Node-pair finite volume/finite element schemes for the Euler equation in cylindrical and spherical coordinates

A numerical scheme is presented for the solution of the compressible Euler equations in both cylindrical and spherical coordinates. The unstructured grid solver is based on a mixed finite volume/finite element approach. Equivalence conditions linking the node-centered finite volume and the linear Lagrangian finite element scheme over unstructured grids are reported and used to devise a common framework for solving the discrete Euler equations in both the cylindrical and the spherical reference systems. Numerical simulations are presented for the explosion and implosion problems with spherical symmetry, which are solved in both the axial–radial cylindrical coordinates and the radial–azimuthal spherical coordinates. Numerical results are found to be in good agreement with one-dimensional simulations over a fine mesh.     

Finite Volume Methods for Multi-Symplectic PDES

We investigate the application of a cell-vertex finite volume discretization to multi-symplectic PDEs. The investigated discretization reduces to the Preissman box scheme when used on a rectangular grid. Concerning arbitrary quadrilateral grids, we show that only methods with parallelogram-like finite volume cells lead to a multi-symplectic discretization; i.e., to a method that preserves a discrete conservation law of symplecticity. One of the advantages of finite volume methods is that they can be easily adjusted to variable meshes. But, although the implementation of moving mesh finite volume methods for multi-symplectic PDEs is rather straightforward, the restriction to parallelogram-like cells implies that only meshes moving with a constant speed are multi-symplectic. To overcome this restriction, we suggest the implementation of reversible moving mesh methods based on a semi-Lagrangian approach. Numerical experiments are presented for a one dimensional dispersive shallow-water system.     

奇异摄动反应扩散方程的后验误差估计及自适应算法

研究了一类奇异摄动半线性反应扩散方程的自适应网格方法.在任意非均匀网格上建立迎风有限差分离散格式,并推导出离散格式的后验误差界,然后用该误差界设计自适应网格移动算法.数值实验结果证明了所提出的自适应网格方法的有效性.     

R-Adaptive Reconnection-Based Arbitrary Lagrangian Eulerian Method-R-ReALE

We present a new R-adaptive Arbitrary Lagrangian Eulerian (ALE) method, based on the reconnection-based ALE - ReALE methodology [5, 41, 42]. The main elements in a standard ReALE method are: an explicit Lagrangian phase on an arbitrary polygonal (in 2D) mesh, followed by a rezoning phase in which a new grid is defined, and a remapping phase in which the Lagrangian solution is transferred onto the new grid. The rezoned mesh is smoothed by using one or several steps toward centroidal Voronoi tessellation, but it is not adapted to the solution in any way. We present a new R-adaptive ReALE method (R-ReALE, where R stands for Relocation). The new method is based on the following design principles. First, a monitor function (or error indicator) based on Hessian of some flow parameter(s), is utilized. Second, the new algorithm uses the equidistribution principle with respect to the monitor function as criterion for defining an adaptive mesh. Third, centroidal Voronoi tessellation is used for the construction of the adaptive mesh. Fourth, we modify the raw monitor function (scale it to avoid extremely small and large cells and smooth it to create a smooth mesh), in order to utilize theoretical results related to centroidal Voronoi tessellation. In the R-ReALE method, the number of mesh cells is chosen at the beginning of the calculation and does not change with time, but the mesh is adapted according to the modified monitor function during the rezone stage at each time step. We present all details required for implementation of the new adaptive R-ReALE method and demonstrate its performance relative to standard ReALE method on a series of numerical examples.     

Simulation of flows with strong shocks with an adaptive conservative scheme

In the present work, numerical simulations of unsteady flows with moving shocks are presented. An unsteady mesh adaptation method, based on error equidistribution criteria, is adopted to capture the most important flow features. The modifications to the topology of the grid are locally interpreted in terms of continuous deformation of the finite volumes built around the nodes. The arbitrary Lagrangian–Eulerian formulation of the Euler equations is then applied to compute the flow variable over the new grid without resorting to any explicit interpolation step. The numerical results show an increase in the accuracy of the solution, together with a strong reduction of the computational costs, with respect to computations with a uniform grid using a larger number of nodes.     

Une méthode de volumes finis pour les équations elliptiques du second ordre

We present a new finite volume method for approximating second order elliptic equations. This method has several advantages: — it allows large mesh distortions which occur in Lagrangian hydrodynamics calculations; — it is convenient for the approximation of crossed derivative terms as those which take place in magnetohydrodynamics will Hall effect; — it generalizes the finite difference method and the finite volume method using Delaunay-Voronoi meshes.     

