their robustness and efficiency in aerodynamic shape optimization. Different analytical test problems and aerodynamic shape optimization problems have been investigated. In comparison with traditional multi-objective EI algorithms and a standard evolutionary multi-objective optimization algorithm, the proposed method is shown to be more robust and efficient in the tests due to its hybrid characteristics, easier handling of sub-optimization problems, and enhanced exploitation capability.

Over-the-air federated learning (AirFL) has developed as a communication-efficient solution to enable distributed machine learning over edge devices by integrating computation and communication into a joint design. However, the aggregation error caused by noise and wireless channel fading may lead to a compromised learning performance in an AirFL system. Potential malicious participants can also cause this compromise such that decoding the individual uploaded model updates at the server side to perform security examination is still imperative for AirFL. In this paper, we investigate a novel multi-objective resource allocation problem to combat against such aggregation error and improve the information transmission efficiency of uplink by jointly optimizing the device selection, power allocation and receive scalar control. The objective of the multi-objective optimization problem (MOP) is to minimize the average mean squared error (MSE) of the over-the-air aggregation for different communication rounds and maximize the long-term energy efficiency (EE) of the system. Considering the complexity and dynamic environment, we present a deep reinforcement learning (DRL) based framework to solve the MOP. The MOP is firstly decomposed into multiple subproblems. Each subproblem is modelled as a neural network, which can be addressed by the proposed learning-based resource allocation (LRA) algorithm. The numerical results illustrate that our proposed approach can effectively tackle the MOP and outperform the benchmark approaches with appropriate training mechanism and reward design.

<p>The unsteady wind pressure fields deriving from the atmospheric boundary layer interacting with the aerodynamics of a tall building can be represented as spatially-correlated multi-variate random processes. Due to its random nature, the fluctuating pressure field can be considered as the superposition of a large number of independent partial pressure fields related to particular mechanisms of the excitation. It is within this context that Proper Orthogonal Decomposition (POD), a powerful statistical tool particularly suitable to deal with a large number of random variables, can be used to enhance the understanding and help with the interpretation of the physical phenomena that can drive the wind-induced dynamic response of tall buildings. The main purpose of the work presented within this technical paper, which makes use of wind pressures measured on a scaled wind tunnel model of a prismatic 180m tall building, was to identify hidden physical phenomena embedded within chaotic wind pressure fields with the idea of potentially simplify the analysis work and reduce the increasing demand for wind tunnel data storage. More specifically, this paper will show how complex fluctuating wind pressure fields – driving the response of tall buildings in both the along- and the cross-wind direction – can be approximated by a few dominant standard modes of the pressure field itself.</p>

Principal Component Analysis (PCA), also known as Proper Orthogonal Decomposition (POD), is one of the prevailing Reduced Order Modeling (ROM) techniques for aerodynamic data analysis. Along with the PCA, its variant for missing data, gappy POD, has recently found its application by transforming problems into missing data problems. In this paper, an alternative based on probability theory, i.e., Probabilistic Principal Component Analysis (PPCA), is employed for aerodynamic data reconstruction problems in lieu of the PCA and the gappy POD. Unlike the PCA, the PPCA generates spanning vectors of low-dimensional space iteratively with an Expectation Maximization (EM) algorithm. The combined algorithm, termed as Expectation-Maximization Probabilistic Principal Component Analysis (EM-PCA), can deal with not only complete data but also incomplete data sets, since the EM algorithm naturally incorporates missing data in its formulation. Therefore, the EM-PCA has potential to be a substitute for both PCA and gappy POD. For illustration, the EM-PCA is applied to complete as well as incomplete snapshot ensembles compiled from aerodynamic simulations, and acquired modes and restored data are verified with those obtained by the PCA and the gappy POD.

The focus of this study is to experimentally quantify the impact of gusts on the performance of a NACA-0012 airfoil and provide a load estimation strategy from modal decomposition coefficients. Experiments are performed in a water tunnel facility at chord-based Reynolds number $Re\approx4\times10^4$ for a range of angles of attack $-6^\circ\le\alpha\le12^\circ$. The gust generator functioned as a bluff body with a controlled shedding phase, producing a von K\'{a}rm\'{a}n vortex street. The generator was tested at two different locations; in the first configuration, the gust generator was positioned inline 1.5 chords upstream of the airfoil leading edge (referred to as the centered-gust case), and, in the second configuration, the gust generator was positioned upstream of the airfoil with a 0.5 chord transverse offset (referred to as the offset-gust case). Aerodynamic load experiments were performed for both configurations, $\alpha$ from $-6^\circ$ to $12^\circ$ with an increment of $1^\circ$. Synchronized load-cell and time-resolved PIV measurements were performed for selected $\alpha$ (i.e., $0^\circ$, $6^\circ$, and $12^\circ$) for both configurations. Proper Orthogonal Decomposition (POD) of the velocity field was used to extract coherent structures and identify dominant modes across different scenarios. Cross-correlation between the time-varying POD coefficients of the dominant modes and $c_l$, together with magnitude-squared coherence, was used to quantify the impact of the flow field on $c_l$. The centered-gust preserved a near-linear $c_l(\alpha)$ but reduced lift-curve slope and significantly increased variance; the offset-gust induced earlier deviation from a linear slope and higher drag values. Time-resolved PIV showed the rise/fall of $c_l$ associated with leading-edge upwash/downwash due to vortical structures convecting over the airfoil. The gust footprint is shown to be captured by a pair of POD modes; coherence between lift-curve and time-varying coefficient approached peak value at the gust frequency ($\sim$2.1~Hz) and the dominant mode led $c_l$ by $t^*\sim$0.12 and 0.18, with correlation peaks of 0.92 and 0.83, for centered and offset-gust configurations, respectively, at $\alpha = 12^\circ$. These findings showed a phase-accurate linkage between a coherent gust and lift response. This enabled the use of multi-time-delay Linear-Stochastic-Estimation to predict $c_l$ from the time-varying POD coefficient. The estimated $c_l$ values using this approach had a Root Mean Square Error (RMSE) of 0.135. The resulting low-order model description enables estimation of $c_l$ from select dominant modes.