2D shape optimization under proximity constraints by CFD and response surface methodology

In this paper we consider two-dimensional CFD-based shape optimization in the presence of obstacles, which introduce nontrivial proximity constraints to the optimization problem. Built on Gregory’s piecewise rational cubic splines, the main contribution of this paper is the introduction of such parametric deformations to a nominal shape that are guaranteed to satisfy the proximity constraints. These deformed shape candidates are then used in the identification of a multivariate polynomial response surface; proximity-constrained shape optimization thus reduces to parametric optimization on this polynomial model, with simple interval bounds on the design variables. We illustrate the proposed approach by carrying out lift and/or drag optimization for the NACA 0012 airfoil containing a rectangular fuel tank: By identifying polynomial response surfaces using a large batch of 1800 design candidates, we conclude that the lift coefficient can be optimized by a linear model, whereas the drag coefficient can be optimized by using a quadratic model. Higher order polynomial models yield no improvement in the optimization.     

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.     

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.     

基于卷积神经网络气动力降阶模型的翼型优化方法

针对非线性大扰动翼型气动力优化问题,提出了基于卷积神经网络气动力降阶模型的优化方法.该方法用不同形状参数下翼型的气动力数据作为训练信号,训练卷积神经网络翼型气动力降阶模型.采用该气动力降阶模型,以最大升阻比为目标,对翼型进行优化,结果表明该方法可用于大扰动下翼型气动力的预测和优化.该文同时还讨论了池化法和径向基法的训练...     

Solution Remapping Technique to Accelerate Flow Convergence for Finite Volume Methods Applied to Shape Optimization Design

A solution remapping technique is applied to transonic airfoil optimization design to provide a fast flow steady state convergence of intermediate shapes for the finite volume schemes in solving the compressible Euler equations. Specifically, once the flow solution for the current shape is obtained, the flow state for the next shape is initialized by remapping the current solution with consideration of mesh deformation. Based on this strategy, the formula of deploying the initial value for the next shape is theoretically derived under the assumption of small mesh deformation. Numerical experiments show that the present technique of initial value deployment can attractively accelerate flow convergence of intermediate shapes and reduce computational time up to 70% in the optimization process.     

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.     

An Improved Hybrid Adjoint Method in External Aerodynamics Using Variational Technique for the Boundary Integral Based Optimal Objective Function Gradient

An improved hybrid adjoint method to the viscous, compressible Reynold-Averaged Navier-Stokes Equation (RANS) is developed for the computation of objective function gradient and demonstrated for external aerodynamic design optimization. In this paper, the main idea is to extend the previous coupling of the discrete and continuous adjoint method by the grid-node coordinates variation technique for the computation of the variation in the gradients of flow variables. This approach in combination with the Jacobian matrices of flow fluxes refrained the objective function from field integrals and coordinate transformation matrix. Thus, it opens up the possibility of employing the hybrid adjoint method to evaluate the subsequent objective function gradient analogous to many shape parameters, comprises of only boundary integrals. This avoids the grid regeneration in the geometry for every surface perturbation in a structured and unstructured grid. Hence, this viable technique reduces the overall CPU cost. Moreover, the new hybrid adjoint method has been successfully applied to the computation of accurate sensitivity derivatives. Finally, for the investigation of the presented numerical method, simulations are carried out on NACA0012 airfoil in a transonic regime and its accuracy and effectiveness related to the new gradient equation have been verified with the Finite Difference Method (FDM). The analysis reveals that the presented methodology for the optimization provides the designer with an indispensable CPU-cost effective tool to reshape the complex geometry airfoil surfaces, useful relative to the state-of-the-art, in a less computing time.     

Sobre la necesidad de controlar el error de discretización de elementos finitos en optimización de forma estructural con algoritmos evolutivos

This work analyzes the influence of the discretization error contained in the Finite Element (FE) analyses of each design configuration proposed by the structural shape optimization algorithms over the behavior of the algorithm. The paper clearly shows that if FE analyses are not accurate enough, the final solution provided by the optimization algorithm will neither be optimal nor satisfy the constraints. The need for the use of adaptive FE analysis techniques in shape optimum design will be shown. The paper proposes the combination of two strategies to reduce the computational cost related to the use of mesh adaptivity in evolutionary optimization algorithms: (a) the use of the algorithm described by Bugeda et al. [1] which reduces the computational cost associated to the adaptive FE analysis of each geometrical configuration and, (b) the successive increase of the required accuracy of the FE analyses in order to obtain a considerable reduction of the computational cost in the early stages of the optimization process.     

