A lattice Boltzmann‐BGK algorithm for a diffusion equation with Robin boundary condition—application to NMR relaxation

We present a lattice Boltzmann‐BGK (LBGK) algorithm for a diffusion equation together with a Robin boundary condition, which we apply in the case of nuclear magnetic resonance relaxation. The boundary condition we employ is independent of the direction of the wall. This makes the algorithm very suitable for complicated geometries, such as porous media. We discuss the effect of lattice topology by using, respectively, an eight‐speed and a four‐speed lattice. The numerical algorithm is compared with analytical results for a square and an equilateral triangle. The eight‐speed lattice performs well in both cases. The four‐speed lattice performs well for the square, but fails in the case of an equilateral triangle. Comparison with a random walk algorithm is also included. The LBGK algorithm presented here can also be used for a convective diffusion problem if the speed of the fluid can be neglected close to the boundary. Copyright © 2008 John Wiley & Sons, Ltd.     

An approach of dynamic mesh adaptation for simulating 3‐dimensional unsteady moving‐immersed‐boundary flows

In this paper, we present an approach of dynamic mesh adaptation for simulating complex 3‐dimensional incompressible moving‐boundary flows by immersed boundary methods. Tetrahedral meshes are adapted by a hierarchical refining/coarsening algorithm. Regular refinement is accomplished by dividing 1 tetrahedron into 8 subcells, and irregular refinement is only for eliminating the hanging points. Merging the 8 subcells obtained by regular refinement, the mesh is coarsened. With hierarchical refining/coarsening, mesh adaptivity can be achieved by adjusting the mesh only 1 time for each adaptation period. The level difference between 2 neighboring cells never exceeds 1, and the geometrical quality of mesh does not degrade as the level of adaptive mesh increases. A predictor‐corrector scheme is introduced to eliminate the phase lag between adapted mesh and unsteady solution. The error caused by each solution transferring from the old mesh to the new adapted one is small because most of the nodes on the 2 meshes are coincident. An immersed boundary method named local domain‐free discretization is employed to solve the flow equations. Several numerical experiments have been conducted for 3‐dimensional incompressible moving‐boundary flows. By using the present approach, the number of mesh nodes is reduced greatly while the accuracy of solution can be preserved.     

A method for computing curved meshes via the linear elasticity analogy,application to fluid dynamics problems

We propose and analyze an algorithm for the robust construction of curved meshes in two and three dimensions. The meshes are made of curved simplexes. The algorithm starts from a mesh made of straight simplexes, and using a linear elasticity analogy applied on well‐chosen data, one can generate a curved mesh. Note that if the initial mesh has a boundary layer, this method allows to conserve it on the final mesh. This algorithm is used on several airfoils in two and three dimensions, including a turbulent M6 wing. Copyright © 2014 John Wiley & Sons, Ltd.     

Algebraic multigrid and incompressible fluid flow

This paper is concerned with the development of algebraic multigrid (AMG) solution methods for the coupled vector–scalar fields of incompressible fluid flow. It addresses in particular the problems of unstable smoothing and of maintaining good vector–scalar coupling in the AMG coarse‐grid approximations. Two different approaches have been adopted. The first is a direct approach based on a second‐order discrete‐difference formulation in primitive variables. Here smoothing is stabilized using a minimum residual control harness and velocity–pressure coupling is maintained by employing a special interpolation during the construction of the inter‐grid transfer operators. The second is an indirect approach that avoids the coupling problem altogether by using a fourth‐order discrete‐difference formulation in a single scalar‐field variable, primitive variables being recovered in post‐processing steps. In both approaches the discrete‐difference equations are for the steady‐state limit (infinite time step) with a fully implicit treatment of advection based on central differencing using uniform and non‐uniform unstructured meshes. They are solved by Picard iteration, the AMG solvers being used repeatedly for each linear approximation. Both classical AMG (C‐AMG) and smoothed‐aggregation AMG (SA‐AMG) are used. In the direct approach, the SA‐AMG solver (with inter‐grid transfer operators based on mixed‐order interpolation) provides an almost mesh‐independent convergence. In the indirect approach for uniform meshes, the C‐AMG solver (based on a Jacobi‐relaxed interpolation) provides solutions with an optimum scaling of the convergence rates. For non‐uniform meshes this convergence becomes mesh dependent but the overall solution cost increases relatively slowly with increasing mesh bandwidth. Copyright © 2006 John Wiley & Sons, Ltd.     

