Valentino Pediroda

Hybridization of a multi-objective genetic algorithm, a neural network and a classical optimizer for a complex design problem in fluid dynamics

Abstract This paper describes the combination of several optimization technologies that can be used to tackle challenging design problems. The approach, that uses a multi-objective genetic algorithm, a neural network, and a gradient-based optimizer, is first outlined with the help of a computationally inexpensive mathematical test function. Then the methodology is applied to the design of a sailing yacht fin keel, coupling the optimization codes to 3D Navier–Stokes simulations. To perform the multi-objective optimization task a parallel computer is employed.

Fictitious Domain approach with hp-finite element approximation for incompressible fluid flow

We consider the application of Fictitious Domain approach combined with least squares spectral elements for the numerical solution of fluid dynamic incompressible equations. Fictitious Domain methods allow problems formulated on a complicated shaped domain ? to be solved on a simpler domain ? containing ? . Least Squares Spectral Element Method has been used to develop the discrete model, as this scheme combines the generality of finite element methods with the accuracy of spectral methods. Moreover the least squares methods have theoretical and computational advantages in the algorithmic design and implementation. This paper presents the formulation and validation of the Fictitious Domain Least Squares Spectral Element approach for the steady incompressible Navier-Stokes equations. The convergence of the approximated solution is verified solving two-dimensional benchmark problems, demonstrating the predictive capability of the proposed formulation.

Sensitivity analysis of dense gas flow simulations to thermodynamic uncertainties

The paper investigates the sensitivity of numerically computed flow fields to uncertainties in thermodynamic models for complex organic fluids. Precisely, the focus is on the propagation of uncertainties introduced by some popular thermodynamic models to the numerical results of a computational fluid dynamics solver for flows of molecularly complex gases close to saturation conditions (dense gas flows). A tensorial-expanded chaos collocation method is used to perform both a priori and a posteriori tests on the output data generated by thermodynamic models for dense gases with uncertain input parameters. A priori tests check the sensitivity of each equation of state to uncertain input data via some reference thermodynamic outputs, such as the saturation curve and the critical isotherm. A posteriori tests investigate how the uncertainties propagate to the computed field properties and aerodynamic coefficients for a flow around an airfoil placed into a transonic dense gas stream.

A Competitive Game Approach for Multi Objective Robust Design Optimization

This paper describes an application of Robust Design methodology in the transonic airfoil design. It has been observed that, minimizing the drag at a single design point (Mach number and angle of attack fixed), it is possible to find solutions characterized by poor offdesign performances (over-optimizing problem). For this reasons, the stability of the performances inside the range of operative conditions is an important objective in the design. Once the operative conditions are defined (range of Mach number and angle of attack), a Multi Objective approach is ne eded; in particular, two are the objectives to be optimized: the mean performances inside the range of operative conditions (optimise mean value of the aerodynamic coefficients) and the stability of the solution (minimize variance of the coefficients). In this Multi Objective optimization problem, we have applied a competitive Game Strategy, based on Nash equilibrium, combined with a particular mono-objective algorithm, the Simplex. The players are in charge of different objectives, corresponding to the two objectives, that have to be optimized by the Simplex algorithm. Since the variables space is split between the two players, each player influences the choices of the other one in the course of the optimisation, until an equilibrium point, corresponding to the best compromise between the objectives, is found. About the optimization test case, the range of operative conditions is Mach=0.73±0.0 5 and angle of attack 2°±0.5, and the original RAE2822 airfoil is parameterized. To reduce the high number of CFD analysis based on Navier -Stokes equations, a statistic extrapolation method, based on an adaptation of DACE, is used to define the required response surfaces. According to our results, the methodology seems to be a promising approach which offers a new possibility to the designer, in particular when a good compromise of performance and stability is required, with cheap computational resources.

TSI metamodels-based multi-objective robust optimization

Purpose – This paper aims to deal with an efficient strategy for robust optimization when a large number of uncertainties are taken into account. Design/methodology/approach – ANOVA analysis is used in order to perform a variance-based decomposition and to reduce stochastic dimension based on an appropriate criterion. A massive use of metamodels allows reconstructing response surfaces for sensitivity indexes in the design variables plan. To validate the proposed approach, a simplified configuration, an inverse problem on a 1D nozzle flow, is solved and the performances compared to an exact Monte Carlo reference solution. Then, the same approach is applied to the robust optimization of a turbine cascade for thermodynamically complex flows. Findings – First, when the stochastic dimension is reduced, the error on the variance between the reduced and the complete problem was found to be roughly estimated by the quantity (1−T¯ TSI )×100, where T¯ TSI is the summation of TSI concerning the variables respecting ...

