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 geometry‐based level set method for curvilinear overset grids with application to ship hydrodynamics

In a previous work (Int. J. Numer. Meth. Fluids 2007; 55 :867–897), we presented a two‐phase level set method to simulate air/water turbulent flows using curvilinear body‐fitted grids for ship hydrodynamics problems. This two‐phase level set method explicitly enforces jump conditions across the interface, thus resulting in a fully coupled representation of the air/water flow. Though the method works well with multiblock curvilinear grids, severe robustness problems were found when attempting to use it with overset grids. The problem was tracked to small unphysical level set discontinuities across the overset grids with large differences in curvature. Though negligible for single‐phase approaches, the problem magnifies with large density differences between the phases, causing computation failures. In this paper, we present a geometry‐based level set method for curvilinear overset grids that overcomes these difficulties. The level set transport and reinitialization equations are not discretized along grid coordinates, but along the upwind streamline and level set gradient directions, respectively. The method is essentially an unstructured approach that is transparent to the differences between overset grids, but still the discretization is under the framework of a finite differences approach. As a result, significant improvements in robustness and to a less extent in accuracy are achieved for the level set function interpolation between overset grids, especially with big differences in grid curvature. Example tests are shown for the case of bow breaking waves around the surface combatant model David Taylor Model Basin (DTMB) 5415 and for the steady‐state ONR Tumblehome DTMB 5613 with superstructure. In the first case, the results are compared against experimental data available and in the second against results of a semi‐coupled method. Copyright © 2011 John Wiley & Sons, Ltd.     

Towards consistent hybrid overset mesh methods for rotorcraft CFD

The overset mesh method chimera is popular within the rotorcraft research community, because the use of multiple, non‐matching grids make the CFD simulations of bodies in relative motion much simpler. Consequently, the relative motion between the helicopter blades and fuselage can be accurately accounted for. In this paper, the method for treating overset grids within CFD codes is presented. It is compatible with multi‐block, structured‐grid solvers. The proposed method is based on hierarchy of overset, non‐matching grids, whose cells are automatically identified as computational or non‐computational and localised with respect to all grids they overlap with. The efficiency of the method relies on the hierarchical, multi‐step approach, for the overset mesh localisation and the use of a tree search. Because of the high efficiency of the algorithm, the search for overlapping cells can be carried out on‐the‐fly, during time‐marching of the unsteady, implicit CFD solver. In addition, the algorithm is suitable for parallel execution. The method has been demonstrated for several flows, ranging from simple aerofoils to rotor‐body interaction. The paper presents and demonstrates the method and shows that it has a low CPU overhead. It also highlights the limitations of the method and suggests remedies for improvement. Copyright © 2013 John Wiley & Sons, Ltd.     

A pressure–velocity coupling approach for high void fraction free surface bubbly flows in overset curvilinear grids

A methodology for improved robustness in the simulation of high void fraction free surface polydisperse bubbly flows in curvilinear overset grids is presented. The method is fully two‐way coupled in the sense that the bubbly field affects the continuous fluid and vice versa. A hybrid projection approach is used in which staggered contravariant velocities at cell faces are computed for transport and pressure–velocity coupling while the momentum equation is solved on a collocated grid arrangement. Conservation of mass is formulated such that a strong coupling between void fraction, pressure, and velocity is achieved within a partitioned approach, solving each field separately. A pressure–velocity projection solver is iterated together with a predictor stage for the void fraction to achieve a robust coupling. The implementation is described for general curvilinear grids detailing particulars in the neighborhood to overset interfaces or a free surface. A balanced forced method to avoid the generation of spurious currents is extended for curvilinear grids. The overall methodology allows simulation of high void fraction flows and is stable even when strong packing forces accounting for bubble collisions are included. Convergence and stability in one‐dimensional (1D) and two‐dimensional (2D) configurations is evaluated. Finally, a full‐scale simulation of the bubbly flow around a flat‐bottom boat is performed demonstrating the applicability of the methodology to complex problems of engineering interest. Copyright © 2015 John Wiley & Sons, Ltd.     

