The independent set perturbation adjoint method: A new method of differentiating mesh‐based fluids models

A new scheme for differentiating complex mesh‐based numerical models (e.g. finite element models), the Independent Set Perturbation Adjoint method (ISP‐Adjoint), is presented. Differentiation of the matrices and source terms making up the discrete forward model is realized by a graph coloring approach (forming independent sets of variables) combined with a perturbation method to obtain gradients in numerical discretizations. This information is then convolved with the ‘mathematical adjoint’, which uses the transpose matrix of the discrete forward model. The adjoint code is simple to implement even with complex governing equations, discretization methods and non‐linear parameterizations. Importantly, the adjoint code is independent of the implementation of the forward code. This greatly reduces the effort required to implement the adjoint model and maintain it as the forward model continues to be developed; as compared with more traditional approaches such as applying automatic differentiation tools. The approach can be readily extended to reduced‐order models. The method is applied to a one‐dimensional Burgers' equation problem, with a highly non‐linear high‐resolution discretization method, and to a two‐dimensional, non‐linear, reduced‐order model of an idealized ocean gyre. Copyright © 2010 John Wiley & Sons, Ltd.     

Adjoint‐based airfoil optimization with discretization error control

In this article, we develop a new airfoil shape optimization algorithm based on higher‐order adaptive DG methods with control of the discretization error. Each flow solution in the optimization loop is computed on a sequence of goal‐oriented h‐refined or hp‐refined meshes until the error estimation of the discretization error in a flow‐related target quantity (including the drag and lift coefficients) is below a prescribed tolerance. Discrete adjoint solutions are computed and employed for the multi‐target error estimation and adaptive mesh refinement. Furthermore, discrete adjoint solutions are employed for evaluating the gradients of the objective function used in the CGs optimization algorithm. Furthermore, an extension of the adjoint‐based gradient evaluation to the case of target lift flow computations is employed. The proposed algorithm is demonstrated on an inviscid transonic flow around the RAE2822, where the shape is optimized to minimize the drag at a given constant lift and airfoil thickness. The effect of the accuracy of the underlying flow solutions on the quality of the optimized airfoil shapes is investigated. Copyright © 2014 John Wiley & Sons, Ltd.     

Examination for adjoint boundary conditions in initial water elevation estimation problems

I present here a method of generating a distribution of initial water elevation by employing the adjoint equation and finite element methods. A shallow‐water equation is employed to simulate flow behavior. The adjoint equation method is utilized to obtain a distribution of initial water elevation for the observed water elevation. The finite element method, using the stabilized bubble function element, is used for spatial discretization, and the Crank–Nicolson method is used for temporal discretizations. In addition to a method for optimally assimilating water elevation, a method is presented for determining adjoint boundary conditions. An examination using the observation data including noise data is also carried out. Copyright © 2009 John Wiley & Sons, Ltd.     

Adjoint‐based design of shock mitigation devices

Unsteady Euler and adjoint Euler solvers have been combined in order to aid in the design of shock mitigation devices. The flowfield is integrated forward in time and stored. The adjoint is then integrated going backwards in time, restoring and interpolating the saved Euler solution to the current point in time. The gradient is obtained from a surface integral formulation during the adjoint run. Comparisons of adjoint‐based and finite‐differencing gradients for different verification cases show less than 10% deviation. The results obtained indicate that this is a very cost‐effective way to obtain the gradients of an objective function with respect to surface design changes. Moreover, as the sensitivity information is determined over a complete surface, the procedure provides considerable insight, and can efficiently facilitate the design of shock mitigation devices such as architecturally appealing blast walls. Copyright © 2009 John Wiley & Sons, Ltd.     

Remarks on the numerical solution of the adjoint quasi‐one‐dimensional Euler equations

We examine the numerical solution of the adjoint quasi‐one‐dimensional Euler equations with a central‐difference finite volume scheme with Jameson‐Schmidt‐Turkel (JST) dissipation, for both the continuous and discrete approaches. First, the complete formulations and discretization of the quasi‐one‐dimensional Euler equations and the continuous adjoint equation and its counterpart, the discrete adjoint equation, are reviewed. The differences between the continuous and discrete boundary conditions are also explored. Second, numerical testing is carried out on a symmetric converging–diverging duct under subsonic flow conditions. This analysis reveals that the discrete adjoint scheme, while being manifestly less accurate than the continuous approach, gives nevertheless more accurate flow sensitivities. Copyright © 2011 John Wiley & Sons, Ltd.     

