Continuous and Discrete Adjoint Approach Based on Lattice Boltzmann Method in Aerodynamic Optimization Part I: Mathematical Derivation of Adjoint Lattice Boltzmann Equations

The significance of flow optimization utilizing the lattice Boltzmann (LB) method becomes obvious regarding its advantages as a novel flow field solution method compared to the other conventional computational fluid dynamics techniques. These unique characteristics of the LB method form the main idea of its application to optimization problems. In this research, for the first time, both continuous and discrete adjoint equations were extracted based on the LB method using a general procedure with low implementation cost. The proposed approach could be performed similarly for any optimization problem with the corresponding cost function and design variables vector. Moreover, this approach was not limited to flow fields and could be employed for steady as well as unsteady flows. Initially, the continuous and discrete adjoint LB equations and the cost function gradient vector were derived mathematically in detail using the continuous and discrete LB equations in space and time, respectively. Meanwhile, new adjoint concepts in lattice space were introduced. Finally, the analytical evaluation of the adjoint distribution functions and the cost function gradients was carried out.     

Topology optimization of unsteady incompressible Navier–Stokes flows

This paper discusses the topology optimization of unsteady incompressible Navier–Stokes flows. An optimization problem is formulated by adding the artificial Darcy frictional force into the incompressible Navier–Stokes equations. The optimization procedure is implemented using the continuous adjoint method and the finite element method. The effects of dynamic inflow, Reynolds number and target flux on specified boundaries for the optimal topology of unsteady Navier–Stokes flows are presented. Numerical examples demonstrate the feasibility and necessity of this topology optimization method for unsteady Navier–Stokes flows.     

Aerodynamic design optimization by using a continuous adjoint method

This paper presents the fundamentals of a continuous adjoint method and the applications of this method to the aerodynamic design optimization of both external and internal flows.General formulation of the continuous adjoint equations and the corresponding boundary conditions are derived.With the adjoint method,the complete gradient information needed in the design optimization can be obtained by solving the governing flow equations and the corresponding adjoint equations only once for each cost function,regardless of the number of design parameters.An inverse design of airfoil is firstly performed to study the accuracy of the adjoint gradient and the effectiveness of the adjoint method as an inverse design method.Then the method is used to perform a series of single and multiple point design optimization problems involving the drag reduction of airfoil,wing,and wing-body configuration,and the aerodynamic performance improvement of turbine and compressor blade rows.The results demonstrate that the continuous adjoint method can efficiently and significantly improve the aerodynamic performance of the design in a shape optimization problem.     

Error estimation and adaptation for functional outputs in time-dependent flow problems

The paper presents an adjoint-based approach for determining global error in the time domain that is relevant to functional outputs from unsteady flow simulations. The algorithm is derived for the unsteady Euler equations that are discretized for second-order accuracy in both space and time and takes into account the effect of dynamic meshes. In addition to error due to temporal resolution, the formulation also takes into account algebraic error arising from partial convergence of the governing equations at each implicit time-step. The resulting error distributions are then used to drive adaptation of the temporal resolution and the convergence tolerances for the governing equations at each time-step. The method is demonstrated in the context of both time-integrated and instantaneous functionals and the results are compared against traditional adaptation methods.     

Optimal nonlocal boundary control of the wide-angle parabolic equation for inversion of a waveguide acoustic field

This paper applies the concept of optimal boundary control for solving inverse problems in shallow water acoustics. To treat the controllability problem, a continuous analytic adjoint model is derived for the Claerbout wide-angle parabolic equation (PE) using a generalized nonlocal impedance boundary condition at the water-bottom interface. While the potential of adjoint methodology has been recently demonstrated for ocean acoustic tomography, this approach combines the advantages of exact transparent boundary conditions for the wide-angle PE with the concept of adjoint-based optimal control. In contrast to meta-heuristic approaches the inversion procedure itself is directly controlled by the waveguide physics and, in a numerical implementation based on conjugate gradient optimization, many fewer iterations are required for assessment of an environment that is supported by the underlying subbottom model. Furthermore, since regularization schemes are particularly important to enhance the performance of full-field acoustic inversion, special attention is devoted to the application of penalization methods to the adjoint optimization formalism. Regularization incorporates additional information about the desired solution in order to stabilize ill-posed inverse problems and identify useful solutions, a feature that is of particular importance for inversion of field data sampled on a vertical receiver array in the presence of measurement noise and modeling uncertainty. Results with test data show that the acoustic field and the bottom properties embedded in the control parameters can be efficiently retrieved.     

