An equivalent multiscale method for 2D static and dynamic analyses of lattice truss materials

A uniform multiscale computational method is developed for 2D static and dynamic analyses of lattice truss materials in elasticity based on the extended multiscale finite element method. A kind of multi-node coarse element is proposed to describe the more complex deformations compared with the original four-node coarse element and the mode base functions are added into the original multiscale base functions to consider the effects of inertial forces for the dynamic problems. The constructions of the displacement and mode base functions are introduced in detail. In addition, the orthogonality of the displacement and mode base functions are also proved, which indicates that the macroscopic displacement DOF and modal DOF are irrelevant and independent of each other. Finally, some numerical experiments are carried out to verify the validity and efficiency of the proposed method by comparison with the reference solution obtained by the standard finite element method on the fine mesh.     

A short and versatile finite element multiscale code for homogenization problems

We describe a multiscale finite element (FE) solver for elliptic or parabolic problems with highly oscillating coefficients. Based on recent developments of the so-called heterogeneous multiscale method (HMM), the algorithm relies on coupled macro- and microsolvers. The framework of the HMM allows to design a code whose structure follows the classical finite elements implementation at the macro level. To account for the fine scales of the problem, elementwise numerical integration is replaced by micro FE methods on sampling domains. We discuss a short and flexible FE implementation of the multiscale algorithm, which can accommodate simplicial or quadrilateral FE and various coupling conditions for the constrained micro simulations. Extensive numerical examples including three dimensional and time dependent problems are presented illustrating the efficiency and the versatility of the computational strategy.     

Efficient structure topology optimization by using the multiscale finite element method

Efficient SIMP and level set based topology optimization schemes are proposed based on the computation framework of the multiscale finite element method (MsFEM). In the proposed optimization schemes, the equilibrium equations are solved on a coarse-scale mesh and the design variables are updated on a fine-scale mesh. To describe more complex deformation, a multi-node coarse element is also presented in the MsFEM computation. In the MsFEM, a multiscale shape function is constructed numerically and employed to obtain the equivalent stiffness matrix and load vector of the multi-node coarse element. In the optimization schemes with the MsFEM, the coarse elements are divided into two categories: homogeneous and heterogeneous. For the homogeneous coarse elements, their multiscale shape functions are constructed only once before the iterations. Since the material distribution is varying locally in most of the iterations, one only needs to reconstruct them of a small part of the coarse elements where the material distribution is changed by comparison with that in the previous iteration step. This will save lots of computational cost. In addition, due to the independence of each coarse element, the constructions of the multiscale shape functions could be easily proceeded in parallel. In this work, the computational accuracy and efficiency of this method is investigated in detail, as well as the speedup ratio and parallel efficiency when using multiple processors to construct the multiscale shape functions simultaneously. Furthermore, several 2D and 3D examples show the effectiveness and efficiency of the proposed optimization schemes based on the MsFEM analysis framework.     

Multiscale-stabilized solutions to one-dimensional systems of conservation laws

We present a variational multiscale formulation for the numerical solution of one-dimensional systems of conservation laws. The key idea of the proposed formulation, originally presented by Hughes [Comput. Methods Appl. Mech. Engrg., 127 (1995) 387–401], is a multiple-scale decomposition into resolved grid scales and unresolved subgrid scales. Incorporating the effect of the subgrid scales onto the coarse scale problem results in a finite element method with enhanced stability properties, capable of accurately representing the sharp features of the solution. In the formulation developed herein, the multiscale split is invoked prior to any linearization of the equations. Special attention is given to the choice of the matrix of stabilizing coefficients and the discontinuity-capturing diffusion. The methodology is applied to the one-dimensional simulation of three-phase flow in porous media, and the shallow water equations. These numerical simulations clearly show the potential and applicability of the formulation for solving highly nonlinear, nearly hyperbolic systems on very coarse grids. Application of the numerical formulation to multidimensional problems is presented in a forthcoming paper.     

