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.     

Finite element and implicit Runge–Kutta implementation of an acoustics–convection upstream resolution algorithm for the time‐dependent two‐dimensional Euler equations

The second of a two‐paper series, this paper details a solver for the characteristics‐bias system from the acoustics–convection upstream resolution algorithm for the Euler and Navier–Stokes equations. An integral formulation leads to several surface integrals that allow effective enforcement of boundary conditions. Also presented is a new multi‐dimensional procedure to enforce a pressure boundary condition at a subsonic outlet, a procedure that remains accurate and stable. A classical finite element Galerkin discretization of the integral formulation on any prescribed grid directly yields an optimal discretely conservative upstream approximation for the Euler and Navier–Stokes equations, an approximation that remains multi‐dimensional independently of the orientation of the reference axes and computational cells. The time‐dependent discrete equations are then integrated in time via an implicit Runge–Kutta procedure that in this paper is proven to remain absolutely non‐linearly stable for the spatially‐discrete Euler and Navier–Stokes equations and shown to converge rapidly to steady states, with maximum Courant number exceeding 100 for the linearized version. Even on relatively coarse grids, the acoustics–convection upstream resolution algorithm generates essentially non‐oscillatory solutions for subsonic, transonic and supersonic flows, encompassing oblique‐ and interacting‐shock fields that converge within 40 time steps and reflect reference exact solutions. Copyright © 2005 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.     

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.     

基于离散共轭方法的高超声速导弹气动外形优化设计

开展了离散共轭方法在高超声速气动外形优化设计中的应用研究。构建了基于NURBS方法的几何外形参数化方法,完成了一种简单高效的动网格方法,建立了基于Euler方程的离散共轭方法,并将这些方法与优化算法等集成起来够构建了适合复杂外形的高超声速气动外形优化设计系统。利用该系统对一种导弹的前体进行了优化设计研究,使其升阻比提高了11.2%,优化后导弹前体形状接近双锥外形,说明双锥形前体有利于减小阻力。算例表明,离散共轭方法在高超声速气动外形优化设计中具有良好的应用前景。     

Aerodynamic design using the truncated Newton algorithm and the continuous adjoint approach

In this paper, the so‐called ‘continuous adjoint‐direct approach’ is used within the truncated Newton algorithm for the optimization of aerodynamic shapes, using the Euler equations. It is known that the direct differentiation (DD) of the flow equations with respect to the design variables, followed by the adjoint approach, is the best way to compute the exact matrix, for use along with the Newton optimization method. In contrast to this, in this paper, the adjoint approach followed by the DD of both the flow and adjoint equations (i.e. the other way round) is proved to be the most efficient way to compute the product of the Hessian matrix with any vector required by the truncated Newton algorithm, in which the Newton equations are solved iteratively by means of the conjugate gradient (CG) method. Using numerical experiments, it is demonstrated that just a few CG steps per Newton iteration are enough. Considering that the cost of solving either the adjoint or the DD equations is approximately equal to that of solving the flow equations, the cost per Newton iteration scales linearly with the (small) number of CG steps, rather than the (much higher, in large‐scale problems) number of design variables. By doing so, the curse of dimensionality is alleviated, as shown in a number of applications related to the inverse design of ducts or cascade airfoils for inviscid flows. Copyright © 2011 John Wiley & Sons, Ltd.     

涡波一体乘波飞行器宽速域气动优化设计研究

涡波一体宽速域乘波飞行器通过在低速引入涡效应,显著改善了传统乘波体在低速状态下的升阻特性,具有在未来宽速域空天飞行器总体气动设计当中得到广泛应用的巨大潜力.但是,该设计方法的研究尚不完善,特别是在基准流场建立过程中忽略了三维效应、低速效应、黏性效应以及头部/前缘的钝化效应,因此其高低速气动特性均有优化设计的空间.针对此问题,本文结合高保真RANS求解器、自由变形参数化方法、鲁棒的结构网格变形方法、离散伴随方法以及序列二次规划算法,发展了基于离散伴随的宽速域飞行器气动优化设计方法.基于上述方法,针对涡波一体乘波飞行器开展了兼顾低速与高超声速气动性能的三维整机气动优化设计研究,获得了宽速域优化构型并对其进行了流动机理分析.结果表明,相较于初始构型,宽速域优化构型可以将飞行器高超声速状态下升阻特性略微提升的同时,显著增强低速状态飞行器背风面的旋涡效应,进而使飞行器低速状态的升力和升阻比均提升10%以上,改善了涡波一体宽速域乘波飞行器的高低速气动性能.     

