TWO-GRID METHOD FOR CHARACTERISTICS FINITE-ELEMENT SOLUTION OF 2D NONLINEAR CONVECTION-DOMINATED DIFFUSION PROBLEM

Introduction Convection diffusionequationgovernssuchphenomenaastheflowofheatwithina movingfluid,thetransportofdissolvednutrientsorcontaminantswithinthegroundwater,andthetransportofasurfactantortracerwithinanincompressibleoilinapetroleum reservoir.Weconsid…     

Analysis of a two-grid method for semiconductor device problem

The mathematical model of a semiconductor device is governed by a system of quasi-linear partial differential equations.The electric potential equation is approximated by a mixed finite element method,and the concentration equations are approximated by a standard Galerkin method.We estimate the error of the numerical solutions in the sense of the Lqnorm.To linearize the full discrete scheme of the problem,we present an efficient two-grid method based on the idea of Newton iteration.The main procedures are to solve the small scaled nonlinear equations on the coarse grid and then deal with the linear equations on the fine grid.Error estimation for the two-grid solutions is analyzed in detail.It is shown that this method still achieves asymptotically optimal approximations as long as a mesh size satisfies H=O(h^1/2).Numerical experiments are given to illustrate the efficiency of the two-grid method.     

Local and parallel finite element algorithms for time-dependent convection-diffusion equations

Local and parallel finite element algorithms based on two-grid discretization for the time-dependent convection-diffusion equations are presented. These algorithms are motivated by the observation that, for a solution to the convection-diffusion problem, low frequency components can be approximated well by a relatively coarse grid and high frequency components can be computed on a fine grid by some local and parallel proce- dures. Hence, these local and parallel algorithms only involve one small original problem on the coarse mesh and some correction problems on the local fine grid. One technical tool for the analysis is the local a priori estimates that are also obtained. Some numerical examples are given to support our theoretical analvsis.     

Local and parallel finite element algorithms based on two-grid discretization for steady Navier-Stokes equations

Local and parallel finite element algorithms based on two-grid discretization for Navier-Stokes equations in two dimension are presented. Its basis is a coarse finite element space on the global domain and a fine finite element space on the subdomain. The local algorithm consists of finding a solution for a given nonlinear problem in the coarse finite element space and a solution for a linear problem in the fine finite element space, then droping the coarse solution of the region near the boundary. By overlapping domain decomposition, the parallel algorithms are obtained. This paper analyzes the error of these algorithms and gets some error estimates which are better than those of the standard finite element method. The numerical experiments are given too. By analyzing and comparing these results, it is shown that these algorithms are correct and high efficient.     

Two-grid methods for semi-linear elliptic interface problems by immersed finite element methods

In this paper, two-grid immersed finite element(IFE) algorithms are proposed and analyzed for semi-linear interface problems with discontinuous diffusion coefficients in two dimension. Because of the advantages of finite element(FE) formulation and the simple structure of Cartesian grids, the IFE discretization is used in this paper. Two-grid schemes are formulated to linearize the FE equations. It is theoretically and numerically illustrated that the coarse space can be selected as coarse asH= O(h~(1/4))(orH=O(h~(1/8))), and the asymptotically optimal approximation can be achieved as the nonlinear schemes. As a result, we can settle a great majority of nonlinear equations as easy as linearized problems. In order to estimate the present two-grid algorithms, we derive the optimal error estimates of the IFE solution in theL pnorm. Numerical experiments are given to verify the theorems and indicate that the present two-grid algorithms can greatly improve the computing efficiency.     

PARTITION OF UNITY FINITE ELEMENT METHOD FOR SHORT WAVE PROPAGATION IN SOLIDS

IntroductionIt is known that standard finite element procedure is unable to simulate the wavepropagation with high oscillations or gradients in space in the media with reasonableefficiency and accuracy due to the nature of polynomial interpolation approxi…     

基于单位分解法的无网格数值流形方法

在数值流形方法和单位分解法的基础上，提出了无网格数值流形方法. 无网格数值流形 方法在分析时采用了双重覆盖系统，即数学覆盖和物理覆盖. 数学覆盖提供的节点形成求解 域的有限覆盖和单位分解函数；而物理覆盖描述问题的几何区域及其域内不连续性. 与原有 的数值流形方法相比，无网格数值流形方法的数学覆盖形状更加灵活，可以用一系列节点的 影响域来建立数学覆盖和单位分解函数，具有无网格方法的特性，从而摆脱了传统的数值流 形方法中网格所带来的困难. 与无网格方法相比，由于采用了有限覆盖技术，试函数的构造 不受域内不连续的影响，克服了原有的无网格方法在处理不连续问题时所遇到的困难. 详细推导了无网格数值流形方法的试函数和求解方程，最后给出了算例，验证了该方法的正 确性.     

Parallel finite element computation of incompressible magnetohydrodynamics based on three iterations

Based on local algorithms, some parallel finite element(FE) iterative methods for stationary incompressible magnetohydrodynamics(MHD) are presented. These approaches are on account of two-grid skill include two major phases: find the FE solution by solving the nonlinear system on a globally coarse mesh to seize the low frequency component of the solution, and then locally solve linearized residual subproblems by one of three iterations(Stokes-type, Newton, and Oseen-type) on subdomains with fine...     

多网格防阻块护栏系统的冲击响应

防阻块薄壁结构是护栏系统中重要的吸能元件,通过防阻块的优化设计提升护栏防护能力,最大限度地减小对乘员的伤害具有重要的意义。应用有限元软件ABAQUS,建立了单跨多网格防阻块波形梁护栏系统受质量块冲击的有限元分析模型,计算了壁厚分别为2、3和4 mm的多网格防阻块护栏系统的冲击响应过程,分析了3种不同壁厚的防阻块在给定冲击速度下,内能的变化情况。通过对比分析发现:冲击过程中,壁厚2 mm的双网格防阻块最终内能质量比明显高于其他情形,能量吸收能力最强;并且壁厚2 mm双网格防阻块情形的质量块加速度峰值比壁厚3 mm单网格防阻块情形小约47.6%,更有利于乘员安全。此外,根据WSTC曲线,模拟工况下,冲击引起的人体头部减速度均在安全忍受范围内。     

Variational multiscale analysis of elastoplastic deformation using meshfree approximation

A variational multiscale method has been presented for efficient analysis of elastoplastic deformation problems. Severe deformation occurs in plastic region and leads to high gradient displacement. Therefore, solution needs to be refined to properly capture local deformation in plastic region. In this work, scale decomposition based on variational formulation is presented. A coarse scale and a fine scale are introduced to represent global and local behavior, respectively. The displacement is decomposed into a coarse and a fine scale. Subsequently the problem is also decomposed into a coarse and a fine scale from the variational formulation. Each scale variable is approximated using meshfree method. Adaptivity can easily and nicely be implemented in meshfree method. As a method of increasing resolution, extrinsic enrichment of partition of unity is used. Each scale problem is solved iteratively and conversed results are obtained consequently. Iteration procedure is indispensable for the elastoplastic deformation analysis. Therefore iterative solution procedure of each scale problem is naturally adequate. The proposed method is applied to the Prandtl’s punch test and shear band problem. The results are compared with those of other methods and the validity of the proposed method is demonstrated.     

Turbulent heat and mass transfer over a rotating disk for the Prandtl or Schmidt numbers much larger than unity: an integral method

Turbulent heat and mass transfer of a rotating disk for Prandtl and Schmidt numbers much larger than unity was modeled using an integral method validated against empirical equations of different authors for Sherwood numbers. As shown, decrease in relative thickness of thermal/diffusion boundary layers with increasing local radii entails additional increase of the exponent at the Reynolds number in expressions for Nusselt and Sherwood numbers in comparison with air flows.     