The dissipative structure of variational multiscale methods for incompressible flows

In this paper, we present a precise definition of the numerical dissipation for the orthogonal projection version of the variational multiscale method for incompressible flows. We show that, only if the space of subscales is taken orthogonal to the finite element space, this definition is physically reasonable as the coarse and fine scales are properly separated. Then we compare the diffusion introduced by the numerical discretization of the problem with the diffusion introduced by a large eddy simulation model. Results for the flow around a surface-mounted obstacle problem show that numerical dissipation is of the same order as the subgrid dissipation introduced by the Smagorinsky model. Finally, when transient subscales are considered, the model is able to predict backscatter, something that is only possible when dynamic LES closures are used. Numerical evidence supporting this point is also presented.     

一种新的基于多尺度似然比检验的SAR图像分割方法*

提出的图像分割新算法利用当图像分辨率改变时,不同目标斑点模式变化方式的不同以及相邻图像尺度间的Markov性,推导得出了多尺度似然比的表达式;该方法同时考虑了多尺度自回归（MAR）模型产生的残差信息和较粗尺度图像的灰度信息,增强了区分度,分割结果更精确;考虑了被分类像素的邻域特性,使其对噪声不敏感,具有稳健性。实验结果表明分割效果是显著的。     

基于格子Boltzmann方法的一维Burgers方程的数值模拟

基于格子Boltzmann方法(LBM)的一维Burgers方程的数值解法,已有2-bit和4-bit模型。文中通过选择合适的离散速度模型构造出恰当的平衡态分布函数; 然后, 利用单松弛的格子Bhatnagar-Gross-Krook模型、Chapman-Enskog展开和多尺度技术, 提出了用于求解一维Burgers方程的3-bit的格子Boltzmann模型,即D1Q3模型,并进行了数值实验。实验结果表明,该方法的数值解与解析解吻合的程度很好,且误差比现有文献中的误差更小,从而验证了格子Boltzamnn模型的有效性。     

扩展的多尺度有限元法基本原理

阐述一种适用于非均质材料力学性能分析的扩展的多尺度有限元法（Extended Multiscale Finite Element Method,EMsFEM）的基本原理．该方法的基本思想是利用数值方法构造能反映胞体单元内部材料非均质影响的多尺度基函数,在此基础上求得粗网格层次的等效单元刚度阵,从而在粗网格尺度上对原问题进行求解,很大程度地减少计算量．以该方法进行的具有周期和随机微观结构的材料计算示例,通过与传统有限元法的结果比较,说明这一方法的有效性．EMsFEM的优势在于,能容易地进行降尺度计算,可较准确地求得单元内部的微观应力应变信息,在非均质材料强度和非线性分析中有很大的应用潜力．     

Unified analysis of enriched finite elements for modeling cohesive cracks

Aiming to provide a justified theoretical ground based on which different enriched finite elements can be systematically compared, we developed in this work a unified framework for modeling cohesive cracks using the variational multiscale method. The kinematics (i.e. coarse and fine scale displacement and strain fields) and statics (i.e. coarse and fine scale equilibrium equations) are thoroughly investigated in both continuum and discrete settings. With respect to the fine scale kinematics and statics adopted in the embedded finite element method (EFEM) and the extended finite element method (XFEM), we mainly discuss four groups of enriched finite elements with non-uniform discontinuity modes, i.e. the kinematically optimal symmetric EFEM and XFEM, as well as, the kinematically and statically optimal non-symmetric EFEM and XFEM. In all these methods, the enrichment parameters can be regarded either as element-wise local or continuous global variables. The enriched finite elements with material/element/node enrichments are then exemplified in an increasing order of their respective capabilities in representing the fine scale kinematics, and the interrelations between them are also discussed. Owing to this unified framework, we are able to clarify some existing points of view related to EFEM and XFEM. It is found that, if the same fine scale statics is used and the same local/global property is assumed for the involved enrichment parameters, XFEM can be regarded as a kinematically enhanced EFEM since it accounts for a more general fine scale kinematics than that (i.e. relative rigid body motions and self-stretching) considered in EFEM. Finally, several simple, but sufficiently representative, numerical examples are presented to show the significance in appropriately reflecting the fine scale kinematics and statics in an enriched finite element for modeling cohesive cracks.     