Adiabatic shock capturing in perfect gas hypersonic flows

This paper considers the streamline‐upwind Petrov/Galerkin (SUPG) method applied to the compressible Euler and Navier–Stokes equations in conservation‐variable form. The spatial discretization, including a modified approach for interpolating the inviscid flux terms in the SUPG finite element formulation, is briefly reviewed. Of particular interest is the behavior of the shock‐capturing operator, which is required to regularize the scheme in the presence of strong, shock‐induced gradients. A standard shock‐capturing operator that has been widely used in previous studies by several authors is presented and discussed. Specific modifications are then made to this standard operator that is designed to produce a more physically consistent discretization in the presence of strong shock waves. The actual implementation of the term in a finite‐dimensional approximation is also discussed. The behavior of the standard and modified scheme is then compared for several supersonic/hypersonic flows. The modified shock‐capturing operator is found to preserve enthalpy in the inviscid portion of the flowfield substantially better than the standard operator. Published in 2009 by John Wiley & Sons, Ltd.     

Euler solutions using flux‐based wave decomposition

The Euler equations are solved for non‐hydrostatic atmospheric flow problems in two dimensions using a high‐resolution Godunov‐type scheme. The Riemann problem is solved using a flux‐based wave decomposition suggested by LeVeque. This paper describes in detail, the design and implementation of the Riemann solver used for computing the Godunov fluxes. The methodology is then validated against benchmark cases for non‐hydrostatic atmospheric flows. Comparisons are made with solutions obtained from the National Center for Atmospheric Research's state‐of‐the‐art numerical model. The method shows promise in simulating non‐hydrostatic flows, which are characterized by steep gradients on the meso‐, micro‐ and urban‐scales. Copyright © 2006 John Wiley & Sons, Ltd.     

Comparison of adjoint‐based and feature‐based grid adaptation for functional outputs

Adjoint‐based and feature‐based grid adaptive strategies are compared for their robustness and effectiveness in improving the accuracy of functional outputs such as lift and drag coefficients. The output‐based adjoint approach strives to improve the adjoint error estimates that relate the local residual errors to the global error in an output function via adjoint variables as weight functions. A conservative adaptive indicator that takes into account the residual errors in both the primal (flow) and dual (adjoint) solutions is implemented for the adjoint approach. The physics‐based feature approach strives to identify and resolve significant features of the flow to improve functional accuracy. Adaptive indicators that represent expansions and compressions in the flow direction and gradients normal to the flow direction are implemented for the feature approach. The adaptive approaches are compared for functional outputs of three‐dimensional arbitrary Mach number flow simulations on mixed‐element unstructured meshes. Grid adaptation is performed with h‐refinement and results are presented for inviscid, laminar and turbulent flows. Copyright © 2006 John Wiley & Sons, Ltd.     

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.     

Optimal shape design for the time‐dependent Navier–Stokes flow

This paper is concerned with the problem of shape optimization of two‐dimensional flows governed by the time‐dependent Navier–Stokes equations. We derive the structures of shape gradients for time‐dependent cost functionals by using the state derivative and its associated adjoint state. Finally, we apply a gradient‐type algorithm to our problem, and numerical examples show that our theory is useful for practical purposes and the proposed algorithm is feasible in low Reynolds number flows. Copyright © 2007 John Wiley & Sons, Ltd.     

Adjoint‐based error estimation and mesh adaptation for hybridized discontinuous Galerkin methods

We present a robust and efficient target‐based mesh adaptation methodology, building on hybridized discontinuous Galerkin schemes for (nonlinear) convection–diffusion problems, including the compressible Euler and Navier–Stokes equations. The hybridization of finite element discretizations has the main advantage that the resulting set of algebraic equations has globally coupled degrees of freedom (DOFs) only on the skeleton of the computational mesh. Consequently, solving for these DOFs involves the solution of a potentially much smaller system. This not only reduces storage requirements but also allows for a faster solution with iterative solvers. The mesh adaptation is driven by an error estimate obtained via a discrete adjoint approach. Furthermore, the computed target functional can be corrected with this error estimate to obtain an even more accurate value. The aim of this paper is twofold: Firstly, to show the superiority of adjoint‐based mesh adaptation over uniform and residual‐based mesh refinement and secondly, to investigate the efficiency of the global error estimate. Copyright © 2014 John Wiley & Sons, Ltd.     

