Dual Formulations of Mixed Finite Element Methods with Applications

Mixed finite element methods solve a PDE using two or more variables. The theory of Discrete Exterior Calculus explains why the degrees of freedom associated to the different variables should be stored on both primal and dual domain meshes with a discrete Hodge star used to transfer information between the meshes. We show through analysis and examples that the choice of discrete Hodge star is essential to the numerical stability of the method. Additionally, we define interpolation functions and discrete Hodge stars on dual meshes which can be used to create previously unconsidered mixed methods. Examples from magnetostatics and Darcy flow are examined in detail.     

机翼机身对接接头非线性有限元分析

接头是飞机结构中常见的结构连接形式,也往往是强度破坏或疲劳破坏的部位.有时为了更大的挖掘接头结构的承载能力,甚至还要考虑结构局部进入塑性区应力分布,属于材料非线性和力边界非线性的耦合问题.试图通过直接求解偏微分方程得到接头接触问题的解析解是很难的,甚至可以说是不可能的.基于非线性有限元分析方法,借助大型通用有限元软件MARC的弹塑性接触分析模块,通过不断细化网格的方法来对机翼机身对接接头进行非线性耦合分析.建立了对接接头的弹塑性接触有限元模型,给出了接头的应力分布以及屈服状况.数值模拟结果与实际情况相符,数值模拟结果为机身机翼对接接头的设计提供了重要的数据.     

An incremental elastic-plastic Finite Element solver in a workstation cluster environment Part I. Formulations and parallel processing

We discuss solution schemes for the incremental elastic-plastic structural problem, discretized by means of the Finite Element method. Attention is focused on their formulation and implementation in a parallel computing environment defined by a cluster of workstations connected by means of a network. The availability of parallel computers allows one to consider possible formulations and solution strategies so far not considered competitive with the classical Newton-like schemes implying the definition of an elastic-plastic tangent stiffness matrix. The solution strategies herein considered are based on the explicit integration of the actual elastic-plastic rate problem. This, in turn, is phrased in terms of two different formulations, whose relative advantages—particularly with respect to their integration in parallel—are discussed. A − gl (displacemen plastic multiplier) formulation of the structural rate theory of plasticity [1], integrated by means of an explicit, element-by-element scheme, seems to be the most promising one.     

八节点四边形等参元非线性有限元分析

应用几何非线性有限元方法 ,研究了大变形问题中的纸张力学特性 ,采用修正的拉格朗日格式 ,推导了八节点四边形等参元用于非线性问题分析时的增量平衡方程 ,并编写了有关有限元计算程序 .应用该程序 ,对纸张在悬臂状态及自重作用下的静变形、无阻尼动态响应以及有阻尼动态响应分别进行计算 ,并与日本理光公司提供的试验数据以及MARC的计算结果进行比较 ,证实了该方法对于纸张的非线性力学特性的分析是有效的 ,可以为办公设备的合理设计提供科学依据     

Unreinforced Masonry Walls: Nonlinear Finite Element Analysis with a Unified Constitutive Model

This paper presents a unified model for characterizing the behavior of masonry structures. The model is based on the Disturbed State Concept (DSC) with modified hierarchical single yield surface (HISS) plasticity, which is called DSC/HISS-CT and which is used to simulate the behavior of masonry structures. The model uses two HISS yield surfaces for compressive and tensile behavior. The proposed constitutive model is verified by comparing numerical predictions with results available from test data; the comparisons are found to be highly satisfactory. A new explicit formula is also presented to estimate the strength of unreinforced masonry structures.     

Error Analysis of Mixed Finite Element Methods for Nonlinear Parabolic Equations

In this paper, we prove a discrete embedding inequality for the Raviart–Thomas mixed finite element methods for second order elliptic equations, which is analogous to the Sobolev embedding inequality in the continuous setting. Then, by using the proved discrete embedding inequality, we provide an optimal error estimate for linearized mixed finite element methods for nonlinear parabolic equations. Several numerical examples are provided to confirm the theoretical analysis.     

