An hp‐adaptive unstructured finite volume solver for compressible flows

The idea of hp‐adaptation, which has originally been developed for compact schemes (such as finite element methods), suggests an adaptation scheme using a mixture of mesh refinement and order enrichment based on the smoothness of the solution to obtain an accurate solution efficiently. In this paper, we develop an hp‐adaptation framework for unstructured finite volume methods using residual‐based and adjoint‐based error indicators. For the residual‐based error indicator, we use a higher‐order discrete operator to estimate the truncation error, whereas this estimate is weighted by the solution of the discrete adjoint problem for an output of interest to form the adaptation indicator for adjoint‐based adaptations. We perform our adaptation by local subdivision of cells with nonconforming interfaces allowed and local reconstruction of higher‐order polynomials for solution approximations. We present our results for two‐dimensional compressible flow problems including subsonic inviscid, transonic inviscid, and subsonic laminar flow around the NACA 0012 airfoil and also turbulent flow over a flat plate. Our numerical results suggest the efficiency and accuracy advantages of adjoint‐based hp‐adaptations over uniform refinement and also over residual‐based adaptation for flows with and without singularities.     

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.     

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.     

Nonlinear corrector for Reynolds-averaged Navier-Stokes equations

The scope of this paper is to present a nonlinear error estimation and correction for Navier-Stokes and Reynolds-averaged Navier-Stokes equations. This nonlinear corrector enables better solution or functional output predictions at fixed mesh complexity and can be considered in a mesh adaptation process. After solving the problem at hand, a corrected solution is obtained by solving again the problem with an added source term. This source term is deduced from the evaluation of the residual of the numerical solution interpolated on the h/2 mesh. To avoid the generation of the h/2 mesh (which is prohibitive for realistic applications), the residual at each vertex is computed by local refinement only in the neighborhood of the considered vertex. One of the main feature of this approach is that it automatically takes into account all the properties of the considered numerical method. The numerical examples point out that it successfully improves solution predictions and yields a sharp estimate of the numerical error. Moreover, we demonstrate the superiority of the nonlinear corrector with respect to linear corrector that can be found in the literature.     

On the use of anisotropic a posteriori error estimators for the adaptative solution of 3D inviscid compressible flows

This paper describes the use of an a posteriori error estimator to control anisotropic mesh adaptation for computing inviscid compressible flows. The a posteriori error estimator and the coupling strategy with an anisotropic remesher are first introduced. The mesh adaptation is controlled by a single‐parameter tolerance (TOL) in regions where the solution is regular, whereas a condition on the minimal element size h_min is enforced across solution discontinuities. This h_min condition is justified on the basis of an asymptotic analysis. The efficiency of the approach is tested with a supersonic flow over an aircraft. The evolution of a mesh adaptation/flow solution loop is shown, together with the influence of the parameters TOL and h_min. We verify numerically that the effect of varying h_min is concordant with the conclusions of the asymptotic analysis, giving hints on the selection of h_min with respect to TOL. Finally, we check that the results obtained with the a posteriori error estimator are at least as accurate as those obtained with anisotropic a priori error estimators. All the results presented can be obtained using a standard desktop computer, showing the efficiency of these adaptative methods. Copyright © 2008 John Wiley & Sons, Ltd.     

Error sensitivity to refinement: a criterion for optimal grid adaptation

Most indicators used for automatic grid refinement are suboptimal, in the sense that they do not really minimize the global solution error. This paper concerns with a new indicator, related to the sensitivity map of global stability problems, suitable for an optimal grid refinement that minimizes the global solution error. The new criterion is derived from the properties of the adjoint operator and provides a map of the sensitivity of the global error (or its estimate) to a local mesh refinement. Examples are presented for both a scalar partial differential equation and for the system of Navier–Stokes equations. In the last case, we also present a grid-adaptation algorithm based on the new estimator and on the \(FreeFem++\) software that improves the accuracy of the solution of almost two order of magnitude by redistributing the nodes of the initial computational mesh.     

Sequential feature‐based mesh movement and adjoint error‐based mesh refinement

Nowadays, aerodynamic computational modeling is carried out on a daily basis in an industrial setting. This is done with the aim of predicting the performance and flow characteristics of new components. However, limited resources in terms of time and hardware force the engineer to employ relatively coarse computational grids, thus achieving results with variable degree of inaccuracy. In this article, a novel combination of feature and adjoint‐based mesh adaptation methods is investigated and applied to typical three‐dimensional turbomachinery cases, such as compressor and fan blades. The proposed process starts by employing feature‐based mesh movement to improve the global flow solution and then adjoint refinement to tune the mesh for each quantity of interest. Comparison of this process with one utilizing only the adjoint refinement procedure shows significant benefits in terms of accuracy of the performance quantity.     

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.     

