The continuous direct-adjoint approach for second order sensitivities in viscous aerodynamic inverse design problems

A novel continuous adjoint approach for the computation of the second order sensitivities of the objective function used in inverse design problems is proposed. In the framework of the Newton method, the proposed approach can be used to efficiently cope with inverse design problems in viscous flows, where the target is a given pressure distribution along the solid walls. It consists of two steps and will, thus, be referred to as the direct-adjoint approach. At the first step, the direct differentiation method is used to compute the first order sensitivities of the flow variables with respect to the design variables and build the gradient of the objective function. At the second step, the adjoint approach is used to compute the second order sensitivities. The final Hessian expression is free of field integrals and its computation requires the solution of N + 1 equivalent flow (system) solutions for N design variables. Since the CPU cost of using the Newton method, with exact gradient and Hessian data at each cycle, becomes prohibitively high, an approach that computes the exact Hessian only once and then updates it in an approximated manner through the BFGS formula, is used instead. The accuracy of the Hessian matrix components, computed using the direct-adjoint approach is demonstrated on the inverse design of a diffuser and a cascade airfoil.     

Aerodynamic Shape Optimization Using First and Second Order Adjoint and Direct Approaches

This paper focuses on discrete and continuous adjoint approaches and direct differentiation methods that can efficiently be used in aerodynamic shape optimization problems. The advantage of the adjoint approach is the computation of the gradient of the objective function at cost which does not depend upon the number of design variables. An extra advantage of the formulation presented below, for the computation of either first or second order sensitivities, is that the resulting sensitivity expressions are free of field integrals even if the objective function is a field integral. This is demonstrated using three possible objective functions for use in internal aerodynamic problems; the first objective is for inverse design problems where a target pressure distribution along the solid walls must be reproduced; the other two quantify viscous losses in duct or cascade flows, cast as either the reduction in total pressure between the inlet and outlet or the field integral of entropy generation. From the mathematical point of view, the three functions are defined over different parts of the domain or its boundaries, and this strongly affects the adjoint formulation. In the second part of this paper, the same discrete and continuous adjoint formulations are combined with direct differentiation methods to compute the Hessian matrix of the objective function. Although the direct differentiation for the computation of the gradient is time consuming, it may support the adjoint method to calculate the exact Hessian matrix components with the minimum CPU cost. Since, however, the CPU cost is proportional to the number of design variables, a well performing optimization scheme, based on the exactly computed Hessian during the starting cycle and a quasi Newton (BFGS) scheme during the next cycles, is proposed.     

Total pressure losses minimization in turbomachinery cascades using the exact Hessian

A method for the design of turbomachinery cascades with minimum total pressure losses, subject to constraints on the minimum blade thickness and flow turning, is presented. It is based on the Newton–Lagrange method which requires the computation of first- and second-order sensitivities of the objective function and the constraints, with respect to the design variables. The computation of the exact Hessian of the function which expresses the difference in total pressure between the inlet to and the outlet from the cascade, is new in the literature. To compute the Hessian, the direct differentiation of the viscous flow equations is used for the first-order sensitivities of the functional and the flow-related constraints, followed by the discrete adjoint method. Since the objective function is defined along boundaries other than those controlled by the design variables, it is challenging to investigate the significance of all terms comprising the exact second-order sensitivity expressions. All these terms were temporarily computed using automatic differentiation and those which proved to be significant are hand-differentiated to minimize CPU cost and memory requirements. Insignificant terms are eliminated, giving rise to the so-called “exact” Hessian matrix. An “exactly” initialized quasi-Newton method was also programmed and tested. In the latter, at the first cycle, the exact gradients and Hessians are computed and used; during the subsequent optimization cycles, the discrete adjoint method provides the exact gradient whereas the Hessian is updated as in quasi-Newton methods. The comparison of the efficiency of the aforementioned methods depends on the number of design variables used; the “exactly” initialized quasi-Newton method constantly outperforms its conventional variant in terms of CPU cost, particularly in non-convex and/or constrained optimization problems.     

Continuous Adjoint Methods for Turbulent Flows,Applied to Shape and Topology Optimization: Industrial Applications

This article focuses on the formulation, validation and application of the continuous adjoint method for turbulent flows in aero/hydrodynamic optimization. Though discrete adjoint has been extensively used in the past to compute objective function gradients with respect to (w.r.t.) the design variables under turbulent flow conditions, the development of the continuous adjoint variant for these flows is not widespread in the literature, hindering, to an extend, the computation of exact sensitivity derivatives. The article initially presents a general formulation of the continuous adjoint method for incompressible flows, under the commonly used assumption of “frozen turbulence”. Then, the necessary addenda are presented in order to deal with the differentiation of both low- and high-Reynolds (with wall functions) number turbulence models; the latter requires the introduction of the so-called “adjoint wall functions”. An approach to dealing with distance variations is also presented. The developed methods are initially validated in \(2D\) cases and then applied to industrial shape and topology optimization problems, originating from the automotive and hydraulic turbomachinery industries.     

