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 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.     

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.     

FEM-based reconstruction of sound pressure field damped by partially absorbing boundary conditions

This paper presents an inverse formulation of the acoustic boundary value problem featuring arbitrary admittance boundary conditions. The problem is discretised by using finite elements to reconstruct the sound pressure field of a cavity based on a preferably small number of measurements. For that, a modal approach is investigated to handle the generally under-determined and ill-conditioned system of equations. The viability of the algorithm is tested to reconstruct sound pressure field inside a two-dimensional sedan passenger compartment.     

Domain versus boundary computation of flow sensitivities with the continuous adjoint method for aerodynamic shape optimization problems

This paper considers the computation of flow sensitivities that arise in the context of design optimization. The scheme is based on the solution of a continuous adjoint problem, for which two complementary, although analytically equivalent, approaches have been routinely used for some time now, yielding expressions for the sensitivities that contain, respectively, boundary and domain integrals. These concepts are clarified in a unified framework and their equivalence at the continuous level is demonstrated through appropriate algebraic manipulations. Equivalence at the discrete level is assessed through numerical testing for various aerodynamic shape‐optimization problems. Copyright © 2011 John Wiley & Sons, Ltd.     

End-to-end differentiable learning of turbulence models from indirect observations

The emerging push of the differentiable programming paradigm in scientific computing is conducive to training deep learning turbulence models using indirect observations. This paper demonstrates the viability of this approach and presents an end-to-end differentiable framework for training deep neural networks to learn eddy viscosity models from indirect observations derived from the velocity and pressure fields. The framework consists of a Reynolds-averaged Navier–Stokes(RANS) solver and a neuralnetwork-represented turbulence model, each accompanied by its derivative computations. For computing the sensitivities of the indirect observations to the Reynolds stress field, we use the continuous adjoint equations for the RANS equations, while the gradient of the neural network is obtained via its built-in automatic differentiation capability. We demonstrate the ability of this approach to learn the true underlying turbulence closure when one exists by training models using synthetic velocity data from linear and nonlinear closures. We also train a linear eddy viscosity model using synthetic velocity measurements from direct numerical simulations of the Navier–Stokes equations for which no true underlying linear closure exists. The trained deep-neural-network turbulence model showed predictive capability on similar flows.     

A continuous adjoint formulation for the computation of topological and surface sensitivities of ducted flows

Topology optimization of fluid dynamic systems is a comparatively young optimal design technique. Its central ingredient is the computation of topological sensitivity maps. Whereas, for finite element solvers, implementations of such sensitivity maps have been accomplished in the past, this study focuses on providing this functionality within a professional finite volume computational fluid dynamics solver. On the basis of a continuous adjoint formulation, we derive the adjoint equations and the boundary conditions for typical cost functions of ducted flows and present first results for two‐ and three‐dimensional geometries. Emphasis is placed on the versatility of our approach with respect to changes in the objective function. We further demonstrate that surface sensitivity maps can also be computed with the implemented functionality and establish their connection with topological sensitivities. Copyright © 2008 John Wiley & Sons, Ltd.     

On the use of the continuous adjoint method to compute nongeometric sensitivities

This work explores an alternative approach to computing sensitivity derivatives of functionals, with respect to a broader range of control parameters. It builds upon the complementary character of Riemann problems that describe the Euler flow and adjoint solutions. In a previous work, we have discussed a treatment of the adjoint boundary problem, which made use of such complementarity as a means to ensure well‐posedness. Here, we show that the very same adjoint solution that satisfies those boundary conditions also conveys information on other types of sensitivities. In essence, then, that formulation of the boundary problem can extend the range of applications of the adjoint method to a host of new possibilities.     

TOPOLOGY SYNTHESIS OF GEOMETRICALLY NONLINEAR COMPLIANT MECHANISMS USING MESHLESS METHODS

This paper presents a new method for topology optimization of geometrical nonlinear compliant mechanisms using the element-free Galerkin method （EFGM）. The EFGM is employed as an alternative scheme to numerically solve the state equations by fully taking advantage of its capability in dealing with large displacement problems. In the meshless method, the imposition of essential boundary conditions is also addressed. The popularly studied solid isotropic material with the penalization （SIMP） scheme is used to represent the nonlinear dependence between material properties and regularized discrete densities. The output displacement is regarded as the objective function and the adjoint method is applied to finding the sensitivity of the design functions. As a result, the optimization of compliant mechanisms is mathematically established as a nonlinear programming problem, to which the method of moving asymptotes （MMA） belonging to the sequential convex programming can be applied. The availability of the present method is finally demonstrated with several widely investigated numerical examples.     