Meshfree analysis of softening elastoplastic solids using variational multiscale method

A meshfree multiscale method is presented for efficient analysis of elastoplastic solids. In the analysis of softening elastoplastic solids, standard finite element methods or meshfree methods typically yield mesh-dependent results. The reason for this well-known effect is the loss of ellipticity of the boundary value problem. In this work, the scale decomposition is carried out based on a variational form of the problem. A coarse scale is designed to represent global behavior and a fine scale to represent local behavior. A fine scale region is detected from the local failure analysis of an acoustic tensor to indicate a region where deformation changes abruptly. Each scale variable is approximated using a meshfree method. Meshfree approximation is well-suited for adaptivity. As a method of increasing the resolution, a partition of unity based extrinsic enrichment is used. In particular, fine scale approximations are designed to appropriately represent local behavior by using a localization angle. Moreover, the regularization effect through the convexification of non-convex potential is embedded to represent fine scale behavior. Each scale problem is solved iteratively. The proposed method is applied to shear band problems. In the results of analysis about pure shear and compression problems, straight shear bands can be captured and mesh-insensitive results are obtained. Curved shear bands can also be captured without mesh dependency in the analysis of indentation problem.     

Global stability of a delayed SEIRS epidemic model with saturation incidence rate

In this paper, an SEIRS epidemic model with a saturation incidence rate and a time delay describing a latent period is investigated. By analyzing the corresponding characteristic equations, the local stability of an endemic equilibrium and a disease-free equilibrium is established. When the basic reproduction number is greater than unity, by means of an iteration technique, sufficient conditions are obtained for the global asymptotic stability of the endemic equilibrium. By comparison arguments, it is proved that if the basic reproduction number is less than unity, the disease-free equilibrium is globally asymptotically stable. Numerical simulations are carried out to illustrate the main theoretical results.     

Local moving least square‐one‐dimensional integrated radial basis function networks technique for incompressible viscous flows

This paper presents a local moving least square‐one‐dimensional integrated radial basis function networks method for solving incompressible viscous flow problems using stream function‐vorticity formulation. In this method, the partition of unity method is employed as a framework to incorporate the moving least square and one‐dimensional integrated radial basis function networks techniques. The major advantages of the proposed method include the following: (i) a banded sparse system matrix which helps reduce the computational cost; (ii) the Kronecker‐ δ property of the constructed shape function which helps impose the essential boundary condition in an exact manner; and (iii) high accuracy and fast convergence rate owing to the use of integration instead of conventional differentiation to construct the local radial basis function approximations. Several examples including two‐dimensional (2D) Poisson problems, lid‐driven cavity flow and flow past a circular cylinder are considered, and the present results are compared with the exact solutions and numerical results from other methods in the literature to demonstrate the attractiveness of the proposed method. Copyright © 2012 John Wiley & Sons, Ltd.     

Over-nonlocal gradient enhanced plastic-damage model for concrete

Classical continuum models exhibit strong mesh dependency during softening. One method to regularize the problem is to introduce a length scale parameter via the nonlocal formulation. However, standard nonlocal enhancement (either by integral or gradient formulation) may serve only as a partial localization limiter for many material models. The “over-nonlocal” formulation, where the weight for the nonlocal value is greater than unity and the excesses compensated by assigning a negative weight to the local value, is able to fully regularize certain material models when standard nonlocal enhancement fails to do so. A plastic-damage model for concrete is formulated with this over-nonlocal enhancement via the gradient approach and the full regularizing capabilities demonstrated.     

