Dynamic finite element with diagonalized consistent mass matrix and elastic-plastic impact calculation

There are some common difficulties encountered in elastic-plastic impact codes such as EPIC^[1,2], NONSAP^[3] etc. Most of these codes use the simple linear functions usually taken from static problems to represent the displacement components. In such finite element formulation, the strain and stress components are constants in every element. In the equations of motion, the stress components in general appear in the form of their space derivatives. Thus, if we use such form functions to represent the displacement components, the effect of internal stresses to the equations of motion vanishes identically. The usual practice to overcome such difficulties is to establish as self-equilibrium system of internal forces acting on various nodal points by means of transforming equations of motion into variational form of energy relation through the application of virtual displacement principle. The nodal acceleration is then calculated from the total force acting on this node from all the neighbouring elements. The transformation of virtual displacement principle into the variational energy form is performed on the bases of continuity conditions of stress and displacement throughout the integrated space. That is to say, on the interface boundary of finite element, the assumed displacement and stress functions should be conformed. However, it is easily seen that, for linear form function of finite element calculation, the displacement continues everywhere, but not the stress components. Thus, the convergence of such kind of finite element computation is open to question. This kind of treatment has never been justified even in approximation sense. Furthermore, the calculation of nodal points needs a rule to calculate the mass matrix. There are two ways to establish mass matrix, namely lumped mass method and consistent mass method [4]. The consistent mass matrix can be obtained naturally through finite element formulation, which is consistent to the assumed form functions. However, the resulting consistent mass matrix is not in diagonalized form, which is inconvenient for numerical computation. For most codes, the lumped mass matrix is used, and in this case, the element mass is distributed in certain assumed proportions to all the nodal points of this element. The lumped mass matrix is diagonalized with diagonal terms composed of the nodal mass. However, the lumped mass assumption has never been justified. All these difficulties are originated from the simple linear form functions usually used in static problems.In this paper, we introduce a new quadratic form function for elastic-plastic impact problems. This quadratic form function possesses diagonalized consistent mass matrix, and non-vanishing effect of internal stress to the equations of motion. Thus with this kind of dynamic finite element, all above-said difficulties can be eliminated.     

轴对称冲击有限元一致质量矩阵迭代解

给出高速冲击动力有限元一致质量矩阵解的迭代过程，即把集总质量矩阵解作为初值进行有限次迭代得到满足工程需要的一致质量矩阵近似解．实际算例说明，一致质量近似解较集总质量解改善了对应力波传播过程的分析，而且在高速冲击计算中能给出与实验接近的计算结果．     

A high‐order mass‐lumping procedure for B‐spline collocation method with application to incompressible flow simulations

This paper presents new developments of the staggered spline collocation method for cost‐effective solution to the incompressible Navier–Stokes equations. Maximal decoupling of the velocity and the pressure is obtained by using the fractional step method of Gresho and Chan, allowing the solution to sparse elliptic problems only. In order to preserve the high‐accuracy of the B‐spline method, this fractional step scheme is used in association with a sparse approximation to the inverse of the consistent mass matrix. Such an approximation is constructed from local spline interpolation method, and represents a high‐order generalization of the mass‐lumping technique of the finite‐element method. A numerical investigation of the accuracy and the computational efficiency of the resulting semi‐consistent spline collocation schemes is presented. These schemes generate a stable and accurate unsteady Navier–Stokes solver, as assessed by benchmark computations. Copyright © 2003 John Wiley & Sons, Ltd.     

A simple method to generate high-order accurate convection operators for explicit schemes based on linear finite elements

A simple method is proposed to generate high-order accurate convection operators for lumped-explicit schemes based on linear or multilinear finite elements. The basic idea is to reduce the truncation error on the first-order spatial derivatives by exploiting the consistent mass matrix of the finite element method in a purely explicit multistep procedure. The effectiveness of the method is demonstrated on pure convection problems in one and two dimensions.     