具有大量椭圆颗粒/孔洞随机分布区域的计算机模拟及其改进三角形自动网格生成算法

针对颗粒／孔洞为椭圆形状并且呈随机分布的多相复合材料给出了一种计算机模拟方法，同时针对这样复杂的二维区域提供了一个改进的快速自动三角形网格剖分算法。最后用实例说明了算法的有效性。此算法可以推广到三维情形、裂纹材料等。     

On the usage of NURBS as interface representation in free‐surface flows

When simulating free‐surface flows using the finite element method, there are many cases where the governing equations require information which must be derived from the available discretized geometry. Examples are curvature or normal vectors. The accurate computation of this information directly from the finite element mesh often requires a high degree of refinement—which is not necessarily required to obtain an accurate flow solution. As a remedy and an option to be able to use coarser meshes, the representation of the free surface using non‐uniform rational B‐splines (NURBS) curves or surfaces is investigated in this work. The advantages of a NURBS parameterization in comparison with the standard approach are discussed. In addition, it is explored how the pressure jump resulting from surface tension effects can be handled using doubled interface nodes. Numerical examples include the computation of surface tension in a two‐phase flow as well as the computation of normal vectors as a basis for mesh deformation methods. For these examples, the improvement of the numerical solution compared with the standard approaches on identical meshes is shown. Copyright © 2011 John Wiley & Sons, Ltd.     

一种基于双前沿推进的边界层网格生成算法

高雷诺数粘性流动模拟对边界层内的网格正交性有特殊要求.对于复杂外形,这类问题的网格自动化生成十分困难.面向该问题,提出一种双前沿推进思想,并形成一种面向复杂几何外形的边界层网格全自动生成算法.结合多种网格技术处理局部几何特征以保证边界层网格的质量.双前沿推进思想同时适用于多块结构网格和混合网格的边界层网格生成.多个模型...     

一种新的求解对流占优问题的自适应网格细化方法

在均匀网格上求解对流占优问题时,往往会产生数值震荡现象,因此需要局部加密网格来提高解的精度。针对对流占优问题,设计了一种新的自适应网格细化算法。该方法采用流线迎风SUPG(Petrov-Galerkin)格式求解对流占优问题,定义了网格尺寸并通过后验误差估计子修正来指导自适应网格细化,以泡泡型局部网格生成算法BLMG为网格生成器,通过模拟泡泡在区域中的运动得到了高质量的点集。与其他自适应网格细化方法相比,该方法可在同一框架内实现网格的细化和粗化,同时在所有细化层得到了高质量的网格。数值算例结果表明,该方法在求解对流占优问题时具有更高的数值精度和更好的收敛性。     

Hybrid mesh generation with embedded surfaces using a multiple marching direction approach

This paper proposes a method for the creation of hybrid meshes with embedded surfaces for viscous flow simulations as an extension of the multiple marching direction approach (AIAA J. 2007; 45 (1):162–167). The multiple marching direction approach enables to place semi‐structured elements around singular points, where valid semi‐structured elements cannot be placed using conventional hybrid mesh generation methods. This feature is discussed first with a couple of examples. Elements sometimes need to be clustered inside a computational domain to obtain more accurate results. For example, solution features, such as shocks, vortex cores and wake regions, can be extracted during the process of adaptive mesh generation. These features can be represented as surface meshes embedded in a computational domain. Semi‐structured elements can be placed around the embedded surface meshes using the multiple marching direction approach with a pretreatment method. Tetrahedral elements can be placed easily instead. A couple of results are presented to demonstrate the capability of the mesh generation method. Copyright © 2008 John Wiley & Sons, Ltd.     