基于改进CORDIC算法的正余弦编码器信号细分技术

为了获取更精确的电机位置等实时反馈信息,提高数控系统等闭环控制系统的反馈精度,主要设计了基于改进坐标旋转数字计算(CORDIC)算法的正余弦编码器高分辨率信号处理方法.首先通过信号调理电路对编码器信号进行差分放大和整形滤波等处理;然后四倍频计数得到粗码信息,并通过基于改进CORDIC算法的电子学细分方法获得精插补位置信息;最后,将粗码和精码信息整合,得到高分辨率和高精度的电机角位置信息.实验结果显示,与算法优化改进前相比,该方法信号测量误差明显减少,精度较高.     

Development of a Variational Multiscale Large-Eddy Simulation Code ‘MISTRAL’ Using Double-Scale Finite Elements

A variational multiscale large-eddy simulation (VMS-LES) code, named MISTRAL, has been developed based upon the finite element method (FEM) for accurate and practical computation of geometrically complicated turbulent flow problems. The numerical strategy of the FEM-based VMS-LES is explained, especially focusing on the double-scale approximation for velocity and pressure in the incompressible Navier-Stokes equations, a pressure stabilization technique and a multiscale turbulence modeling. A unique technique is also employed in the time integration to realize an efficient inversion of the multiscale mass matrix and to form the multiscale pressure Poisson equation used in the approximate projection method for divergence-free constraint of velocity. As a numerical demonstration, a 2D driven cavity flow problem has been solved with the MISTRAL code in a wide range of Reynolds number (Re=1000 to 50000). The results are compared with reference data to quantitatively estimate the accuracy (magnitude of errors in terms of L ² norms) of the proposed VMS-LES method.     

Efficient domain decomposition methods for elliptic problems arising from flows in heterogeneous porous media

In this paper we study domain decomposition methods for solving some elliptic problem arising from flows in heterogeneous porous media. Due to the multiple scale nature of the elliptic coefficients arising from the heterogeneous formations, the construction of efficient domain decomposition methods for these problems requires a coarse solver which is adaptive to the fine scale features, [4]. We propose the use of a multiscale coarse solver based on a finite volume – finite element formulation. The resulting domain decomposition methods seem to induce a convergence rate nearly independent of the aspect ratio of the extreme permeability values within the substructures. A rigorous convergence analysis based on the Schwarz framework is carried out, and we demonstrate the efficiency and robustness of the preconditioner through numerical experiments which include problems with multiple scale coefficients, as well as problems with continuous scales. Communicated by: G. Wittum     

A scalable multiscale LATIN method adapted to nonsmooth discrete media

The simulation of discrete systems often leads to large scale problems, for instance if they result of a discretization technique, or a modeling at a small scale.A multiscale analysis may involve an homogenized macroscopic problem, as well as a coarse space mechanism to accelerate convergence of the numerical scheme. A multilevel domain decomposition technique is used herein as both a numerical strategy to simulate the behaviour of a non smooth discrete media, and to provide a macroscopic numerical behaviour of the same system.Several generic formulations for such systems are discussed in this article. A multilevel domain decomposition is tested and several choices of the embedded coarse space are discussed, in particular with respect of the emergence of weak interfaces, characteristics of the discrete media substructuration. The application problem is the quasi-static simulation of a large scale tensegrity grid.     

Feature sensitive multiscale editing on surfaces

A novel editing method for large triangular meshes is presented. We detect surface features, such as edge and corners, by computing local zero and first surface moments, using a robust and noise resistant method. The feature detection is encoded in a finite element matrix, passed to an algebraic multigrid (AMG) algorithm. The AMG algorithm generates a matrix hierarchy ranging from fine to coarse representations of the initial fine grid matrix. This hierarchy comes along with a corresponding multiscale of basis functions, which reflect the surface features on all hierarchy levels. We consider either these basis functions or distinct sets from an induced multiscale domain decomposition as handles for surface manipulation. We present a multiscale editor which enables Boolean operations on this domain decomposition and simply algebraic operations on the basis functions. Users can interactively design their favorite surface handles by simple grouping operations on the multiscale of domains. Several applications on large meshes underline the effectiveness and flexibility of the presented tool.     