Hypersonic airfoils of maximum lift-to-drag ratio

The problem of determining the slender, hypersonic airfoil shape which produces the maximum lift-to-drag ratio for a given profile area, chord, and free-stream conditions is considered. For the estimation of the lift and the drag, the pressure distribution on a surface which sees the flow is approximated by the tangent-wedge relation. On the other hand, for surfaces which do not see the flow, the Prandtl-Meyer relation is used. Finally, base drag is neglected, while the skin-friction coefficient is assumed to be a constant, average value. The method used to determine the optimum upper and lower surfaces is the calculus of variations. Depending on the value of the governing parameter, the optimum airfoil shapes are found to be of three types. For low values of the governing parameter, the optimum shape is a flat plate at an angle of attack followed by slightly concave upper and lower surfaces. The next type of solution has a finite thickness over the entire chord with the upper surface inclined so that the flow is an expansion. Finally, for the last type of solution, the upper surface begins with a portion which sees the flow and is followed by an inclined portion similar to that above. For all of these solutions, the lower surface sees the flow. Results are presented for the optimum dimensionless airfoil shape, its dimensions, and the maximum lift-to-drag ratio. To calculate an actual airfoil shape requires an iteration procedure due to the assumption on the skin-friction coefficient. However, simple results can be obtained by assuming an approximate value for the skin-friction coefficient.This research was supported in part by the Air Force Office of Scientific Research, Office of Aerospace Research, U.S. Air Force, under AFOSR Grant No. 69-1744.     

变可信度模型在气动优化中的应用

在气动外形优化中, 采用近似模型管理结构（AMF）方法,对变可信度模型进行组织和管理．这样能够充分利用低可信度模型,将主要计算量集中在低可信度模型的优化迭代过程中．同时,采用高可信度模型监控优化过程,使最终的优化解收敛到高可信度模型上．最后,设计了零阶变可信度气动特性优化管理结构与搜索算法,对某飞翼型无人机的翼型进行了气动优化．优化外形的气动性能与初始外形比有所提高．实际结果表明所提出的方法具有良好的可行性和适用性．     

Shape optimization of an airfoil in the presence of compressible and viscous flows

The objective of this article is to present a step-by-step problem-solving procedure of shape optimization. The procedure is carried out to design an airfoil in the presence of compressible and viscous flows using a control theory approach based on measure theory. An optimal shape design (OSD) problem governed by full Navier-Stokes equations is given. Then, a weak variational form is derived from the linearized governing equations. During the procedure, because the measure theory (MT) approach is implemented using fixed geometry versus moving geometry, a proper bijective transformation is introduced. Finally, an approximating linear programming (LP) problem of the original shape optimization problem is obtained by means of MT approach that is not iterative and does not need any initial guess to proceed. Illustrative examples are provided to demonstrate efficiency of the proposed procedure.     

Airfoil design optimization using the Navier-Stokes equations

A design optimization technique is presented which couples a computationally efficient Navier-Stokes code with a numerical optimization algorithm. The design method improves the aerodynamic performance of an airfoil subject to specified design objectives and constraints. Recent advances in computers and compputational fluid dynamics have permitted the use of the Navier-Stokes equations in the design procedure to include the nonlinear, rotational, viscous physics of transonic flows. Using numerical optimization guarantees that a better design will be produced even with strict design constraints. The method is demonstrated with several examples at transonic flow conditions.     

Shape functional optimization with restrictions boosted with machine learning techniques

Shape optimization is a widely used technique in the design phase of a product. Current ongoing improvement policies require a product to fulfill a series of conditions from the perspective of mechanical resistance, fatigue, natural frequency, impact resistance, etc. All these conditions are translated into equality or inequality restrictions which must be satisfied during the optimization process that is necessary in order to determine the optimal shape. This article describes a new method for shape optimization that considers any regular shape as a possible shape, thereby improving on traditional methods limited to straight profiles or profiles established a priori. Our focus is based on using functional techniques and this approach is, based on representing the shape of the object by means of functions belonging to a finite-dimension functional space. In order to resolve this problem, the article proposes an optimization method that uses machine learning techniques for functional data in order to represent the perimeter of the set of feasible functions and to speed up the process of evaluating the restrictions in each iteration of the algorithm. The results demonstrate that the functional approach produces better results in the shape optimization process and that speeding up the algorithm using machine learning techniques ensures that this approach does not negatively affect design process response times.     

Advances in concrete arch dams shape optimization

This paper presents an efficient methodology to find the optimum shape of arch dams. In order to create the geometry of arch dams a new algorithm based on Hermit Splines is proposed. A finite element based shape sensitivity analysis for design-dependent loadings involving body force, hydrostatic pressure and earthquake loadings is implemented. The sensitivity analysis is performed using the concept of mesh design velocity. In order to consider the practical requirements in the optimization model such as construction stages, many geometrical and behavioral constrains are included in the model in comparison with previous researches. The optimization problem is solved via the sequential quadratic programming (SQP) method. The proposed methods are applied successfully to an Iranian arch dam, and good results are achieved. By using such methodology, efficient software for shape optimization of concrete arch dams for practical and reliable design now is available.     