TWO-GRID CHARACTERISTIC FINITE VOLUME METHODS FOR NONLINEAR PARABOLIC PROBLEMS＊

In this work, two-grid characteristic finite volume schemes for the nonlinear parabolic problem are considered. In our algorithms, the diffusion term is discretized by the finite volume method, while the temporal differentiation and advection terms are treated by the characteristic scheme. Under some conditions about the coefficients and exact solution, optimal error estimates for the numerical solution are obtained. Furthermore, the two- grid characteristic finite volume methods involve solving a nonlinear equation on coarse mesh with mesh size H, a large linear problem for the Oseen two-grid characteristic finite volume method on a fine mesh with mesh size h = O（H2） or a large linear problem for the Newton two-grid characteristic finite volume method on a fine mesh with mesh size h = 0（I log hll/2H3）. These methods we studied provide the same convergence rate as that of the characteristic finite volume method, which involves solving one large nonlinear problem on a fine mesh with mesh size h. Some numerical results are presented to demonstrate the efficiency of the proposed methods.     

求解流固耦合问题的一种四步分裂有限元算法

基于arbitrary Lagrangian Eulerian (ALE) 有限元方法,发展了一种求解流固耦合问题的弱耦合算法．将半隐式四步分裂有限元格式推广至求解ALE描述下的Navier-Stokes（N-S）方程,并在动量方程中引入迎风流线（streamline upwind/Petrov-Galerkin, SUPG）稳定项以消除对流引发的速度场数值振荡;采用Newmark-β法对结构方程进行时间离散;运用经典的Galerkin有限元法求解修正的Laplace方程以实现网格更新,每个计算步施加网格总变形量防止结构长时间、大位移运动时的网格质量恶化．运用上述算法对弹性支撑刚性圆柱体的流致振动问题进行了数值模拟,计算结果与已有结果相吻合,初步验证了该算法的正确性和有效性．     

基于FEMLIP的全风化边坡失稳破坏全过程的数值模拟研究

边坡坡角和强度是影响边坡稳定性的重要因素,而边坡失稳往往伴随着大变形的发生,其变形从数十米至数千米不等.目前,传统有限元法在处理大变形问题时常常因网格畸变而导致计算终止.因此,为了实现边坡失稳破坏全过程的模拟,并研究边坡坡角和强度对边坡稳定性的影响,基于Lagrange(拉格朗日)积分点有限元法（FEMLIP）,采用C语言编写了能够模拟边坡失稳滑塌全过程的Ellipsis程序,并通过一个典型案例对该方法的正确性和可行性进行了验证.采用该方法分析了边坡在不同坡角和强度条件下的稳定性和滑坡过程.研究结果表明,Lagrange积分点有限元法可以较准确地模拟边坡的潜在滑移面,并且可以模拟边坡失稳后的滑坡发展过程,为边坡滑坡大变形分析提供了一种新的数值计算方法.     

A reduced domain strategy for local mesh movement application in unstructured grids

Automatic control of mesh movement is mandatory in many fluid flow and fluid-solid interaction problems. This paper presents a new strategy, called reduced domain strategy (RDS), which enhances the efficiency of node connectivity-based mesh movement methods and moves the unstructured grid locally and effectively. The strategy dramatically reduces the grid computations by dividing the unstructured grid into two active and inactive zones. After any local boundary movement, the grid movement is performed only within the active zone. To enhance the efficiency of our strategy, we also develop an automatic mesh partitioning scheme. This scheme benefits from a new quasi-structured mesh data ordering, which determines the boundary of active zone in the original unstructured grid very easily. Indeed, the new partitioning scheme eliminates the need for sequential reordering of the original unstructured grid data in different mesh movement applications. We choose the spring analogy method and apply our new strategy to perform local mesh movements in two boundary movement problems including a multi-element airfoil with moving slat or deforming main body section. We show that the RDS is robust and cost effective. It can be readily employed in different node connectivity-based mesh movement methods. Indeed, the RDS provides a flexible local grid deformation tool for moving grid applications.     

扩散方程保正的有限体积格式

在大变形网格上数值求解多介质扩散方程时, 如何构造具有保正性的扩散格式一直是人们关注的难题. 本文将简要综述与保正性相关的扩散格式的研究历史, 并为解决这一难题提出新的设计途径,构造出新的具有较高精度的单元中心型守恒保正格式, 它们可兼顾网格几何变形和物理量变化. 本文将给出数值实验结果, 验证新格式在变形的网格上保持非负性.     

Design of Finite Element Tools for Coupled Surface and Volume Meshes

Many problems with underlying variational structure involve a coupling of volume with surface effects.A straight-forward approach in a finite element discretiza- tion is to make use of the surface triangulation that is naturally induced by the volume triangulation.In an adaptive method one wants to facilitate"matching"local mesh modifications,i.e.,local refinement and/or coarsening,of volume and surface mesh with standard tools such that the surface grid is always induced by the volume grid. We describe the concepts behind this approach for bisectional refinement and describe new tools incorporated in the finite element toolbox ALBERTA.We also present several important applications of the mesh coupling.     

