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).     

Diffusion-based coarse graining in hybrid continuum–discrete solvers: Theoretical formulation and a priori tests

Coarse graining is an important ingredient in many multi-scale continuum–discrete solvers such as CFD–DEM (computational fluid dynamics–discrete element method) solvers for dense particle-laden flows. Although CFD–DEM solvers have become a mature technique that is widely used in multiphase flow research and industrial flow simulations, a flexible and easy-to-implement coarse graining algorithm that can work with CFD solvers of arbitrary meshes is still lacking. In this work, we proposed a new coarse graining algorithm for continuum–discrete solvers for dense particle-laden flows based on solving a transient diffusion equation. Via theoretical analysis we demonstrated that the proposed method is equivalent to the statistical kernel method with a Gaussian kernel, but the current method is much more straightforward to implement in CFD–DEM solvers. A priori numerical tests were performed to obtain the solid volume fraction fields based on given particle distributions, the results obtained by using the proposed algorithm were compared with those from other coarse graining methods in the literature (e.g., the particle centroid method, the divided particle volume method, and the two-grid formulation). The numerical tests demonstrated that the proposed coarse graining procedure based on solving diffusion equations is theoretically sound, easy to implement and parallelize in general CFD solvers, and has improved mesh-convergence characteristics compared with existing coarse graining methods. The diffusion-based coarse graining method has been implemented into a CFD–DEM solver, the results of which are presented in a separate work.     

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.     

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.     

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.     

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.     

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.     

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.     

The implementation of an aeronautical CFD flow code onto distributed memory parallel systems

The parallelization of an industrially important in‐house computational fluid dynamics (CFD) code for calculating the airflow over complex aircraft configurations using the Euler or Navier–Stokes equations is presented. The code discussed is the flow solver module of the SAUNA CFD suite. This suite uses a novel grid system that may include block‐structured hexahedral or pyramidal grids, unstructured tetrahedral grids or a hybrid combination of both. To assist in the rapid convergence to a solution, a number of convergence acceleration techniques are employed including implicit residual smoothing and a multigrid full approximation storage scheme (FAS). Key features of the parallelization approach are the use of domain decomposition and encapsulated message passing to enable the execution in parallel using a single programme multiple data (SPMD) paradigm. In the case where a hybrid grid is used, a unified grid partitioning scheme is employed to define the decomposition of the mesh. The parallel code has been tested using both structured and hybrid grids on a number of different distributed memory parallel systems and is now routinely used to perform industrial scale aeronautical simulations. Copyright © 2000 John Wiley & Sons, Ltd.     

Sensitivity of shear rate in artificial grafts using automatic differentiation

An accurate numerical simulation of blood requires the solution of incompressible Navier–Stokes equations coupled with specific constitutive models. We consider a generalized Newtonian fluid model in which viscosity depends on shear rate, accounting for the shear‐thinning behavior of blood. Previous work on the design of an artificial graft indicated that there is an influence of the fluid model on the solution of the partial differential equation‐constrained shape optimization problem. Therefore, we carry out a sensitivity analysis of the actual implementation of the flow solver using automatic differentiation (AD). We compare the sensitivities of shear rate with respect to viscosity for different configurations and validate the truncation‐error‐free sensitivities obtained from AD with those based on divided differencing and, if available, with analytic derivatives. Copyright © 2009 John Wiley & Sons, Ltd.     

Charge-conserving grid based methods for the Vlasov–Maxwell equations

In this article, we introduce numerical schemes for the Vlasov–Maxwell equations relying on different kinds of grid-based Vlasov solvers, as opposite to PIC schemes, which enforce a discrete continuity equation. The idea underlying these schemes relies on a time-splitting scheme between configuration space and velocity space for the Vlasov equation and on the computation of the discrete current in a form that is compatible with the discrete Maxwell solver.     

间断Galerkin有限元和有限体积混合计算方法研究

通过局部坐标变换而建立的非正交单元间断Galerkin(DG)有限元计算方法计算精度高， 但计算量大、内存需求大；而非结构网格有限体积方法虽然准确计算热流的问题目 前还没有完全解决，但其具有计算速度快和内存需求小的优点. 该研究是将有 限元和有限体积方法的优点结合，发展有限元和有限体积的混合方法. 在物面 附近黏性占主导作用的区域内采用有限元方法进行计算，在远离物面的区域采用快速的有限 体积方法进行计算，在有限元和有限体积方法结合处要保证通量守恒. 通过算例说明有 限元和有限体积混合方法既能保证黏性区域的热流计算精度和流场结构的分辨率，又能 降低内存需求和提高计算效率，使有限元方法应用于复杂外形(实际工程问题)的计 算成为可能.     

Edge‐based finite element method for shallow water equations

This paper describes an edge‐based implementation of the generalized residual minimum (GMRES) solver for the fully coupled solution of non‐linear systems arising from finite element discretization of shallow water equations (SWEs). The gain in terms of memory, floating point operations and indirect addressing is quantified for semi‐discrete and space–time analyses. Stabilized formulations, including Petrov–Galerkin models and discontinuity‐capturing operators, are also discussed for both types of discretization. Results illustrating the quality of the stabilized solutions and the advantages of using the edge‐based approach are presented at the end of the paper. Copyright © 2001 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.     