Optimal control of shallow water flow using time-delay function

This paper presents an optimal control system that includes a time-delay function for application to flood control setups with a retardation area. This system consists of the present and past controls that express flow behaviour in the retardation area. Optimal control theory is used to obtain a control discharge that satisfies the state equation including the time-delay function and minimizes the performance function. The optimal control and the delayed control discharges are obtained by the solution of an adjoint equation. The weighted gradient method is employed as a minimization algorithm. The Galerkin finite element procedure is employed to discretize the state and adjoint equations in the spatial direction. The bubble function interpolation, originated by the authors' group, using a stabilized term, is employed for the discretization in space. The flood flow in the Tsurumi river is presented as a numerical model. We show in this paper that floods can be controlled by means of a time-delay function.     

An adjoint method for the calculation of remote sensitivities in supersonic flow

This paper presents an adjoint method for the calculation of remote sensitivities in supersonic flow. The goal is to develop a set of discrete adjoint equations and their corresponding boundary conditions in order to quantify the influence of geometry modifications on the pressure distribution at an arbitrary location within the domain of interest. First, this paper presents the complete formulation and discretization of the discrete adjoint equations. The special treatment of the adjoint boundary condition to obtain remote sensitivities or sensitivities of pressure distributions at points remotely located from the wing surface are discussed. Secondly, we present results that demonstrate the application of the theory to a three-dimensional remote inverse design problem using a low sweep biconvex wing and a highly swept blunt leading edge wing. Lastly, we present results that establish the added benefit of using an objective function that contains the sum of the remote inverse and drag minimization cost functions.     

Optimal control applied to water flow using second order adjoint method

This paper presents an optimal control applied to water flow using the first and second order adjoint equations. The gradient of the performance function with respect to control variables is analytically obtained by the first order adjoint equation. It is not necessary to compute the Hessian matrix directly using the second order adjoint equation. Two numerical studies have been performed to show the adaptability of the present method. The performance of the second order adjoint method is compared with that of the weighted gradient method, Broyden–Fletcher–Goldfarb–Shanno method and Lanczos method. The precise forms of the adjoint equations and the gradient to use for the minimisation algorithm are derived. The computation by the Lanczos method is shown as superior to those of the other methods discussed in this paper. The message passing interface library is used for the communication of parallel computing.     

On estimation of entrance boundary parameters from downstream measurements using adjoint approach

An estimation of entrance boundary conditions from the downstream measurements is considered in variational statement form for two‐dimensional supersonic laminar flow. The adjoint problem is used for the calculation of the discrepancy gradient in space of control parameters. The numerical tests presented demonstrate the successful estimation of boundary parameters of spatial distribution by using gradient methods. Copyright © 2001 John Wiley & Sons, Ltd.     

A stabilized finite element method for shape optimization in low Reynolds number flows

A gradient‐based optimization procedure based on a continuous adjoint approach is formulated and implemented for steady low Reynolds number flows. A stabilized finite element formulation is proposed to solve the adjoint equations. The accuracy of the gradients from the adjoint approach is verified against the ones computed from a simple finite difference procedure. The validation of the formulation and its implementation is carried out via flow past an elliptical bump whose eccentricity is used as a design parameter. Shape design studies for the elliptical bump are then carried on with a more complex 4th order Bézier parametrization of the bump. Results for, both, optimal design and inverse problems are presented. Using different initial guesses, multiple optimal shapes are obtained. A multi‐objective function with additional constraints on the volume and the drag coefficient of the bump is utilized. It is seen that as more constraints are added to the objective function the design space is constrained and the multiple optimal shapes become progressively similar to each other. The study demonstrates the usefulness of this tool in obtaining multiple engineering solutions to a given design problem and also providing a framework to impose multiple constraints simultaneously. Copyright © 2007 John Wiley & Sons, Ltd.     