A contact force model considering constant external forces for impact analysis in multibody dynamics

The adjoint method is an elegant approach for the computation of the gradient of a cost function to identify a set of parameters. An additional set of differential equations has to be solved to compute the adjoint variables, which are further used for the gradient computation. However, the accuracy of the numerical solution of the adjoint differential equation has a great impact on the gradient. Hence, an alternative approach is the discrete adjoint method, where the adjoint differential equations are replaced by algebraic equations. Therefore, a finite difference scheme is constructed for the adjoint system directly from the numerical time integration method. The method provides the exact gradient of the discretized cost function subjected to the discretized equations of motion.     

Second order adjoint sensitivity analysis of multibody systems described by differential–algebraic equations

The optimization strategies employing second order sensitivity information has higher accuracy, but its computation is complex. In this paper, an adjoint variable method applied for the second order design sensitivity analysis of multibody design problems is developed. Based on Lagrange equations of multibody system dynamics, a general objective function, constraint conditions, initial and end conditions, the adjoint variable equations for first order sensitivity analysis and design sensitivity formulations are derived firstly. Then, second order sensitivity analysis formulations, as well as the detailed computation steps, are given based on the previous results. For simplification, the second derivative of the objective function with respect to design variables is translated into an initial value problem of an ordinary differential equation with one variable. Finally, a numerical example of slider–crank mechanism validates the accuracy and efficiency of the method for second order sensitivity analysis.     

A Newton method for solving continuous multiple material minimum compliance problems

This paper presents an implementation of an active-set line-search Newton method intended for solving large-scale instances of a class of multiple material minimum compliance problems. The problem is modeled with a convex objective function and linear constraints. At each iteration of the Newton method, one or two linear saddle point systems are solved. These systems involve the Hessian of the objective function, which is both expensive to compute and completely dense. Therefore, the linear algebra is arranged such that the Hessian is not explicitly formed. The main concern is to solve a sequence of closely related problems appearing as the continuous relaxations in a nonlinear branch and bound framework for solving discrete minimum compliance problems. A test-set consisting of eight discrete instances originating from the design of laminated composite structures is presented. Computational experiments with a branch and bound method indicate that the proposed Newton method can, on most instances in the test-set, take advantage of the available starting point information in an enumeration tree and resolve the relaxations after branching with few additional function evaluations. Discrete feasible designs are obtained by a rounding heuristic. Designs with provably good objective functions are presented.     

Fast convergence of inviscid fluid dynamic design problems

A new procedure for robust and efficient design optimization of inviscid flow problems has been developed and implemented on a wide variety of test problems. The methodology involves the use of an accurate flow solver to calculate the objective function and an approximate, dissipative flow solver, which is used only in the solution of the discrete quasi-time-dependent adjoint problem. The resulting design sensitivities are very robust even in the presence of noise or other non-smoothness associated with objective functions in many high-speed flow problems. The design problem is solved using what we term progressive optimization, whereby a sequence of a partially converged flow solution, followed by a partially converged adjoint solution followed by an optimization step is performed. This procedure is performed using a sequence of progressively finer grids for the solution of the flow field, while only using coarser grids for the adjoint equation solution.This approach has been tested on numerous inverse and direct (constrained) design problems involving two- and three-dimensional transonic nozzles and airfoils as well as supersonic blunt bodies. The methodology is shown to be robust and highly efficient, with a converged design optimization produced in no more than the amount of computational work to perform from 0.5 to 2.5 fine-mesh flow analyses.     

An implicit, exact dual adjoint solution method for turbulent flows on unstructured grids

An implicit algorithm for solving the discrete adjoint system based on an unstructured-grid discretization of the Navier-Stokes equations is presented. The method is constructed such that an adjoint solution exactly dual to a direct differentiation approach is recovered at each time step, yielding a convergence rate which is asymptotically equivalent to that of the primal system. The new approach is implemented within a three-dimensional unstructured-grid framework and results are presented for inviscid, laminar, and turbulent flows. Improvements to the baseline solution algorithm, such as line-implicit relaxation and a tight coupling of the turbulence model, are also presented. By storing nearest-neighbor terms in the residual computation, the dual scheme is computationally efficient, while requiring twice the memory of the flow solution. The current implementation allows for multiple right-hand side vectors, enabling simultaneous adjoint solutions for several cost functions or constraints with minimal additional storage requirements, while reducing the solution time compared to serial applications of the adjoint solver. The scheme is expected to have a broad impact on computational problems related to design optimization as well as error estimation and grid adaptation efforts.     

