On the Zienkiewicz four-time-level scheme for the numerical integration of vibration problems

The Zienkiewicz Weighted Residual Method produces a very general (three-parameter) four-time-level scheme for the numerical integration of vibration problems. This paper obtains the values of these parameters which minimize the incremental phase error and gives a general method for testing the stability and artificial damping with any other choice of the parameters.     

A non-linear transformation applied to boundary layer effect and thin-body effect in BEM for 2D potential problems

The accurate numerical evaluation of nearly singular integrals plays an important role in many engineering applications. In general, these include evaluating the solution near the boundary or treating problems with thin domains, which are respectively named the boundary layer effect and the thin-body effect in the boundary element method. Although many methods of evaluating nearly singular integrals have been developed in recent years with varying degrees of success, questions still remain. In this article, a general non-linear transformation for evaluating nearly singular integrals over curved two-dimensional (2D) boundary elements is employed and applied to treat boundary layer effect and thin-body effect occurring in 2D potential problems. The introduced transformation can remove or damp out the rapid variations of nearly singular kernels and extremely high accuracy of numerical results can be achieved without increasing other computational efforts. Extensive numerical experiments indicate that the proposed transformation will be more efficient, in terms of the necessary integration points and central processing unit-time, compared to previous transformation methods, especially for dealing with thin-body problems.     

An analysis of solidification problems by the boundary element method

The problem of interest in this paper is the calculation of the motion of the solid–liquid interface and the time-dependent temperature field during solidification of a pure metal. An iterative implicit algorithm has been developed for this purpose using the boundary element method (BEM) with time-dependent Green's functions and convolution integrals. The BEM approach requires discretization of only the surface of the solidifying body. Thus, the numerical method closely follows the physics of the problems and is intuitively very appealing. The formulation and the numerical scheme presented here are general and can be applied to a broad range of moving boundary problems. Emphasis is given to two-dimensional problems. Comparison with existing semi-analytical solutions and other numerical solutions from the literature reveals that the method is fast, accurate and without major time step limitations.     

Direct determination of asymptotic structural postbuckling behaviour by the finite element method

Application of the finite element method to Koiter's asymptotic postbuckling theory often leads to numerical problems. Generally it is believed that these problems are due to locking of non-linear terms of different orders. A general method is given here that explains the reason for the numerical problems and eliminates these problems. The reason for the numerical problems is that the postbuckling stresses are inaccurately determined. By including a local stress contribution, the postbuckling stresses are calculated correctly. The present method gives smooth postbuckling stresses and shows a quick convergence of the postbuckling coefficients. © 1998 John Wiley & Sons, Ltd.     

NUMERICAL SOLUTION OF TRANSIENT,FREE SURFACE PROBLEMS IN POROUS MEDIA

The focus of this paper is the development of numerical schemes for tracking the moving fluid surface during the filling of a porous medium (e.g., polymer injection into a porous mold cavity). Performing a mass balance calculation on an arbitrarily deforming control volume, leads to a general governing filling equation. From this equation, a general, fully time implicit, numerical scheme based on a finite volume space discretization is derived. Two numerical schemes are developed: (1) a fully deforming grid scheme, which explicitly tracks the location of the filling front, and (2) a fixed grid scheme, that employs an auxiliary variable to locate the front. The validity of the two schemes is demonstrated by solving a variety of one- and two-dimensional problems; both approaches provide predictions with similar accuracy and agree well with available analytical solutions.     

数值流形方法的数学推导及其应用

以往的数值流形方法都是以最小势能原理或变分原理为基础来建立求解方程的。但在实际工程中科技人员所遇到的有些实际问题,其控制方程所对应的泛函往往是难以找到的,在这些情况下就无法应用变分方法来建立数值流形方法的求解方程,而必须寻找较为一般的方法来推导数值流形方法的求解方程。因此,研究了如何从加权残数法出发建立数值流形方法的求解方程。在此过程中,通过建立弹性力学方程的数值流形方法,可以看出,通过选取适当的权函数,该方法最终的求解方程将转化为以最小势能原理或以变分原理为基础的离散形式。为了说明方法的有效性,求解了岩石试件中含单裂隙双边受拉的问题,并给出了裂隙尖端的应力强度因子和应力场的变化关系。     

An iterative algorithm for solving inverse problems in structural dynamics

The pulse-spectrum technique (PST), an iterative numerical algorithm, is presented and extended to solve the inverse problems arising from the dynamic structural identification and structural design problems. A simple one-dimensional shear beam model is used to demonstrate the applicability of PST. Numerical simulations are carried out to test the feasibility and to study the general characteristics of this technique without the real measurement and design data. It is found that the PST is not only quite robust in providing accurate results but also an excellent numerical method according to the four practical criteria for evaluating numerical methods.     

