Multi-fidelity design optimization of transonic airfoils using physics-based surrogate modeling and shape-preserving response prediction

A computationally efficient design methodology for transonic airfoil optimization has been developed. In the optimization process, a numerically cheap physics-based low-fidelity surrogate (the transonic small-disturbance equation) is used in lieu of an accurate, but computationally expensive, high-fidelity (the compressible Euler equations) simulation model. Correction of the low-fidelity model is achieved by aligning its corresponding airfoil surface pressure distribution with that of the high-fidelity model using a shape-preserving response prediction technique. The resulting method requires only a single high-fidelity simulation per iteration of the design process. The method is applied to airfoil lift maximization in two-dimensional inviscid transonic flow, subject to constraints on shock-induced pressure drag and airfoil cross-sectional area. The results showed that more than a 90% reduction in high-fidelity function calls was achieved when compared to direct high-fidelity model optimization using a pattern-search algorithm.     

A phase field approach to shape optimization in Navier–Stokes flow with integral state constraints

We consider the shape optimization of an object in Navier–Stokes flow by employing a combined phase field and porous medium approach, along with additional perimeter regularization. By considering integral control and state constraints, we extend the results of earlier works concerning the existence of optimal shapes and the derivation of first order optimality conditions. The control variable is a phase field function that prescribes the shape and topology of the object, while the state variables are the velocity and the pressure of the fluid. In our analysis, we cover a multitude of constraints which include constraints on the center of mass, the volume of the fluid region, and the total potential power of the object. Finally, we present numerical results of the optimization problem that is solved using the variable metric projection type (VMPT) method proposed by Blank and Rupprecht, where we consider one example of topology optimization without constraints and one example of maximizing the lift of the object with a state constraint, as well as a comparison with earlier results for the drag minimization.     

On the optimal shape of a hydrofoil

On the basis of the results on domain shape optimization for elliptic systems developed in Ref. 1, the problem of the optimal shape of a hydrofoil, moving slowly in a viscous incompressible fluid, is formulated and studied. The otpimization problem consists in finding the shape of the hydrofoil having minimal drag, while satisfying certain constraints on the volume and hydrodynamic lift.     

Coupled Aerostructural Design Optimization Using the Kriging Model and Integrated Multiobjective Optimization Algorithm

The paper develops and implements a highly applicable framework for the computation of coupled aerostructural design optimization. The multidisciplinary aerostructural design optimization is carried out and validated for a tested wing and can be easily extended to complex and practical design problems. To make the framework practical, the study utilizes a high-fidelity fluid/structure interface and robust optimization algorithms for an accurate determination of the design with the best performance. The aerodynamic and structural performance measures, including the lift coefficient, the drag coefficient, Von-Mises stress and the weight of wing, are precisely computed through the static aeroelastic analyses of various candidate wings. Based on these calculated performance, the design system can be approximated by using a Kriging interpolative model. To improve the design evenly for aerodynamic and structure performance, an automatic design method that determines appropriate weighting factors is developed. Multidisciplinary aerostructural design is, therefore, desirable and practical. The authors acknowledge the support of a Korea Research Foundation Grant funded by the Korean Government and the second stage of Brain Korea 21st project.     

一个新的边界层方程及其在形状控制中的应用

本文给出固壁边界上(即一个二维流形上) 的流体速度梯度和压力的二阶偏微分方程, 从而也给出边界上法向应力, 以及流体中运动物体所受的阻力和升力的计算公式. 本方法的创新在于边界上法向速度梯度不是通过在边界层内速度梯度的数值微分达到, 而是通过它与其他变量一起作为一组偏微分方程的解而得到, 证明边界层方程组的适定性问题, 并且给出解关于边界形状的Gâteaux 导数所满足的偏微分方程. 本文将本方法应用于飞机外形的形状最优控制, 给出阻力泛函关于形状第一变分的可计算形式. 数值例子表明, 用本方法得到的阻力精度比通用程序得到要高.     

One nonlinear variational problem of the theory of cavitating profiles

In this paper we study the limiting values of the lift and drag coefficients of profiles in the Helmholtz-Kirchhoff (infinite cavity) flow. The coefficients are based on the wetted arc length of profile surfaces. Namely, for a given value of the lift coefficient we find minimum and maximum values of the drag coefficient. Thereby we determine maximum and minimum values of the lift-to-drag ratios.     

