Perspective on the geometric conservation law and finite element methods for ALE simulations of incompressible flow

This paper takes a fresh look at the geometric conservation law (GCL) from the perspective of the finite element method (FEM) for incompressible flows. The GCL arises naturally in the context of Arbitrary Lagrangian Eulerian (ALE) formulations for solving problems on deforming domains. GCL compliance is traditionally interpreted as a consistency criterion for applying an unsteady flow solution algorithm to simulate exactly a uniform flow on a deforming domain. We introduce an additional requirement: the time integrator must maintain its fixed mesh accuracy when applied to deforming meshes. A review of the literature shows that while many authors use an ALE FEM, few of them discuss the GCL issues. We show how a fixed mesh unsteady FEM using high order time integrator (up to fifth order in time) can be transposed to solve problems on deforming meshes and preserve its fixed mesh high order temporal accuracy. An appropriate construction of the divergence of the mesh velocity guarantees GCL compliance while a separate construction of the mesh velocity itself allows the time-integrator to deliver its fixed mesh high order temporal accuracy on deforming domains. Analytical error analysis of problems with closed form solutions provides insight on the behavior of the time integrators. It also explains why high order temporal accuracy is achieved with a conservative formulation of the incompressible Navier–Stokes equations, while only first order time accuracy is observed with the non-conservative formulation and all time-integrators investigated here. We present thorough time-step and grid refinement studies for simple problems with closed form solutions and for a manufactured solution with a non-trivial flow on a deforming mesh. In all cases studied, the proposed reconstructions of the mesh velocity and its divergence for the conservative formulation lead to optimal time accuracy on deforming grids.     

一种基于预估-修正策略的非定常流网格自适应加密/稀疏方法

给出一种非定常流动数值模拟的网格自适应处理方法.在"求解流动方程-自适应调整网格"的流程中,引入预估-修正步.根据自适应周期内每个时间步上的流场预估解,计算单元上的事后误差估算值.建立考虑解演变的网格自适应指示器,并进行多层次单元加密-稀疏的动态网格自适应处理.在自适应网格上重新计算流场.每个自适应周期中,流动演变区域的网格获得加密;而前一个周期中的特征现象已离开区域的网格被稀疏.应用边界非协调的当地DFD(Domain-Free Discretization)方法求解流动方程.为验证网格自适应处理方法,针对静止圆柱和自推进游鱼的流动进行了数值实验.     

Efficient mesh motion using radial basis functions with data reduction algorithms

Mesh motion using radial basis functions has been demonstrated previously by the authors to produce high quality meshes suitable for use within unsteady and aeroelastic CFD codes. In the aeroelastic case the structural mesh may be used as the set of control points governing the deformation, which is efficient since the structural mesh is usually small. However, as a stand alone mesh motion tool, where the surface mesh points control the motion, radial basis functions may be restricted by the size of the surface mesh, as an update of a single volume point depends on all surface points. In this paper a method is presented that allows an arbitrary deformation to be represented to within a desired tolerance by using a significantly reduced set of surface points intelligently identified in a fashion that minimises the error in the interpolated surface. This method may be used on much larger cases and is successfully demonstrated here for a 10⁶ cell mesh, where the initial solve phase cost reduces by a factor of eight with the new scheme and the mesh update by a factor of 55. It has also been shown that the number of surface points required to represent the surface is only geometry dependent (i.e. grid size independent), and so this reduction factor actually increases for larger meshes.     

Output error estimation for summation-by-parts finite-difference schemes

The paper develops a posteriori error estimates of integral output functionals for summation-by-parts finite-difference methods. The error estimates are based on the adjoint-weighted residual method and take advantage of a variational interpretation of summation-by-parts discretizations. The estimates are computed on a fixed grid and do not require an embedded grid or explicit interpolation operators. For smooth boundary-value problems containing first and second derivatives the error estimates converge to the exact error as the mesh is refined. The theory is verified using linear boundary-value problems and the Euler equations.     

部分流泵整机非定常流动数值模拟

本文在设计工况下采用滑移网格技术对部分流泵进行了整机非定常流动数值模拟,分别分析了叶轮、蜗壳内的非定常流动规律。计算表明,部分流泵内流的非定常特性极其强烈,只有采用非定常计算方法才能反映其内流的实质。本文的计算为进一步研究部分流泵的内流现象、提高效率、减少水利损失提供了一定的理论依据。     

基于Riemann解的二维流体力学Lagrange有限点无网格方法

在高维流体力学计算中,对于多介质大变形等一类问题,采用有网格方法常遇到较大的困难.针对二维问题,研究了一种无网格方法——Lagrange有限点方法:在求解区域上设置适当的离散点集,视其中每一点为流体力学Lagrange点;对于点集的任一点,确定邻点集合,并基于该点同邻点集合的联系,应用Godunov方法将流体力学Lagrange方程进行离散;考虑到算法的稳健性,方法中可设置较多邻点并采用最小二乘法.将该方法应用于典型的数值算例,取得了良好效果.     

