Dual-mixed finite element analysis of crystalline sub-micron gold

An extended crystal plasticity model is applied to crystalline sub-micron gold in order to study the mechanical response. Numerical results for different crystal sizes are presented and discussed. The governing equations are discretized and, subsequently, solved via a dual-mixed finite element formulation [1, 2]. The evolution equation of the dislocation density is taken as a global field relation additionally to the balance of linear momentum, whereas the flow rule is solved locally at the Gauß point level [3,4]. (© 2014 Wiley-VCH Verlag GmbH & Co. KGaA, Weinheim)     

Spectral meshless radial point interpolation (SMRPI) method to two‐dimensional fractional telegraph equation

H. Ammari In this article, an innovative technique so‐called spectral meshless radial point interpolation (SMRPI) method is proposed and, as a test problem, is applied to a classical type of two‐dimensional time‐fractional telegraph equation defined by Caputo sense for (1 < α≤2). This new methods is based on meshless methods and benefits from spectral collocation ideas, but it does not belong to traditional meshless collocation methods. The point interpolation method with the help of radial basis functions is used to construct shape functions, which play as basis functions in the frame of SMRPI method. These basis functions have Kronecker delta function property. Evaluation of high‐order derivatives is not difficult by constructing operational matrices. In SMRPI method, it does not require any kind of integration locally or globally over small quadrature domains, which is essential of the finite element method (FEM) and those meshless methods based on Galerkin weak form. Also, it is not needed to determine strict value for the shape parameter, which plays an important role in collocation method based on the radial basis functions (Kansa's method). Therefore, computational costs of SMRPI method are less expensive. Two numerical examples are presented to show that SMRPI method has reliable rates of convergence. Copyright © 2015 John Wiley & Sons, Ltd.     

Acoustic simulations with higher order finite and infinite elements

The efficiency of finite element based simulations of Helmholtz problems is primarily affected by two facts. First, the numerical solution suffers from the so-called pollution effect, which leads to very high element resolutions at higher frequencies. Furthermore, the spectral properties of the resulting system matrices, and hence the convergence of iterative solvers, deteriorate with increasing wave numbers. In this contribution the influence of different types of polynomial basis functions on the efficiency and stability of interior as well as exterior acoustic simulations is analyzed. The current investigations show that a proper choice for the polynomial shape approximation may significantly increase the performance of Krylov subspace methods. In particular, the efficiency of higher order finite and infinite elements based on Bernstein polynomial shape approximation and the corresponding iterative solution strategies is assessed for practically relevant numerical examples including the sound radiation from rolling vehicle tires. (© 2008 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim)     

求解Brinkman方程的修正弱有限元方法

本文针对Brinkman方程引入了一种修正弱Galerkin(MWG)有限元方法.我们通过具有两个离散弱梯度算子的变分形式来逼近模型. 在MWG方法中, 分别用次数为$k$和$k-1$的不连续分段多项式来近似速度函数$u$和压力函数$p$. MWG方法的主要思想是用内部函数的平均值代替边界函数. 因此, 与WG方法相比, MWG方法在不降低准确性的同时, 具有更少的自由度, 对于任意次数不超过$k-1$ 的多项式,MWG方法均可以满足稳定性条件. MWG 方法具有高度的灵活性, 它允许在具有一定形状正则性的任意多边形或多面体上使用不连续函数. 针对$H^1$和$L^22$范数下的速度和压力近似解, 建立了最优阶误差估计. 数值算例表明了该方法的准确性, 收敛性和稳定性.     