Comparison of Galerkin and control volume finite element for advection–diffusion problems

The control volume finite element method (CVFEM) was developed to combine the local numerical conservation property of control volume methods with the unstructured grid and generality of finite element methods (FEMs). Most implementations of CVFEM include mass‐lumping and upwinding techniques typical of control volume schemes. In this work we compare, via numerical error analysis, CVFEM and FEM utilizing consistent and lumped mass implementations, and stabilized Petrov–Galerkin streamline upwind schemes in the context of advection–diffusion processes. For this type of problem, we find no apparent advantage to the local numerical conservation aspect of CVFEM as compared to FEM. The stabilized schemes improve accuracy and degree of positivity on coarse grids, and also reduce iteration counts for advection‐dominated problems. Published in 2005 by John Wiley & Sons, Ltd.     

A nonlinear,transient finite element method for coupled solvent diffusion and large deformation of hydrogels

Hydrogels are capable of coupled mass transport and large deformation in response to external stimuli. In this paper, a nonlinear, transient finite element formulation is presented for initial boundary value problems associated with swelling and deformation of hydrogels, based on a nonlinear continuum theory that is consistent with classical theory of linear poroelasticity. A mixed finite element method is implemented with implicit time integration. The incompressible or nearly incompressible behavior at the initial stage imposes a constraint to the finite element discretization in order to satisfy the Ladyzhenskaya–Babuska–Brezzi (LBB) condition for stability of the mixed method, similar to linear poroelasticity as well as incompressible elasticity and Stokes flow; failure to choose an appropriate discretization would result in locking and numerical oscillations in transient analysis. To demonstrate the numerical method, two problems of practical interests are considered: constrained swelling and flat-punch indentation of hydrogel layers. Constrained swelling may lead to instantaneous surface instability for a soft hydrogel in a good solvent, which can be regulated by assuming a stiff surface layer. Indentation relaxation of hydrogels is simulated beyond the linear regime under plane strain conditions, in comparison with two elastic limits for the instantaneous and equilibrium states. The effects of Poisson’s ratio and loading rate are discussed. It is concluded that the present finite element method is robust and can be extended to study other transient phenomena in hydrogels.     

含无限刚性体结构的动力学特征与工程应用

除显含无限刚性段的振动体系外,结合动力问题有限单元法中的一致质量矩阵提出了隐性无限刚性的概念,以及由此引起的振动惯性耦合问题．结合地震作用下高层建筑与桩-土相互作用时的动力响应,提出了对于实际工程问题利用分支模态二步分析法建立计算模型时应注意的问题．     

Velocity–pressure integrated versus penalty finite element methods for high-Reynolds-number flows

Velocity–pressure integrated and consistent penalty finite element computations of high-Reynolds-number laminar flows are presented. In both methods the pressure has been interpolated using linear shape functions for a triangular element which is contained inside the biquadratic flow element. It has been shown previously that the pressure interpolation method, when used in conjunction with the velocity-pressure integrated method, yields accurate computational results for high-Reynolds-number flows. It is shown in this paper that use of the same pressure interpolation method in the consistent penalty finite element method yields computational results which are comparable to those of the velocity–pressure integrated method for both the velocity and the pressure fields. Accuracy of the two finite element methods has been demonstrated by comparing the computational results with available experimental data and/or fine grid finite difference computational results. Advantages and disadvantages of the two finite element methods are discussed on the basis of accuracy and convergence nature. Example problems considered include a lid-driven cavity flow of Reynolds number 10 000, a laminar backward-facing step flow and a laminar flow through a nest of cylinders.     

Compatible dynamic finite element with diagonalized consistent mass matrix

In this paper, the compatible dynamic finite elements with diagonalized consistent mass matrix are studied. In previous papers^[1,2], the author studied the dynamic finite elements with diagonalized consistent mass matrix, but all of them are incompatible elements. In this paper, the compatible form functions are obtained not only for the tetrahedron elements, but also for the triangular ring elements, with diagonalized consistent mass matrices. This kind of finite elements can be used for the treatment of impact problems, vibration problems, and problems involving time coordinates, including the linear and nonlinear problems.     