气动/几何约束条件下翼型优化设计的最优控制理论方法

基于最优控制理论原理和Navier-Stokes方程,研究了气动/几何约束条件下多设计变量的翼型气动优化设计问题.根据给定的目标函数表达形式,在计算坐标下详细推导了相应的共轭方程及边界条件,以及梯度方程的具体数学形式.通过合理数学变换,得到了物理空间上适应于CFD数值求解的共轭方程直观表达形式,并发展了有效数值求解目的.通过将流动方程、共轭方程、目标函数敏感性导数和优化算法相结合,发展了一种新的气动优化设计目的.相关设计算例表明该目的在设计理论、适用性以及时间费用等方面具有显著特色和优点,且设计结果更为可靠.     

Adjoint-based sensitivity analysis of quantities of interest of complex combustion models

An adjoint-based approach for the sensitivity analysis of complex reaction mechanisms is presented. It builds purely on the evaluation of the governing equations. No adjoint equations have to be derived explicitly. Instead, the required adjoint operator is constructed numerically. The approach can be utilised for various kinetic models and in existing codes with minimal implementation effort. All dependencies on the state and on model parameters are fully evaluated without simplifications. Sensitivities are calculated more efficiently and more robustly compared with the often-used brute-force method. The approach is demonstrated for a homogeneous (zero-dimensional) reactor with different complex reaction mechanisms including several reaction types.     

Adjoint-based optimization of PDEs in moving domains

In this investigation we address the problem of adjoint-based optimization of PDE systems in moving domains. As an example we consider the one-dimensional heat equation with prescribed boundary temperatures and heat fluxes. We discuss two methods of deriving an adjoint system necessary to obtain a gradient of a cost functional. In the first approach we derive the adjoint system after mapping the problem to a fixed domain, whereas in the second approach we derive the adjoint directly in the moving domain by employing methods of the noncylindrical calculus. We show that the operations of transforming the system from a variable to a fixed domain and deriving the adjoint do not commute and that, while the gradient information contained in both systems is the same, the second approach results in an adjoint problem with a simpler structure which is therefore easier to implement numerically. This approach is then used to solve a moving boundary optimization problem for our model system.     

An adjoint method for shape optimization in unsteady viscous flows

A new method for shape optimization for unsteady viscous flows is presented. It is based on the continuous adjoint approach using a time accurate method and is capable of handling both inverse and direct objective functions. The objective function is minimized or maximized subject to the satisfaction of flow equations. The shape of the body is parametrized via a Non-Uniform Rational B-Splines (NURBS) curve and is updated by using the gradients obtained from solving the flow and adjoint equations. A finite element method based on streamline-upwind Petrov/Galerkin (SUPG) and pressure stabilized Petrov/Galerkin (PSPG) stabilization techniques is used to solve both the flow and adjoint equations. The method has been implemented and tested for the design of airfoils, based on enhancing its time-averaged aerodynamic coefficients. Interesting shapes are obtained, especially when the objective is to produce high performance airfoils. The effect of the extent of the window of time integration of flow and adjoint equations on the design process is studied. It is found that when the window of time integration is insufficient, the gradients are most likely to be erroneous.     

Adjoint-based sensitivity and ignition threshold mapping in a turbulent mixing layer

Successful ignition in non-premixed turbulent flows remains a fundamental challenge in combustion systems. Current design strategies typically rely on iterative testing to map the spatial distribution of ignition probability. We propose to accelerate this by formulating the adjoint of the perturbed and linearised governing equations in such a way that sensitivity of an ignition indicator can be obtained with a cost comparable to the flow solution. A space–time discrete adjoint method for multi-component chemically reacting flows is developed, and the gradient formed via the corresponding adjoint solution is used to identify regions favourable to ignition in a direct numerical simulation of non-premixed turbulent free shear flow. This approach requires a specific definition of an ignition metric, although this can be problematic because ignition either succeeds or fails after some period and thus gradients for some metrics become ill-defined near the ignition threshold. To this end, a quantity of interest is designed to provide short-time sensitivity in conjunction with an indicator function over a long-time period that informs whether successful ignition occurred. The gradients are used in a line-search algorithm to map the ignition boundary under specific constraints. Finally, parametric sensitivity is evaluated at different flow realisations to analyse factors governing local sensitivity in unsteady chemically reacting flows.     