On the immersed boundary‐lattice Boltzmann simulations of incompressible flows with freely moving objects

For simulating freely moving problems, conventional immersed boundary‐lattice Boltzmann methods encounter two major difficulties of an extremely large flow domain and the incompressible limit. To remove these two difficulties, this work proposes an immersed boundary‐lattice Boltzmann flux solver (IB‐LBFS) in the arbitrary Lagragian–Eulerian (ALE) coordinates and establishes a dynamic similarity theory. In the ALE‐based IB‐LBFS, the flow filed is obtained by using the LBFS on a moving Cartesian mesh, and the no‐slip boundary condition is implemented by using the boundary condition‐enforced immersed boundary method. The velocity of the Cartesian mesh is set the same as the translational velocity of the freely moving object so that there is no relative motion between the plate center and the mesh. This enables the ALE‐based IB‐LBFS to study flows with a freely moving object in a large open flow domain. By normalizing the governing equations for the flow domain and the motion of rigid body, six non‐dimensional parameters are derived and maintained to be the same in both physical systems and the lattice Boltzmann framework. This similarity algorithm enables the lattice Boltzmann equation‐based solver to study a general freely moving problem within the incompressible limit. The proposed solver and dynamic similarity theory have been successfully validated by simulating the flow around an in‐line oscillating cylinder, single particle sedimentation, and flows with a freely falling plate. The obtained results agree well with both numerical and experimental data. Copyright © 2016 John Wiley & Sons, Ltd.     

黏性边界层网格自动生成

高雷诺数黏性流动在壁面附近存在边界层,在计算模拟中自动生成可靠且有效的计算网格仍然是计算流体力学存在的瓶颈.三棱柱/四面体混合网格技术在一定程度上缓解了这个困难.然而,对于复杂外形的情况,在边界层内自动高效生成高质量的三棱柱单元仍然十分困难.常用的层推进法在凹凸区域及角点处生成的边界层网格单元质量较差,边界层网格最外层尺寸不均匀.针对这些问题,发展了一种黏性边界层网格自动生成方法,通过顶点周围边的二面角识别物面网格特征确定多生长方向,预估并调整生长高度处理相交情况.同时提出一种三维前沿尺寸调节方式,提高了边界层网格单元的正交性,保证了边界层网格与远场网格尺寸的光滑过渡.通过ONERA M6翼型以及带发动机短舱的DLR-F6翼身组合体等外形的网格生成实例及绕流数值模拟,将计算值与标准实验值进行对比,结果表明:该方法能够自动高效地生成满足数值计算需求的混合网格.     

面向复杂几何模型的并行四面体网格生成方法

面向大规模工程计算等数值模拟领域，提出了一种支持复杂几何模型的大规模四面体网格并行生成方法。该方法以复杂几何模型作为输入，首先采用串行网格生成方法生成初始四面体网格，然后通过两级区域分解方法将初始网格分解为多个子网格并分配到相应的进程中，进程间并行地提取出子网格的表面网格，并基于几何模型对面网格进行贴体加密，最后对加密后的面网格采用Delaunay方法重新生成四面体网格，该方法可以更好地适应高性能计算机体系结构，较好地克服了并行方法中并行性能和网格质量不能兼顾的问题。对三峡大坝模型进行测试和验证，证明该方法具有良好的并行效率和可扩展性，可以在数万处理器核上并行生成数十亿高质量四面体网格。     

A structured but non‐uniform Cartesian grid‐based model for the shallow water equations