A new coupled level set and volume-of-fluid method to capture free surface on an overset grid system

We present a new coupled level set and volume-of-fluid (CLSVOF) method for free surface flow simulations on an overset grid system. The coupled method takes advantages of the strengths of the level set (LS) method and the volume-of-fluid (VOF) method, and is superior to either single method. The novelty of the present method lies in that we develop the methodology for an overset grid system of embedding, overlapping and moving structured grids. The new methodology accurately captures interface and greatly preserves mass on an overset grid system by demonstrating the 3D sphere advection test. The method is coupled to a well validated Reynolds-Averaged Navier–Stokes incompressible flow solver. The method is validated with the dam-breaking flow interacting with a 3D obstacle (square structure/circular cylinder) by comparing the numerical results with available experimental and numerical studies. The water impact of a sphere case is further performed to demonstrate the capabilities of the new method on a complicated moving overset grid system.     

Computations of two passing‐by high‐speed trains by a relaxation overset‐grid algorithm

This paper presents a relaxation algorithm, which is based on the overset grid technology, an unsteady three‐dimensional Navier–Stokes flow solver, and an inner‐ and outer‐relaxation method, for simulation of the unsteady flows of moving high‐speed trains. The flow solutions on the overlapped grids can be accurately updated by introducing a grid tracking technique and the inner‐ and outer‐relaxation method. To evaluate the capability and solution accuracy of the present algorithm, the computational static pressure distribution of a single stationary TGV high‐speed train inside a long tunnel is investigated numerically, and is compared with the experimental data from low‐speed wind tunnel test. Further, the unsteady flows of two TGV high‐speed trains passing by each other inside a long tunnel and at the tunnel entrance are simulated. A series of time histories of pressure distributions and aerodynamic loads acting on the train and tunnel surfaces are depicted for detailed discussions. Copyright © 2004 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。     

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     

Continuous and discrete adjoint approaches for aerodynamic shape optimization with low Mach number preconditioning

Discrete and continuous adjoint approaches for use in aerodynamic shape optimization problems at all flow speeds are developed and assessed. They are based on the Navier–Stokes equations with low Mach number preconditioning. By alleviating the large disparity between acoustic waves and fluid speeds, the preconditioned flow and adjoint equations are numerically solved with affordable CPU cost, even at the so‐called incompressible flow conditions. Either by employing the adjoint to the preconditioned flow equations or by preconditioning the adjoint to the ‘standard’ flow equations (under certain conditions the two formulations become equivalent, as proved in this paper), efficient optimization methods with reasonable cost per optimization cycle, even at very low Mach numbers, are derived. During the mathematical development, a couple of assumptions are made which are proved to be harmless to the accuracy in the computed gradients and the effectiveness of the optimization method. The proposed approaches are validated in inviscid and viscous flows in external aerodynamics and turbomachinery flows at various Mach numbers. Copyright © 2007 John Wiley & Sons, Ltd.     

THE PROGRESS OF THE OVERLAPPING GRID TECHNIQUES

从网格装配和插值计算两个主要方面对现有的重叠网格方法进行了综述。首先,从挖洞方法和建立嵌入网格关系环节的寻点技术出发归纳和介绍了网格装配方法;其次,介绍了数值迭代过程中的插值计算方法,并特别讨论了插值守恒性以及插值计算精度等问题;另外,对重叠网格方法的并行计算和应用成果也作了介绍;最后,通过总结认为重叠网格方法在改进网格装配方法、改善插值和并行计算效率等方面仍需进一步研究。     

A highly automated parallel Chimera method for overset grids based on the implicit hole cutting technique

A parallel automated Chimera overset grid algorithm for complex configurations is presented in this paper. The algorithm is based on the implicit hole cutting (IHC) algorithm presented by Lee and Baeder, which is enhanced by several methodologies to eliminate shortcomings in the original method. The resulting overall algorithm is more general than the IHC algorithm, while preserving the high degree of automation of the original method. The approach is well suited for handling steady flow problems. The effectiveness of the presented method is demonstrated with several aerodynamic test cases. Copyright © 2010 John Wiley & Sons, Ltd.     