A multi-block ADI finite-volume method for incompressible Navier–Stokes equations in complex geometries

An efficient second-order accurate finite-volume method is developed for a solution of the incompressible Navier–Stokes equations on complex multi-block structured curvilinear grids. Unlike in the finite-volume or finite-difference-based alternating-direction-implicit (ADI) methods, where factorization of the coordinate transformed governing equations is performed along generalized coordinate directions, in the proposed method, the discretized Cartesian form Navier–Stokes equations are factored along curvilinear grid lines. The new ADI finite-volume method is also extended for simulations on multi-block structured curvilinear grids with which complex geometries can be efficiently resolved. The numerical method is first developed for an unsteady convection–diffusion equation, then is extended for the incompressible Navier–Stokes equations. The order of accuracy and stability characteristics of the present method are analyzed in simulations of an unsteady convection–diffusion problem, decaying vortices, flow in a lid-driven cavity, flow over a circular cylinder, and turbulent flow through a planar channel. Numerical solutions predicted by the proposed ADI finite-volume method are found to be in good agreement with experimental and other numerical data, while the solutions are obtained at much lower computational cost than those required by other iterative methods without factorization. For a simulation on a grid with O(10⁵) cells, the computational time required by the present ADI-based method for a solution of momentum equations is found to be less than 20% of that required by a method employing a biconjugate-gradient-stabilized scheme.     

Uncertainty quantification of kinetic models using adjoint-driven active subspace algorithms

Simulations of chemically reacting flows are particularly sensitive to kinetic models, which in turn depend on a large number of parameters. Therefore, quantifying parametric uncertainties is essential in order to assess the applicability and performance of such models. The size of the parameter space renders conventional methods in quantifying uncertainties impractical. The active subspace methodology has proven to be very effective in such applications and has been adopted here to analyze ignition delay time in an isochoric adiabatic reactor. This method has been augmented with an adjoint-based algorithm to provide low-cost access to gradient information, necessary for dimensionality reduction. In addition, the predictions of the active subspace method have been compared to a linear approximation using the adjoint model (LAAM), where depending on the case, a satisfactory estimation is achieved at a fraction of the cost. Moreover, a strategy is presented for constructing a multi-dimensional response surface for cases, where contrary to previous studies, a single dominant direction is insufficient in predicting the corresponding PDF. Finally, the uncertainties with respect to initial conditions are analyzed, highlighting the limitations of the linear approach (LAAM) in cases with strong nonlinearities.     

升力系数不变时跨音速机翼阻力优化设计

基于共轭方程的优化设计理论,应用三维欧拉方程进行了升力系数不变时跨音速机翼阻力优化设计研究,根据给定的目标函数推导了在物理空间上表述的共轭方程及边界条件,研究了共轭方程的数值求解方法及目标函数对设计变量的敏感性导数求解问题,发展了一种跨音速机翼阻力优化设计方法,应用该设计方法进行了跨音速机翼阻力优化设计研究,优化后机翼表面的激波强度减弱很多,有效减少了波阻.     

基于梯度响应面模型的优化设计

本文主要研究一种梯度响应面模型及其在气动优化设计中的应用. 目前应用广泛的多项式响应面模型是连续可导的, 采用梯度信息构造完全二阶多项式响应面模型, 所需样本数与设计参数个数呈线性关系. 首先通过改进实验设计方法, 快速生成满足精度要求的样本并确定梯度响应面模型. 随后通过函数实验验证梯度响应面模型的精度, 及该模型在多极值函数最值搜索中的有效性. 最后由伴随方法快速求解气动优化设计目标函数的梯度信息, 并开展基于梯度响应面模型和复合形法的叶片压力反设计和效率优化设计. 结果表明: 基于梯度响应面模型的优化方法在全局最优及提高优化效率两方面均有出色表现, 基于该优化方法的气动优化设计能够显著改善叶片的气动性能. 关键词： 气动优化设计 响应面模型 伴随方法 复合形法     

Iterative optimization based on an objective functional in frequency-space with application to jet-noise cancellation

In many technical applications, like supersonic jets, noise with a characteristic spectrum including certain dominant frequencies (e.g. jet-screech) is prevalent, and the elimination of sharp peaks in the acoustic spectrum is the aim of active or passive flow/noise control efforts. A mathematical framework for the optimization of control strategies is introduced that uses a cost objective in frequency-space coupled to constraints in form of partial differential equations in the time domain. An iterative optimization scheme based on direct and adjoint equations arises, which has been validated on two examples, the one-dimensional Burgers equation and the two-dimensional compressible Navier–Stokes equations. In both cases, the iterative scheme has proven effective and efficient in targeting and removing specified frequency bands in the acoustic spectrum. It is expected that this technique will find use in acoustic and other applications where the elimination or suppression of distinct frequency components is desirable.     