The residual formulation of the method of manufactured solutions for computationally efficient solution verification

The method of manufactured solutions (MMS) is a solution verification methodology for determining whether the implementation of a discretization method is achieving its theoretical order of accuracy. This methodology combines the advantages of grid refinement studies and comparison with exact solution, by modifying the governing equations solved within a code by adding a source term to drive the solution towards a predetermined analytic function. By solving the modified equations on a sequence of grids and comparing the differences between the converged solution and manufactured solution, the order of accuracy of the implementation can be investigated. However, in its current form, converged solutions on a sequence of grids are required which can be quite costly and difficult to obtain. In this paper, by comparing the MMS to the method for determining the theoretical order of accuracy of a discretization method, the residual formulation of the MMS is developed. This new formulation only requires that the residual of the discretized governing equations to be calculated and not the solution to the discretized equations, thus avoiding the computational cost and difficulties inherent in obtaining converged solutions. Furthermore, since only the residuals are interrogated, individual components of the flow solver can be tested, in isolation, allowing the MMS to be used more effectively in locating errors within the code. This new approach is demonstrated to yield the same order of accuracy as the original MMS using three different cases—one-dimensional porous media equation, one-dimensional St Venant equations and two-dimensional unstructured Navier–Stokes simulations.     

黏性阻尼系统时域响应灵敏度及其一致性研究

时域响应灵敏度分析是时域梯度优化算法的基础. 灵敏度分析通常只涉及对设计变量的微分运算, 但时域响应灵敏度问题还涉及时间域的离散化. 因此, 微分和离散的先后顺序可能对时域响应灵敏度结果产生影响. 针对黏性阻尼系统时域响应灵敏度求解问题, 基于改进精细积分方法, 分别推导了先微分后离散和先离散后微分两种伴随变量方法. 其中, 先微分后离散法首先对由伴随变量构造的增广函数微分, 再利用改进精细积分方法在各离散时间点求解时域响应灵敏度; 而先离散后微分方法则首先在各离散时间点引入残值方程构造增广函数, 再对各增广函数进行微分以求解时域响应灵敏度. 通过数值算例验证了所提出方法的有效性和准确性, 并与传统基于Newmark的方法进行比较. 结果表明, 积分方案、数值离散误差以及离散和微分的先后顺序共同影响灵敏度的一致性误差. 综合考虑精度、效率和一致性问题, 基于改进精细积分的先微分后离散伴随变量法表现更优, 最适合应用于黏性阻尼系统时域梯度优化算法.      

Large eddy simulation of natural convection along a vertical isothermal surface

Large eddy simulations of natural convection along a vertical isothermal surface have been carried out using a parallel CFD code SMAFS (Smoke Movement And Flame Spread) developed by the first author to study the dynamics of the natural convection flow and the associated convective heat transfer, with sub-grid scale turbulence modeled using the Smagorinsky model. In the computation, the filtered governing equations are discretized using finite volume method, with the variables at the cell faces in the finite volume discrete equations approximated by a second order bounded QUICK scheme and the diffusion term computed based on central difference scheme. The computation was time marched explicitly, with momentum equations solved based on a second order fractional-step Adams–Bashford scheme and enthalpy computed using a second order Runge–Kutta scheme. The Poisson equation for pressure from the continuity equation was solved using a multi-grid solver. The results including the temperature and velocity profiles of the boundary layer and the local heat transfer rate are analyzed. Comparison is made with experimental data and good agreement is found.     

Anisotropic mesh adaptation: towards user‐independent,mesh‐independent and solver‐independent CFD. Part I: general principles

The present paper is the lead article in a three‐part series on anisotropic mesh adaptation and its applications to structured and unstructured meshes. A flexible approach is proposed and tested on two‐dimensional, inviscid and viscous, finite volume and finite element flow solvers, over a wide range of speeds. The directional properties of an interpolation‐based error estimate, extracted from the Hessian of the solution, are used to control the size and orientation of mesh edges. The approach is encapsulated into an edge‐based anisotropic mesh optimization methodology (MOM), which uses a judicious sequence of four local operations: refinement, coarsening, edge swapping and point movement, to equi‐distribute the error estimate along all edges, without any recourse to remeshing. The mesh adaptation convergence of the MOM loop is carefully studied for a wide variety of test cases. The mesh optimization generic coupling of MOM with finite volume and finite element flow solvers is shown to yield the same final mesh no matter what the starting point is. It is also shown that on such optimized meshes, the need for computational fluid dynamics (CFD) stabilization artifices, such as upwinding or artificial viscosity, are drastically reduced, if not altogether eliminated, in most well‐posed formulations. These two conclusions can be considered significant steps towards mesh‐independent and solver‐independent CFD. The structure of the three‐part series is thus, 1, general principles; 2, methodology and applications to structured and unstructured grids; 3, applications to three‐dimensional flows. Copyright © 2000 John Wiley & Sons, Ltd.     

An Anisotropic A posteriori Error Estimator for CFD

In this article, a robust anisotropic adaptive algorithm is presented, to solve compressible-flow equations using a stabilized CFD solver and automatic mesh generators. The association includes a mesh generator, a flow solver, and an a posteriori error-estimator code. The estimator was selected among several choices available (Almeida et al. (2000). Comput. Methods Appl. Mech. Engng , 182 , 379-400; Borges et al. (1998). "Computational mechanics: new trends and applications". Proceedings of the 4th World Congress on Computational Mechanics , Bs.As., Argentina) giving a powerful computational tool. The main aim is to capture solution discontinuities, in this case, shocks, using the least amount of computational resources, i.e. elements, compatible with a solution of good quality. This leads to high aspect-ratio elements (stretching). To achieve this, a directional error estimator was specifically selected. The numerical results show good behavior of the error estimator, resulting in strongly-adapted meshes in few steps, typically three or four iterations, enough to capture shocks using a moderate and well-distributed amount of elements.