Two-Grid Method for Nonlinear Reaction-Diffusion Equations by Mixed Finite Element Methods

In this paper, we investigate a scheme for nonlinear reaction-diffusion equations using the mixed finite element methods. To linearize the mixed method equations, we use the two-grid algorithm. First, we solve the original nonlinear equations on the coarse grid, then, we solve the linearized problem on the fine grid used Newton iteration once. It is shown that the algorithm can achieve asymptotically optimal approximation as long as the mesh sizes satisfy H=O(h^frac12)H=mathcal{O}(h^{frac{1}{2}}). As a result, solving such a large class of nonlinear equations will not much more difficult than the solution of one linearized equation.     

三维互连阻抗的混合边界积分方程提取算法

基于混合边界元三维互连阻抗提取方法,针对其离散线性方程组的特点,提出有效的稀疏矩阵组织和矩阵行列调整技术,以及一种预条件迭代求解技术,这些技术结合起来形成了一种有效的三维互连阻抗提取算法.该算法在保证计算精度的同时,速度优于MIT最新的提取算法FastImp.最后通过2个典型互连结构的数值实验验证了该算法的有效性.     

Real-Time Nonlinear Finite Element Computations on GPU - Application to Neurosurgical Simulation

Application of biomechanical modeling techniques in the area of medical image analysis and surgical simulation implies two conflicting requirements: accurate results and high solution speeds. Accurate results can be obtained only by using appropriate models and solution algorithms. In our previous papers we have presented algorithms and solution methods for performing accurate nonlinear finite element analysis of brain shift (which includes mixed mesh, different non-linear material models, finite deformations and brain-skull contacts) in less than a minute on a personal computer for models having up to 50.000 degrees of freedom. In this paper we present an implementation of our algorithms on a Graphics Processing Unit (GPU) using the new NVIDIA Compute Unified Device Architecture (CUDA) which leads to more than 20 times increase in the computation speed. This makes possible the use of meshes with more elements, which better represent the geometry, are easier to generate, and provide more accurate results.     

A Posteriori Error Estimates of Two-Grid Finite Element Methods for Nonlinear Elliptic Problems

In this article, we study the residual-based a posteriori error estimates of the two-grid finite element methods for the second order nonlinear elliptic boundary value problems. Computable upper and lower bounds on the error in the \(H^1\)-norm are established. Numerical experiments are also provided to illustrate the performance of the proposed estimators.     

非直线排列剥叶元件的疲劳寿命分析

阐述了整杆式甘蔗收获机械非直线排列剥叶元件疲劳分析的总体思路．利用计算机仿真分析的方法分析了剥叶元件装夹参数及其注塑加工过程中产生的缺陷对元件寿命的影响，提出了不同工况剥叶元件疲劳分析的有限元计算模型．通过仿真分析，方便、快捷地得出剥叶元件疲劳寿命预估值，并用试验证明了其有效性．     

Parallel Formulations of Decision-Tree Classification Algorithms

Classification decision tree algorithms are used extensively for data mining in many domains such as retail target marketing, fraud detection, etc. Highly parallel algorithms for constructing classification decision trees are desirable for dealing with large data sets in reasonable amount of time. Algorithms for building classification decision trees have a natural concurrency, but are difficult to parallelize due to the inherent dynamic nature of the computation. In this paper, we present parallel formulations of classification decision tree learning algorithm based on induction. We describe two basic parallel formulations. One is based on Synchronous Tree Construction Approach and the other is based on Partitioned Tree Construction Approach. We discuss the advantages and disadvantages of using these methods and propose a hybrid method that employs the good features of these methods. We also provide the analysis of the cost of computation and communication of the proposed hybrid method. Moreover, experimental results on an IBM SP-2 demonstrate excellent speedups and scalability.     