Projected Newton methods and optimization of multicommodity flows

A superlinearly convergent Newton like method for linearly constrained optimization problems is adapted for solution of multicommodity network flow problems of the type arising in communication and transportation networks. We show that the method can be implemented approximately by making use of conjugate gradient iterations without the need to compute explicitly the Hessian matrix. Preliminary computational results suggest that this type of method is capable of yielding highly accurate solutions of nonlinear multicommodity flow problems far more efficiently than any of the methods available at present.     

A continuous adjoint method with objective function derivatives based on boundary integrals, for inviscid and viscous flows

A continuous adjoint formulation for inverse design problems in external aerodynamics and turbomachinery is presented. The advantage of the proposed formulation is that the objective function gradient does not depend upon the variation of field geometrical quantities, such as metrics variations in the case of structured grids. The final expression for the objective function gradient includes only boundary integrals which can readily be calculated in both structured and unstructured grids; this is feasible in design problems where the objective function is either a boundary integral (pressure deviation along the solid walls) or a field integral (the entropy generation over the flow domain). The formulation governs inviscid and viscous flows; it takes into account the streamtube thickness variation terms in quasi-3D cascade designs or rotational terms in rotating blade design problems. The application of the method is illustrated through a number of design problems concerning isolated airfoils, a 3D duct, 2D, quasi-3D and 3D, stationary and rotating turbomachinery blades.     

A hybrid meta heuristic algorithm for bi-objective minimum cost flow (BMCF) problem

In this paper we study the bi-objective minimum cost flow (BMCF) problem which can be categorized as multi objective minimum cost flow problems. Generally, the exact computation of the efficient frontier is intractable and there may exist an exponential number of extreme non-dominated objective vectors. Hence, it is better to employ an approximate method to compute solutions within reasonable time. Therefore, we propose a hybrid meta heuristic algorithm (memetic algorithm hybridized with simulated annealing MA/SA) to develop an efficient approach for solving this problem. In order to show the efficiency of the proposed MA/SA some problems have been generated and solved by both the MA/SA and an exact method. It is perceived from this evaluation that the proposed MA/SA outputs are very close to the exact solutions. It is shown that when the number of arcs and nodes exceed 30 (large problems) the MA/SA model will be more preferred because of its strongly shorter computational time in comparison with exact methods.     

Computations of optimal cylinder flow control in weakly conductive fluids

A nonlinear adjoint-based optimal control approach of cylinder wake by electromagnetic force has been investigated numerically in the paper. A cost functional representing the balance of the regulated quantities with different weights and interaction parameter N (Lorentz force) has been constituted, where the regulated quantities related with flow and force are taken as targets of regulation and the Lorentz force, (as interaction parameter N), is taken as a control input. Based on the cost functional and Navier-Stokes equations, the corresponding adjoint equations have been derived and the sensitivity of the cost functional is found to be a simple function of the adjoint stream function in the adjoint field. For the different regulations, the forms of optimal control rules are similar while the adjoint equations are different. The receding-horizon predictive control setting is employed to discuss the optimal control problems. Under the action of optimal N(t), the flow separation is suppressed fully, so that the oscillations of drag and lift are suppressed and the total drag coefficient decreases dramatically. For the different regulations, the control effects have some differences due to the different values of optimal inputs corresponding to the different adjoint flow fields.     

On the numerical implementation of continuous adjoint sensitivity for transient heat conduction problems using an isogeometric approach

A crucial problem of continuous adjoint shape sensitivity analysis is the numerical implementation of its lengthy formulations. In this paper, the numerical implementation of continuous adjoint shape sensitivity analysis is presented for transient heat conduction problems using isogeometric analysis, which can serve as a tutorial guide for beginners. Using the adjoint boundary and loading conditions derived from the design objective and the primary state variable fields, the numerical analysis procedure of the adjoint problem, which is solved backward in time, is demonstrated. Following that, the numerical integration algorithm of the shape sensitivity using a boundary approach is provided. Adjoint shape sensitivity is studied with detailed explanations for two transient heat conduction problems to illustrate the numerical implementation aspects of the continuous adjoint method. These two problems can be used as benchmark problems for future studies.     

Some implementation issues associated with multidimensional interval Newton methods

This paper presents the development of an optimal interval Newton method for systems of nonlinear equations. The context of solving such systems of equations is that of optimization of nonlinear mathematical programs. The modifications of the interval Newton method presented in this paper provide computationally effective enhancements to the general interval Newton method. The paper demonstrates the need to compute an optimal step length in the interval Newton method in order to guarantee the generation of a sequence of improving solutions. This method is referred to as the optimal Newton method and is implemented in multiple dimensions. Secondly, the paper demonstrates the use of the optimal interval Newton method as a feasible direction method to deal with non-negativity constraints. Also, included in this implementation is the use of a matrix decomposition technique to reduce the computational effort required to compute the Hessian inverse in the interval Newton method. The methods are demonstrated on several problems. Included in these problems are mathematical programs with perturbations in the problem coefficients. The numerical results clearly demonstrate the effectiveness and efficiency of these approaches.     

