Characteristics‐based boundary conditions for the Euler adjoint problem

Over the last decade, the adjoint method has been consolidated as one of the most versatile and successful tools for aerodynamic design. It has become a research area on its own, spawning a large variety of applications and a prolific literature. Yet, some relevant aspects of the method remain relatively less explored in the literature. Such is the case with the adjoint boundary problem. In particular for Euler flows, both fluid dynamic and adjoint equations entail complementary Riemann problems, and these yield boundary conditions that are fully consistent with well‐posedness. In the literature, this approach has been pursued solely in terms of Riemann variables. This work formulates the adjoint boundary problem so as to correspond precisely to that imposed on the flow, as it is given in terms of primitive variables. Test results have shown to be in agreement with the traditional approach for external flow problems. Copyright © 2012 John Wiley & Sons, Ltd.     

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.     

Aerodynamic shape optimization for unsteady three-dimensional flows

This paper presents an adjoint method for the optimum shape design of unsteady flows. The goal is to develop a set of discrete unsteady adjoint equations and the corresponding boundary condition for the non-linear frequency domain method. First, this paper presents the complete formulation of the time dependent optimal design problem. Second, we present the non-linear frequency domain adjoint equations for three-dimensional flows. Third, we present results that demonstrate the application of the theory to a three-dimensional wing.     

A uniformly convergent diffrence scheme for the singular perturbation of a self adjoint elliptic partial differential equation

In this paper, we consider a self adjoint elliptic first boundary value problem with a small parameter affecting the highest derivative.In the paper, we set up a new scheme by the asymptotic analysis method, compare asymptotic behavior between the solution of the difference equation and the solution of the differential equation, and show uniform convergence of the new scheme.     

Optimal flow control for Navier–Stokes equations: drag minimization

Optimal control and shape optimization techniques have an increasing role in Fluid Dynamics problems governed by partial differential equations (PDEs). In this paper, we consider the problem of drag minimization for a body in relative motion in a fluid by controlling the velocity through the body boundary. With this aim, we handle with an optimal control approach applied to the steady incompressible Navier–Stokes equations. We use the Lagrangian functional approach and we consider the Lagrangian multiplier method for the treatment of the Dirichlet boundary conditions, which include the control function itself. Moreover, we express the drag coefficient, which is the functional to be minimized, through the variational form of the Navier–Stokes equations. In this way, we can derive, in a straightforward manner, the adjoint and sensitivity equations associated with the optimal control problem, even in the presence of Dirichlet control functions. The problem is solved numerically by an iterative optimization procedure applied to state and adjoint PDEs which we approximate by the finite element method. Copyright © 2007 John Wiley & Sons, Ltd.     

Numerical solution of steady free‐surface flows by the adjoint optimal shape design method

Numerical solution of flows that are partially bounded by a freely moving boundary is of great importance in practical applications such as ship hydrodynamics. Free‐boundary problems can be reformulated into optimal shape design problems, which can in principle be solved efficiently by the adjoint method. In this work we investigate the suitability of the adjoint shape optimization method for solving steady free‐surface flows. The asymptotic convergence behaviour of the method is determined for free‐surface flows in 2D and 3D. It is shown that the convergence behaviour depends sensitively on the occurrence of critical modes. The convergence behaviour is moreover shown to be mesh‐width independent, provided that proper preconditioning is applied. Numerical results are presented for 2D flow over an obstacle in a channel. The observed convergence behaviour is indeed mesh‐width independent and conform the derived asymptotic estimates. Copyright © 2003 John Wiley & Sons, Ltd.     

On the approximate solution of elliptic,self adjoint boundary value problems

Summary Integral representations for the solutions to linear elliptic self adjoint boundary value problems are derived in terms of two functions which are generalisations of the single and double layer potentials used in the theory of harmonic functions. The generalised potentials are constructed in terms of a fundamental solution which is an approximation to the exact kernel of the boundary value problem in question. The representations so obtained are shown to provide a basis from which strict approximations to the solutions of boundary value problems can be developed. In particular the structure of the integral equation representing the given boundary value problem is precisely determined.     