Versatile anisotropic mesh adaptation methodology applied to pure quantity of interest error estimator. Steady,laminar incompressible flow

We introduce a new flexible mesh adaptation approach to efficiently compute a quantity of interest by the finite element method. Efficiently, we mean that the method provides an evaluation of that quantity up to a predetermined accuracy at a lower computational cost than other classical methods. The central pillar of the method is our scalar error estimator based on sensitivities of the quantity of interest to the residuals. These sensitivities result from the computation of a continuous adjoint problem. The mesh adaptation strategy can drive anisotropic mesh adaptation from a general scalar error contribution of each element. The full potential of our error estimator is then reached. The proposed method is validated by evaluating the lift, the drag, and the hydraulic losses on a 2D benchmark case: the flow around a cylinder at a Reynolds number of 20.     

Collocated discrete least‐squares (CDLS) meshless method: Error estimate and adaptive refinement

Meshless methods are new approaches for solving partial differential equations. The main characteristic of all these methods is that they do not require the traditional mesh to construct a numerical formulation. They require node generation instead of mesh generation. In other words, there is no pre‐specified connectivity or relationships among the nodes. This characteristic make these methods powerful. For example, an adaptive process which requires high computational effort in mesh‐dependent methods can be very economically solved with meshless methods. In this paper, a posteriori error estimate and adaptive refinement strategy is developed in conjunction with the collocated discrete least‐squares (CDLS) meshless method. For this, an error estimate is first developed for a CDLS meshless method. The proposed error estimator is shown to be naturally related to the least‐squares functional, providing a suitable posterior measure of the error in the solution. A mesh moving strategy is then used to displace the nodal points such that the errors are evenly distributed in the solution domain. Efficiency and effectiveness of the proposed error estimator and adaptive refinement process are tested against two hyperbolic benchmark problems, one with shocked and the other with low gradient smooth solutions. These experiments show that the proposed adaptive process is capable of producing stable and accurate results for the difficult problems considered. Copyright © 2007 John Wiley & Sons, Ltd.     

Using adjoint error estimation techniques for elastohydrodynamic lubrication line contact problems

The use of an adjoint technique for goal‐based error estimation described by Hartit et al. (Int. J. Numer. Meth. Fluids 2005; 47 :1069–1074) is extended to the numerical solution of free boundary problems that arise in elastohydrodynamic lubrication (EHL). EHL systems are highly nonlinear and consist of a thin‐film approximation of the flow of a non‐Newtonian lubricant which separates two bodies that are forced together by an applied load, coupled with a linear elastic model for the deformation of the bodies. A finite difference discretization of the line contact flow problem is presented, along with the numerical evaluation of an exact solution for the elastic deformation, and a moving grid representation of the free boundary that models cavitation at the outflow in this one‐dimensional case. The application of a goal‐based error estimate for this problem is then described. This estimate relies on the solution of an adjoint problem; its effectiveness is demonstrated for the physically important goal of the total friction through the contact. Finally, the application of this error estimate to drive local mesh refinement is demonstrated. Copyright © 2010 John Wiley & Sons, Ltd.     

A posteriori error estimation and anisotropy detection with the dual‐weighted residual method

In this work we develop a new framework for a posteriori error estimation and detection of anisotropies based on the dual‐weighted residual (DWR) method by Becker and Rannacher. The common approach for anisotropic mesh adaptation is to analyze the Hessian of the solution. Eigenvalues and eigenvectors indicate dominant directions and optimal stretching of elements. However, this approach is firmly linked to energy norm error estimation. Here, we extend the DWR method to anisotropic finite elements allowing for the direct estimation of directional errors with regard to given output functionals. The resulting meshes reflect anisotropic properties of both the solution and the functional. For the optimal measurement of the directional errors, the coarse meshes need some alignment with the dominant anisotropies. Numerical examples will demonstrate the efficiency of this method on various three‐dimensional problems including a well‐known Navier–Stokes benchmark. Copyright © 2009 John Wiley & Sons, Ltd.     

An estimation of the sensitivity of numerical error norm using adjoint model

A posteriori estimation of the numerical error sensitivity to the local truncation error is addressed using adjoint model endowed with the information on the error field. The numerical error is estimated from the solution of the linear tangent model (LTM) or from a Richardson extrapolation. The local truncation error used in the LTM is obtained by the action of a high‐order finite‐difference stencil on the field computed by the main (low‐order accuracy) algorithm. Copyright © 2009 John Wiley & Sons, Ltd.     

Adaptive mesh refinement and load balancing based on multi-level block-structured Cartesian mesh

We developed a framework for a distributed-memory parallel computer that enables dynamic data management for adaptive mesh refinement and load balancing. We employed simple data structure of the building cube method (BCM) where a computational domain is divided into multi-level cubic domains and each cube has the same number of grid points inside, realising a multi-level block-structured Cartesian mesh. Solution adaptive mesh refinement, which works efficiently with the help of the dynamic load balancing, was implemented by dividing cubes based on mesh refinement criteria. The framework was investigated with the Laplace equation in terms of adaptive mesh refinement, load balancing and the parallel efficiency. It was then applied to the incompressible Navier–Stokes equations to simulate a turbulent flow around a sphere. We considered wall-adaptive cube refinement where a non-dimensional wall distance y⁺ near the sphere is used for a criterion of mesh refinement. The result showed the load imbalance due to y⁺ adaptive mesh refinement was corrected by the present approach. To utilise the BCM framework more effectively, we also tested a cube-wise algorithm switching where an explicit and implicit time integration schemes are switched depending on the local Courant-Friedrichs-Lewy (CFL) condition in each cube.     