Reducing memory requirements of unsteady adjoint by synergistically using check-pointing and compression

The unsteady adjoint method used in gradient-based optimization in 2D and, particularly, 3D industrial problems modeled by unsteady PDEs may have significant storage requirements and/or computational cost. The reason for this is that the backward in time integration of the adjoint equations requires the previously computed instantaneous flow fields to be available at each time-step. This article proposes remedies to this problem, by extending/upgrading relevant techniques proposed by the group of authors as well as other researchers. Their applicability is wide, even if these remedies are herein demonstrated in shape optimization problems in unsteady fluid mechanics. Check-pointing is in widespread use as it reduces the memory footprint and CPU cost of the optimization with a controllable computational overhead. Alternatively, flow field time-series can be stored in a lossless or lossly compressed form. The novelty of this article is the development of a Compressed Coarse-grained Check-Pointing strategy for second-order accurate schemes in time, by optimally combining check-pointing and lossy compression. The latter includes (a) the incremental Proper Generalized Decomposition (iPGD) algorithm and (b) a hybridization of the iPGD with the ZFP and Zlib algorithms. This is implemented within OpenFOAM, which is used to solve the flow and adjoint equations and conduct the optimization, and assessed in 2D/3D aerodynamic shape optimization problems on unstructured grids. Effectiveness in data reduction, computational cost, and reconstruction accuracy are compared, vis-à-vis also to the “standard” binomial check-pointing technique after adjusting it to second-order accurate schemes in time.     

Recent Development on the Conservation Property of Chimera

An estimate on the conservation error due to the non-conservative data interpolation scheme for overset grids is given in this paper. It is shown that the conservation error is a first-order term if second-order conservative schemes are employed for the Chimera grids and if discontinuities are located away from overlapped grid interfaces. Therefore in the limit of global grid refinement, valid numerical solutions should be obtained with a data interpolation scheme. In one demonstration case the conservation error in the original Chimera scheme was shown to affect flow even without discontinuities on coarse to medium grids. The conservative Chimera scheme was shown to give significantly better solutions than the original Chimera scheme on these grids with other factors being the same.     

Backward uncertainty propagation in shape optimization

We aim at quantifying the impact of state uncertainties in shape optimization. This provides confidence bounds for the optimal solution. The approach is presented for inverse designs where the target is assumed uncertain. No sampling of a large dimensional space is necessary, and the approach uses what is already available in a deterministic gradient‐based inversion algorithm. Our proposal is based on the introduction of directional quantile‐based extreme scenarios knowing the probability density function of the target data. We use these scenarios to define a matrix having the structure of the covariance matrix of the optimization parameters. We compare this construction to another one using the gradient of the functional by an adjoint method. The paper goes beyond inverse design and shows how to apply the method to general optimization problems. The ingredients of the paper are illustrated on a model problem with the Burgers equation and on the optimization of the shape of an aircraft. Overall, the computational complexity is comparable with the deterministic case. Copyright © 2015 John Wiley & Sons, Ltd.     

A combined level set/ghost cell immersed boundary representation for floating body simulations

A six degrees of freedom (6DOF) algorithm is implemented in the open‐source CFD code REEF3D. The model solves the incompressible Navier–Stokes equations. Complex free surface dynamics are modeled with the level set method based on a two‐phase flow approach. The convection terms of the velocities and the level set method are treated with a high‐order weighted essentially non‐oscillatory discretization scheme. Together with the level set method for the free surface capturing, this algorithm can model the movement of rigid floating bodies and their interaction with the fluid. The 6DOF algorithm is implemented on a fixed grid. The solid‐fluid interface is represented with a combination of the level set method and ghost cell immersed boundary method. As a result, re‐meshing or overset grids are not necessary. The capability, accuracy, and numerical stability of the new algorithm is shown through benchmark applications for the fluid‐body interaction problem. Copyright © 2016 John Wiley & Sons, Ltd.     

An unstructured/structured multi‐layer hybrid grid method and its application

A multi‐layer hybrid grid method is constructed to simulate complex flow field around 2‐D and 3‐D configuration. The method combines Cartesian grids with structured grids and triangular meshes to provide great flexibility in discretizing a domain. We generate the body‐fitted structured grids near the wall surface and the Cartesian grids for the far field. In addition, we regard the triangular meshes as an adhesive to link each grid part. Coupled with a tree data structure, the Cartesian grid is generated automatically through a cell‐cutting algorithm. The grid merging methodology is discussed, which can smooth hybrid grids and improve the quality of the grids. A cell‐centred finite volume flow solver has been developed in combination with a dual‐time stepping scheme. The flow solver supports arbitrary control volume cells. Both inviscid and viscous flows are computed by solving the Euler and Navier–Stokes equations. The above methods and algorithms have been validated on some test cases. Computed results are presented and compared with experimental data. Copyright © 2006 John Wiley & Sons, Ltd.     