A Nonlinear Generalized Thermoelasticity Model of Temperature-Dependent Materials Using Finite Element Method

In this article, a general finite element method (FEM) is proposed to analyze transient phenomena in a thermoelastic model in the context of the theory of generalized thermoelasticity with one relaxation time. The exact solution of the nonlinear model of the thermal shock problem of a generalized thermoelastic half-space of temperature-dependent materials exists only for very special and simple initial- and boundary problems. In view of calculating general problems, a numerical solution technique is to be used. For this reason, the FEM is chosen. The results for the temperature increment, the stress components, and the displacement component are illustrated graphically with some comparisons.     

Constitutive parameter sensitivity in elasto-plasticity

The paper presents equations and algorithms for numerical computation of elasto-plastic and elasto-viscoplastic constitutive parameter sensitivity problems. The general integration idea is based on the return-mapping algorithm. Two viscoplastic constitutive models are discussed in details: the overstress (Perzyna) model and the power law strain and strain-rate hardening model. The use of the consistent tangent operator is shown to be essential for the accuracy of the sensitivity analysis. A possible discontinuity of sensitivity at the transition (yield limit) point is discussed. It is concluded that in principle the nonuniqueness of the sensitivity solutions at such points does not invalidate the general idea of sensitivity calculations. A number of numerical examples illustrate the theoretical considerations.The paper was supported by the Polish National Committee for Scientific Research (KBN) with grant no. 3 P404 018 04.Dedicated to J. C. Simo     

A Meshless Numerical Method for Kirchhoff Plates under Arbitrary Loadings

This paper describes the combination of the method of fundamental solutions (MFS) and the dual reciprocity method (DRM) as a meshless numerical method to solve problems of Kirchhoff plates under arbitrary loadings. In the solution procedure, a arbitrary distributed loading is first approximated by either the multiquadrics (MQ) or the augmented polyharmonic splines (APS), which are constructed by splines and monomials. The particular solutions of multiquadrics, splines and monomials are all derived analytically and explicitly. Then, the complementary solutions are solved formally by the MFS. Furthermore, the boundary conditions of lateral displacement, slope, normal moment, and effective shear force are all given explicitly for the particular solutions of multiquadrics, splines and polynomials as well as the kernels of MFS. Finally, numerical experiments are carried out to validate these analytical formulas. In these numerical experiments, homogeneous problems are first considered to find the best location of the MFS sources by the way proposed by Tsai, Lin, Young and Atluri (2006). Then the corresponding nonhomogeneous problems are solved by the DRM based on both the MQ and APS. The numerical results demonstrate that the MQ is in general more accurate than the thin plate spline, or the first order APS, but less accurate than the high order APSs. Overall, this paper derives a meshless numerical method for solving problems of Kirchhoff plates under arbitrary loadings with all kinds of boundary conditions by both the MQ and APS.     

A classification and survey of numerical methods for boundary value problems in ordinary differential equations

The objective of the paper is a survey and classification of available numerical methods for boundary value problems with an emphasis on two point boundary value problems. In the paper a general two point boundary value problem is stated first and later available numerical methods are classified into five main groups according to their approach in numerical solution. In the next sections of the paper, each of these main groups are described and appropriate subgroups are defined and discussed.     

Boundary element‐free method for fracture analysis of 2‐D anisotropic piezoelectric solids

This paper considers a 2‐D fracture analysis of anisotropic piezoelectric solids by a boundary element‐free method. A traction boundary integral equation (BIE) that only involves the singular terms of order 1/r is first derived using integration by parts. New variables, namely, the tangential derivative of the extended displacement (the extended displacement density) for the general boundary and the tangential derivative of the extended crack opening displacement (the extended displacement dislocation density), are introduced to the equation so that solution to curved crack problems is possible. This resulted equation can be directly applied to general boundary and crack surface, and no separate treatments are necessary for the upper and lower surfaces of the crack. The extended displacement dislocation densities on the crack surface are expressed as the product of the characteristic terms and unknown weight functions, and the unknown weight functions are modelled using the moving least‐squares (MLS) approximation. The numerical scheme of the boundary element‐free method is established, and an effective numerical procedure is adopted to evaluate the singular integrals. The extended ‘stress intensity factors’ (SIFs) are computed for some selected example problems that contain straight or curved cracks, and good numerical results are obtained. Copyright © 2006 John Wiley & Sons, Ltd.     

Interactions between inclusions and various types of cracks

The problems of a crack inside, outside, penetrating or lying along the interface of an anisotropic elliptical inclusion are considered in this paper. Because the crack may be represented by a distribution of dislocation, integrating the analytical solutions of dislocation problems along the crack and applying the technique of numerical solution on the singular integral equation, we can obtain the general solutions to the problems of interactions between cracks and anisotropic elliptical inclusions. Since there are no analytical solutions existing for the general cases of interactions between cracks and inclusions, the comparison is made with the numerical results obtained by other methods or with the analytical results for the special cases which can be reduced from the present problems. These results show that our solutions are correct and universal     

Multiple pole residue approach for 3D BEM analysis of mathematical degenerate and non‐degenerate materials