直接基于单元平衡的变截面梁反应分析方法

固定形状的单元位移插值函数不能合理地近似变截面梁内部的位移变化,从而影响了传统梁单元用于计算变截面梁的精度.采用直接基于单元平衡的思想给出了计算变截面梁反应的有限元方法,解决了单元位移插值函数局限性所带来的问题.导出了变截面梁单元的单元刚度矩阵、单元等效节点荷载和单元一致质量矩阵.在此基础上,利用编制的程序进行了算例验证与分析.算例验证了本文理论的正确性,表明本文方法具有很高的计算精度.     

ELEMENT－BY－ELEMENT MATRIX DECOMPOSITION ANDSTEP－BY－STEP INTEGRATION METHOD FOR TRANSIENTDYNAMIC PROBLEMS

ＥＬＥＭＥＮＴ－ＢＹ－ＥＬＥＭＥＮＴＭＡＴＲＩＸＤＥＣＯＭＰＯＳＩＴＩＯＮＡＮＤＳＴＥＰ－ＢＹ－ＳＴＥＰＩＮＴＥＧＲＡＴＩＯＮＭＥＴＨＯＤＦＯＲＴＲＡＮＳＩＥＮＴＤＹＮＡＭＩＣＰＲＯＢＬＥＭＳＷａｎｇＨｕａｉｚｈｏｎｇ（王怀忠）（ＲｅｃｅｉｖｅｄＪｕ...     

柔性梁大变形计算的一种新方法

研究了柔性梁大变形问题。常规Lagrangian有限元格式在处理大变形问题时,由于其单元插值函数不满足位移场的协调性要求,从而需要划分较多的单元,才能得到较好的结果。本文首先推导了Lagrangian坐标描述下的位移场变量满足的协调关系式,利用此关系式给出了位移场协调的非线性单元插值函数。基于虚功原理导出了梁大变形问题的非线性控制方程,数值计算结果证明了本文方法的正确性和有效性。     

Inverse finite element formulations for transient heat conduction problems

Transient heat conduction problems are normally simulated by the conventional consistent and lumped finite element methods. The discretization error and the physical reality violation in such problems are noticeable and unwanted responses are observed in the results when using the consistent formulations. Although in utilizing the lumped formulations, the violation of physical reality becomes reduced; however, emerging the discretization error would also become obvious to the degree of being quantifiable. In using the inverse finite element method without considering the element shape functions, the element matrices will be obtained by minimizing the governing equation and its generalized discretized corresponding formula. The results obtained by using this method indicates that the reduction in both the discretization error as well as the violation of physical reality would be realized.     

Some improvements on free surface simulation by the particle finite element method

The Lagrangian method has become increasingly popular in numerical simulation of free surface problems. In this paper, after a brief review of a recent Lagrangian method, namely the particle finite element method, some issues are discussed and some improvements are made. The least‐square finite element method is adopted to simplify the solving of the Navier–Stokes equations. An adaptive time method is derived to obtain suitable time steps. A mass correction procedure is imported to improve the mass conservation in long time calculations and time discretization scheme is adopted to decrease the pressure oscillations during the calculations. Finally, the method is used to simulate a series of examples and the results are compared with the commercial FLOW3D code. Copyright © 2008 John Wiley & Sons, Ltd.     

The finite dynamic annular element for the vibration analysis of variable thickness discs

According to the gyro-periodicity of dynamic displacements, the two-dimensional problems of circular plates with variable thickness are simplified into one-dimensional ones in this paper. Taking the expanded form of frequency power series of the dynamic displacement functions as the dynamic shape functions of the finite annular element, the mass and stiffness matrices as well as their one-order revised matrices are given succinctly. The dynamic method is used to analyse the vibration characteristics of a bladed disc assembly and is compared with conventional finite element method and experiment, and is proved to be superior to other numerical methods.     

