Hypergeometric Summation Techniques for High Order Finite Elements

The goal of this paper is to discuss the application of computer algebra methods in the design of a high order finite element solver. The finite element method is nowadays the most popular method for the computer simulation of partial differential equations. The performance of iterative solution methods depends on the condition number of the system matrix, which itself depends on the chosen basis functions. A major goal is to design basis functions minimizing the condition number, and which can be implemented efficiently. A related goal is the application of symbolic summation techniques to derive cheap recurrence relations allowing a simple and efficient implementation of basis functions. (© 2006 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim)     

基于剪切效应纤维梁单元的结构非线性有限元数值模拟

基于Euler-Bernoulli梁理论的经典纤维模型忽略了剪切变形给截面带来的影响,为了得到更加精确的梁单元模型,该文基于考虑剪切效应的纤维梁单元,根据Timoshenko梁理论,推导了该纤维梁单元的刚度矩阵,并结合弹塑性增量理论,同时考虑了几何非线性和材料非线性的双重影响,建立了压弯剪复杂应力状态下结构非线性有限元...     

不可压缩流问题低次元稳定有限体积数值方法研究

分析了R^d,d=2,3维不可压缩流Stokes问题低次元稳定有限体积方法,它主要利用局部压力投影方法对两种流行但不满足inf-sup条件的有限元配对(P_1-P_0和P_1-P_1)在有限体积方法的框架下进行稳定;利用有限元与有限体积方法的等价性进行有限体积方法理论分析.结果表明不可压缩流Stokes问题在f∈Hd,d=2,3维不可压缩流Stokes问题低次元稳定有限体积方法,它主要利用局部压力投影方法对两种流行但不满足inf-sup条件的有限元配对(P_1-P_0和P_1-P_1)在有限体积方法的框架下进行稳定;利用有限元与有限体积方法的等价性进行有限体积方法理论分析.结果表明不可压缩流Stokes问题在f∈H¹情况下,本文方法得到的解与稳定有限元方法解之间具有O(h1情况下,本文方法得到的解与稳定有限元方法解之间具有O(h²)阶超收敛阶结果,且稳定有限体积方法取得了与稳定有限元方法相同的收敛速度,与稳定有限元方法比较,稳定有限体积方法计算简单高效,同时保持物理守恒,因此在实际应用中具有很好的潜力。     

Modelling of Wave Propagation using Spectral Finite Elements

For the analysis of wave propagation at high frequencies, the spectral finite element method (SFEM) is under investigation. In contrast to the conventional finite element method high-order shape functions are used. They are composed of Lagrange polynomials with nodes at the Gauß-Lobatto-Legendre points. The Gauß-Lobatto-Legendre integration scheme is applied in order to obtain a diagonal mass matrix. So, the resulting system equations can be solved efficiently. In the numerical examples, spectral finite elements with shape functions of different order are applied to a plane strain problem. The numerical examples cover structures without and with stiffness discontinuities. It is shown that the results agree well with analytical solutions. (© 2010 Wiley-VCH Verlag GmbH & Co. KGaA, Weinheim)     

二阶椭圆方程的混合有限元分析

本文给出二阶椭圆方程的一种新的混合框架,并给出两种混合有限元法.     

Analysis of Lamb waves with Spectral Finite Elements

For the analysis of the elastic wave propagation at high frequencies, the spectral finite element method (SFEM) is under investigation. The SFEM uses high-order shape functions which are composed of Lagrange polynomials with nodes at the Gauss-Lobatto quadrature points. In this way we obtain a diagonal mass matrix which makes an explicit time scheme more efficient. In a numerical example a computation by Montjoie of an elastic wave propagation within a Reissner-Mindlin (RM) model is presented. (© 2011 Wiley-VCH Verlag GmbH & Co. KGaA, Weinheim)     

Pointwise Stabilization for a Chain of Coupled Vibrating Strings

For the system representing a chain of coupled vibrating strings,we show that the associated semigroup satisfies the assumptionof spectrum-determined growth, and hence obtain conditions forenergy to decay strongly or exponentially. We examine in detailthe three-string case, and our results include those obtainedby others for the two-string case. Permanent address: Beijing Institute of Information and Control,Beijing, China.     