In this paper we develop an alternative boundary element method (BEM) formulation for the analysis of anisotropic three‐dimensional (3D) elastic solids. Our implementation is based on the derivation of explicit expressions for the fundamental solution displacements and tractions, of general validity for any class of anisotropic materials, by means of Stroh formalism and Cauchy's residue theory. The resulting fundamental solution remains valid for mathematical degenerate cases when Stroh's eigenvalues are coincident, meanwhile it does not exhibit numerical instabilities for quasi‐degenerate cases when Stroh's eigenvalues are nearly equal. A multiple pole residue approach is followed, leading to general explicit expressions to evaluate the traction fundamental solution for poles of m‐multiplicity. Despite the existence of general displacement solutions in the literature, and for the sake of completeness, the same approach as for the traction solution is considered to derive the displacement fundamental solution as well. Based on these solutions, an explicit BEM approach for the numerical solution of 3D linear elastic problems for solids with general anisotropic behavior is presented. The analysis of cracked anisotropic solids is also considered. Details on the numerical implementation and its validation for degenerate cases are discussed. Copyright © 2010 John Wiley & Sons, Ltd.     

Loading,unloading and reloading of a generalized plane plastica

This article presents a shooting-type numerical procedure for the analysis of a planar, continuous and flexible member where the material is permitted to unload and reload plastically. A kinematic type of strainhardening rule is employed, and large displacements, large rotations and general cross section shapes are admitted. Though numerical and semi-analytical solutions exist in the literature for specific problems where the internal and external loads increase monotonically (see, for example, References 1–4), little attention has been paid to problems where these loads (specifically internal loads) are permitted to decrease or reverse direction. In the present work a comparison is made with the solution to one such problem found in the literature, and the solution to four additional problems is given to show the versatility and usefulness of the method.     

Numerical implementation of a CTO-based implicit approach for the BEM solution of usual and sensitivity problems in elasto-plasticity

Boundary element method (BEM) formulations for usual and sensitivity problems in small strain elastoplasticity, using the concept of the consistent tangent operator (CTO), have been recently proposed by Bonnet and Mukherjee. ‘Usual' problems here refer to analysis of nonlinear problems in structural and solid continua, for which Simo and Taylor first proposed the use of the CTO within the context of the finite element method (FEM). It was shown by Bonnet and Mukherjee that the sensitivity of the strain increment, associated with an infinitesimal variation of some design parameter, solves a linear problem which is governed by the (converged value of the) same global CTO as the one that appears in the usual problem.

This paper presents a general numerical implementation of the above formulations. Numerical results for the usual and sensitivity problems are presented for a two-dimensional (plane strain) example. Sensitivities are calculated with respect to a material parameter that characterizes isotropic strain hardening. The crucial role of the CTO, in providing accurate numerical results for the mechanical variables as well as their sensitivities, is examined in this paper.     

Patch based stress recovery for plate structures

In this paper we address the application of recovery procedures in advanced problems in structural mechanics. The attention is focused on the recovery by compatibility in patches procedure (RCP) and shear deformable plate structures. The formulation of RCP procedure is extended to shear deformable plate problems (Reissner–Mindlin theory) and is applied to recover stresses from mixed and hybrid stress finite elements. These elements offer new possibilities, for recovery procedures in general, which deserve to be discussed. A comprehensive investigation on which finite element solution can be used as input for the recovery procedures is given through standard benchmark problems, obtained for several values of the thickness on structured and unstructured meshes. The numerical results confirm the effectiveness of the recovery procedure extended to plates problems.     

Boundary integral equation method for shape optimization of elastic structures

A general method for shape design sensitivity analysis as applied to plane elasticity problems is developed with a direct boundary integral equation formulation, using the material derivative concept and adjoint variable method. The problem formulation is very general and a complete consideration is given to describing the boundary variation by including the tangential component of the velocity field. The method is then applied to obtain the sensitivity formula for a general stress constraint imposed over a small part of the boundary. The accuracy of the design sensitivity analysis is studied with a fillet and an elastic ring design problem. Among the several numerical implementations tested, the second order boundary elements with a cubic spline representation of the moving boundary have shown the best accuracy. A smooth characteristic function is found to be better than a plateau function for localization of the stress constraint. Optimal shapes for the two problems are presented to show numerical applications.     

CONDIF: A modified central-difference scheme for convective flows

For most practical purposes, the central-difference scheme (CDS) would be ideal only if it were unconditionally stable. It is a simple and second-order scheme which is easy to implement. It does not introduce any second-order ‘diffusion’ like truncation error. However, for grid Peclet numbers larger than 2, the CDS leads to over- and under-shoots and is unstable. This paper presents a method, called CONDIF, which eliminates this undesirable feature of the CDS. It modifies the CDS by introducing a controlled amount of numerical diffusion based on the local gradients. The numerical diffusion can be adjusted to be negligibly low for most problems. CONDIF has been used to solve a number of test problems which have been widely used for comparative study of numerical schemes in the published literature. For all these problems the CONDIF results are significantly more accurate than those obtained from the hybrid scheme when the Peclet number is very high (→∞) and the flow is at large angles (→45 degrees) to the grid. In general the computational effort for CONDIF is comparable (within 20 per cent) to that for the hybrid scheme. However, in one instance the rate of convergence was found to be significantly slower.