Implicit plane stress algorithm for multilayer J2‐plasticity using the Prager–Ziegler translation rule

A totally implicit algorithm for plane stress multilayer plasticity is presented. The algorithm presents the plane stress version of the 3D/plane strain model for multilayer plasticity presented in a previous work. As in the 3D/plane strain case, the model is consistent with the principle of maximum dissipation and it may be considered as an extension of classical J₂‐plasticity for anisotropic non‐linear kinematic behaviour preserving Masing's rules. In order to obtain the asymptotic second‐order convergence of the Newton algorithm, the consistent tangent moduli are also given. Copyright © 2003 John Wiley & Sons, Ltd.     

The Governing Parameter Method for implicit integration of viscoplastic constitutive relations for isotropic and orthotropic metals

A general algorithm of implicit stress integration in viscoplasticity, based on the governing parameter method (GPM) is briefly presented. It is assumed that the associative viscoplastic constitutive relations are governed by the Perzyna formulation with a generalization suggested by Simo and Hughes. The algorithm is first applied to isotropic metals obeying the von Mises yield condition with mixed hardening and then, to orthotropic metals with a generalized Hill's yield condition including a mixed hardening assumption. Derivation of consistent tangent moduli is presented for both viscoplastic material models. The proposed computational procedures are efficient, since they reduce the problem of stress integration to the solution of one nonlinear equation, can use large time steps and are applicable to 2-D, 3-D, shell and beam structures. The tangent elastic viscoplastic matrix provides high convergence rate in the overall equilibrium iterations. Numerical examples illustrate the main characteristics of the developed computational procedures.     

Classification of stress modes in assumed stress fields of hybrid finite elements

A classification method is presented to classify stress modes in assumed stress fields of hybrid finite element based on the eigenvalue examination and the concept of natural deformation modes. It is assumed that there only exist m (=n−r) natural deformation modes in a hybrid finite element which has n degrees of freedom and r rigid-body modes. For a hybrid element, stress modes in various assumed stress fields proposed by different researchers can be classified into m stress mode groups corresponding to m natural deformation modes and a zero-energy stress mode group corresponding to rigid-body modes by the m natural deformation modes. It is proved that if the flexibility matrix [H] is a diagonal matrix, the classification of stress modes is unique. Each stress mode group, except the zero-energy stress mode group, contains many stress modes that are interchangeable in an assumed stress field and do not cause any kinematic deformation modes in the element. A necessary and sufficient condition for avoiding kinematic deformation modes in a hybrid element is also presented. By means of the m classified stress mode groups and the necessary and sufficient condition, assumed stress fields with the minimum number of stress modes can be constructed and the resulting elements are free from kinematic deformation modes. Moreover, an assumed stress field can be constructed according to the problem to be solved. As examples, 2-D, 4-node plane element and 3-D, 8-node solid element are discussed. © 1997 John Wiley & Sons, Ltd.     

A posteriori error analysis and adaptive processes in the finite element method: Part I—error analysis

This is a paper presented in two parts dealing respectively with error analysis and adaptive processes applied to finite element calculations. Part I contains the basic theory and methods of deriving error estimates for second-order problems. Part II of the paper deals with the strategy for adaptive refinement and concentrates on the p-convergent methods. It is shown that an extremely high rate of convergence is reached in practical problems using such procedures. Applications to realistic stress analysis and potential problems are presented.     

A posteriori error analysis and adaptive processes in the finite element method: Part II—adaptive mesh refinement

This is a paper presented in two parts dealing respectively with error analysis and adaptive processes applied to finite element calculations. Part I contains the basic theory and methods of deriving error estimates for second-order problems. Part II of the paper deals with the strategy for adaptive refinement and concentrates again on the p-convergent methods. It is shown that an extremely high rate of convergence is reached in practical problems using such procedures. Applications to realistic stress analysis and potential problems are presented.     

Efficient and accurate stress recovery procedure and a posteriori error estimator for the stable generalized/extended finite element method

This paper presents a new stress recovery technique for the generalized/extended finite element method (G/XFEM) and for the stable generalized FEM (SGFEM). The recovery procedure is based on a locally weighted L² projection of raw stresses over element patches; the set of elements sharing a node. Such projection leads to a block-diagonal system of equations for the recovered stresses. The recovery procedure can be used with GFEM and SGFEM approximations based on any choice of elements and enrichment functions. Here, the focus is on low-order 2D approximations for linear elastic fracture problems. A procedure for computing recovered stresses at re-entrant corners of any internal angle is also presented. The proposed stress recovery technique is used to define a Zienkiewicz-Zhu (ZZ) a posteriori error estimator for the G/XFEM and the SGFEM. The accuracy, computational cost, and convergence rate of recovered stresses together with the quality of the ZZ estimator, including its effectivity index, are demonstrated in problems with smooth and singular solutions.     

