Numerical simulation of fluid flows using an unstructured finite volume method with adaptive tri‐tree grids

A tri‐tree grid generation procedure is developed together with a finite volume method on the unstructured grid for solving the Navier–Stokes equations. A hierarchic numbering system for the data structure is used. The grid is adapted by adding and removing cell elements dependent on the vorticity magnitude. A special treatment is developed to ensure good quality triangular elements around the cylinder boundary. The adopted finite volume method is based on the cell‐centred scheme. The pressure–velocity coupling is treated using the SIMPLE algorithm. A modified QUICK scheme for unstructured grids is derived. The developed method is used to simulate the flow past a single and multiple cylinders at low Reynolds number. The obtained results are in good agreement with the published data. Copyright © 2002 John Wiley & Sons, Ltd.     

Local a posteriori error estimates and adaptive control of pollution effects

Local a posteriori error estimators are derived for linear elliptic problems over general polygonal domains in 2d. The estimators lead to a sharp upper bound for the energy error in a local region of interest. This upper bound consists of H¹‐type local error indicators in a slightly larger subdomain, plus weighted L²‐type local error indicators outside this subdomain, which account for the pollution effects. This constitutes the basis of a local adaptive refinement procedure. Numerical experiments show a superior performance than the standard global procedure as well as the generation of locally quasi‐optimal meshes. © 2003 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq 19: 421–442, 2003     

The hp‐version of the BEM with quasi‐uniform meshes for a three‐dimensional crack problem: The case of a smooth crack having smooth boundary curve

The article considers a three‐dimensional crack problem in linear elasticity with Dirichlet boundary conditions. The crack in this model problem is assumed to be a smooth open surface with smooth boundary curve. The hp‐version of the boundary element method with weakly singular operator is applied to approximate the unknown jump of the traction which is not L₂‐regular due to strong edge singularities. Assuming quasi‐uniform meshes and uniform distributions of polynomial degrees, we prove an a priori error estimate in the energy norm. The estimate gives an upper bound for the error in terms of the mesh size h and the polynomial degree p. It is optimal in h for any given data and quasi‐optimal in p for sufficiently smooth data. © 2007 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq, 2008     

An adaptive discretization of shallow‐water equations based on discontinuous Galerkin methods

In this paper, we present a discontinuous Galerkin formulation of the shallow‐water equations. An orthogonal basis is used for the spatial discretization and an explicit Runge–Kutta scheme is used for time discretization. Some results of second‐order anisotropic adaptive calculations are presented for dam breaking problems. The adaptive procedure uses an error indicator that concentrates the computational effort near discontinuities like hydraulic jumps. Copyright © 2006 John Wiley & Sons, Ltd.     

基于二重网格的动网格松弛法改进

相比传统的弹簧法等方法,基于球松弛算法的动网格松弛法在复杂边界大变形条件下可以得到质量更高的边界网格以及更大的极限变形量,但该方法在时间效率上还有提升的空间.引入二重网格,采用动网格松弛法进行稀疏网格的网格变形,将边界位移传递到整个网格计算域;再利用二重网格映射,将稀疏网格位移映射到原有计算网格的节点上.算例表明,改进...     

Upwind-biased FORCE schemes with applications to free-surface shallow flows

In this paper we develop numerical fluxes of the centred type for one-step schemes in conservative form for solving general systems of conservation laws in multiple-space dimensions on structured meshes. The proposed method is an extension of the multidimensional FORCE flux developed by Toro et al. (2009) [14]. Here we introduce upwind bias by modifying the shape of the staggered mesh of the original FORCE method. The upwind bias is evaluated using an estimate of the largest eigenvalue, which in any case is needed for selecting a time step. The resulting basic flux is first-order accurate and monotone. For the linear advection equation, the proposed UFORCE method reproduces exactly the upwind Godunov method. Extension to non-linear systems has been done empirically via the two-dimensional inviscid shallow water equations. Second order of accuracy in space and time on structured meshes is obtained in the framework of finite volume methods. The proposed method improves the accuracy of the solution for small Courant numbers and intermediate waves associated with linearly degenerate fields (contact discontinuities, shear waves and material interfaces). It achieves comparable accuracy to that of upwind methods with approximate Riemann solvers, though retaining the simplicity and efficiency of centred methods. The performance of the schemes is assessed on a suite of test problems for the two-dimensional shallow water equations.     

UNIFORM SUPERCONVERGENCE OF GALERKIN METHODS FOR SINGULARLY PERTURBED PROBLEMS

In this paper, we are concerned with uniform superconvergence of Galerkin methods for singularly perturbed reaction-diffusion problems by using two Shishkin-type meshes. Based on an estimate of the error between spline interpolation of the exact solution and its numerical approximation, an interpolation post-processing technique is applied to the original numerical solution. This results in approximation exhibit superconvergence which is uniform in the weighted energy norm. Numerical examples are presented to demonstrate the effectiveness of the interpolation post-processing technique and to verify the theoretical results obtained in this paper.     

Small perturbations of polynomial meshes

We show that the property of being a (weakly) admissible mesh for multivariate polynomials is preserved by small perturbations on real and complex Markov compacts. Applications are given to smooth transformations of polynomial meshes and to polynomial interpolation.     

各向异性网格下抛物方程一个新的非协调混合元收敛性分析

本文将 Crouzeix-Raviart 型非协调线性三角形元应用到抛物方程,建立了一个新的混合元格式.在抛弃传统有限元分析的必要工具 Ritz 投影算子的前提下,直接利用单元的插值性质和导数转移技巧, 分别得到了各向异性剖分下关于原始变量u 的H^-1-模和积分意义下L²-模以及通量p=-▽u 在L²-模下的最优阶误差估计.数值结果与我们的理论分析是相吻合的.