Optimal Control Formulations of Vibration Reduction Problems

Design of controls to move a flexible body such as a robot arm while minimizing unwanted vibrations has been described in many papers and presented in many forms. For the vibration reduction issue alone, it is shown that almost all the proposed designs can be formulated as optimal controls of either the fixed final time or the minimum time type. Furthermore, it is shown that under reasonable assumptions the two types have the same solution and are thus equivalent. Continuous time, tapped-delay-line input shaping filters, and discrete controls are considered. It is shown that the discrete equivalent of the general vibration reduction problem is a convex problem for the fixed final time case and quasi-convex for the free final time problem. The two formulations are compared in terms of computation complexity as well as practical implementation issues.      

磁悬浮系统的非线性控制

本文主要针对普通线性化控制方法的不足,为了使控制系统满足全局稳定。建立磁悬浮系统非线性模型,利用状态反馈法将系统线性化,然后按照设计指标计算出控制算法,最后进行仿真和试验验证,得出了必要的结论。     

Nonlinear adaptive control of a turbogenerator

The synergetic approach to the problem of synthesis of nonlinear adaptive control for electric power generation based on the introduction of invariant manifolds into the state space of studied systems is developed. The problem of adaptive control of the power supply system operating with high power buses is considered. A fundamentally new adaptive controller ensuring the flexibility of the turbogenerator to the variation of the load parameters is synthesized on the basis of the presented method.     

一类非线性系统的故障诊断方法

针对一类非线性系统,提出了一种新的故障诊断方法;首先,对未知的系统正常模式和系统故障模式分别进行确定学习,将系统正常模式和各种故障模式以空间分布的常数神经网络权值方式储存,建立模式库;然后,根据已有模式库中的模式构造一系列估计器,将估计器的状态与实际系统状态进行比较,构造残差,以此来检测和分离各种故障;最后,以弹簧减震器系统为例,用仿真结果证明了文中设计的故障诊断方法的可行性和有效性.     

A Study of Viscous Flux Formulations for a p-Multigrid Spectral Volume Navier Stokes Solver

In this paper, we improve the Navier–Stokes flow solver developed by Sun et al. based on the spectral volume method (SV) in the following two aspects: the development of a more efficient implicit/p-multigrid solution approach, and the use of a new viscous flux formula. An implicit preconditioned LU-SGS p-multigrid method developed for the spectral difference (SD) Euler solver by Liang is adopted here. In the original SV solver, the viscous flux was computed with a local discontinuous Galerkin (LDG) type approach. In this study, an interior penalty approach is developed and tested for both the Laplace and Navier–Stokes equations. In addition, the second method of Bassi and Rebay (also known as BR2 approach) is also implemented in the SV context, and also tested. Their convergence properties are studied with the implicit BLU-SGS approach. Fourier analysis revealed some interesting advantages for the penalty method over the LDG method. A convergence speedup of up to 2-3 orders is obtained with the implicit method. The convergence was further enhanced by employing a p-multigrid algorithm. Numerical simulations were performed using all the three viscous flux formulations and were compared with existing high order simulations (or in some cases, analytical solutions). The penalty and the BR2 approaches displayed higher accuracy than the LDG approach. In general, the numerical results are very promising and indicate that the approach has a great potential for 3D flow problems.     

Nonlinear Modeling and Control of a Unicycle

A unicycle system is composed of a unicycle and a rider. This system is inherently unstable, but together with a skilled rider can be autonomously controlled and stabilized. A dynamical investigation, a control design and a numerical solution of a nonlinear unicycle autonomous model are presented. The use of a nonlinear model for the control design is shown in this paper to be of great importance. A three-rigid-body physical model was selected for the dynamical study of the system. In a linearized model important physical characteristics of the unicycle system disappear (e.g. interactions between the longitudinal and lateral systems are being neglected), and therefore it is not recommended to be used for the control design. A nonlinear control law, which replaces the rider in stabilizing the model, was derived in the present work, using a nonlinear unicycle model. A simulation study shows good performance of this controller. Time spectral element methods are developed and used for integrating the nonlinear equations of motion. The approach employs the time discontinuous Galerkin method which leads to A-stable high order accurate time integration schemes.