Analysis,design and optimization of structures using force method and genetic algorithm

In the first part of this paper, the energy formulation of the force method is presented and analysis is performed using genetic algorithm. Two simple examples are provided to show the accuracy of the approach. In the second part, an efficient method is developed for designing structures with prescribed stress ratios for its members. The genetic algorithm performed very well and designs with specified stress ratios were achieved with a good convergence rate. A unit value of c_i for all the members of a structure corresponds to the well known fully stressed design. In the third part, minimum weight design is formulated by the additional conditions being imposed on the design process. Again, genetic algorithm showed to be a powerful means for optimization. Copyright © 2005 John Wiley & Sons, Ltd.     

On the convergence of the lp ‐norm algorithm for polynomial perceptron having different error signal distributions

Abstract

The convergence property of the l_p ‐norm algorithm for polynomial‐perceptron having different error signal distributions will be analyzed in this paper. To see the effect of error signal on the convergence rate, two types of activation functions are considered in the analysis: one is of a linear type and the other is of a sigmoidal type. Different activation functions yield different ranges of output signal and, in turn, yield different error signal distributions. Linear activation function causes the error signal to be distributed in an uncertain way, while sigmoidal activation function causes it to be distributed in a tightly bounded region. Based on this difference the convergence property of the l_p ‐norm algorithm, 1 ≤ p ≤ 2, is investigated in this paper. Expressions of average learning gains are obtained in terms of the power metric p, the error probability, and the upper bound of the error signal distribution. Analytic results indicate that it is of particular value in using the l_p ‐norm algorithm for the perceptron using sigmoidal activation functions. Computer simulation of an adaptive equalizer using this algorithm confirms the theoretical analysis.     

Two stress update algorithms for large strains: accuracy analysis and numerical implementation

Two algorithms for the stress update (i.e., time integration of the constitutive equation) in large-strain solid mechanics are compared from an analytical point of view. The order of the truncation error associated to the numerical integration is deduced for each algorithm a priori, using standard numerical analysis. This accuracy analysis has been performed by means of a convected frame formalism, which also allows a unified derivation of both algorithms in spite of their inherent differences. Then the two algorithms are adapted from convected frames to a fixed Cartesian frame and implemented in a small-strain finite element code. The implementation is validated by means of a set of simple deformation paths (simple shear, extension, extension and compression, extension and rotation) and two benchmark tests in non-linear mechanics (the necking of a circular bar and a shell under ring loads). In these numerical tests, the observed order of convergence is in very good agreement with the theoretical order of convergence, thus corroborating the accuracy analysis. © 1997 John Wiley & Sons, Ltd.     

A priori and a posteriori analysis of the meshless Galerkin boundary node method for three-dimensional elasticity

The meshless Galerkin boundary node method is presented in this paper for boundary-only analysis of three-dimensional elasticity problems. In this method, boundary conditions can be implemented directly and easily despite the employed moving least-squares shape functions lack the delta function property, and the resulting system matrices are symmetric and positive definite. A priori error estimates and the consequent rate of convergence are presented. A posteriori error estimates are also provided. Reliable and efficient error estimators and an efficient and convergent adaptive meshless algorithm are then derived. Numerical examples showing the efficiency of the method, confirming the theoretical properties of the error estimates, and illustrating the capability of the adaptive algorithm, are reported.     

求解无限域动力刚度矩阵的双渐近算法

采用连分式算法可以有效地求解无限域动力刚度表示的比例边界有限元方程, 它具有收敛范围广、收敛速度快等优点. 该文在高频渐近连分式算法的基础上考虑了低频渐近, 发展了一种针对矢量波动方程的双渐近算法. 随着展开阶数的增加, 双渐近算法可以在全频域范围内快速逼近准确解. 引入了系数矩阵?X(i)来增强连分式算法的数值稳定性. 通过在高频极限、低频极限时满足动力刚度表示的比例边界有限元方程, 建立了递推关系以求得动力刚度矩阵. 通过二维半无限楔形体、三维均质弹性半空间数值算例表明, 双渐近算法比单渐近算法更稳定、优越.     