Error estimation and adaptation for functional outputs in time-dependent flow problems

The paper presents an adjoint-based approach for determining global error in the time domain that is relevant to functional outputs from unsteady flow simulations. The algorithm is derived for the unsteady Euler equations that are discretized for second-order accuracy in both space and time and takes into account the effect of dynamic meshes. In addition to error due to temporal resolution, the formulation also takes into account algebraic error arising from partial convergence of the governing equations at each implicit time-step. The resulting error distributions are then used to drive adaptation of the temporal resolution and the convergence tolerances for the governing equations at each time-step. The method is demonstrated in the context of both time-integrated and instantaneous functionals and the results are compared against traditional adaptation methods.     

Metric construction by length distribution tensor and edge based error for anisotropic adaptive meshing

Metric tensors play a key role to control the generation of unstructured anisotropic meshes. In practice, the most well established error analysis enables to calculate a metric tensor on an element basis. In this paper, we propose to build a metric field directly at the nodes of the mesh for a direct use in the meshing tools. First, the unit mesh metric is defined and well justified on a node basis, by using the statistical concept of length distribution tensors. Then, the interpolation error analysis is performed on the projected approximate scalar field along the edges. The error estimate is established on each edge whatever the dimension is. It enables to calculate a stretching factor providing a new edge length distribution, its associated tensor and the corresponding metric. The optimal stretching factor field is obtained by solving an optimization problem under the constraint of a fixed number of edges in the mesh. Several examples of interpolation error are proposed as well as preliminary results of anisotropic adaptation for interface and free surface problem using a level set method.     

基于质心Voronoi图重构的UDSM边折叠简化

城市数字表面模型网格(UDSM)的相邻网格常常出现曲率剧变,而这些位置是UDSM的细节部分,简化过程中应当尽量保持。针对该情况,引入了质心Voronoi图重划分网格,将曲率较小的表面的点云密度大大降低。重划分的网格表面细节与周围的平滑表面的三角网格尺寸悬殊,在该基础上使用二次误差矩阵边折叠进行LOD构建时网格发生明显变化,范围大大减少。算法在时间性能与网格误差与直接边折叠相近的前提下,更多地保存简化后的网格细节。     

Output-based space–time mesh adaptation for the compressible Navier–Stokes equations

This paper presents an output-based adaptive algorithm for unsteady simulations of convection-dominated flows. A space–time discontinuous Galerkin discretization is used in which the spatial meshes remain static in both position and resolution, and in which all elements advance by the same time step. Error estimates are computed using an adjoint-weighted residual, where the discrete adjoint is computed on a finer space obtained by order enrichment of the primal space. An iterative method based on an approximate factorization is used to solve both the forward and adjoint problems. The output error estimate drives a fixed-growth adaptive strategy that employs hanging-node refinement in the spatial domain and slab bisection in the temporal domain. Detection of space–time anisotropy in the localization of the output error is found to be important for efficiency of the adaptive algorithm, and two anisotropy measures are presented: one based on inter-element solution jumps, and one based on projection of the adjoint. Adaptive results are shown for several two-dimensional convection-dominated flows, including the compressible Navier–Stokes equations. For sufficiently-low accuracy levels, output-based adaptation is shown to be advantageous in terms of degrees of freedom when compared to uniform refinement and to adaptive indicators based on approximation error and the unweighted residual. Time integral quantities are used for the outputs of interest, but entire time histories of the integrands are also compared and found to converge rapidly under the proposed scheme. In addition, the final output-adapted space–time meshes are shown to be relatively insensitive to the starting mesh.     

A Method of Lines Based on Immersed Finite Elements for Parabolic Moving Interface Problems

This article extends the finite element method of lines to a parabolic initial boundary value problem whose diffusion coefficient is discontinuous across an interface that changes with respect to time. The method presented here uses immersed finite element (IFE) functions for the discretization in spatial variables that can be carried out over a fixed mesh (such as a Cartesian mesh if desired), and this feature makes it possible to reduce the parabolic equation to a system of ordinary differential equations (ODE) through the usual semi-discretization procedure. Therefore, with a suitable choice of the ODE solver, this method can reliably and efficiently solve a parabolic moving interface problem over a fixed structured (Cartesian) mesh. Numerical examples are presented to demonstrate features of this new method.     