Robust design in aerodynamics using third‐order sensitivity analysis based on discrete adjoint. Application to quasi‐1D flows

In this paper, the second‐order second moment approach, coupled with an adjoint‐based steepest descent algorithm, for the solution of the so‐called robust design problem in aerodynamics is proposed. Because the objective function for the robust design problem comprises first‐order and second‐order sensitivity derivatives with respect to the environmental parameters, the application of a gradient‐based method , which requires the sensitivities of this function with respect to the design variables, calls for the computation of third‐order mixed derivatives. To compute these derivatives with the minimum CPU cost, a combination of the direct differentiation and the discrete adjoint variable method is proposed. This is presented for the first time in the relevant literature and is the most efficient among other possible schemes on condition that the design variables are much more than the environmental ones; this is definitely true in most engineering design problems. The proposed approach was used for the robust design of a duct, assuming a quasi‐1D flow model; the coordinates of the Bézier control points parameterizing the duct shape are used as design variables, whereas the outlet Mach number and the Darcy–Weisbach friction coefficient are used as environmental ones. The extension to 2D and 3D flow problems, after developing the corresponding direct differentiation and adjoint variable methods and software, is straightforward. Copyright © 2011 John Wiley & Sons, Ltd.     

A method to carry out shape optimization with a large number of design variables

A new method for shape optimization with relatively large number of design variables is proposed. It is well known that gradient‐based methods converge to a local optimum. As a result, utilization of a richer design space does not necessarily lead to a better design. This is demonstrated via the design of an airfoil for maximum lift for Re = 1000 and α = 4° flow. The airfoil is represented by fourth‐order non‐uniform rational B‐splines, and the control points are used as design variables. Starting with a NACA0012 airfoil, it is found that the optimal airfoil obtained with 13 control points has far superior aerodynamic performance than the ones obtained with 39 and 61 control points. For effective utilization of a richer design space, it is proposed that the number of design variables be increased gradually. The method is demonstrated by designing high lift airfoils for Re = 1000 and 1 × 10⁴. The objective function is the maximization of the time‐averaged lift coefficient for α = 4°. The optimization cycle with 27 control points is initiated with the optimal airfoil obtained with 13 control points. The process is continued with gradual increase in the number of design variables. Beyond a certain number of control points, the optimization leads to a spontaneous appearance of corrugations on the upper surface of the airfoil. The corrugations are responsible for the generation of small vortices that add to the suction on the upper surface of the airfoil and lead to enhanced lift. A stabilized finite element method is used to solve the unsteady flow and adjoint equations. Copyright © 2012 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.     

A stabilized finite element formulation for high‐speed inviscid compressible flows using finite calculus

This paper aims at the development of a new stabilization formulation based on the finite calculus (FIC) scheme for solving the Euler equations using the Galerkin FEM on unstructured triangular grids. The FIC method is based on expressing the balance of fluxes in a space–time domain of finite size. It is used to prevent the creation of instabilities typically present in numerical solutions due to the high convective terms and sharp gradients. Two stabilization terms, respectively called streamline term and transverse term, are added via the FIC formulation to the original conservative equations in the space–time domain. An explicit fourth‐order Runge–Kutta scheme is implemented to advance the solution in time. The presented numerical test examples for inviscid flows prove the ability of the proposed stabilization technique for providing appropriate solutions especially near shock waves. Although the derived methodology delivers precise results with a nearly coarse mesh, a mesh refinement technique is coupled to the solution process for obtaining a suitable mesh particularly in the high‐gradient zones. Copyright © 2014 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.     

Convergence acceleration of continuous adjoint solvers using a reduced‐order model

A novel acceleration technique using a reduced‐order model is presented to speed up convergence of continuous adjoint solvers. The acceleration is achieved by projecting to an improved solution within an iterative process solely using early solution results. This is achieved by forming basis vectors from early iteration adjoint solutions to perform model order reduction of the adjoint equations. The reduced‐order model of the adjoint equations is then substituted into the full‐order discretized governing equations to determine weighting coefficients for each basis vector. With these coefficients, a linear combination of the basis vectors is used to project to an improved solution. The method is applied to 3 inviscid quasi‐1D nozzle flow cases including fully subsonic flow, subsonic inlet to supersonic outlet flow, and transonic flow with a shock. Significant cost reductions are achieved for a single application as well as repeated applications of the convergence acceleration technique.     

Computational study of curved shock–vortex interaction

The interaction between a curved shock wave and a compressible vortex is numerically investigated. The investigation concentrates on the local deformation of the shock structure due to the shock–vortex interaction. The essentially non‐oscillatory (ENO) scheme is used to solve the unsteady two‐dimensional Euler equations. A curved shock wave is obtained by the diffraction of an initially planar shock wave around a right‐angled corner and then allowed to interact with a strong compressible vortex superimposed on the flow. The same vortex affects the shock wave differently depending on the placement of the vortex because of the varying strength of the shock wave. This effect could range from a non‐symmetric deformation of the shock wave to a local disruption in the shock structure depending on the strength of the shock wave in the interaction region. This process leading to a local disruption in the shock structure is analyzed in detail. It is shown that such a disruption in the shock structure can be predicted by simple one‐dimensional considerations. Copyright © 1999 John Wiley & Sons, Ltd.