Numerical Solution of Time Fractional Burgers Equation by Cubic B-spline Finite Elements

Mediterranean Journal of Mathematics - We present some numerical examples which support numerical results for the time fractional Burgers equation with various boundary and initial conditions...     

Solid-Shell Finite Elements for Tesselated Geometries

Sandwich structures made of sheets of composite materials are in widespread use, particularly in the transportation industry. Finite Element simulation of the thin, tesselated structures with complex, three-dimensional material behaviour is a challenging task for the underlying element technology. In particular, frequently used linear isoparametric approaches exhibit unphysical stiffening phenomena. Recent developments in solid-shell finite element technology aim to overcome these undesirable effects. Here, they are applied to an example of a corrugated sandwich core under transversal compression. A study of convergence is conducted with respect to commercially available shell and solid finite elements, and their ability to reproduce the bending dominated deformation state. (© 2012 Wiley-VCH Verlag GmbH & Co. KGaA, Weinheim)     

Generalized Park-Sheen Finite Elements for Adaptivity

The P1-nonconforming finite element method on simply connected Lipschitz domains partitioned into quadrilaterals was introduced by Park and Sheen in 2003. In order to apply mesh-adaptive refinement strategies, the concept is generalized to arbitrary triangulations into quadrilaterals and triangles of multiple connected Lipschitz domains. An explicit a priori analysis is given for second-order elliptic PDEs with inhomogeneous Dirichlet boundary conditions. (© 2012 Wiley-VCH Verlag GmbH & Co. KGaA, Weinheim)     

四段式有限元法分析热耦合的流体-固体相互作用问题

给出了一种流(体)-热-结构综合的分析方法,固体中的热传导耦合了粘性流体中的热对流,因而在固体中产生热应力.应用四段式有限元法和流线逆风Petrov-Galerkin法分析热粘性流动,应用Galerkin法分析固体中的热传导和热应力.应用二阶半隐式Crank-Nicolson格式对时间积分,提高了非线性方程线性化后的计算效率.为了简化所有有限元公式,采用3节点的三角形单元,对所有的变量:流体的速度分量、压力、温度和固体的位移,使用同阶次的插值函数.这样做的主要优点是,使流体-固体介面处的热传导连接成一体.数个测试问题的结果表明,这种有限元法是有效的,且能加深对流(体)-热-结构相互作用现象的理解.     

Bubble—函数稳定化混合有限元分析

本文针对混合结构抽象问题，基于「９」的非标准稳定化有限元方法的一般框架研究了ｂｕｂｂｌｅ－函数稳定化方法，该逼近代格式使得Ｂａｂｕｓｋａ－Ｂｒｅｚｚｉ条件是不必要的。     

Hamilton系统的有限元研究

该文对Hamilton系统的连续有限元法证明了两个优美的性质:在任何情形m次有限元总是能量守恒的, 它对线性系统也是辛的,且对非线性系统每次步进是高精度O(h^2m+1)近似辛的. 在长时间计算中时空平面上轨道和周期的偏离随时间线性增长. 数值实验表明其偏离比其他算法小.     

对流-扩散型问题Bubble-稳定有限元分析

本文针对形如σu α·u-kΔu=f对流—扩散型的模型问题,发展耦合局部bubble-函数的有限元方法,我们就α=0和σ=0两种情形证明了方法的与“影响因素”σ和pedlet-数无关稳定性及全局最佳收敛阶。     