基于工业CT图像重构的网格模型的保特征简化方法

对基于工业CT图像重构的网格模型进行网格简化时,大多数现有网格模型简化算法会丢失特征,出现网格质量不好的问题。因此提出一种网格模型保特征简化方法,该方法用三角形折叠法对原始模型进行简化,当简化后模型的平均二面角角度误差达到允许误差后,再使用边折叠法对模型进行简化。在三角形折叠法中提出了利用被折叠三角形的法向量、各个顶点的高斯曲率及其在周边三角形上的投影确定该三角形的折叠点,利用局部体积误差与二面角角度误差的无因次化和确定折叠代价的方法;在边折叠法中提出了将二面角角度误差引入到二次误差测度(QEM)法的折叠代价中的改进QEM法。实验结果表明:与其他算法相比,该方法能够生成保特征、高质量、低几何误差的网格模型。     

Moving mesh methods for blowup in reaction–diffusion equations with traveling heat source

In this paper moving mesh methods are used to simulate the blowup in a reaction–diffusion equation with traveling heat source. The finite-time blowup occurs if the speed of the movement of the heat source remains sufficiently low, and the blowup procedure is not fixed at one point not like that for stationary heat source. As time goes to the blowup time, the blowup profile converges to a stationary state. In the simulation a new moving mesh algorithm is designed to deal with the difficulty caused by the delta function in the traveling heat source. The convergence rates are verified and new blowup figures are generated from the numerical experiments.     

A high order Discontinuous Galerkin – Fourier incompressible 3D Navier–Stokes solver with rotating sliding meshes

We present the development of a sliding mesh capability for an unsteady high order (order ? 3) h/p Discontinuous Galerkin solver for the three-dimensional incompressible Navier–Stokes equations. A high order sliding mesh method is developed and implemented for flow simulation with relative rotational motion of an inner mesh with respect to an outer static mesh, through the use of curved boundary elements and mixed triangular–quadrilateral meshes.A second order stiffly stable method is used to discretise in time the Arbitrary Lagrangian–Eulerian form of the incompressible Navier–Stokes equations. Spatial discretisation is provided by the Symmetric Interior Penalty Galerkin formulation with modal basis functions in the x–y plane, allowing hanging nodes and sliding meshes without the requirement to use mortar type techniques. Spatial discretisation in the z-direction is provided by a purely spectral method that uses Fourier series and allows computation of spanwise periodic three-dimensional flows. The developed solver is shown to provide high order solutions, second order in time convergence rates and spectral convergence when solving the incompressible Navier–Stokes equations on meshes where fixed and rotating elements coexist.In addition, an exact implementation of the no-slip boundary condition is included for curved edges; circular arcs and NACA 4-digit airfoils, where analytic expressions for the geometry are used to compute the required metrics.The solver capabilities are tested for a number of two dimensional problems governed by the incompressible Navier–Stokes equations on static and rotating meshes: the Taylor vortex problem, a static and rotating symmetric NACA0015 airfoil and flows through three bladed cross-flow turbines. In addition, three dimensional flow solutions are demonstrated for a three bladed cross-flow turbine and a circular cylinder shadowed by a pitching NACA0012 airfoil.     

An adaptive method for computing invariant manifolds in non-autonomous, three-dimensional dynamical systems

We present a computational method for determining the geometry of a class of three-dimensional invariant manifolds in non-autonomous (aperiodically time-dependent) dynamical systems. The presented approach can be also applied to analyse the geometry of 3D invariant manifolds in three-dimensional, time-dependent fluid flows. The invariance property of such manifolds requires that, at any fixed time, they are given by surfaces in R³. We focus on a class of manifolds whose instantaneous geometry is given by orientable surfaces embedded in R³. The presented technique can be employed, in particular, to compute codimension one (invariant) stable and unstable manifolds of hyperbolic trajectories in 3D non-autonomous dynamical systems which are crucial in the Lagrangian transport analysis. The same approach can also be used to determine evolution of an orientable ‘material surface’ in a fluid flow. These developments represent the first step towards a non-trivial 3D extension of the so-called lobe dynamics — a geometric, invariant-manifold-based framework which has been very successful in the analysis of Lagrangian transport in unsteady, two-dimensional fluid flows. In the developed algorithm, the instantaneous geometry of an invariant manifold is represented by an adaptively evolving triangular mesh with piecewise C² interpolating functions. The method employs an automatic mesh refinement which is coupled with adaptive vertex redistribution. A variant of the advancing front technique is used for remeshing, whenever necessary. Such an approach allows for computationally efficient determination of highly convoluted, evolving geometry of codimension one invariant manifolds in unsteady three-dimensional flows. We show that the developed method is capable of providing detailed information on the evolving Lagrangian flow structure in three dimensions over long periods of time, which is crucial for a meaningful 3D transport analysis.     

