An enhanced finite volume method to model 2D linear elastic structures

This paper details the evaluation and enhancement of the vertex-centred finite volume method for the purpose of modelling linear elastic structures undergoing bending. A matrix-free edge-based finite volume procedure is discussed and compared with the traditional isoparametric finite element method via application to a number of test-cases. It is demonstrated that the standard finite volume approach exhibits similar disadvantages to the linear Q4 finite element formulation when modelling bending. An enhanced finite volume approach is proposed to circumvent this and a rigorous error analysis conducted. It is demonstrated that the developed finite volume method is superior to both standard finite volume and Q4 finite element methods, and provides a practical alternative to the analysis of bending-dominated solid mechanics problems.     

最佳等参元

等参元及其参数变换的插值方法。是有限元分析的有力工具之一,在工程计算中,得到广泛的应用. 在有限元分析中,当采用等参元时,一旦单元的等参坐标变换的Jacobi矩阵发生奇异,就要中止计算,下机修改原有的单元剖分,直到所有单元的Jacobi矩阵均非奇异. [1]突破原等参元的规定,给出了八节点Serendipity等参元的修改公式;[2]也给出了类似的修改公式.上述均以数值例子说明新公式的优点.而[3—6]完整、系统地给出     

三维Poisson方程特征值的四种有限元解及比较

应用三维EQ1rot元、三维Crouzeix-Raviart元、八节点等参数元、四面体线性元计算三维Poisson方程的近似特征值.计算结果表明:三维EQ1rot元和三维Crouzeix-Raviart元特征值下逼近准确特征值,八节点等参数元、四面体线性元特征值上逼近准确特征值,三维EQr1ot元和三维Crouzeix-Raviart元外推特征值下逼近准确特征值.计算结果还表明三维Crouzeix-Raviart元是一种计算效率较高的非协调元.     

非矩形区域上半导体问题的交替方向有限元方法

Transient behavior of semiconductor with heat-conduction on nonrectangular is studied using isoparametric elements and an approximation to the Jacobian of the isoparametric map.Concentration and heat-conduction equations are solved by alternating-direction methods and electric potential equation is approximated by finite element method.Optimal order error estimates in L^2 are demonstrated using the theory and technique of a prior estimate of differential equation.     

The invertibility of the isoparametric mapping for pyramidal and prismatic finite elements

Summary. We consider the isoparametric transformation, which maps a given reference element onto a global element given by its vertices, for multi-linear finite elements on pyramids and prisms. We present easily computable conditions on the position of the vertices, which ensure that the isoparametric transformation is bijective. Received May 7, 1999 / Revised version received April 28, 2000 / Published online December 19, 2000     

惩罚函数法在三维等参杂交元的应用

提出在等参杂交元中用惩罚函数法引入平衡约束条件,具体讨论了惩罚函数法在三维等参杂交元中的运用,并提出采用分项罚数的方法,建立最佳的罚平衡杂交元模型．罚平衡法可以在不增加自由度的前提下,有效地扼制寄生应力．数值实验表明,新建立的单元,可以有效地抑制单元畸变对计算精度的影响,从而大幅度提高畸变网格下的计算精度,方法带有普遍性．     

Implementing and using high-order finite element methods

We present theoretical analyses of and detailed timings for two programs which use high-order finite element methods to solve the Navier- Strokes equations in two and three dimensions. The analyses show that algorithms popular in low-order finite element implementations are not always appropriate for high-order methods. The timings show that with the proper algorithms high-order finite element methods are viable for solving the Navier-Stokes equations. We show that it is more efficient, both in time and storage, not to precompute element matrices as the degree of approximating functions increases. We also study the cost of assembling the stiffness matrix versus directly evaluating bilinear forms in two and three dimensions. We show that it is more efficient not to assemble the full stiffness matrix for high-order methods in some cases. We consider the computational issues with regard to both Euclidean and isoparametric elements. We show that isoparametric elements may be used with higher-order elements without increasing the order of computational complexity.     

Isoparametric finite-element approximation of a Steklov eigenvalue problem

We study the isoparametric variant of the finite-element method(FEM) for an approximation of Steklov eigenvalue problems forsecond-order, selfadjoint, elliptic differential operators.Error estimates for eigenfunctions and eigenvalues are derived.We prove the same estimate for eigenvalues as that obtainedin the case of conforming finite elements provided that theboundary of the domain is well approximated. Some algorithmicaspects arising from the FE isoparametric discretization ofthe Steklov problems are analysed. We finish this paper withnumerical results confirming the considered theory.     

Non-linear seismic response of concrete gravity dams to near-fault ground motions including dam-water-sediment-foundation interaction

This study focuses on non-linear seismic response of a concrete gravity dam subjected to near-fault and far-fault ground motions including dam-water-sediment-foundation rock interaction. The elasto-plastic behavior of the dam concrete is idealized using Drucker–Prager yield criterion based on associated flow rule assumption. Water in the reservoir is represented by 9-noded isoparametric quadrilateral fluid finite elements while the dam, the foundation rock and the sediment layer are modeled by using 8-noded isoparametric quadrilateral solid finite elements. The program NONSAP modified for elasto-plastic analysis of fluid-structure systems using the Lagrangian fluid finite element is employed in the response calculations. The fluid element includes the effects of surface waves and sloshing behavior of fluids. Non-linear seismic analyses of the selected concrete dam subjected to both near-fault and far-fault ground motions are performed. The results obtained from linear and non-linear analyses are compared with each other.     