Quantification of thermodynamic uncertainties in real gas flows

A tensorial-expanded chaos collocation method is developed to take into account uncertainties on thermodynamic properties of complex organic substances. Precisely, we analyse the effect of uncertainties introduced by several thermodynamic models on the numerical results provided by a computational fluid dynamics solver for flows of molecularly complex gases close to saturation condition (dense gas flows). The tensorial-expanded chaos collocation method is used to perform both a priori and a posteriori tests on the output data generated by three popular thermodynamic models for dense gases with uncertain input parameters. A priori tests check the sensitivity of each equation of state to uncertain input data via some reference thermodynamic outputs, such as the saturation curve and the critical isotherm. A posteriori tests investigate how uncertainties propagate to the computed field properties and aerodynamic coefficients for a flow around an airfoil placed into a transonic dense gas stream.

Application of Evolutionary Algorithms and Statistical Analysis in the Numerical Optimization of an Axial Compressor

The purpose of this work is to optimize the stator shape of an axial compressor, in order to maximize the global efficiency of the machine, fixing the rotor shape. We have used a 3D parametric mesh and the CFX-Tascflow code for the flow simulation. To find out the most important variables in this problem, we have run a preliminary series of designs, whose results have been analyzed by a statistic tool. This analysis has helped us to choose the most appropriate variables and their ranges in order to implement the optimization algorithm more efficiently and rapidly. For the simulation of the fluid flow through the machine, we have used a cluster of 12 processors.

A Fast and Robust Adaptive Methodology for Airfoil Design Under Uncertainties based on Game Theory and Self- Organising-Map Theory

Robust Design Optimization is the most appropriate approach to face problems characterized by uncertainties in the operating conditions, that represent a crucial point of aeronautical research activities. The Robust Design methodology illustrated in this paper is based on the multi-objective approach: applying the statistical definition of stability, the method finds, at the same time, optimised solutions for performances and stability. Game Theory is an innovative and efficient numerical methodology that can be applied to solve this kind of multi-objective optimization problems. A Competitive Game Strategy is applied in this paper by linking a mono-objective algorithm, like Downhill Simplex, with a statistical analysis methodology, based on t-Student or on the correlation matrix, that allow to find the optimal variables decomposition between the players (objectives) in the course of the optimization. An alternative to this statistical procedure is given by the innovative Self-OrganisingMaps (SOM) theory, used to find correlations between input or output variables and based on non-linear ordered regression for topology data mapping. The test case used to compare the different methodologies, after a preliminary test on mathematical functions, is the optimization of a symmetric airfoil in transonic and Eulerian flow field with uncertainties in the free stream Mach Number; once the most efficient algorithm is chosen, it is applied to the most demanding optimization of a RAE2822 airfoil in transonic and viscous flow field with uncertainties in the free stream Mach Number and in the angle of attack. In these optimization cases, an adaptive Response Surface Methodology, called DACE, has been used in order to reduce the number of computations required.

Multi criteria decision aiding techniques to select designs after robust design optimization

Robust Design Optimization is the most appropriate approach to face problems characterized by uncertainties on operating conditions, which are peculiarity of aeronautical research activities. The Robust Design methodology illustrated in this paper is based on multi-objective approach. When a Pareto approach is used, a Multi Criteria Decision Method is needed for selecting the final optimal solution. This method is tested on an aeronautic case: the design of a transonic airfoil with uncertainties on free Mach number and angle of attack. The final solution is compared with a well known airfoil: the new design performs as the original one, especially concerning lift and drag stability.

Three-dimensional inverse design of axial compressor stator blade using neural-networks and direct Navier–Stokes solver

In this article we describe a new method for the aerodynamic optimisation and inverse design problem resolution. This method is based on the coupling of a classical optimiser with a neural-network. A Navier–Stokes flow solver is used for an accurate computation of the objective function. At first the neural-network, which has been trained by an initial small database, is used to obtain, by the interpolation of the design sensitivities, a new design point, which is then computed by the Navier–Stokes solver in order to update the neural-network training database for further iterative step. Since the neural-network provides the optimiser with the derivatives, the objective function has to be evaluated only once at every step. By this method, the computational effort is significantly reduced with respect to the classical optimisation methods based on the design sensitivities, that are computed directly by the flow solver. The method proposed has been positively tested on the inverse design of a three-dimensional axial compressor blade, and a summary of the results is provided.