基于微分/代数方程的多体系统动力学设计灵敏度分析的伴随变量方法

基于多体系统动力学微分/代数方程数学模型和通用积分形式的目标函数,建立了多体系统动力学设计灵敏度分析的伴随变量方法,避免了复杂的设计灵敏度计算,对于设计变量较多的多体系统灵敏度分析具有较高的计算效率.文中给出了通用公式以及具体的计算过程和验证方法,并将目标函数及其导数积分形式的计算转化为微分方程的初值问题,进一步提高了计算效率和精度.文末通过一曲柄-滑块机构算例对算法的有效性进行了验证.     

改进的正则化模型在图像恢复中的应用

目的 由拟合项与正则项组成的海森矩阵,如果不具有特殊结构,其逆矩阵计算比较困难,为克服此缺点,提出一种海森矩阵可分块对角化的牛顿投影迭代算法。方法 首先,用L₂范数描述拟合项,用自变量是有界变差函数的复合函数刻画正则项,建立能量泛函正则化模型。其次,引入势函数,将正则化模型转化为增广能量泛函。再次,构造预条件矩阵,使得海森矩阵可分块对角化。最后,为防止牛顿投影迭代算法收敛到局部最优解,采用回溯线性搜索算法和改进的Barzilai-Borwein步长更新准则使得算法全局收敛。结果 针对图像去模糊正则化模型容易使边缘平滑和产生阶梯效应“两难”问题,提出一种新的正则化模型和牛顿投影迭代算法。仿真结果表明,“两难”问题通过本文算法得到了很好的解决。结论 与其他正则化图像去模糊模型相比,本文算法明显改善图像的质量,如有效地保护图像的边缘,抑制阶梯效应,相对偏差和误差较小,较高的峰值信噪比和结构相似测度。     

A hybrid numerical method and its application to inviscid compressible flow problems

An improved version of the artificially upstream flux vector scheme, is developed to efficiently compute inviscid compressible flow problems. This numerical scheme, named AUFSR (Tchuen et al. 2011), is obtained by hybridizing the AUFS scheme with Roe’s solver. This approach handles difficulties encountered by the AUFS scheme, in the case where the flux vector does not check the homogeneous property. The present scheme for multi-dimensional flows introduces a certain amount of numerical dissipation to shear waves, as Roe’s splitting. The AUFSR scheme is not only robust for shock-capturing, but also accurate for resolving shear layers. Numerical results for 1D Riemann problems and several 2D problems are investigated to show the capability of the method to accurately compute inviscid compressible flow when compared to AUFS, and Roe solvers.     

Computing the Isolated Roots by Matrix Methods

Two main approaches are used, nowadays, to compute the roots of a zero-dimensional polynomial system. The first one involves Gröbner basis computation, and applies to any zero-dimensional system. But, it is performed withexact arithmetic and, usually, large numbers appear during the computation. The other approach is based on resultant formulations and can be performed with floating point arithmetic. However, it applies only to generic situations, leading to singular problems in several systems coming from robotics and computational vision, for instance.In this paper, reinvestigating the resultant approach from the linear algebra point of view, we handle the problem of genericity and present a new algorithm for computing the isolated roots of an algebraic variety, not necessarily of dimension zero. We analyse two types of resultant formulations, transform them into eigenvector problems, and describe special linear algebra operations on the matrix pencils in order to reduce the root computation to a non-singular eigenvector problem. This new algorithm, based on pencil decompositions, has a good complexity even in the non-generic situations and can be executed with floating point arithmetic.     

Aerodynamic design optimization on unstructured grids with a continuous adjoint formulation

A continuous adjoint approach for obtaining sensitivity derivatives on unstructured grids is developed and analyzed. The derivation of the costate equations is presented, and a second-order accurate discretization method is described. The relationship between the continuous formulation and a discrete formulation is explored for inviscid, as well as for viscous flow. Several limitations in a strict adherence to the continuous approach are uncovered, and an approach that circumvents these difficulties is presented. The issue of grid sensitivities, which do not arise naturally in the continuous formulation, is investigated and is observed to be of importance when dealing with geometric singularities. A method is described for modifying inviscid and viscous meshes during the design cycle to accommodate changes in the surface shape. The accuracy of the sensitivity derivatives is established by comparing with finite-difference gradients and several design examples are presented.