pFOSM: An efficient algorithm for aerodynamic robust design based on continuous adjoint and matrix-vector products

Abstract This article proposes a novel robust design tool based on the Method of Moments (MoM), demonstrated in problems governed by laminar flows of incompressible fluids. The objective function of the robust design problem consists of the weighted sum of the mean value and standard deviation of the Quantity of Interest (QoI), computed based on the value and gradient of the latter with respect to the uncertain variables. First-order derivatives with respect to the uncertain and design variables are computed using the continuous adjoint method. Using a first-order MoM to compute the objective function and by minimizing it with a gradient-based approach leads to the need of computing up to second-order derivatives of the QoI with respect to the design and uncertain variables. This can lead to a high computational cost in problems with a great number of uncertain variables. The proposed robust design algorithm avoids computing second-order derivatives by computing matrix-vector products instead, using a combination of adjoint and direct differentiation. This makes the cost per gradient computation independent of the numbers of both the design and uncertain variables. The proposed method is demonstrated through the constrained shape optimization of an isolated 2D airfoil under uncertain farfield flow conditions.

[1]  Christian B Allen,et al.  CFD‐based optimization of aerofoils using radial basis functions for domain element parameterization and mesh deformation , 2008 .

[2]  K. Giannakoglou,et al.  Robust design in aerodynamics using third‐order sensitivity analysis based on discrete adjoint. Application to quasi‐1D flows , 2012 .

[3]  M. Eldred,et al.  Comparison of Non-Intrusive Polynomial Chaos and Stochastic Collocation Methods for Uncertainty Quantification , 2009 .

[4]  Tanmoy Chatterjee,et al.  A Critical Review of Surrogate Assisted Robust Design Optimization , 2019 .

[5]  Costas Papadimitriou,et al.  Aerodynamic shape optimization for minimum robust drag and lift reliability constraint , 2016 .

[6]  Pénélope Leyland,et al.  A Multilevel Monte Carlo Evolutionary Algorithm for Robust Aerodynamic Shape Design , 2017 .

[7]  Arthur C. Taylor,et al.  Approach for uncertainty propagation and robust design in CFD using sensitivity derivatives , 2001 .

[9]  Gerhart I. Schuëller,et al.  Computational methods in optimization considering uncertainties – An overview , 2008 .

[10]  Zhao Huan,et al.  Effective robust design of high lift NLF airfoil under multi-parameter uncertainty , 2017 .

[11]  Aerodynamic Shape Optimization Using the Adjoint-Based Truncated Newton Method , 2018, Computational Methods in Applied Sciences.

[12]  Kyriakos C. Giannakoglou,et al.  Noise reduction in car aerodynamics using a surrogate objective function and the continuous adjoint method with wall functions , 2015 .

[13]  Kyriakos C. Giannakoglou,et al.  Third‐order sensitivity analysis for robust aerodynamic design using continuous adjoint , 2013 .

[14]  K. Giannakoglou,et al.  Aerodynamic design using the truncated Newton algorithm and the continuous adjoint approach , 2012 .

[15]  C. Allen,et al.  A Geometric Comparison of Aerofoil Shape Parameterisation Methods , 2016 .

[16]  K. Giannakoglou,et al.  Continuous Adjoint Methods for Turbulent Flows, Applied to Shape and Topology Optimization: Industrial Applications , 2016 .

[17]  Luis F. Gonzalez,et al.  Robust design optimisation using multi-objectiveevolutionary algorithms , 2008 .

[18]  D. Xiu,et al.  Modeling uncertainty in flow simulations via generalized polynomial chaos , 2003 .

[19]  Kyriakos C. Giannakoglou,et al.  Direct, adjoint and mixed approaches for the computation of Hessian in airfoil design problems , 2008 .

[20]  Charles Hirsch,et al.  Assessment of intrusive and non-intrusive non-deterministic CFD methodologies based on polynomial chaos expansions , 2010 .

[21]  Régis Duvigneau.  Aerodynamic Shape Optimization with Uncertain Operating Conditions using Metamodels , 2007 .

[22]  D. Spalding,et al.  A calculation procedure for heat, mass and momentum transfer in three-dimensional parabolic flows , 1972 .

[23]  R. M. Hicks,et al.  Wing Design by Numerical Optimization , 1977 .

[24]  Robert W. Walters,et al.  Uncertainty analysis for fluid mechanics with applications , 2002 .

[25]  Hans Janssen,et al.  Monte-Carlo based uncertainty analysis: Sampling efficiency and sampling convergence , 2013, Reliab. Eng. Syst. Saf..

[26]  R. Caflisch,et al.  Quasi-Monte Carlo integration , 1995 .

[27]  K. Giannakoglou,et al.  The continuous adjoint approach to the k–ω SST turbulence model with applications in shape optimization , 2015 .

[28]  H. Sobieczky.  Parametric Airfoils and Wings , 1999 .

[29]  Joaquim R. R. A. Martins,et al.  Aerodynamic Shape Optimization of the CRM Configuration Including Buffet-Onset Conditions , 2016 .

[30]  Sriram Shankaran,et al.  Robust Optimization for Aerodynamic Problems Using Polynomial Chaos and Adjoints , 2012 .

[31]  M. J. D. Powell,et al.  A fast algorithm for nonlinearly constrained optimization calculations , 1978 .

[32]  Athanasios G. Liatsikouras,et al.  AERODYNAMIC SHAPE OPTIMIZATION UNDER FLOW UNCERTAINTIES USING NON-INTRUSIVE POLYNOMIAL CHAOS AND EVOLUTIONARY ALGORITHMS , 2017 .

[33]  Stephen J. Wright,et al.  Numerical Optimization , 2018, Fundamental Statistical Inference.

[34]  Gao Zhenghong,et al.  Review of Robust Aerodynamic Design Optimization for Air Vehicles , 2019 .

[35]  Thomas Gerstner,et al.  Numerical integration using sparse grids , 2004, Numerical Algorithms.

[36]  Sang-Hoon Lee,et al.  Robust design with arbitrary distributions using Gauss-type quadrature formula , 2009 .

[37]  Hossein Zare-Behtash,et al.  State-of-the-art in aerodynamic shape optimisation methods , 2018, Appl. Soft Comput..

[38]  Charles Hirsch,et al.  Implementation of Intrusive Polynomial Chaos in CFD Codes and Application to 3D Navier-Stokes , 2013, Uncertainty Quantification in Computational Fluid Dynamics.

[39]  Kyriakos C. Giannakoglou,et al.  On the proper treatment of grid sensitivities in continuous adjoint methods for shape optimization , 2015, J. Comput. Phys..

[40]  Bernhard Sendhoff,et al.  Robust Optimization - A Comprehensive Survey , 2007 .

[41]  Marin D. Guenov,et al.  Novel Uncertainty Propagation Method for Robust Aerodynamic Design , 2011 .

[42]  Dongbin Xiu,et al.  High-Order Collocation Methods for Differential Equations with Random Inputs , 2005, SIAM J. Sci. Comput..