Fracture analysis of a steel sandwich with hardened core

2-D elasticity was employed to solve a thermal stress problem of a three layer laminated beam, in which each layer has different thermal expansion coefficient. The thermal stresses arise as a result that initially the layers of the beam were bonded at some high temperature T _Hsch and then quenched to some low temperature T _l . Elastic stress field can be derived and expressed in closed form by following the algorithm presented. A numerical example is presented to show that stress concentration due to this thermal effect at the ends of a laminated beam is high and steep enough to cause failure in an ultra-high carbon content and hardened steel.     

Automatic adaptive refinement for shell analysis using nine-node assumed strain element

An automatic adaptive refinement procedure for the analysis of shell structures using the nine-node degenerated solid shell element is suggested. The basic adaptive refinement principle and the effects of singularities and boundary layers on the convergence rate of the nine-node element used are discussed. A new stress recovery procedure based on the patch convective co-ordinate system concept is developed for the construction of a continuous smoothed stress field over the shell domains. The stress recovery procedure is easy to implement, requires a modest computational effort and needs only local patch information. It can be applied to shells with non-uniform thickness as well as to multi-layered shell structures. The smoothed recovered stress obtained is then used with the Zienkiewicz and Zhu error estimator for a posteriori error estimation during the adaptive refinement analysis. Numerical results which are in good agreement with theoretical predictions are obtained and they indicate that the current adaptive refinement procedure can eliminate the effect of singularities inside the problem domains so that a near-optimal convergence rate is achieved in all the numerical examples. This also indicates that the stress recovery procedure can produce an accurate stress field and as a result the error estimator can reflect the error distribution of the finite element solution. Even though in the current study only one type of element is used in the analysis, the whole adaptive refinement scheme can be readily applied to any other types of degenerated solid element. © 1997 John Wiley & Sons, Ltd.     

汽车车内噪声主动控制变步长NFB-LMS算法

为规避最小均方（Least Mean Square,LMS）算法不能同时提高收敛速度和降低稳态误差的固有缺陷,以及已有变步长LMS算法存在收敛速度慢和稳态误差估计精度差的问题,文中提出了一种基于变步长归一化频域块（Normalized Frequency-domain Block,NFB） LMS算法的汽车车内噪声主动控制方法。为了比较,应用传统的LMS算法、基于反正切函数的变步长LMS算法和变步长NFB-LMS算法分别进行实测汽车车内噪声的主动控制。结果表明,与其他两个算法相比,变步长NFB-LMS算法的收敛速度提高了70%以上,稳态误差减小了90%以上。变步长NFB-LMS算法在处理车内噪声信号时具有很高的效率,为进行汽车车内噪声主动控制提供了一种新方法。     

Multi-scale modeling, stress and failure analyses of 3-D woven composites

The very complex, multi-level hierarchical construction of textile composites and their structural components commonly manifests via significant property variation even at the macro-level. The concept of a “meso-volume” (introduced by this author in early 1990s) is consistently applied in this work to 3-D stress/strain and failure analyses of 3-D woven composites at several levels of structural hierarchy. The meso-volume is defined as homogeneous, anisotropic block of composite material with effective elastic properties determined through volumetrically averaged 3-D stress and strain fields computed at a lower (“finer”) level of structural hierarchy and application of generalized Hooke’s law to the averaged fields. The meso-volume can represent a relatively large, homogenized section of a composite structural component, a lamina in laminated composite structure, a homogenized assembly of several textile composite unit cells, a single homogenized unit cell, a resin-impregnated yarn, a single carbon fiber, even a carbon nanotube assembly. When composed together, distinct meso-volumes constitute a 3-D Mosaic model at the respective hierarchy level. A multi-scale methodology presented in this paper first illustrates 3-D stress/strain analysis of the Mosaic unidirectional composite, computation of its effective elastic properties and their further use in 3-D stress/strain analysis of the Mosaic model of 3-D woven composite Unit Cell. The obtained 3-D stress/strain fields are then volumetrically averaged within the Unit Cell, and its effective elastic properties are computed. The predicted effective elastic properties of 3-D woven composite are compared with experimental data and show very good agreement. Further, those effective elastic properties are used in 3-D simulations of three-point bending tests of 3-D woven composite; theoretical predictions for central deflection show excellent agreement with experimental data. Finally, a 3-D progressive failure analysis of generic 3-D Mosaic structure is developed using ultimate strain criterion and illustrated on the 3-D woven composite Unit Cell. The predicted strength values are compared to experimental results. The presented comparisons of theoretical and experimental results validate the adequacy and accuracy of the developed material models, mathematical algorithms, and computational tools.     