Efficient Aerodynamic Shape Optimization in MDO Context

The aerospace industry is increasingly relying on advanced numerical flow simulation tools in the early aircraft design phase. Today's flow solvers, which are based on the solution of the compressible Euler and Navier-Stokes equations, are able to predict aerodynamic behaviour of aircraft components under different flow conditions quite well [1]. Within the next few years numerical shape optimization will play a strategic role for future aircraft design. It offers the possibility of designing or improving aircraft components with respect to a pre-specified figure of merit, subject to geometrical and physical constraints. Here, aero-structural analysis is necessary to reach physically meaningful optimum wing designs. The use of single disciplinary optimizations applied in sequence is not only inefficient but in some cases is known to lead to wrong, non-optimal designs [2]. Although multidisciplinary optimizations (MDO) are possible with classical approaches for sensitivity evaluations by means of finite differences, these methods are extremely expensive in terms of calculation time, requiring the reiterated solution of the coupled problem for every design variable. However, adjoint approaches allow the evaluation of these sensitivities in an efficient way and lead to high accuracy. Firstly, we present the development and application of a continuous adjoint approach for single disciplinary aerodynamic shape design. This approach was previously developed at the German Aerospace Center (DLR) [3] and was the starting point for the extension to aero-structural wing designs. Secondly, we describe the adjoint approach and its implementation for the evaluation of the sensitivities for coupled aero-structure optimization problems [4] and its application to the drag reduction of the AMP wing by constant lift while taking into account the static deformation of this wing caused by the aerodynamic forces (see figures). Finally, we show the application of the coupled aero-structural adjoint approach for the Breguet formula of aircraft range, where in addition to the lift to drag ratio the weight of the AMP wing is taken into account (see figures). (© 2005 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim)     

线性约束三次规划问题的全局最优性必要条件和最优化算法

讨论了带线性不等式约束三次规划问题的最优性条件和最优化算法. 首先, 讨论了带有线性不等式约束三次规划问题的 全局最优性必要条件. 然后, 利用全局最优性必要条件, 设计了解线性约束三次规划问题的一个新的局部最优化算法(强局部最优化算法). 再利用辅助函数和所给出的新的局部最优化算法, 设计了带有线性不等式约束三 规划问题的全局最优化算法. 最后, 数值算例说明给出的最优化算法是可行的、有效的.     

A Hybrid Method for Low Reynolds Number Flow Past an Asymmetric Cylindrical Body

Low Reynolds number fluid flow past a cylindrical body of arbitrary shape in an unbounded, two-dimensional domain is a singular perturbation problem involving an infinite logarithmic expansion in the small parameter ε, representing the Reynolds number. We apply a hybrid asymptotic–numerical method to compute the drag coefficient, C _D and lift coefficient C _L to within all logarithmic terms. The hybrid method solution involves a matrix M , depending only on the shape of the body, which we compute using a boundary integral method. We illustrate the hybrid method results on an elliptic object and on a more complicated profile.     

Analytical modeling of active magnetic bearing geometry

The aim of this paper is to present a new graphical approach to the shape design of the active magnetic bearing (AMB) stator. The AMB is a tool to levitate the rotor without contact. The standard design method uses a computer-aided design (CAD) software in the modeling process. Therefore the designed AMB shape consists of graphical primitives like lines and arcs with fixed properties. For the advanced interdisciplinary analysis of the AMB construction the shape generation and modifications ought to be done automatically. The proposed method is based on mathematical analysis and representation of the AMB stator by curves. Second and third order Bézier curves given in polynomial and rational form are compared to the circle and arc based arcs. The fitting quality is considered for the selection of the appropriate arc representation. The obtained shapes are ready to be used in the magnetic field analysis and optimization procedures to find an optimal form of the AMB construction. The author’s experience in modeling and vector graphics was a motivation to look at the AMB construction from mathematical and programming point of view. The AMB components are modeled with parametric curves under constraints defined by the AMB static and dynamic properties. Such a described 2D or 3D model can be generated automatically in a programming way for a wide range of AMB configurations in further research. Selected configurations are presented to show features of the proposed method and realized algorithm. The selected features of the proposed solution as well as feedback from industry are discussed.     