In large‐scale shallow flow simulations, local high‐resolution predictions are often required in order to reduce the computational cost without losing the accuracy of the solution. This is normally achieved by solving the governing equations on grids refined only to those areas of interest. Grids with varying resolution can be generated by different approaches, e.g. nesting methods, patching algorithms and adaptive unstructured or quadtree gridding techniques. This work presents a new structured but non‐uniform Cartesian grid system as an alternative to the existing approaches to provide local high‐resolution mesh. On generating a structured but non‐uniform Cartesian grid, the whole computational domain is first discretized using a coarse background grid. Local refinement is then achieved by directly allocating a specific subdivision level to each background grid cell. The neighbour information is specified by simple mathematical relationships and no explicit storage is needed. Hence, the structured property of the uniform grid is maintained. After employing some simple interpolation formulae, the governing shallow water equations are solved using a second‐order finite volume Godunov‐type scheme in a similar way as that on a uniform grid. Copyright © 2010 John Wiley & Sons, Ltd.     

适应复杂外形的三维粘性混合网格生成算法

提出一类适应复杂外形的粘性混合网格生成算法。表面网格由前沿推进三角形曲面网格程序获得,边界层布置各向异性的三棱柱体网格,远物面区域采用Delaunay方法生成四面体网格。针对模型的复杂几何特征,综合采用了各种网格处理技术,以保证边界层网格的质量,并避免算法失效问题。网格实例及计算结果表明了本文算法的实用性及和效性。     

A stable and accurate projection method on a locally refined staggered mesh

In this paper, a robust projection method on a locally refined mesh is proposed for two‐ and three‐dimensional viscous incompressible flows. The proposed method is robust not only when the interface between two meshes is located in a smooth flow region but also when the interface is located in a flow region with large gradients and/or strong unsteadiness. In numerical simulations, a locally refined mesh saves many grid points in regions of relatively small gradients compared with a uniform mesh. For efficiency and ease of implementation, we consider a two‐level blocked structure, for which both of the coarse and fine meshes are uniform Cartesian ones individually. Unfortunately, the introduction of the two‐level blocked mesh results in an important but difficult issue: coupling of the coarse and fine meshes. In this paper, by properly addressing the issue of the coupling, we propose a stable and accurate projection method on a locally refined staggered mesh for both two‐ and three‐dimensional viscous incompressible flows. The proposed projection method is based on two principles: the linear interpolation technique and the consistent discretization of both sides of the pressure Poisson equation. The proposed algorithm is straightforward owing to the linear interpolation technique, is stable and accurate, is easy to extend from two‐ to three‐dimensional flows, and is valid even when flows with large gradients cross the interface between the two meshes. The resulting pressure Poisson equation is non‐symmetric on a locally refined mesh. The numerical results for a series of exact solutions for 2D and 3D viscous incompressible flows verify the stability and accuracy of the proposed projection method. The method is also applied to some challenging problems, including turbulent flows around particles, flows induced by impulsively started/stopped particles, and flows induced by particles near solid walls, to test the stability and accuracy. Copyright © 2010 John Wiley & Sons, Ltd.     

Validation of adaptive unstructured hexahedral mesh computations of flow around a wind turbine airfoil

In this paper we investigate local adaptive refinement of unstructured hexahedral meshes for computations of the flow around the DU91 wind turbine airfoil. This is a 25% thick airfoil, found at the mid‐span section of a wind turbine blade. Wind turbine applications typically involve unsteady flows due to changes in the angle of attack and to unsteady flow separation at high angles of attack. In order to obtain reasonably accurate results for all these conditions one should use a mesh which is refined in many regions, which is not computationally efficient. Our solution is to apply an automated mesh adaptation technique. In this paper we test an adaptive refinement strategy developed for unstructured hexahedral meshes for steady flow conditions. The automated mesh adaptation is based on local flow sensors for pressure, velocity, density or a combination of these flow variables. This way the mesh is refined only in those regions necessary for high accuracy, retaining computational efficiency. A validation study is performed for two cases: attached flow at an angle of 6° and separated flow at 12°. The results obtained using our adaptive mesh strategy are compared with experimental data and with results obtained with an equally sized non‐adapted mesh. From these computations it can be concluded that for a given computing time, adapted meshes result in solutions closer to the experimental data compared to non‐adapted meshes for attached flow. Finally, we show results for unsteady computations. Copyright © 2005 John Wiley & Sons, Ltd.     