高速列车紊态外流场的数值模拟研究

高速列车是近地运行的细长、庞大物体,它的空气绕流问题有其特殊性,本文以不可压缩粘性流体的Navier-Stokes方程和k-ε两方程紊流模型为基础,采用有限元方法求解了高速列车三维紊态外流场,针对有限元法应用于流场计算时常出现的问题,采用分离式解法,非对称矩阵一维变带宽压缩存储及带宽极小化等方法,最大限度地降低计算存储量;并采用罚函数法,集中质量矩阵,缩减积分法,带参数迭代法以及 引入松弛因子等技术,提出了一套用有限元法计算非线性问题的求解方法,提高了收敛速度的计算严谨,计算方法和计算结果对列车空气动力学的深入研究有一定的帮助。     

On the coupled vibration of an ideal fluid with a linear elastic structure

The purpose of this paper is to analyse theoretically and numerieally the coupled vibration of an ideal fiuid with a linear elastie structure.It is proved in the paper that the natural frequencies of the coupled vibration do exist and are all real positive. The paper presents an efficient method to transform a coupled fluid-structure system to the structure with added mass and the ribrational analysis of the former is replaced by the latter in vacuum only. Numerical solution is outlined for the transformed problem and a compact frequecy equation is derived in which fluid variables do not appear. This simplifies the analysis significanily. A convergent proof has been given to guarantee the reliability of the solution. The paper also offers a general algorithm combined with Ritz method, boundary element method, and finite element method to analyse the transformed problem. Based on this algorithm, one can apply a known structural analysing program, with a little modification, to solve many different kinds     

IMPROVEMENT OF EXPLICIT ALGORITHMS STABILITY WITH VISCO-ELASTIC ARTIFICIAL BOUNDARY ELEMENTS 1)

黏弹性人工边界单元是目前常用的处理半无限空间波动问题的数值模拟方法,可有效吸收计算区域内产生的外行波动.黏弹性人工边界单元具有与内部介质不同的质量密度、刚度和阻尼,受其影响,对整体模型进行显式时域逐步积分时,在边界区域易发生失稳现象,影响整体系统显式积分的计算效率. 针对该问题目前尚无行之有效的解决方法.本文针对二维黏弹性人工边界单元,建立可代表整体系统典型特征的侧边子系统和角点子系统,利用传递矩阵谱半径分析方法,基于传统中心差分格式,推导得到局部子系统稳定性条件的解析解.在此基础上通过研究解析解中各物理参数对稳定性条件的影响,给出通过增加人工边界单元的质量密度,以改善采用黏弹性人工边界单元时显式算法稳定性的方法.均匀和成层半空间波动问题算例分析表明,将内部单元质量密度设置为人工边界单元质量密度的上限,可以在保证黏弹性人工边界计算精度的前提下,有效改善整体系统显式时域逐步积分的数值稳定性,大幅提高计算效率.     

A variational multiscale Newton–Schur approach for the incompressible Navier–Stokes equations

In the following paper, we present a consistent Newton–Schur (NS) solution approach for variational multiscale formulations of the time‐dependent Navier–Stokes equations in three dimensions. The main contributions of this work are a systematic study of the variational multiscale method for three‐dimensional problems and an implementation of a consistent formulation suitable for large problems with high nonlinearity, unstructured meshes, and non‐symmetric matrices. In addition to the quadratic convergence characteristics of a Newton–Raphson‐based scheme, the NS approach increases computational efficiency and parallel scalability by implementing the tangent stiffness matrix in Schur complement form. As a result, more computations are performed at the element level. Using a variational multiscale framework, we construct a two‐level approach to stabilizing the incompressible Navier–Stokes equations based on a coarse and fine‐scale subproblem. We then derive the Schur complement form of the consistent tangent matrix. We demonstrate the performance of the method for a number of three‐dimensional problems for Reynolds number up to 1000 including steady and time‐dependent flows. Copyright © 2009 John Wiley & Sons, Ltd.     

Transient temperature fields and residual stress fields of metallic materials under welding

Based on the situation of welding thermal conduction and thermo-elasto-plasticityresearch,Ms paper explores some problems in this field.First,the boundary elementmethod for nonlinear problems is improved by linearization of nonlinear problems and usedin welding thermal conduction analysis.Second,the thermo-elasto-plastic finite elementmethod is used for the welding stress calculation,in which the phase transformation isconsidered by the"equivalent linear expansion coefficient method".The comparison of the calculated results with experimental data shows that themethods provided in this paper are available.