A Class of Polynomial Interior Point Algorithms for the Cartesian P-Matrix Linear Complementarity Problem over Symmetric Cones

In this paper, we present a new class of polynomial interior point algorithms for the Cartesian P-matrix linear complementarity problem over symmetric cones based on a parametric kernel function, which determines both search directions and the proximity measure between the iterate and the center path. The symmetrization of the search directions used in this paper is based on the Nesterov and Todd scaling scheme. By using Euclidean Jordan algebras, we derive the iteration bounds that match the currently best known iteration bounds for large- and small-update methods.     

径向基函数参数化翼型的气动力降阶模型优化

基于小扰动和弱非线性假设,提出了一种基于气动力降阶模型和径向基函数参数化的翼型优化方法.其主要方法是用径向基函数参数化翼型扰动;通过CFD辨识参数扰动对翼型气动力影响的降阶模型核函数;基于叠加法建立了参数变化对翼型气动力影响的降阶模型;最后基于该气动力降阶模型计算并优化翼型升阻特性.NACA0012翼型优化的结果表明基于气动力降阶模型的优化方法是可行的,可以极大地提高翼型优化速度.     

Global optimality principles for polynomial optimization over box or bivalent constraints by separable polynomial approximations

In this paper we present necessary conditions for global optimality for polynomial problems with box or bivalent constraints using separable polynomial relaxations. We achieve this by first deriving a numerically checkable characterization of global optimality for separable polynomial problems with box as well as bivalent constraints. Our necessary optimality conditions can be numerically checked by solving semi-definite programming problems. Then, by employing separable polynomial under-estimators, we establish sufficient conditions for global optimality for classes of polynomial optimization problems with box or bivalent constraints. We construct underestimators using the sum of squares convex (SOS-convex) polynomials of real algebraic geometry. An important feature of SOS-convexity that is generally not shared by the standard convexity is that whether a polynomial is SOS-convex or not can be checked by solving a semidefinite programming problem. We illustrate the versatility of our optimality conditions by simple numerical examples.     

Reduced Functions, Gradients and Hessians from Fixed-Point Iterations for State Equations

In design optimization and parameter identification, the objective, or response function(s) are typically linked to the actually independent variables through equality constraints, which we will refer to as state equations. Our key assumption is that it is impossible to form and factor the corresponding constraint Jacobian, but one has instead some fixed-point algorithm for computing a feasible state, given any reasonable value of the independent variables. Assuming that this iteration is eventually contractive, we will show how reduced gradients (Jacobians) and Hessians (in other words, the total derivatives) of the response(s) with respect to the independent variables can be obtained via algorithmic, or automatic, differentiation (AD). In our approach the actual application of the so-called reverse, or adjoint differentiation mode is kept local to each iteration step. Consequently, the memory requirement is typically not unduly enlarged. The resulting approximating Lagrange multipliers are used to compute estimates of the reduced function values that can be shown to converge twice as fast as the underlying state space iteration. By a combination with the forward mode of AD, one can also obtain extra-accurate directional derivatives of the reduced functions as well as feasible state space directions and the corresponding reduced or projected Hessians of the Lagrangian. Our approach is verified by test calculations on an aircraft wing with two responses, namely, the lift and drag coefficient, and two variables, namely, the angle of attack and the Mach number. The state is a 2-dimensional flow field defined as solution of the discretized Euler equation under transonic conditions.     

Natural laminar flow shape optimization in transonic regime with competitive Nash game strategy

The Natural Laminar Flow (NLF) airfoil/wing design optimization is an efficient method which can reduce significantly turbulence skin friction by delaying transition location at high Reynolds numbers. However, the reduction of the friction drag is competitively balanced with the increase of shock wave induced drag in transonic regime. In this paper, a distributed Nash Evolutionary Algorithms (EAs) is presented and extended to multi-level parallel computing, namely multi-level parallel Nash EAs. The proposed improved methodology is used to solve NLF airfoil shape design optimization problem. It turns out that the optimization method developed in this paper can easily capture a Nash Equilibrium (NE) between transition delaying and wave drag increasing. Results of numerical experiments demonstrate that both wave drag and friction drag performances of a NE are greatly improved. Moreover, performance of the NE is equivalent to that of cooperative Pareto-optimum solutions, but it is more efficient in terms of CPU time. The successful application validates efficiency of algorithms in solving complex aerodynamic optimization problem.     