Parallel computation of viscous incompressible flows using Godunov‐projection method on overlapping grids

The Godunov‐projection method is implemented on a system of overlapping structured grids for solving the time‐dependent incompressible Navier–Stokes equations. This projection method uses a second‐order fractional step scheme in which the momentum equation is solved to obtain the intermediate velocity field which is then projected on to the space of divergence‐free vector fields. The Godunov procedure is applied to estimate the non‐linear convective term in order to provide a robust discretization of this terms at high Reynolds number. In order to obtain the pressure field, a separate procedure is applied in this modified Godunov‐projection method, where the pressure Poisson equation is solved. Overlapping grids are used to discretize the flow domain, as they offer the flexibility of simplifying the grid generation around complex geometrical domains. This combination of projection method and overlapping grid is also parallelized and reasonable parallel efficiency is achieved. Numerical results are presented to demonstrate the performance of this combination of the Godunov‐projection method and the overlapping grid. Copyright © 2002 John Wiley & Sons, Ltd.     

A fast and reliable overset unstructured grids approach

A cell-centred overset unstructured grids approach is developed.In this approach,the intergrid boundary is initially established based on the wall distance from the cell centre,and is then optimized.To accelerate the intergrid-boundary definition much more,a neighbor-toneighbor donor search algorithm based on advancing-front method is modified with the help of minimum cuboid boxes.To simplify the communications between different grid cell types and to obtain second-order spatial accuracy,a new interpolation method is constructed based on linear reconstruction,which employs only one layer of fringe cells along the intergrid boundary.For unsteady flows with relative motion,the intergrid boundary can be redefined fast and automatically.Several numerical results show that the present dynamic overset unstructured grids approach is accurate and reliable.     

The shallow flow equations solved on adaptive quadtree grids

This paper describes an adaptive quadtree grid‐based solver of the depth‐averaged shallow water equations. The model is designed to approximate flows in complicated large‐scale shallow domains while focusing on important smaller‐scale localized flow features. Quadtree grids are created automatically by recursive subdivision of a rectangle about discretized boundary, bathymetric or flow‐related seeding points. It can be fitted in a fractal‐like sense by local grid refinement to any boundary, however distorted, provided absolute convergence to the boundary is not required and a low level of stepped boundary can be tolerated. Grid information is stored as a tree data structure, with a novel indexing system used to link information on the quadtree to a finite volume discretization of the governing equations. As the flow field develops, the grids may be adapted using a parameter based on vorticity and grid cell size. The numerical model is validated using standard benchmark tests, including seiches, Coriolis‐induced set‐up, jet‐forced flow in a circular reservoir, and wetting and drying. Wind‐induced flow in the Nichupté Lagoon, México, provides an illustrative example of an application to flow in extremely complicated multi‐connected regions. Copyright © 2001 John Wiley & Sons, Ltd.     

Comparison of derivative free Newton‐based and evolutionary methods for shape optimization of flow problems

The object of this study is to investigate two derivative free optimization techniques, i.e. Newton‐based method and an evolutionary method for shape optimization of flow geometry problems. The approaches are compared quantitatively with respect to efficiency and quality by using the minimization of the pressure drop of a pipe conjunction which can be considered as a representative test case for a practical three‐dimensional flow configuration. The comparison is performed by using CONDOR representing derivative free Newton‐based techniques and SIMPLIFIED NSGA‐II as the representative of evolutionary methods (EM). For the shape variation the computational grid employed by the flow solver is deformed. To do this, the displacement fields are scaled by design variables and added to the initial grid configuration. The displacement vectors are calculated once before the optimization procedure by means of a free form deformation (FFD) technique. The simulation tool employed is a parallel multi‐grid flow solver, which uses a fully conservative finite‐volume method for the solution of the incompressible Navier–Stokes equations on a non‐staggered, cell‐centred grid arrangement. For the coupling of pressure and velocity a pressure‐correction approach of SIMPLE type is used. The possibility of parallel computing and a multi‐grid technique allow for a high numerical efficiency. Copyright © 2006 John Wiley & Sons, Ltd.