An adaptive Rothe method for the wave equation

The adaptive Rothe method approaches a time-dependent PDE as an ODE in function space. This ODE is solved virtually using an adaptive state-of-the-art integrator. The actual realization of each time-step requires the numerical solution of an elliptic boundary value problem, thus perturbing the virtual function space method. The admissible size of that perturbation can be computed a priori and is prescribed as a tolerance to an adaptive multilevel finite element code, which provides each time-step with an individually adapted spatial mesh. In this way, the method avoids the well-known difficulties of the method of lines in higher space dimensions. During the last few years the adaptive Rothe method has been applied successfully to various problems with infinite speed of propagation of information. The present study concerns the adaptive Rothe method for hyperbolic equations in the model situation of the wave equation. All steps of the construction are given in detail and a numerical example (diffraction at a corner) is provided for the 2D wave equation. This example clearly indicates that the adaptive Rothe method is appropriate for problems which can generally benefit from mesh adaptation. This should be even more pronounced in the 3D case because of the strong Huygens' principle. Accepted: 12 August 1997     

基于Java 3D技术和Swing技术的3D建模开发

该文介绍了基于Java技术中的Java3D技术和Swing技术的3D建模的开发。和其他技术开发3D模型相比,利用Java技术的面向对象技术开发的3D模型有着很多优势。比如,此3D模型支持多平台操作,适合编写非常复杂的应用程序。利用Java3D技术实现了盛放3D模型的基础类的开发,同时通过和Swing技术的合理结合实现了3D组件的用户图形界面化、用户操作简易化的特色。利用Java技术还克服了编程代码冗长、繁杂不利于管理这个技术难点。同时利用了Java技术的事件监听处理功能实现了对3D模型的编辑功能。优化了编程工作。文中以此3D组件在石油数值模拟软件中的应用为例说明了该组件具有的特色。     

Motion detail preserving optical flow estimation

A common problem of optical flow estimation in the multiscale variational framework is that fine motion structures cannot always be correctly estimated, especially for regions with significant and abrupt displacement variation. A novel extended coarse-to-fine (EC2F) refinement framework is introduced in this paper to address this issue, which reduces the reliance of flow estimates on their initial values propagated from the coarse level and enables recovering many motion details in each scale. The contribution of this paper also includes adaptation of the objective function to handle outliers and development of a new optimization procedure. The effectiveness of our algorithm is demonstrated by Middlebury optical flow benchmarkmarking and by experiments on challenging examples that involve large-displacement motion.     

Stochastic mixed multiscale finite element methods and their applications in random porous media

We present a framework for stochastic mixed multiscale finite element methods (mixed MsFEMs) for elliptic equations with heterogeneous random coefficients. The use of some global information is necessary in multiscale simulations when there is no scale separation for the heterogeneity. The methods in the proposed framework for the stochastic mixed MsFEMs use some global information. The media properties in a stochastic environment drastically vary among realizations and, thus, many global fields are needed for multiscale simulation. The computations of these global fields on a fine grid can be very expensive. One can utilize upscaling methods to compute the global information on an intermediate coarse grid that reduces the computational cost. We investigate two approaches of stochastic mixed MsFEMs in the framework. First approach entails no stochastic interpolation and the second approach uses stochastic interpolation. If the random media have deterministic features that play significant roles in the flow, we can use the deterministic features of the random media as the global information. This reduces the computational cost of the simulations. We make convergence analysis of the stochastic mixed MsFEMs and investigate their applications to incompressible two-phase flows in random porous media. The numerical results demonstrate the effectiveness of the proposed methods and confirm the convergence.