Path-following energy optimization in unilateral contact problems

Path-following (load incrementation) methods are studied in this paper for elastostatic analysis problems with unilateral contact relations in the framework of a large displacement theory by means of the parametric optimization techniques. Finite element discretization yields sparse polynomial optimization problems with equality and inequality constraints. For such sparse problems generically appearing singularities along the path of solutions are completely classified. Perturbations involving only a minimal number of parameters are shown to be sufficient to guarantee these generic situations. This clarifies stability and uniqueness questions for the solution along the examined path.     

Deriving compact extended formulations via LP-based separation techniques

The best formulations for some combinatorial optimization problems are integer linear programming models with an exponential number of rows and/or columns, which are solved incrementally by generating missing rows and columns only when needed. As an alternative to row generation, some exponential formulations can be rewritten in a compact extended form, which have only a polynomial number of constraints and a polynomial, although larger, number of variables. As an alternative to column generation, there are compact extended formulations for the dual problems, which lead to compact equivalent primal formulations, again with only a polynomial number of constraints and variables. In this this paper we introduce a tool to derive compact extended formulations and survey many combinatorial optimization problems for which it can be applied. The tool is based on the possibility of formulating the separation procedure by an LP model. It can be seen as one further method to generate compact extended formulations besides other tools of geometric and combinatorial nature present in the literature.     

A Pseudo-Global Optimization Approach with Application to the Design of Containerships

In this paper, we present a new class of pseudo-global optimization procedures for solving formidable optimization problems in which the objective and/or constraints might be analytically complex and expensive to evaluate, or available only as black-box functions. The proposed approach employs a sequence of polynomial programming approximations that are constructed using the Response Surface Methodology (RSM), and embeds these within a branch-and-bound framework in concert with a suitable global optimization technique. The lower bounds constructed in this process might only be heuristic in nature, and hence, this is called a pseudo-global optimization approach. We develop two such procedures, each employing two alternative branching techniques, and apply these methods to the problem of designing containerships. The model involves five design variables given by the design draft, the depth at side, the speed, the overall length, and the maximum beam. The constraints imposed enforce the balance between the weight and the displacement, a required acceptable length to depth ratio, a restriction on the metacentric height to ensure that the design satisfies the Coast Guard wind heel criterion, a minimum freeboard level as governed by the code of federal regulations (46 CFR 42), and a lower bound on the rolling period to ensure sea-worthiness. The objective function seeks to minimize the required freight rate that is induced by the design in order to recover capital and operating costs, expressed in dollars per metric ton per nautical mile. The model formulation also accommodates various practical issues in improving the representation of the foregoing considerations, and turns out to be highly nonlinear and nonconvex. A practical test case is solved using the proposed methodology, and the results obtained are compared with those derived using a contemporary commercialized design optimization tool. The prescribed solution yields an improved design that translates to an estimated increase in profits of about $18.45 million, and an estimated 27% increase in the return on investment, over the life of the ship.     

An Inverse Reliability-based Approach for Designing under Uncertainty with Application to Robust Piston Design

In this work, we propose an optimization framework for designing under uncertainty that considers both robustness and reliability issues. This approach is generic enough to be applicable to engineering design problems involving nonconvex objective and constraint functions defined in terms of random variables that follow any distribution. The problem formulation employs an Inverse Reliability Strategy that uses percentile performance to address both robustness objectives and reliability constraints. Robustness is achieved through a design objective that evaluates performance variation as a percentile difference between the right and left trails of the specified goals. Reliability requirements are formulated as Inverse Reliability constraints that are based on equivalent percentile performance levels. The general proposed approach first approximates the formulated problem via a Gaussian Kriging model. This is then used to evaluate the percentile performance characteristics of the different measures inherent in the problem formulation for various design variable settings via a Most Probable Point of Inverse Reliability search algorithm. By using these percentile evaluations in concert with the response surface methodology, a polynomial programming approximation is generated. The resulting problem formulation is finally solved to global optimality using the Reformulation–Linearization Technique (RLT) approach. We demonstrate this overall proposed approach by applying it to solve the problem of reducing piston slap, an undesirable engine noise due to the secondary motion of a piston within a cylinder.