Efficient and robust algorithms for solution of the adjoint compressible Navier–Stokes equations with applications

The complete discrete adjoint equations for an unstructured finite volume compressible Navier–Stokes solver are discussed with respect to the memory and time efficient evaluation of their residuals, and their solution. It is seen that application of existing iteration methods for the non‐linear equation—suitably adjointed—has a property of guaranteed convergence provided that the non‐linear iteration is well behaved. For situations where this is not the case, in particular for strongly separated flows, a stabilization technique based on the Recursive Projection Method is developed. This method additionally provides the dominant eigenmodes of the problem, allowing identification of flow regions that are unstable under the basic iteration. These are found to be regions of separated flow. Finally, an adjoint‐based optimization with 96 design variables is performed on a wing–body configuration. The initial flow has large regions of separation, which are significantly diminished in the optimized configuration. Copyright © 2008 John Wiley & Sons, Ltd.     

A methodology for the development of discrete adjoint solvers using automatic differentiation tools

A methodology for the rapid development of adjoint solvers for computational fluid dynamics (CFD) models is presented. The approach relies on the use of automatic differentiation (AD) tools to almost completely automate the process of development of discrete adjoint solvers. This methodology is used to produce the adjoint code for two distinct 3D CFD solvers: a cell-centred Euler solver running in single-block, single-processor mode and a multi-block, multi-processor, vertex-centred, magneto-hydrodynamics (MHD) solver. Instead of differentiating the entire source code of the CFD solvers using AD, we have applied it selectively to produce code that computes the transpose of the flux Jacobian matrix and the other partial derivatives that are necessary to compute sensitivities using an adjoint method. The discrete adjoint equations are then solved using the Portable, Extensible Toolkit for Scientific Computation (PETSc) library. The selective application of AD is the principal idea of this new methodology, which we call the AD adjoint (ADjoint). The ADjoint approach has the advantages that it is applicable to any set of governing equations and objective functions and that it is completely consistent with the gradients that would be computed by exact numerical differentiation of the original discrete solver. Furthermore, the approach does not require hand differentiation, thus avoiding the long development times typically required to develop discrete adjoint solvers for partial differential equations, as well as the errors that result from the necessary approximations used during the differentiation of complex systems of conservation laws. These advantages come at the cost of increased memory requirements for the discrete adjoint solver. However, given the amount of memory that is typically available in parallel computers and the trends toward larger numbers of multi-core processors, this disadvantage is rather small when compared with the very significant advantages that are demonstrated. The sensitivities of drag and lift coefficients with respect to different parameters obtained using the discrete adjoint solvers show excellent agreement with the benchmark results produced by the complex-step and finite-difference methods. Furthermore, the overall performance of the method is shown to be better than most conventional adjoint approaches for both CFD solvers used.     

非结构网格二阶有限体积法中黏性通量离散格式精度分析与改进

非结构网格二阶有限体积离散方法广泛应用于计算流体力学工程实践中,研究非结构网格二阶精度有限体积离散方法的计算精度具有现实意义. 计算精度主要受到网格和计算方法的影响,本文从单元梯度重构方法、黏性通量中的界面梯度计算方法两个方面考察黏性流动模拟精度的影响因素. 首先从理论上分析了黏性通量离散中的“奇偶失联”问题,并通过基于标量扩散方程的制造解方法验证了“奇偶失联”导致的精度下降现象,进一步通过引入差分修正项消除了“奇偶失联”并提高了扩散方程计算精度;其次,在不同类型、不同质量的网格上进行基于扩散方程的制造解精度测试,考察单元梯度重构方法、界面梯度计算方法对扩散方程计算精度的影响,结果显示,单元梯度重构精度和界面梯度计算方法均对扩散方程计算精度起重要作用;最后对三个黏性流动算例(二维层流平板、二维湍流平板和二维翼型近尾迹流动)进行网格收敛性研究,初步验证了本文的结论,得到了计算精度和网格收敛性均较好的黏性通量计算格式.      

Adjoint Equation-Based Methods for Control Problems in Incompressible, Viscous Flows

A review of adjoint equation-based methodologies for viscous,incompressible flow control and optimization problems is given and illustrated by a drag minimization example. A number of approaches to ameliorating the high storage and CPU costs associated with straightforward implementations of adjoint equation based methodologies are discussed. Other issues, including the relative merits of the differentiate-then-discretize and discretize-then-differentiate approaches to deriving discrete adjoint equations, the incorporation of side constraints into adjoint equation-based methodologies, and inaccuracies that occur due to differentiations at the boundary, are also discussed. This revised version was published online in July 2006 with corrections to the Cover Date.     