Precise integration method for solving singular perturbation problems

This paper presents a precise method for solving singularly perturbed boundary-value problems with the boundary layer at one end. The method divides the interval evenly and gives a set of algebraic equations in a matrix form by the precise integration relationship of each segment. Substituting the boundary conditions into the algebraic equations, the coefficient matrix can be transformed to the block tridiagonal matrix. Considering the nature of the problem, an efficient reduction method is given for solving singular perturbation problems. Since the precise integration relationship introduces no discrete error in the discrete process, the present method has high precision. Numerical examples show the validity of the present method.     

Body-Fitted Adjoint Formulation of the Euler Equations

The shape optimization problem governed by the Euler equations is posed in a fixed reference plane. The boundary control is exerted by a parametric mapping from the physical plane to the reference fixed plane. The adjoint equations are derived in such fixed plane. By using this approach remeshing is unnecessary; furthermore, as in many practical applications the parametric mapping can be easily differentiated, the computation of mesh sensitivities is avoided.     

Analytical solution to one-dimensional consolidation in unsaturated soils

This paper presents an analytical solution of the one-dimensional consolidation in unsaturated soil with a finite thickness under vertical loading and confinements in the lateral directions. The boundary contains the top surface permeable to water and air and the bottom impermeable to water and air. The analytical solution is for Fredlund＇s one-dimensional consolidation equation in unsaturated soils. The transfer relationship between the state vectors at top surface and any depth is obtained by using the Laplace transform and Cayley-Hamilton mathematical methods to the governing equations of water and air, Darcy＇s law and Fick＇s law. Excess pore-air pressure, excess pore-water pressure and settlement in the Laplace-transformed domain are obtained by using the Laplace transform with the initial conditions and boundary conditions. By performing inverse Laplace transforms, the analytical solutions are obtained in the time domain. A typical example illustrates the consolidation characteristics of unsaturated soil from an- alytical results. Finally, comparisons between the analytical solutions and results of the finite difference method indicate that the analytical solution is correct.     

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.     

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.     

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.     

A local domain‐free discretization method to simulate three‐dimensional compressible inviscid flows

In this paper, the domain‐free discretization method (DFD) is extended to simulate the three‐dimensional compressible inviscid flows governed by Euler equations. The discretization strategy of DFD is that the discrete form of governing equations at an interior point may involve some points outside the solution domain. The functional values at the exterior‐dependent points are updated at each time step by extrapolation along the wall normal direction in conjunction with the wall boundary conditions and the simplified momentum equation in the vicinity of the wall. Spatial discretization is achieved with the help of the finite element Galerkin approximation. The concept of ‘osculating plane’ is adopted, with which the local DFD can be easily implemented for the three‐dimensional case. Geometry‐adaptive tetrahedral mesh is employed for three‐dimensional calculations. Finally, we validate the DFD method for three‐dimensional compressible inviscid flow simulations by computing transonic flows over the ONERA M6 wing. Comparison with the reference experimental data and numerical results on boundary‐conforming grid was displayed and the results show that the present DFD results compare very well with the reference data. Copyright © 2009 John Wiley & Sons, Ltd.     

An Introduction to the Adjoint Approach to Design

Optimal design methods involving the solution of an adjoint system of equations are an active area of research in computational fluid dynamics, particularly for aeronautical applications. This paper presents an introduction to the subject, emphasising the simplicity of the ideas when viewed in the context of linear algebra. Detailed discussions also include the extension to p.d.e.'s, the construction of the adjoint p.d.e. and its boundary conditions, and the physical significance of the adjoint solution. The paper concludes with examples of the use of adjoint methods for optimising the design of business jets. This revised version was published online in July 2006 with corrections to the Cover Date.     

Spectral Element Discretization of the Circular Driven Cavity. Part IV: The Navier-Stokes Equations

This paper deals with the spectral element discretization of the Navier-Stokes equations in a disk with discontinuous boundary data, which is known as the driven cavity problem. The numerical treatment does not involve any regularization of these data. Relying on a variational formulation in the primitive variables of velocity and pressure, we describe a discretization of these equations and derive error estimates in appropriate weighted Sobolev spaces. We propose an algorithm to solve the nonlinear discrete system and present numerical experiments to verify its efficiency.