Airfoil design optimization based on lattice Boltzmann method and adjoint approach

We present a new aerodynamic design method based on the lattice Boltzmann method (LBM) and the adjoint approach. The flow field and the adjoint equation are numerically simulated by the GILBM (generalized form of interpolation supplemented LBM) on non-uniform meshes. The first-order approximation for the equilibrium distribution function on the boundary is proposed to diminish the singularity of boundary conditions. Further, a new treatment of the solid boundary in the LBM is described particularly for the airfoil optimization design problem. For a given objective function, the adjoint equation and its boundary conditions are derived analytically. The feasibility and accuracy of the new approach have been perfectly validated by the design optimization of NACA0012 airfoil.     

An agglomeration‐based adaptive discontinuous Galerkin method for compressible flows

In this work, we exploit the possibility to devise discontinuous Galerkin discretizations over polytopic grids to perform grid adaptation strategies on the basis of agglomeration coarsening of a fine grid obtained via standard unstructured mesh generators. The adaptive agglomeration process is here driven by an adjoint‐based error estimator. We investigate several strategies for converting the error field estimated solving the adjoint problem into an agglomeration factor field that is an indication of the number of elements of the fine grid that should be clustered together to form an agglomerated element. As a result the size of agglomerated elements is optimized for the achievement of the best accuracy for given grid size with respect to the target quantities. To demonstrate the potential of this strategy we consider problem‐specific outputs of interest typical of aerodynamics, eg, the lift and drag coefficients in the context of inviscid and viscous flows test cases.     

广义变分原理的结构形状优化伴随法灵敏度分析

提出了一种利用伴随变量进行结构形状优化灵敏度分析的新方法. 基于广义变分原理， 考虑了形状优化中位移边界条件的变化对结构响应的影响. 新方法弥补了Arora 等人所提出的形状优化灵敏度分析变分原理中的缺陷，为采用伴随法进行灵敏度分析提供了 新的框架.     

Saint-Venant problem of three-dimensional linear viscoelasticity in the Hamiltonian system

The traditional Saint-Venant problem of three-dimensional viscoelasticity is discussed under the Hamiltonia system with the use of the Laplace integral transformation, and the original problem is transformed into finding eigenvalues and eigenvectors of the Hamiltonia operator matrix. Since local effect near the boundary is usually neglected, all solutions of Saint-Venant problems can be obtained directly by the combinations of zero eigenvectors. Moreover, the adjoint relationships of the symplectic orthogonality of zero eigenvectors in the Laplace domain are generalized to the time domain. Therefore the problem can be discussed directly in the eigenvector space of the time domain, and the iterative application of Laplace transformation is not needed. Simply by applying the adjoint relationships of the symplectic orthogonality, an effective method for boundary condition is given. Based on this method, some typical examples are discussed, in which the whole character of total creep and relaxation of viscoelasticity is clearly revealed.     

What is the Optimal Shape of a Pipe?

We consider an incompressible fluid in a three-dimensional pipe, following the Navier–Stokes system with classical boundary conditions. We are interested in the following question: is there any optimal shape for the criterion “energy dissipated by the fluid”? Moreover, is the cylinder the optimal shape? We prove that there exists an optimal shape in a reasonable class of admissible domains, but the cylinder is not optimal. For that purpose, we define the first order optimality condition, thanks to the adjoint state and we prove that it is impossible that the adjoint state be a solution of this over-determined system when the domain is the cylinder. At last, we show some numerical simulations for that problem.     

Optimum Aerodynamic Design Using the Navier–Stokes Equations

This paper describes the formulation of optimization techniques based on control theory for aerodynamic shape design in viscous compressible flow, modeled by the Navier–Stokes equations. It extends previous work on optimization for inviscid flow. The theory is applied to a system defined by the partial differential equations of the flow, with the boundary shape acting as the control. The Fréchet derivative of the cost function is determined via the solution of an adjoint partial differential equation, and the boundary shape is then modified in a direction of descent. This process is repeated until an optimum solution is approached. Each design cycle requires the numerical solution of both the flow and the adjoint equations, leading to a computational cost roughly equal to the cost of two flow solutions. The cost is kept low by using multigrid techniques, in conjunction with preconditioning to accelerate the convergence of the solutions. The power of the method is illustrated by designs of wings and wing–body combinations for long range transport aircraft. Satisfactory designs are usually obtained with 20–40 design cycles. Received 5 February 1997 and accepted 30 May 1997     

Identification of shallow sea models

In this paper we consider a parameter estimation procedure for shallow sea models. The method is formulated as a minimization problem. An adjoint model is used to calculate the gradient of the criterion which is to be minimized. In order to obtain a robust estimation method, the uncertainty of the open boundary conditions can be taken into acoount by allowing random noise inputs to act on the open boundaries. This method avoids the possibility that boundary errors are interpreted by the estimation procedure as parameter fluctuations. We apply the parameter estimation method to identify a shallow sea model of the entire European continental shelf. First, a space-varying bottom friction coefficient is estimated simultaneously with the depth. The second application is the estimation of the parameterization of the wind stress coefficient as a function of the wind velocity. Finally, an uncertain open boundary condition is included. It is shown that in this case the parameter estimation procedure does become more robust and produces more realistic estimates. Furthermore, an estimate of the open boundary conditions is also obtained.     