基于对偶数理论的资料同化新方法

针对变分资料同化中目标泛函梯度计算精度不高且复杂等问题, 提出了一种基于对偶数理论的资料同化新方法, 主要优点是: 能避免复杂的伴随模式开发及其逆向积分, 只需在对偶数空间通过正向积分就能同时计算出目标泛函和梯度向量的值. 首先利用对偶数理论把梯度分析过程转换为对偶数空间中目标泛函计算过程, 简单、高效和高精度地获得梯度向量值; 其次结合典型的最优化方法, 给出了非线性物理系统资料同化问题的新求解算法; 最后对Lorenz 63混沌系统、包含开关的不可微物理模型和抛物型偏微分方程分别进行了资料同化数值实验, 结果表明: 新方法能有效和准确地估计出预报模式的初始条件或物理参数值.     

叶型优化在非定常流动条件下效果的数值分析

本文对叶型优化在非定常流动条件下的效果进行数值研究。叶型优化基于比较成熟的在定常流动条件下的反问题解法，非定常流动则考虑到动静叶栅之间的相互作用，非定常流动的求解在Ｎ－Ｓ方程的基础上采用分区计算的方法来完成。数值计算的结果表明，定常流动条件下优化得到的叶型在非定常流动条件下同样具有较好的气动性能。     

二冷配水优化建模与混合自适应PSO算法求解

针对连铸二冷区生产环境复杂且存在着大量水雾干扰的情况,建立了连铸水量优化模型并提出了一种混合的自适应粒子群算法来求解连铸二冷水优化问题。依据冶金过程中的工艺要求建立了二冷水量优化模型,并在经典的PSO算法基础上提出了适合该问题求解了混合自适应PSO算法。由于连铸过程存在着偏微分方程约束,传统的优化方法容易陷入局部最优解,不能达到很好的动态优化效果。研究了粒子群算法,基于种群的多样性,不断的自适应的更新粒子群算法中参数,将禁忌搜索的方法和传统的粒子群算法结合,增强了算法的局部搜索能力和全局寻找全局最优的能力。将该算法应用到连铸二冷水动态优化中,实验结果表面该算法能够快速有效的求解该优化问题。该方法用于连铸二冷水优化是可行的、有效的。     

Time accurate anisotropic goal-oriented mesh adaptation for unsteady flows

We present a new algorithm for combining an anisotropic goal-oriented error estimate with the mesh adaptation fixed point method for unsteady problems. The minimization of the error on a functional provides both the density and the anisotropy (stretching) of the optimal mesh. They are expressed in terms of state and adjoint. This method is used for specifying the mesh for a time sub-interval. A global fixed point iterates the re-evaluation of meshes and states over the whole time interval until convergence of the space–time mesh. Applications to unsteady blast-wave and acoustic-wave Euler flows are presented.     

Adjoint wall functions: A new concept for use in aerodynamic shape optimization

The continuous adjoint method for the computation of sensitivity derivatives in aerodynamic optimization problems of steady incompressible flows, modeled through the k–ε turbulence model with wall functions, is presented. The proposed formulation leads to the adjoint equations along with their boundary conditions by introducing the adjoint to the friction velocity. Based on the latter, an adjoint law of the wall that bridges the gap between the solid wall and the first grid node off the wall is proposed and used during the solution of the system of adjoint (to both the mean flow and turbulence) equations. Any high Reynolds turbulence model, other than the k–ε one used in this paper, could also profit from the proposed adjoint wall function technique. In the examined duct flow problems, where the total pressure loss due to viscous effects is used as objective function, emphasis is laid on the accuracy of the computed sensitivity derivatives, rather than the optimization itself. The latter might rely on any descent method, once the objective function gradient has accurately been computed.