Mesh adaptation using C1 interpolation of the solution in an unstructured finite volume solver

The goal of this paper is to show the effectiveness of a newly developed estimate of the truncation error calculated based on C¹ interpolation of the solution weighted by the adjoint solution as the adaptation indicator for an unstructured finite volume solver. We will show that adjoint‐based mesh adaptation based on the corrected functional using the new developed truncation error estimate is capable of adapting the mesh to improve the accuracy of the functional and the convergence rate. Both discrete and continuous adjoint solutions are used for adaptation. Results are significantly better with new truncation error estimate than with previously used estimates.     

Water quality control by bank placement based on optimal control and finite element method

This paper presents a method for quality control by bank placement based on an optimal control theory and the finite element method. The shallow water equation is employed for the analysis of the flow condition and the advection‐diffusion equation is used for the analysis of pollutant concentration. The optimal control theory is utilized to obtain a control value for the objective state value. The shear‐slip mesh update method which is suitable for the rotational problem of body is employed. To solve the optimization problem, the time domain decomposition method is applied as a technique of storage requirements reduction. The Sakawa–Shindo method is employed as a minimization technique. The Crank–Nicolson method is applied to the temporal discretization. A method for optimal control of bank placement has been presented. Copyright © 2003 John Wiley & Sons, Ltd.     

A FE‐based algorithm for the inverse natural convection problem

Several numerical algorithms for solving inverse natural convection problems are revisited and studied. Our aim is to identify the unknown strength of a time‐varying heat source via a set of coupled nonlinear partial differential equations obtained by the so‐called finite element consistent splitting scheme (CSS) in order to get a good approximation of the unknown heat source from both the measured data and model results, by minimizing a functional that measures discrepancies between model and measured data. Viewed as an optimization problem, the solutions are obtained by means of the conjugate gradient method. A second‐order CSS in time involving the direct problem, the adjoint problem, the sensitivity problem and a system of sensitivity functions is used in order to enhance the numerical accuracy obtained for the unknown heat source function. A spatial discretization of all field equations is implemented using equal‐order and mixed finite element methods. Numerical experiments validate the proposed optimization algorithms that are in good agreement with the existing results. Copyright © 2010 John Wiley & Sons, Ltd.     

Two‐objective optimization strategies using the adjoint method and game theory in inverse natural convection problems

This paper considers the problem of estimating the strengths of two time‐varying heat sources simultaneously, from measurements of the temperature inside the square domain in a porous medium, when prior knowledge of the source functions is not available. This problem is an inverse natural convection problem. In order to circumvent this problem, we define several optimization criteria (objective functionals) that measure discrepancies between model and measured data, where objective functionals depend on two heat sources and use multi‐criteria optimization to identify Nash equilibria, which are solutions to the non‐cooperative game according to game theory. Two non‐cooperative game strategies are considered: competitive (Nash) game and hierarchical (modified Stackelberg) game. The methodology that we employ relies on a combination of mixed finite element space approximations, finite difference time discretizations, adjoint equation and sensitivity equation techniques, and nonlinear conjugate gradient algorithms for the solutions of estimating two heat sources. Applying the Sobolev gradient for the noise removal is investigated. The performance of the present technique of inverse analysis is evaluated, by means of several numerical experiments, and is found to be very accurate as well as efficient. Copyright © 2012 John Wiley & Sons, Ltd.     

Design optimization of unsteady airfoils with continuous adjoint method

In this paper, a new unsteady aerodynamic design method is presented based on the Navier-Stokes equations and a continuous adjoint approach. A basic framework of time-accurate unsteady airfoil optimization which adopts time-averaged aerodynamic coefficients as objective functions is presented. The time-accurate continuous adjoint equation and its boundary conditions are derived. The flow field and the adjoint equation are simulated numerically by the finite volume method (FVM). Feasibility and accuracy of the approach are perfectly validated by the design optimization results of the plunging NACA0012 airfoil.     

Numerical control of two‐dimensional shock waves in dual solution domain by instant temperature disturbances

The search for the temperature disturbance causing transition between regular and Mach reflections in the dual solution domain is addressed in an optimization statement. The gradient of the discrepancy between the current and target flow fields was calculated using adjoint equations. The control was determined by gradient‐based optimization. The flow field simulation is verified via a posteriori error estimates using the solution of an additional adjoint problem. Copyright © 2012 John Wiley & Sons, Ltd.