Application of second‐order adjoint technique for conduit flow problem

This paper presents the way to obtain the Newton gradient by using a traction given by the perturbation for the Lagrange multiplier. Conventionally, the second‐order adjoint model using the Hessian/vector products expressed by the product of the Hessian matrix and the perturbation of the design variables has been researched (Comput. Optim. Appl. 1995; 4 :241–262). However, in case that the boundary value would like to be obtained, this model cannot be applied directly. Therefore, the conventional second‐order adjoint technique is extended to the boundary value determination problem and the second‐order adjoint technique is applied to the conduit flow problem in this paper. As the minimization technique, the Newton‐based method is employed. The Broyden–Fletcher–Goldfarb–Shanno (BFGS) method is applied to calculate the Hessian matrix which is used in the Newton‐based method and a traction given by the perturbation for the Lagrange multiplier is used in the BFGS method. Copyright © 2007 John Wiley & Sons, Ltd.     

A Discrete Adjoint Approach for the Optimization of Unsteady Turbulent Flows

In this paper we present a discrete adjoint approach for the optimization of unsteady, turbulent flows. While discrete adjoint methods usually rely on the use of the reverse mode of Automatic Differentiation (AD), which is difficult to apply to complex unsteady problems, our approach is based on the discrete adjoint equation directly and can be implemented efficiently with the use of a sparse forward mode of AD. We demonstrate the approach on the basis of a parallel, multigrid flow solver that incorporates various turbulence models. Due to grid deformation routines also shape optimization problems can be handled. We consider the relevant aspects, in particular the efficient generation of the discrete adjoint equation and the parallel implementation of a multigrid method for the adjoint, which is derived from the multigrid scheme of the flow solver. Numerical results show the efficiency of the approach for a shape optimization problem involving a three dimensional Large Eddy Simulation (LES).     

A Free Boundary Problem Arising from Segregation of Populations with High Competition

In this work, we show how to obtain a free boundary problem as the limit of a fully nonlinear elliptic system of equations that models population segregation of the Gause–Lotka–Volterra type. We study the regularity of the solutions. In particular, we prove Lipschitz regularity across the free boundary. The problem is motivated by the work done by Caffarelli, Karakhanyan and Fang-Hua Lin for the linear case.     

Mode decomposition of nonlinear eigenvalue problems and application in flow stability

Direct numerical simulations are carried out with different disturbance forms introduced into the inlet of a flat plate boundary layer with the Mach number 4.5. According to the biorthogonal eigenfunction system of the linearized Navier-Stokes equations and the adjoint equations, the decomposition of the direct numerical simulation results into the discrete normal mode is easily realized. The decomposition coefficients can be solved by doing the inner product between the numerical results and the eigenfunctions of the adjoint equations. For the quadratic polynomial eigenvalue problem, the inner product operator is given in a simple form, and it is extended to an Nth-degree polynomial eigenvalue problem. The examples illustrate that the simplified mode decomposition is available to analyze direct numerical simulation results.     

流体流动的边界滑移问题研究进展

最近十几年来,随着现代微/纳米测试以及分子动力学模拟技术的出现和发展,流体流动的边界滑移问题研究获得了突 飞猛进的发展.边界滑移相关研究大体可分为3个方面: 实验、分子动力学模拟和理论数值分析,前两者主要以发现边界滑移现象、探索边界滑移的产生机理以及各因素对边 界滑移的影响规律为主要研究目的,而后者主要研究边界滑移的物理模型、相关问题的计算方法以及边界滑移对流体系 统流体动力学行为的影响.本文首先简要回顾了液体流动的边界滑移及其相关问题的早期研究历史,随后对边界滑移问题 的研究现状进行了综述,最后展望了该领域今后的研究重点及其应用前景.