Simulation of Lamb wave interaction with defects in laminated plates by SEM

For the simulation of the interaction of elastic waves in CFRP Plates with inhomogeneities and defects the spectral finite element method (SEM) is under investigation. The SEM uses high-order shape functions which are composed of Lagrange polynomials with nodes at the Gauss-Lobatto quadrature (GLq) points. In this way we obtain a diagonal mass matrix which makes an explicit time scheme more efficient. In a numerical example based on the first order shear deformation theory (FSDT) a computation by FEAP of an interaction with an inhomogeneity is presented. (© 2012 Wiley-VCH Verlag GmbH & Co. KGaA, Weinheim)     

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)     

Parallel Solution Methods for Porous Media Models in Biomechanics

We present a biomechanical application of our parallel finite element model for coupled problems in solid mechanics. This programming framework provides a very lean and flexible interface, which allows to realize time-dependent nonlinear simulations. In this context, a special variant of a stabilized local Gauß-Seidel preconditioner is introduced, which can be successfully employed to large scale computations. Finally, the efficiency of the implemented algorithm is shown by a numerical example considering the axial compression of a L4-L5 motion segment of the spine. (© 2005 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim)     

On an implicit particle method for simulation of forming processes

Task is the simulation of forming processes using particle methods. We implemented some mesh-free methods (the element free Galerkin method [1] and others) and the finite element method in one programme system which permits a direct comparison. For the mesh-free methods a moving least squares approximation is applied. The shape functions are not zero or one at the nodes, thus essential boundary conditions cannot be imposed directly [2]. We use a penalty method to enforce essential boundary conditions and contact conditions. The contact algorithm (normal contact of nodes to C¹-continuous surfaces) is checked by means of the element free Galerkin method and the FEM on the basis of numerical examples which deal with forming processes. (© 2005 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim)     

Monoton einschließend konvergente iterationsprozesse vom gauß-seidel-typ zur lösung nichtlinearer gleichungssysteme im RN und anwendungen

In this paper we describe a class of Gauß-Seidel iterations for the solution of large sparse nonlinear systems of equations. For suitable starting vectors every method of this class is monotonically convergent with respect to both sides. The result will be applied to the numerical solution of nonlinear boundary-value problems and boundary layer problems.     

Poisson方程的混合无网格方法(英文)

以Poisson方程的混合变分形式为基础,采用移动最小二乘方法建立插值形函数空间,给出了Poisson方程的混合无网格方法,理论上证明了Poisson方程混合无网格解的存在唯一性,并给出了误差估计.本质边界条件的处理采用Lagrange乘子法.数值算例表明,在应用相同阶次的基函数条件下,利用混合无网格方法求解Poisson方程所得的解的梯度值优于传统的无网格方法及有限元法.     

一种新的声椭球无限单元

提出了一种新的声椭球无限单元．这种声无限单元基于一种新的声压表达式,这种声压表达式能够更准确地代表着椭球声场的声传播模式．这种新方法的形函数类似于Burnett方法,而权函数定义为形函数和一个附加因子的乘积．因为仅需要一维的数值积分,这种新方法的代码生成十分容易,就像处理一维单元一样．耦合标准的有限元程序,这种声无限单元理论上能够高效地求解任何形状的声源的声辐射和声散射现象．简要地推导了这种新方法,并给出了这种方法详尽的推导结果．为更有效地检验该无限元方法的可行性,文中例子仅考虑无限元求解的精度,而不包括相应的有限元．使用这种新方法,精确地推导出了摆动球的理论计算公式．而长旋转椭球的例子则表明了这种方法优于边界元方法和其他声椭球无限元方法．这些例子表明了这种新方法是切实可行的．     

Follower loads for high order finite elements

Finite element modelling of hydrostatic compaction where the applied pressure acts normal to the deformed surface requires a geometric nonlinear formulation and follower load terms [1, 5, 7]. These concepts are applied to high order [6] (p-FEM) elements with hierarchic shape functions. Applying the blending function method allows to precisely describe curved boundaries on coarse meshes. High order elements exhibit good performance even for high aspect ratios and strong distortion and therefore allow an efficient discretization of thin-walled structures. Since high order finite elements are less prone to locking effects a pure displacement-based formulation can be chosen. After introducing the basic concept of the p-version the application of follower loads to geometrically nonlinear high order elements is presented. For the numerical solution the displacement based formulation is linearized yielding the basis for a Newton-Raphson iteration. The accuracy and performance of the high order finite element scheme is demonstrated by a numerical example. (© 2005 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim)     

固体中短波传播的单位分解有限元法

提出了固体中短波传播数值模拟的单位分解有限元法．有限元空间由形成单位分解的标准等参有限元形函数乘以定义为局部子空间基函数的特殊形函数构成．特殊形函数使试空间中包含了关于波动方程的已有知识,因而在单个单元内能近似地再现高度振荡性质．数值例题显示了所提出单位分解有限元在计算精度和效率上的良好性能．     

Dynamic response of circular and annular circular plates using spectral element method

This paper presents development of the spectral element method (SEM) for analysis of circular and annular circular plates vibration under impact load. A novel formulation is proposed in frequency domain for conducting spectral element matrix. Several numerical examples are presented in order to demonstrate performance of the presented method compared to other numerical methods in literature. The presented formulation was applied in order to analyze vibration of annular circular plates with various thicknesses and various internal-to-external radius ratios. In addition, annular circular plate displacements subjected to an impact load were calculated using the presented SEM.     

A local radial basis function collocation method for band structure computation of phononic crystals with scatterers of arbitrary geometry

A numerical algorithm based on the local radial basis function collocation method (LRBFCM) is developed to efficiently compute the derivatives of primary field quantities. Instead of a direct calculation of the derivatives by partial differentiation of the shape functions as in traditional numerical approaches, the derivative calculation in the present work is performed using a simple finite difference scheme with an introduced fictitious node. The developed algorithm is geometrically very flexible and can be easily applied to the continuity and boundary conditions of arbitrary geometries, which require an accurate derivative computation of the primary field quantities. The developed LRBFCM are applied to phononic crystals with scatterers of arbitrary geometry, which has not yet been reported before to the authors’ knowledge. A few examples for anti-plane elastic wave propagation are modelled to validate the developed LRBFCM. A comparison with finite element modelling shows that the present method is efficient and flexible.     

A Modified Weak Galerkin Finite Element Method for Sobolev Equation

For Sobolev equation, we present a new numerical scheme based on a modified weak Galerkin finite element method, in which differential operators are approximated by weak forms through the usual integration by parts. In particular, the numerical method allows the use of discontinuous finite element functions and arbitrary shape of element. Optimal order error estimates in discrete $H^1$ and $L^2$ norms are established for the corresponding modified weak Galerkin finite element solutions. Finally, some numerical results are given to verify theoretical results.     

A Zienkiewicz-type finite element applied to fourth-order problems

This paper deals with convergence analysis and applications of a Zienkiewicz-type (Z-type) triangular element, applied to fourth-order partial differential equations. For the biharmonic problem we prove the order of convergence by comparison to a suitable modified Hermite triangular finite element. This method is more natural and it could be applied to the corresponding fourth-order eigenvalue problem. We also propose a simple postprocessing method which improves the order of convergence of finite element eigenpairs. Thus, an a posteriori analysis is presented by means of different triangular elements. Some computational aspects are discussed and numerical examples are given.     

精确有限元法

本文提出构造有限单元的新方法——精确有限元法.它可以求解在任意边界条件下任意变系数正定或非正定偏微分方程。文中给出它的收敛性证明和计算偏微分方程的一般格式。用精确元法所得到的单元是一个非协调元,单元之间的相容条件容易处理.与相同自由度普通有限元相比,由精确元法所得到的解的高阶导数具有较高的收敛精度.文末给出数值算例,所得到的结果均收敛于精确解,并有较好的数值精度.     

An optimal rational basis function in the finite element method for convection-diffusion problems in one space variables

A new fixed grid Galerkin finite element technique using a rational basis function of order (1, 1) to solve convection-dominated phenomenon is described. A special feature of the method is that it employs rational basis functions, which are biased in the upstream direction and thus replace the equivalent upwinding techniques. Optimality of the rational basis function results in an accurate and stable numerical scheme which is used to avoid numerical oscillations at high Peclet numbers. Numerical examples are discussed which illustrate the efficiency of the method.