Fully anisotropic goal-oriented mesh adaptation for 3D steady Euler equations

This paper studies the coupling between anisotropic mesh adaptation and goal-oriented error estimate. The former is very well suited to the control of the interpolation error. It is generally interpreted as a local geometric error estimate. On the contrary, the latter is preferred when studying approximation errors for PDEs. It generally involves non local error contributions. Consequently, a full and strong coupling between both is hard to achieve due to this apparent incompatibility. This paper shows how to achieve this coupling in three steps.First, a new a priori error estimate is proved in a formal framework adapted to goal-oriented mesh adaptation for output functionals. This estimate is based on a careful analysis of the contributions of the implicit error and of the interpolation error. Second, the error estimate is applied to the set of steady compressible Euler equations which are solved by a stabilized Galerkin finite element discretization. A goal-oriented error estimation is derived. It involves the interpolation error of the Euler fluxes weighted by the gradient of the adjoint state associated with the observed functional. Third, rewritten in the continuous mesh framework, the previous estimate is minimized on the set of continuous meshes thanks to a calculus of variations. The optimal continuous mesh is then derived analytically. Thus, it can be used as a metric tensor field to drive the mesh adaptation. From a numerical point of view, this method is completely automatic, intrinsically anisotropic, and does not depend on any a priori choice of variables to perform the adaptation.3D examples of steady flows around supersonic and transsonic jets are presented to validate the current approach and to demonstrate its efficiency.     

A Relaxed Dimensional Factorization preconditioner for the incompressible Navier–Stokes equations

In this paper we introduce a Relaxed Dimensional Factorization (RDF) preconditioner for saddle point problems. Properties of the preconditioned matrix are analyzed and compared with those of the closely related Dimensional Splitting (DS) preconditioner recently introduced by Benzi and Guo [7]. Numerical results for a variety of finite element discretizations of both steady and unsteady incompressible flow problems indicate very good behavior of the RDF preconditioner with respect to both mesh size and viscosity.     

流动数值模拟中一种并行自适应有限元算法

给出了一种流动数值模拟中的基于误差估算的并行网格自适应有限元算法.首先,以初网格上获得的当地事后误差估算值为权,应用递归谱对剖分方法划分初网格,使各子域上总体误差近似相等,以解决负载平衡问题.然后以误差值为判据对各子域内网格进行独立的自适应处理.最后应用基于粘接元的区域分裂法在非匹配的网格上求解N-S方程.区域分裂情形下N-S方程有限元解的误差估算则是广义Stokes问题误差估算方法的推广.为验证方法的可靠性,给出了不可压流经典算例的数值结果.     

Unsteady flow through in-vitro models of the glottis

The unsteady two-dimensional flow through fixed rigid in vitro models of the glottis is studied in some detail to validate a more accurate model based on the prediction of boundary-layer separation. The study is restricted to the flow phenomena occurring within the glottis and does not include effects of vocal-fold movement on the flow. Pressure measurements have been carried out for a transient flow through a rigid scale model of the glottis. The rigid model with a fixed geometry driven by an unsteady pressure is used in order to achieve a high accuracy in the specification of the geometry of the glottis. The experimental study is focused on flow phenomena as they might occur in the glottis, such as the asymmetry of the flow due to the Coanda effect and the transition to turbulent flow. It was found that both effects need a relatively long time to establish themselves and are therefore unlikely to occur during the production of normal voiced speech when the glottis closes completely during part of the oscillation cycle. It is shown that when the flow is still laminar and symmetric the prediction of the boundary-layer model and the measurement of the pressure drop from the throat of the glottis to the exit of the glottis agree within 40%. Results of the boundary-layer model are compared with a two-dimensional vortex-blob method for viscous flow. The difference between the results of the simpiflied boundary-layer model and the experimental results is explained by an additional pressure difference between the separation point and the far field within the jet downstream of the separation point. The influence of the movement of the vocal folds on our conclusions is still unclear.     

气冷涡轮级叶栅非定常流场数值模拟

采用具有三阶精度TVD性质的有限差分格式、自由型曲面网格技术、分区算法以及双时间步长的方法,对某型涡轮级叶栅流场进行了非定常NS方程数值求解,考察了在有、无冷气喷射条件下涡轮级气动性能的非定常变化。结果表明,上游静叶栅是否喷射冷气对下游动叶栅超音速区域的影响具有明显的区别,有冷气喷射时动叶栅前缘气动负荷降低,级效率下降约1%,但是不同动叶通道内气动性能随时间周期性变化的幅度明显减小了。