Front tracking for two-phase flow in reservoir simulation by adaptive mesh

In this article, an algorithm for the numerical approximation of two-phase flow in porous media by adaptive mesh is presented. A convergent and conservative finite volume scheme for an elliptic equation is proposed, together with the finite difference schemes, upwind and MUSCL, for a hyperbolic equation on grids with local refinement. Hence, an IMPES method is applied in an adaptive composite grid to track the front of a moving solution. An object-oriented programmation technique is used. The computational results for different examples illustrate the efficiency of the proposed algorithm. © 1997 John Wiley & Sons, Inc. Numer Methods Partial Differential Eq 13: 673–697, 1997     

Numerical solution of transonic and subsonic flow of compressible inviscid and viscous fluid

The work presents two numerical solutions of compressible flows problems with high and very low Mach numbers. Both problems are numerically solved by finite volume method and the explicit MacCormack scheme using a grid of quadrilateral cells. Moved grid of quadrilateral cells is considered in the form of conservation laws using Arbitrary Lagrangian–Eulerian method. In the first case, inviscid transonic flow through cascade DCA 8% is presented and the numerical results are compared to experimental data. The second case, numerical solution of unsteady viscous flow in the channel for upstream Mach number M_∞=0.012 and frequency of the wall motions 100 Hz is presented. The unsteady case can represent a simplified model of airflow coming from the trachea, through the glottal region with periodically vibrating vocal folds to the human vocal tract. (© 2008 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim)     

Multigrid fictitious boundary and grid deformation methods for flow-induced rotation of wing

Numerical simulations of flow-induced rotation of wing by multigrid fictitious boundary and grid deformation methods are presented. The flow is computed by a special ALE formulation with a multigrid finite element solver. The solid wing is allowed to move freely through the computational mesh which is adaptively aligned by a special mesh deformation method. The advantage of this approach is that no expensive remeshing has to be performed. (© 2008 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim)     

A robust iterative scheme for finite volume discretization of diffusive flux on highly skewed meshes

A new approach for diffusive flux discretization on a nonorthogonal mesh for finite volume method is proposed. This approach is based on an iterative method, Deferred correction introduced by M. Peric [J.H. Fergizer, M. Peric, Computational Methods for Fluid Dynamics, Springer, 2002]. It converges on highly skewed meshes where the former approach diverges. A convergence proof of our method is given on arbitrary quadrilateral control volumes. This proof is founded on the analysis of the spectral radius of the iteration matrix. This new approach is applied successfully to the solution of a Poisson equation in quadrangular domains, meshed with highly skewed control volumes. The precision order of used schemes is not affected by increasing skewness of the grid. Some numerical tests are performed to show the accuracy of the new approach.     

Two-level multiscale finite element methods for the steady navier-stokes problem

In this article, on the basis of two-level discretizations and multiscale finite element method, two kinds of finite element algorithms for steady Navier-Stokes problem are presented and discussed. The main technique is first to use a standard finite element discretization on a coarse mesh to approximate low frequencies, then to apply the simple and Newton scheme to linearize discretizations on a fine grid. At this process, multiscale finite element method as a stabilized method deals with the lowest equal-order finite element pairs not satisfying the inf-sup condition. Under the uniqueness condition, error analyses for both algorithms are given. Numerical results are reported to demonstrate the effectiveness of the simple and Newton scheme.     

TWO-LEVEL MULTISCALE FINITE ELEMENT METHODS FOR THE STEADY NAVIER-STOKES PROBLEM

In this article, on the basis of two-level discretizations and multiscale finite element method, two kinds of finite element algorithms for steady Navier-Stokes problem are presented and discussed. The main technique is first to use a standard finite element discretization on a coarse mesh to approximate low frequencies, then to apply the simple and Newton scheme to linearize discretizations on a fine grid. At this process, multiscale finite element method as a stabilized method deals with the lowest equal-order finite element pairs not satisfying the inf-sup condition. Under the uniqueness condition, error analyses for both algorithms are given. Numerical results are reported to demonstrate the effectiveness of the simple and Newton scheme.     

Solidification modelling with a control volume method on domains subjected to viscoplastic deformation

The rise in importance of semi-solid based products has created a need for accurate modelling approaches to coupled solidification and deformation. Current approaches to solidification modelling, using the finite element method (FEM), are principally founded on capacitance methods. Unfortunately they suffer from a major drawback in that energy is not correctly transported through elements, so providing a source of inaccuracy. This paper is concerned with the development and application of a control volume capacitance method (CVCM) to problems where viscoplastic deformation and solidification are combined. The approach adopted is founded on the theory that describes energy transfer through a control volume (CV) moving relative to the deforming mass. This essentially arbitrary Lagrangian–Eulerian (ALE) method facilitates the accurate treatment of discontinuities. The CV approach is tested against known analytical solutions and is shown to be accurate, stable and computationally competitive.     

Two-grid methods for characteristic finite volume element solution of semilinear convection-diffusion equations

Two-grid methods for characteristic finite volume element solutions are presented for a kind of semilinear convection-dominated diffusion equations. The methods are based on the method of characteristics, two-grid method and the finite volume element method. The nonsymmetric and nonlinear iterations are only executed on the coarse grid (with grid size H). And the fine-grid solution (with grid size h) can be obtained by a single symmetric and linear step. It is proved that the coarse grid can be much coarser than the fine grid. The two-grid methods achieve asymptotically optimal approximation as long as the mesh sizes satisfy H = O(h^1/3).