Stability Analysis of an Orbiting Plate with Finite Elements

This paper contributes to the analysis of rotating equilibria of non–driven systems (sometimes called relative equilibria), which are characterized by the fact that their angular momentum is conserved and non–zero. Interesting applications are usually found in space dynamics, in particular if large earth orbiting structures such as the space elevator are considered. For flexible structures relative equilibria can be found with Finite Element software packages. However, in this case the stability analysis is a non–trivial task since one usually has only limited account to the internal data of commercial FE solvers. Therefore, in [5] a new finite element based stability test was developed and applied to one–dimensional structural elements. Here, we consider a large–scale flexible plate orbiting around the earth in order to demonstrate that this method works well also for two dimensional shell–elements. (© 2008 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim)     

Numerical Solutions for Ito Coupled System

In this paper, the classical fourth-order Runge-Kutta method and Heun’s method are applied to initial value problems for system of ordinary differential equations in nonlinear cases which we reach it by Painleve analysis, focusing our interest in Ito coupled nonlinear partial differential system. The equations are solved by scheme of one step methods. Numerical results for the velocity in three dimensions are obtained and reported graphically for various temperatures to show interesting aspects of the solution.     

Modeling Techniques for Coupled,Rotating Beams

Beams are parts of many industrial applications, like robot links, rolls in paper industry and turbo charger. In this work, a rotordynamical problem, the powertrain for a mill stand, is under consideration. Torsional and bending vibrations are used to describe the dynamical behavior. There are several methods for deriving the dynamical equations of motion. In this paper, the Projection Equation, a synthetical method, is used, leading to partial differential equations for the distributed parameter system. A simplification can be done by using the Ritz approximation method. This method requires the fulfillment of the geometric boundary conditions. For our example, a combination of rigid body modes and elastic modes is chosen. Also models for the gear box system and bearings are included. The solutions for the overall example are nonlinear ordinary differential equations which can be integrated numerically. The system is excited by constant torques and forces. Simulation results for this elastic multibody system are presented. (© 2010 Wiley-VCH Verlag GmbH & Co. KGaA, Weinheim)     

Numerical Analysis of the Synchronization of Coupled Chemical Oscillators

A reaction–diffusion model describing a system of coupled oscillators is constructed and investigated. The oscillators in this study are chemical oscillators that represent an oscillatory heterogeneous catalytic reaction in a granular catalyst layer. The oscillators are arranged serially in the reagent stream and are coupled through the gaseous phase. The dynamic behavior of the system is investigated as a function of the main external parameter — the partial pressure of one of the reagents in the gaseous phase. Existence regions of regular and chaotic oscillations are identified. Synchronization conditions are established for the oscillations in such a chain of coupled chemical oscillators.     

Numerical Model for Electro-Mechanical Analysis Based on Finite Element Method

In this paper, we present a numerical procedure that can be used to model the electro-mechanical coupled behavior of the dielectric actuator domain. The equation describing the electrostatical part is given by the reduced form of the Maxwell equation and the electrostatic potential [1]. The mechanical problem is described by the constitutive equations and equilibrium equations. Using the finite element method, this technique is to divide a whole problem into sub-problems. The complexity of the original problem is therefore reduced by focusing only on the most relevant areas. A finite element analysis is then performed by applying the electrostatic Maxwell pressure as Neumann boundary conditions to compute the displacements. Once the displacement is computed, the electrostatic domain or the conductor is updated. Electrostatic analysis is performed on the updated geometry and the finite element method is then used to determine the change in potential due to geometric perturbations. Once the surface charge densities are known, the new electrostatic Maxwell pressure is computed. The mechanical and electrostatic analysis is repeated until an equilibrium state is computed. The procedure is demonstrated in the paper by the solution of some two-dimensional and three-dimensional problems. (© 2013 Wiley-VCH Verlag GmbH & Co. KGaA, Weinheim)     

Approaches to Model Nanotube-Reinforced Polymers using Micromechanics and Finite Elements

In this work, the applicability of new analytical methods for modeling polymer nanocomposites is examined. The influence of an interphase between nanotube and matrix could be captured by this modeling strategy. However, it is shown that there are some difficulties in calculation of composites including extremely weak fibers with stiff coatings. Furthermore, finite element calculations are shown for comparison. (© 2008 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim)