Convergence of a finite element/ALE method for the Stokes equations in a domain depending on time

We consider the approximation of the unsteady Stokes equations in a time dependent domain when the motion of the domain is given. More precisely, we apply the finite element method to an Arbitrary Lagrangian Eulerian (ALE) formulation of the system. Our main results state the convergence of the solutions of the semi-discretized (with respect to the space variable) and of the fully-discrete problems towards the solutions of the Stokes system.     

Simulation and analysis of rotary forging a ring workpiece using finite element method

The methods of dealing with some key problems in analyzing a rotary forging process with a finite element method are given. The presented mechanical model of the finite element analysis is in accordance with the actual conditions of the rotary forging process. A three-dimensional rigid–plastic finite element analysis code is developed in FORTRAN language and used to analyze the rotary forging process of a ring workpiece. Velocity fields and stress–strain fields of both contact and non-contact zones of the ring workpiece in the rotary forging are obtained. The deformation mechanism and metal flow laws of the contact zone surface of the ring workpiece in the rotary forging process are revealed. The pressure distributions of the contact surface along the radial and tangential directions and effects of rotary forging parameters on deformation characteristics are given.     

人工机械心瓣启闭过程的ALE有限元分析

本文采用任意拉格朗日－欧拉（ＡＬＥ）有限单元法，将血液视为不可压缩粘性流体，同时将人工机械瓣简化为定轴转动刚体，建立了机械心瓣－血液耦合运动二维计算模型，解决了人工机械瓣启闭过程的计算问题．对Ｓｔ．Ｊｕｄｅ瓣启闭过程的数值分析表明：１．Ｓｔ．Ｊｕｄｅ瓣启闭过程包括四个阶段，即：Ⅰ．开启相；Ⅱ．开启保持相；Ⅲ．关闭相；Ⅳ．关闭保持相．２．Ｓｔ．Ｊｕｄｅ瓣关闭时回流明显，伴有显著水击效应．３．在启闭过程中，高剪力区位于两瓣叶间及瓣环附近．本文的研究突破了以非耦合的方法研究心瓣启闭过程的局限性．     

Stochastic shape sensitivity in powder metallurgy processing

Stochastic shape sensitivity in forming process of powder metallurgy materials is analyzed. For this purpose the rigid-poroplastic material model has been assumed. The theoretical formulation for stochastic shape sensitivity is described which presents probabilistic distributions taking into account random initial and boundary conditions. The control volume approach is discussed. Stochastic finite element equations for rigid – poroplastic materials are solved for the first two probabilistic moments. Numerical simulations were performed to illustrate shape sensitivity problems in the process of compression of rigid-poroplastic cylinder. The differences in deterministic and stochastic sensitivities are presented. The results derived can be used for the subsequent quantitative stochastic shape design as well as stochastic shape optimization.     

Geometrically nonlinear analysis with a 4-node membrane element formulated by the quadrilateral area coordinate method

Recently, a 4-node quadrilateral membrane element AGQ6-I, has been successfully developed for analysis of linear plane problems. Since this model is formulated by the quadrilateral area coordinate method (QACM), a new natural coordinate system for developing quadrilateral finite element models, it is much less sensitive to mesh distortion than other 4-node isoparametric elements and free of various locking problems that arise from irregular mesh geometries. In order to extend these advantages of QACM to nonlinear applications, the total Lagrangian (TL) formulations of element AGQ6-I was established in this paper, which is also the first time that a plane QACM element being applied in the implicit geometrically nonlinear analysis. Numerical examples of geometrically nonlinear analysis show that the presented formulations can prevent loss of accuracy in severely distorted meshes, and therefore, are superior to those of other 4-node isoparametric elements. The efficiency of QACM for developing simple, effective and reliable serendipity plane membrane elements in geometrically nonlinear analysis is demonstrated clearly.     

The finite element-Galerkin method for singular self-adjoint differential equations

We investigate the finite element-Galerkin method for singular self-adjoint second-order differential expressions. The weak formulation of the problem involves integration by parts, which allows the use of the usual piecewise linear functions. Our analysis shows that the method produces the solution corresponding to a particular self-adjoint realization of the differential expression. We also propose two algorithms to approximate the solution of any self-adjoint realization. Numerical examples are given to illustrate the analysis as well as the proposed algorithms.     

A stabilized mixed formulation for finite strain deformation for low-order tetrahedral solid elements

A stabilized mixed finite element formulation for four-noded tetrahedral elements is introduced for robustly solving small and large deformation problems. The uniqueness of the formulation lies within the fact that it is general in that it can be applied to any type of material model without requiring specialized geometric or material parameters. To overcome the problem of volumetric locking, a mixed element formulation that utilizes linear displacement and pressure fields was implemented. The stabilization is provided by enhancing the rate of deformation tensor with a term derived using a bubble function approach. The element was implemented through a user-programmable element of the commercial finite element software ANSYS. Using the ANSYS platform, the performance of the element was evaluated by comparing the predicted results with those obtained using mixed quadratic tetrahedral elements and hexahedral elements with a B-bar formulation. Based on the quality of the results, the new element formulation shows significant potential for use in simulating complex engineering processes.     

3D hp-adaptive finite element simulations of bend,step, and magic-T electromagnetic waveguide structures

Metallic rectangular waveguides are often the preferred choice on telecommunication systems and medical equipment working on the upper microwave and millimeter wave frequency bands due to the simplicity of its geometry, low losses, and the capacity to handle high powers. Waveguide translational symmetry is interrupted by the unavoidable presence of bends, transitions, and junctions, among others. This paper employs a 3D hp self-adaptive grid-refinement finite element strategy for the solution of these relevant electromagnetic waveguide problems. These structures often incorporate dielectrics, metallic screws, round corners, and so on, which may facilitate its construction or improve its design, but significantly difficult its modeling when employing semi-analytical techniques. The hp-adaptive finite element method enables accurate modeling of these structures even in the presence of complex materials and geometries. Numerical results demonstrate the suitability of the hp-adaptive method for modeling these waveguide structures, delivering errors below 0.5% with a limited number of unknowns. Solutions of waveguide problems obtained with the self-adaptive hp-FEM are of comparable accuracy to those obtained with semi-analytical techniques such as the Mode Matching method, for problems where the latest methods can be applied. At the same time, the hp-adaptive FEM enables accurate modeling of more complex waveguide structures.     

Discrete finite element characterizations of source fields for volume and boundary constraints in electromagnetic problems

The consideration of electromagnetic field sources in potential formulations necessitates the definition of source fields. Such source fields are first defined for both volume and boundary constraints in static electromagnetic models. Then, automatic procedures are proposed to conveniently and efficiently characterize discrete source fields, with regard to their use in finite element formulations, their supports, their direct expression requiring no pre-computation, and their associated constraints. Two application examples are proposed to illustrate the approach.     

A finite element method for interface problems in domains with smooth boundaries and interfaces

This paper is concerned with the analysis of a finite element method for nonhomogeneous second order elliptic interface problems on smooth domains. The method consists in approximating the domains by polygonal domains, transferring the boundary data in a natural way, and then applying a finite element method to the perturbed problem on the approximate polygonal domains. It is shown that the error in the finite element approximation is of optimal order for linear elements on a quasiuniform triangulation. As such the method is robust in the regularity of the data in the original problem.     

2D internal flux compatibility equation of the flux Green element method for transient nonlinear potential problems

This article presents the derivation and implementation of the normal directional flux compatibility equation (relationship) at internal nodes when the Green element formulation that consistently provides accurate estimates of the primary variable, and its normal directional derivative (normal flux) is applied in 2D heterogeneous media to steady and transient potential problems. Such a relationship is required to resolve the closure problem due to having fewer integral equations than the number of unknowns at internal nodes. The derivation of the relationship is based on Stokes' theorem, which transforms the contour integral of the normal directional fluxes into a surface integral that is identically zero. The numerical discretization of the compatibility equation is demonstrated with four numerical examples using the six‐node quadratic triangular and the four and eight‐node rectangular elements. The incorporation of triangular elements into the current formulation demonstrates that the internal compatibility equation can be successfully implemented on irregular grids. The direct calculation of the fluxes significantly enhances the accuracy of the formulation, so that high accuracy, exceeding that of the finite element method, is achieved with very coarse spatial discretization. © 2009 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq 2010     

两种人工机械心瓣启闭过程的ALE有限元分析*

本文采用任意拉格朗日-欧拉(ALE)有限单元法,将血液视为不可压缩粘性流体,同时将人工机械瓣简化为定轴转动刚体,建立了机械心瓣/血液耦合运动的二维计算模型。在此基础上。计算了顺向人工机械瓣(DDM)的开闭过程,并与St.Jude瓣(SJM)进行了对比。由本文研究可得以下主要结论:1.DDM瓣与SJM相比,开启较为迅速,关闭则较为柔和。2.DDM瓣的峰值回流量较之SJM瓣小。本文研究表明DDM瓣由于更好利用了天然心瓣的关闭机理,而具有较高的耐久性潜力。     

蒸汽沉淀化学反应方程混合元法的离散格式研究

蒸汽沉淀化学反应过程有着极其广泛的应用,其数学模型归结为一个包含流速场,温度场,压力场和气体溶质场的非线性偏微分方程组．用混合有限元方法研究蒸汽沉淀化学反应方程组,导出其半离散化和全离散化的混合元格式,并证明这些格式的解的存在性和收敛性(误差估计)．用混合元法处理究蒸汽沉淀化学反应方程组,可以同时求出流速场,温度场,压力场和气体溶质场的数值解. 因此该研究既具有重要的理论意义,又具有广泛的应用前景．     

A finite element method for contact/impact

Ideas from the analysis of differential-algebraic equations are applied to the numerical solution of frictionless contact/impact problems in solid mechanics. Index-one and two formulations for dynamic contact–impact within the context of the finite element method are derived. The resulting equations are shown to stabilize the kinematic fields at the contact interface, at the expense of a small energy loss, which is shown to decrease consistently with mesh refinement. This energy dissipation is shown to be necessary for the establishment of persistent contact. A Newmark-type time integration scheme is derived from the proposed formulation, and shown to yield excellent results in modeling the transition to contact/impact.     

Augmented high order finite volume element method for elliptic PDEs in non-smooth domains: Convergence study

The accuracy of a finite element numerical approximation of the solution of a partial differential equation can be spoiled significantly by singularities. This phenomenon is especially critical for high order methods. In this paper, we show that, if the PDE is linear and the singular basis functions are homogeneous solutions of the PDE, the augmentation of the trial function space for the Finite Volume Element Method (FVEM) can be done significantly simpler than for the Finite Element Method. When the trial function space is augmented for the FVEM, all the entries in the matrix originating from the singular basis functions in the discrete form of the PDE are zero, and the singular basis functions only appear in the boundary conditions. That is to say, there is no need to integrate the singular basis functions over the elements and the sparsity of the matrix is preserved without special care. FVEM numerical convergence studies on two-dimensional triangular grids are presented using basis functions of arbitrary high order, confirming the same order of convergence for singular solutions as for smooth solutions.     

Numerical modelling of linear and nonlinear diffusion equations by compact finite difference method

In this work, accurate solutions to linear and nonlinear diffusion equations were introduced. A combination of a sixth-order compact finite difference scheme in space and a low-storage third-order total variation diminishing Runge-Kutta scheme in time have been used for treatment of these equations. The computed results with the use of this technique have been compared with the exact solution to show the accuracy of it. Here, the approximate solution to the diffusion equations has been obtained easily and elegantly with neither transforming nor linearizing the equation. The present method is seen to be a very good alternative method to some existing techniques for realistic problems.     

On best possible order of convergence estimates in the collocation method and Galerkin's method for singularly perturbed boundary value problems for systems of first-order ordinary differential equations

The collocation method and Galerkin method using parabolic splines are considered. Special adaptive meshes whose number of knots is independent of the small parameter of the problem are used. Unimprovable estimates in the -norm are obtained. For the Galerkin method these estimates are quasioptimal, while for the collocation method they are suboptimal.

     

Three-dimensional nonlinear finite element analysis of dovetail joints in aeroengine discs

Three-dimensional nonlinear finite element analysis is made of the dovetail region in aeroengine compressor disc assemblies using contact elements. The study is devoted to examining the effect of the critical geometrical features, such as flank length, flank angle, fillet radii and skew angle upon the resulting stress field. Frictional conditions at the interface between the disc and the blade are also examined. The finite element predictions were validated using three-dimensional photoelastic stress freezing results. Comparisons with the two-dimensional finite element analysis made earlier by Papanikos and Meguid (Fatigue Fract. Eng. Mater. Struct. 17 (5) (1994) 539–550) of the same geometry reveal certain inadequacies. Specifically, the earlier analysis underestimates the maximum equivalent stress along the interface by as much as 40%. This could have serious implications concerning the safety margins of the disc assembly.     

New intergrid transfer operator in multigrid method for P1-nonconforming finite element method

For a second-order elliptic boundary value problem, We develop an intergrid transfer operator in multigrid method for the P₁-nonconforming finite element method. This intergrid transfer operator needs smaller computation than previous intergrid transfer operators. Multigrid method with this operator converges well.