Meshfree isoparametric finite point interpolation method (IFPIM) with weak and strong forms for evaporative laser drilling

An isoparametric finite point interpolation method (IFPIM) with weak and strong forms has been developed to analyze evaporative laser drilling. The method is based on isoparametric finite point representation of the unknowns in the influence domain. The local influence domains are mapped onto a master domain where the shape functions and their derivatives are known. The solution in the master domain is approximated by a linear combination of shape functions. The present method employs a simple strong form in the domain and a weak form on the boundary. Three different types of boundary conditions considered are of essential, convection, and laser irradiation type. The problem is geometrically nonlinear because the domain is not known a priori due to material removal in drilling. An iterative scheme is used to solve the nonlinear problem. The material removal is handled by redistributing points in the domain. This renders the point distribution non-uniform as in random distribution. The numerical results show excellent agreement with those by FEM and BEM in terms of groove shape, temperature and heat flux distributions, and amount of material removal. The results are superior to those from the isoparametric finite point interpolation methods with only strong forms.     

A flexible approach to the simulation of hybrid multibody systems

The present works deals with the incorporation of both flexible beam and shell structures into the realm of flexible multibody dynamics. Geometrically exact beam formulations based on classical Simo-Reissner kinematics are suitable for modelling beam-type flexible components in the context of finite-deformation multibody dynamics. So geometrically exact shell formulations are based on Reissner-Mindlin kinematics. In [2], a flexible framework for dealing with flexible structural elements in a multibody context is described. A specific isoparametric finite element discretization of a shell formulation leads to semi-discrete equations of motions assuming the form of differential-algebraic equations (DAEs). A compatible isoparametric formulation of beams has already been developed in [1]. The uniform DAE framework makes possible the incorporation of alternative finite element formulations. In addition to that, various time-stepping schemes such as energy-momentum methods or variational integrators can be applied. (© 2010 Wiley-VCH Verlag GmbH & Co. KGaA, Weinheim)     

Isoparametric, Finite Element Methods for Nonlinear Dirichlet Problem

In this paper, the author solves the nonlinear Dirichlet problem with nonhomo- genous boundary condition by use of the isoparametric finite element method, and obtains the optimal error estimate.     

Fourteen-Node Mixed Stiffness Element and Its Computational Comparisons with Twenty-Node Isoparametric Element

In this paper, a family of 3-dimensional elements different from isoparametric serendipity is developed according to the variational principle and the convergence criteria of the mixed stiffness finite element method. For the new family, which is named mixed stiffness elements, the number of nodes on the quadratic element is not 20 but 14. Theoretical analysis and various computational comparisons have found the mixed stiffness element superior over the isoparametric serendipity element, especially a substantial improvement in computational efficiency can be achieved by replacing the 20 node-isoparametric element with the 14-node mixed stiffness element.     

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．     

On Error Estimate of Space-Time Finite Element Methods for Parabolic Equations in a Time-Dependent Domain

We consider linear parabolic equations in a space-time domain with curved boundaries and nonhomogeneous Dirichlet boundary conditions and discuss their approximations with isoparametric space-time finite elements. A general error estimate is proved and applied to some elements of practical interest.     

定常Satokes方程的窄边四边形有限元法

本文提出了一个试探函数不满足divν＝0的定常Stokes方程的窄边四边形有限元法，利用窄边四边形等参有限元插值定理和Falk的思想，得到了H^1(Ω）模误差的最优收敛估计阶O（h）。     

Ruled surfaces of finite type in 3- dimensional minkowski space

We show that a ruled surface M in E₁ ³ is of finite type if and only if M is minimal, or M is a part of a circular cylinder, or M is a part of a hyperbolic cylinder, or M is an isoparametric surface with null rules. We also give a complete classification of isoparametric ruled surfaces with null rules.     

On mean curvature flow of singular Riemannian foliations: Noncompact cases

In this paper we investigate the mean curvature flow (MCF) of a regular leaf of a closed generalized isoparametric foliation as initial datum, generalizing previous results of Radeschi and the first author. We show that, under bounded curvature conditions, any finite time singularity is a singular leaf, and the singularity is of type I. The new techniques also allow us to discuss the existence of basins of attraction, how cylinder structures can affect convergence of basic MCF of immersed submanifolds and assure convergence of MCF of non-closed leaves of generalized isoparametric foliation on compact manifold.     

Cascadic Multigrid Method for Isoparametric Finite Element with Numerical Integration

The purpose of this paper is to study the cascadic multigrid method for the secondorder elliptic problems with curved boundary in two-dimension which are discretized by the isoparametric finite element method with numerical integration. We show that the CCG method is accurate with optimal complexity and traditional multigrid smoother (likesymmetric Gauss-Seidel, SSOR or damped Jacobi iteration) is accurate with suboptimal complexity.     

抛物方程的一种广义差分法(有限体积法)

广义差分法自1982年被提出,至今已获得很大发展（见[1]或[10],这种方法在国际上被称为有限体积（元）法（见[8],[9]）,它的主要优点是保持物理量的局部守恒性.文[3],[5]分别将三角形网格上的椭圆型方程的广义差分法（有限体积法）（见[2],[4]）推广到抛物型方程.我们知道三角形网格与四边形网格是两种基本的分割空间区域的方法,实践上使用哪一种网格,要根据空间区域的几何形状而定.文[7],[6]讨论了一般四边形网上椭圆型方程的广义差分法.本文以抛物方程为模型,取试探函数空间为一般四边形剖分上的等参双线性元,检验函数空间为对偶剖分上的分片常数,导出了一种新的有效的广义差分算法（有限体积算法）,证明了半离散与全离散格式的最佳H1误差估计.遇到的主要困难是双线性形式a（uh,Πh＊uh）