固体中短波传播单位分解有限元法的解析积分

针对固体中短波传播数值模拟的单位分解有限元法中单元矩阵积分的被积函数的强烈振荡特性,应用直角坐标系下标准有限元形函数和单元内的波动方向知识提出了一种单元矩阵的解析积分方案。它对于平面三,六,四,八和九节点的直边单位分解有限单元是完全解析的,对于与这些单元相应的曲边单元则是半解析的。数值结果显示所提出的积分方案在计算效率上比高斯-勒让德积分有大幅度提高。     

Enriched meshless manifold method for two-dimensional crack modeling

The meshless manifold method is based on the partition of unity method and the finite cover approximation theory which provides a unified framework for solving problems dealing with both continuum with and without discontinuities. The meshless manifold method employs two cover systems. The mathematical cover system provides the nodes for forming finite covers of the solution domain and the partition of unity functions. And the physical cover system describes geometry of the domain and the discontinuous surfaces in the domain. The shape functions are derived by the partition of unity and the finite covers approximation theory. In meshless manifold method, the mathematical finite cover approximation theory is used to model cracks that lead to interior discontinuities in the displacement. Therefore, the discontinuity is treated mathematically instead of empirically by the existing methods. However, one cover of a node is divided into two irregular sub-covers when the meshless manifold method is used to model the discontinuity. As a result, the method sometimes causes numerical errors at the tip of a crack. To improve the precision of the meshless manifold method, the enriched methods are introduced in this work for crack problems.     

TWO-GRID ERROR ESTIMATES FOR THE STREAM FUNCTION FORM OF NAVIER-STOKES EQUATIONS

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基于适合分析T样条的高阶数值流形方法

数值流形方法是一种非常灵活的数值计算方法,连续体的有限单元方法和块体系统的非连续变形分析方法只是这一数值方法的特例.数值流形方法中高阶位移函数的构造可通过提高权函数的阶次来实现,这种方法往往需要沿单元边界配置适当的边内节点,这些结点的出现增加了前处理的复杂性,特别是对于大型复杂的空间问题.另一方面,在数值流形方法中可通过缩小单元尺寸(h加密)来提高求解精度.当模拟裂纹扩展时,这种细化策略可用来克服裂纹尖端的奇异性.一个传统的解决方案是细化整个网格,但这会导致计算效率的显著降低.将适合分析的T样条(analysis-suitable T-spline,AST)引入数值流形方法中来建立高阶数值流形方法的分析格式,有效的避免了该问题的出现.AST样条基函数具有线性无关,单位分解,局部加密等许多重要性质,使得其非常适合用于工程设计及分析.在引入AST样条后,可通过改变数学覆盖的构造形式建立不同阶次的数值流形方法分析格式;AST样条自身的局部加密性质也使得数值流形方法中的数学网格局部加密更容易实现.算例结果表明:随着AST样条基函数阶次的提高,数值流形方法的计算结果有了明显的改善;基于AST样条基函数的数值流形方法在保持计算精度的前提下降低了自由度的数量.     

Diffusion-based coarse graining in hybrid continuum–discrete solvers: Theoretical formulation and a priori tests

Coarse graining is an important ingredient in many multi-scale continuum–discrete solvers such as CFD–DEM (computational fluid dynamics–discrete element method) solvers for dense particle-laden flows. Although CFD–DEM solvers have become a mature technique that is widely used in multiphase flow research and industrial flow simulations, a flexible and easy-to-implement coarse graining algorithm that can work with CFD solvers of arbitrary meshes is still lacking. In this work, we proposed a new coarse graining algorithm for continuum–discrete solvers for dense particle-laden flows based on solving a transient diffusion equation. Via theoretical analysis we demonstrated that the proposed method is equivalent to the statistical kernel method with a Gaussian kernel, but the current method is much more straightforward to implement in CFD–DEM solvers. A priori numerical tests were performed to obtain the solid volume fraction fields based on given particle distributions, the results obtained by using the proposed algorithm were compared with those from other coarse graining methods in the literature (e.g., the particle centroid method, the divided particle volume method, and the two-grid formulation). The numerical tests demonstrated that the proposed coarse graining procedure based on solving diffusion equations is theoretically sound, easy to implement and parallelize in general CFD solvers, and has improved mesh-convergence characteristics compared with existing coarse graining methods. The diffusion-based coarse graining method has been implemented into a CFD–DEM solver, the results of which are presented in a separate work.