Design problem solved by optimal control theory

In this paper we present an application to airfoil design of an optimum design method based on optimal control theory. The method used here transforms the design problem by way of a change of variable into an optimal control problem for a distributed system with Neumann boundary control. This results in a set of variational inequalities which is solved by adding a penalty term to the differential equation. This is in turn solved by a finite element method.     

General shape optimization capability

Structural optimization is almost as old as the finite element method (FEM). Whereas FEM found its way to real-life applications very quickly, structural optimization remained a topic of interest in the research community for many years. However, there have been a number of attemps recently to develop general purpose program systems for property optimization. For shape optimization, there is no general purpose code currently available that can solve realistic problems. This paper will describe a method of calculating shape sensitivities within , in a simple manner, without resort to external programs. Once the shape sensitivities are obtained, the shape optimization process can proceed in a manner similar to property optimization. The key concept is the use of natural design variables to define the shape changes in a given structure. The design variables are the magnitudes of enforced displacements applied to the structure. The displacements produced by these variables are added to the initial shape to obtain a new shape. This approach can be computationally intensive and since one shape variable is dependent of another, multiple CPU's can be used to significantly reduce the solution time.

Two examples are solved to demonstrate the capability of these techniques. The first is a cantilever beam with holes loaded by a point load at the free end. The shape of the holes as well as the thickness of the beam are selected as the design variables. The second example is the shape optimization of a connecting rod subjected to several different loading and boundary conditions.     

An efficient sensitivity computation strategy for the evolutionary structural optimization (ESO) of continuum structures subjected to self-weight loads

This work presents a modified version of the evolutionary structural optimization procedure for topology optimization of continuum structures subjected to self-weight forces. Here we present an extension of this procedure to deal with maximum stiffness topology optimization of structures when different combinations of body forces and fixed loads are applied. Body forces depend on the density distribution over the design domain. Therefore, the value and direction of the loading are coupled to the shape of the structure and they change as the material layout of the structure is modified in the course of the optimization process. It will be shown that the traditional calculation of the sensitivity number used in the ESO procedure does not lead to the optimum solution. Therefore, it is necessary to correct the computation of the element sensitivity numbers in order to achieve the optimum design. This paper proposes an original correction factor to compute the sensitivities and enhance the convergence of the algorithm. The procedure has been implemented into a general optimization software and tested in several numerical applications and benchmark examples to illustrate and validate the approach, and satisfactorily applied to the solution of 2D, 3D and shell structures, considering self-weight load conditions. Solutions obtained with this method compare favourably with the results derived using the SIMP interpolation scheme.     

A study of design velocity field computation for shape optimal design

Design velocity field computation is an important step in computing shape design sensitivity coefficients and updating a finite element mesh in the shape design optimization process. Applying an inappropriate design velocity field for shape design sensitivity analysis and optimization will yield inaccurate sensitivity results or a distorted finite element mesh, and thus fail in achieving an optimal solution. In this paper, theoretical regularity and practical requirements of the design velocity field are discussed. The crucial step of using the design velocity field to update the finite element mesh in the design optimization process is emphasized. Available design velocity field computation methods in the literature are summarized and their applicability for shape design sensitivity analysis and optimization is discussed. Five examples are employed to discuss applicability of these methods. It was found that a combination of isoparametric mapping and boundary displacement methods is ideal for the design velocity field computation.     

离散变量的分级优化方法及其应用

在工程优化设计中,绝大多数实际问题的设计变量往往限定取离散值,为了求得问题的真正最优解,就必须采用离散变量的优化方法进行求解.本文根据离散变量数学规划的特性,提出了一种分级优化搜索算法.这种方法的基本思想是在约束集合内,寻求一可行的离散初始点,然后在该点的邻域内,进行分级寻优搜索,以求得一个改进的新离散点,随之,以该点作为初始点,重复执行分级寻优搜索过程,直至求得问题的最优解.通过对工程实例的计算,证明本文所提出的新方法具有快速、简便的特点,能有效地应用各种工程优化设计问题.     

Enhancement of a viscous-inviscid-interaction airfoil analysis code using the parallel direct search algorithm

The coupling mechanism for an existing viscous-inviscid-interaction (VII) code, developed for the analysis of two-dimensional, turbulent, attached flow around airfoils, is enhanced using the parallel direct search (PDS) optimization algorithm. It is demonstrated that this parallel processing implemented optimization scheme leads to faster convergence of the VII code, and therefore, requires less computational time when the number of optimization (or design) variables is low, and a moderate number of processors are available. As the number of design variables increases, more processors are required to maintain this advantage. Results are presented for the NACA-0012 and the RAE-2822 airfoils. The quality of the results obtained is satisfactory and confirms that the enhanced VII code can be an acceptable alternative to reduced Navier-Stokes solvers as an airfoil analysis tool.