Sparse self‐stress matrices for the finite element force method

A basic problem in the finite element force method is that of obtaining a sparse and banded self‐stress matrix and a sparse and banded structure flexibility matrix.Traditionally the self‐stress matrix is obtained through the application of algebraic procedures to the equilibrium matrix. The self‐stress matrix for an indeterminate structure is not unique, and it is possible to obtain another self‐stress matrix from an existing one through algebraic operations and grouping of redundants. The purpose of this paper is to describe and test an algorithm, called REDUC, which combines the vectors of the self‐stress matrix obtained from the LU procedure of the force method. The rows of the transpose of this matrix are combined by using a special form of the Gaussian elimination technique. A plane frame example is presented to demonstrate the algorithm at work. The algorithm REDUC is applied to a plane truss and physical interpretation of the resulting self‐stress matrix highlights the grouping of redundants, improved sparsity and bandwidth. Improvements in the conditioning and bandwidth of the structure flexibility matrix are also observed. The algorithm yields results similar to those of the turn‐back LU procedure, but requires less computation time and programming effort. Copyright © 2001 John Wiley & Sons, Ltd.     

New a posteriori error estimator and adaptive mesh refinement strategy for 2-D crack problems

A posteriori error estimation and adaptive refinement technique for fracture analysis of 2-D/3-D crack problems is the state-of-the-art. The objective of the present paper is to propose a new a posteriori error estimator based on strain energy release rate (SERR) or stress intensity factor (SIF) at the crack tip region and to use this along with the stress based error estimator available in the literature for the region away from the crack tip. The proposed a posteriori error estimator is called the K-S error estimator. Further, an adaptive mesh refinement (h-) strategy which can be used with K-S error estimator has been proposed for fracture analysis of 2-D crack problems. The performance of the proposed a posteriori error estimator and the h-adaptive refinement strategy have been demonstrated by employing the 4-noded, 8-noded and 9-noded plane stress finite elements. The proposed error estimator together with the h-adaptive refinement strategy will facilitate automation of fracture analysis process to provide reliable solutions.     

Error estimation of stress intensity factors for mixed‐mode cracks

This paper presents an a posteriori error estimator for mixed‐mode stress intensity factors in plane linear elasticity. A surface integral over an arbitrary crown is used for the separate calculation of the combined mode's stress intensity factors. The error in the quantity of interest is based on goal‐oriented error measures and estimated through an error in the constitutive relation. Copyright © 2006 John Wiley & Sons, Ltd.     

Error estimation based on Superconvergent Patch Recovery using statically admissible stress fields

In this paper we investigate an approach for a posteriori error estimation based on recovery of an improved stress field. The qualitative properties of the recovered stress field necessary to obtain a conservative error estimator, i.e. an upper bound on the true error, are given. A specific procedure for recovery of an improved stress field is then developed. The procedure can be classified as Superconvergent Patch Recovery (SPR) enhanced with approximate satisfaction of the interior equilibrium and the natural boundary conditions. Herein the interior equilibrium is satisfied a priori within each nodal patch. Compared to the original SPR-method, which usually underestimates the true error, the present approach gives a more conservative estimate. The performance of the developed error estimator is illustrated by investigating two plane strain problems with known closed-form solutions. © 1998 John Wiley & Sons, Ltd.     

A pollution controlled hybrid interior error estimator for linear elastostatics

A hybrid error estimator using a priori interior region estimates in an a posteriori framework is presented for linear elastostatics problems in FEA. It is shown that local rates of convergence are augmented by this technique and global rates are not adversely affected. The effects of pollution for this estimator are explained and a pollution error estimator is derived using the concept of error loads. It is shown that pollution error estimation can improve the performance of both the conventional a posteriori and the hybrid techniques. A series of numerical results are presented which demonstrate the superior performance of the proposed method over previously published interior error estimation techniques. © 1998 John Wiley & Sons, Ltd.