基于AFT-Delaunay的二维解耦并行网格生成算法

面向平面任意几何区域网格生成,提出了一种将波前法AFT(Advancing Front Technique)与Delaunay法相结合的解耦并行网格生成算法。算法主要思想是沿着求解几何区域惯性轴,采用扩展的AFT-Delaunay算法生成高质量三角形网格墙,递归地将几何区域动态划分成多个彼此解耦的子区域;采用OpenMP多线程并行技术,将子区域分配给多个CPU并行生成子区域网格;子区域内部的网格生成复用AFT-Delaunay算法,保证了生成网格的质量、效率和一致性要求。本算法优先生成几何边界与交界面网格,有利于提高有限元计算精度;各个子区域的网格生成彼此完全解耦,因此并行网格生成过程无需通信。该方法克服了并行交界面网格质量恶化难题,且具有良好的并行加速比,能够全自动、高效率地并行生成高质量的三角网格。     

An immersed discontinuous Galerkin method for compressible Navier-Stokes equations on unstructured meshes

We introduce an immersed high-order discontinuous Galerkin method for solving the compressible Navier-Stokes equations on non–boundary-fitted meshes. The flow equations are discretised with a mixed discontinuous Galerkin formulation and are advanced in time with an explicit time marching scheme. The discretisation meshes may contain simplicial (triangular or tetrahedral) elements of different sizes and need not be structured. On the discretisation mesh, the fluid domain boundary is represented with an implicit signed distance function. The cut-elements partially covered by the solid domain are integrated after tessellation with the marching triangle or tetrahedra algorithms. Two alternative techniques are introduced to overcome the excessive stable time step restrictions imposed by cut-elements. In the first approach, the cut-basis functions are replaced with the extrapolated basis functions from the nearest largest element. In the second approach, the cut-basis functions are simply scaled proportionally to the fraction of the cut-element covered by the solid. To achieve high-order accuracy, additional nodes are introduced on the element faces abutting the solid boundary. Subsequently, the faces are curved by projecting the introduced nodes to the boundary. The proposed approach is verified and validated with several two- and three-dimensional subsonic and hypersonic low Reynolds number flow applications, including the flow over a cylinder, a space capsule, and an aerospace vehicle.     

Fictitious domain methods for two‐phase flow energy balance computations in nuclear components

This paper is dedicated to the numerical simulation of nuclear components (cores and steam generators) by fictitious domain methods. The fictitious domain approach consists in immersing the physical domain under study in a Cartesian domain, called the fictitious domain, and in performing the numerical resolution on this fictitious domain. The calculation times are then efficiently reduced by the use of fast solvers. In counterpart, one has to handle with an immersed boundary, generally non‐aligned with the Cartesian mesh, which can be non‐trivial. The two fictitious domain methods compared here on industrial simulations and developed by Ramière et al. deal with an approximate immersed interface directly derived from the uniform Cartesian mesh. All the usual immersed boundary conditions (Dirichlet, Robin, Neumann), possibly mixed, are handled through a unique formulation of the fictitious problem. This kind of approximation leads to first‐order methods in space that exhibit a good ratio of the precision of the approximate solution over the CPU time, which is very important for industrial simulations. After a brief recall of the fictitious domain method with spread interface (Ramière et al., CMAME 2007) and the fictitious domain method with immersed jumps (Ramière et al., JCP 2008), we will focus on the numerical results provided by these methods applied to the energy balance equation in a steam generator. The advantages and drawbacks of each method will be pointed out. Generally speaking, the two methods confirm their very good efficiency in terms of precision, convergence, and calculation time in an industrial context. Copyright © 2011 John Wiley & Sons, Ltd.