不可压Navier-Stokes方程基于误差估算的网格自适应解法

首先导出了广义Stokes方程Petrov—Galerkin有限元数值解的当地事后误差估算公式;以非连续二阶鼓包（bump）函数空间为速度、压强误差的近似空间,该估算基于求解当地单元上的广义Stokes问题。然后,证明了误差估算值与精确误差之间的等价性。最后,将误差估算方法应用于Navier—Stokes环境,以进行不可压粘流计算中的网格自适应处理。数值实验中成功地捕获了多强度物理现象,验证了本文所发展的方法。     

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.     

一种新的并行自适应网格有限元算法

给出了一种新的适用于流体力学问题的并行自适应有限元算法。首先，基于初始稀网格上获得的事后误差估算值，应用反复谱对剖分方法对初网格进行划分，使各子域上总体误差近似相等，从而解决并行自适应计算中的负载平衡问题。然后在各处理器上独立地求解整体问题，并进行指定子域上的网格自适应处理。最后将各子域上的自适应网格组合成一个整体网格，应用基于粘接元技术的区域分裂法在该网格上获得最终解。文末给出了数值实验结果。     

一种基于径向基函数的两步法网格变形策略

基于径向基函数(RBF) 的网格变形方法是一种可靠的网格变形技术,对于任意拓扑的网格都能获得高质量的变形网格. 缩减控制点的RBF 网格变形方法可以大幅提高网格变形效率,但也存在变形后物面误差较大、边界层网格交错的问题. 在缩减控制点方法的基础上,提出了一种适合于带有边界层的黏性网格变形的方法,该方法从物面中选择两组控制点,利用其中一组控制点粗略计算变形后网格位置及变形误差,利用第二组控制点与变形误差插值得到更为精确的变形网格. 利用该方法完成带襟翼的NLR 7301 二维构型和带发动机短舱的DLR F6 翼身组合体的网格变形问题,结果表明该方法可以较大幅度降低变形网格的物面误差,并且有效避免边界层网格交错问题.      

Mesh adaptation framework for embedded boundary methods for computational fluid dynamics and fluid-structure interaction

Embedded Boundary Methods (EBMs) are often preferred for the solution of Fluid-Structure Interaction (FSI) problems because they are reliable for large structural motions/deformations and topological changes. For viscous flow problems, however, they do not track the boundary layers that form around embedded obstacles and therefore do not maintain them resolved. Hence, an Adaptive Mesh Refinement (AMR) framework for EBMs is proposed in this paper. It is based on computing the distance from an edge of the embedding computational fluid dynamics mesh to the nearest embedded discrete surface and on satisfying the y⁺ requirements. It is also equipped with a Hessian-based criterion for resolving flow features such as shocks, vortices, and wakes and with load balancing for achieving parallel efficiency. It performs mesh refinement using a parallel version of the newest vertex bisection method to maintain mesh conformity. Hence, while it is sufficiently comprehensive to support many discretization methods, it is particularly attractive for vertex-centered finite volume schemes where dual cells tend to complicate the mesh adaptation process. Using the EBM known as FIVER, this AMR framework is verified for several academic FSI problems. Its potential for realistic FSI applications is also demonstrated with the simulation of a challenging supersonic parachute inflation dynamics problem.     

Incorporating spatially variable bottom stress and Coriolis force into 2D,a posteriori,unstructured mesh generation for shallow water models

An enhanced version of our localized truncation error analysis with complex derivatives (LTEA?CD ) a posteriori approach to computing target element sizes for tidal, shallow water flow, LTEA+CD , is applied to the Western North Atlantic Tidal model domain. The LTEA + CD method utilizes localized truncation error estimates of the shallow water momentum equations and builds upon LTEA and LTEA?CD‐based techniques by including: (1) velocity fields from a nonlinear simulation with complete constituent forcing; (2) spatially variable bottom stress; and (3) Coriolis force. Use of complex derivatives in this case results in a simple truncation error expression, and the ability to compute localized truncation errors using difference equations that employ only seven to eight computational points. The compact difference molecules allow the computation of truncation error estimates and target element sizes throughout the domain, including along the boundary; this fact, along with inclusion of locally variable bottom stress and Coriolis force, constitute significant advancements beyond the capabilities of LTEA. The goal of LTEA + CD is to drive the truncation error to a more uniform, domain‐wide value by adjusting element sizes (we apply LTEA + CD by re‐meshing the entire domain, not by moving nodes). We find that LTEA + CD can produce a mesh that is comprised of fewer nodes and elements than an initial high‐resolution mesh while performing as well as the initial mesh when considering the resynthesized tidal signals (elevations). Copyright © 2008 John Wiley & Sons, Ltd.