MLPG approximation to the <Emphasis Type="Italic">p</Emphasis>-Laplace problem

Meshless local Petrov-Galerkin (MLPG) method is discussed for solving 2D, nonlinear, elliptic p-Laplace or p-harmonic equation in this article. The problem is transferred to corresponding local boundary integral equation (LBIE) using Divergence theorem. The analyzed domain is divided into small circular sub-domains to which the LBIE is applied. To approximate the unknown physical quantities, nodal points spread over the analyzed domain and MLS approximation, are utilized. The method is a meshless method, since it does not require any background interpolation and integration cells and it dose not depend on geometry of domain. The proposed scheme is simple and computationally attractive. Applications are demonstrated through illustrative examples.     

结构复杂行为分析的有限质点法研究综述

有限质点法是一种结构分析的新方法,它以向量力学理论和数值计算为基础,以点值描述和途径单元为基本概念,以清晰的物理模型和质点运动控制方程描述结构行为。该方法的计算不需组集单元的刚度矩阵,也不需迭代求解控制方程式。与传统方法相比,在结构的动力、几何非线性、材料非线性、屈曲或褶皱失效、机构运动、接触和碰撞等复杂行为分析中有较大的优势。该文首先介绍有限质点法的基本理论,在此基础上着重阐述这种新的数值分析方法在空间结构复杂行为研究领域的优势及应用,并对该方法的发展趋势作出展望。     

Subatmospheric vapor pressures evaluated from internal-energy measurements

Vapor pressures were evaluated from measured internal-energy changes in the vapor+liquid two-phase region, ΔU ⁽²⁾. The method employed a thermodynamic relationship between the derivative quantity (ϖU ⁽²⁾/ϖV) _T and the vapor pressure (p _σ) and its temperature derivative (ϖp/ϖT)_σ. This method was applied at temperatures between the triple point and the normal boiling point of three substances: 1,1,1,2-tetrafluoroethane (R134a), pentafluoroethane (R125), and difluoromethane (R32). Agreement with experimentally measured vapor pressures near the normal boiling point (101.325 kPa) was within the experimental uncertainty of approximately ±0.04 kPa (±0.04%). The method was applied to R134a to test the thermodynamic consistency of a publishedp-p-T equation of state with an equation forp _σ for this substance. It was also applied to evaluate publishedp _σ data which are in disagreement by more than their claimed uncertainty.     

Buckling of non-uniform beams by a direct functional perturbation method

This study is divided into two parts. In the first, the buckling load (P) of heterogeneous columns is found by applying the Functional Perturbation Method (FPM) directly to the Buckling (eigenvalue) Differential Equation (BDE). The FPM is based on considering P and the transverse deflection (W) as functionals of heterogeneity, i.e., the elastic bending stiffness “K” (or the compliance S=1/K). The BDE is expanded functionally, yielding a set of successive differential equations for each order of the (Fréchet) functional derivatives of P and W. The obtained differential equations differ only in their RHS, and therefore a single modified Green function is needed for solving all orders. Consequently, an approximated value for the buckling load is obtained for any given morphology. Both deterministic and stochastic examples of simply supported columns are solved and discussed. Results are compared with solutions found in the literature for validation.In the second part, the Optimized DFPM (ODFPM) is presented. It is based on finding a new material property (which is a function of K or S) around which the DFPM solution is more accurate. The new material property is found by requiring that the second order perturbation term in the Fréchet series is minimized. As a result, a non-linear differential equation is obtained which relates the new material property with K through morphology. An exact solution for this equation is found, in a power form K^f, where f depends on morphology. Calculating P with respect to this new property yields more accurate results for the statistical characteristics of P.     

Solution of generalized density evolution equation via a family of <Emphasis Type="Italic">δ</Emphasis> sequences

The traditional probability density evolution equations of stochastic systems are usually in high dimensions. It is very hard to obtain the solutions. Recently the development of a family of generalized density evolution equation (GDEE) provides a new possibility of tackling nonlinear stochastic systems. In the present paper, a numerical method different from the finite difference method is developed for the solution of the GDEE. In the proposed method, the formal solution is firstly obtained through the method of characteristics. Then the solution is approximated by introducing the asymptotic sequences of the Dirac δ function combined with the smart selection of representative point sets in the random parameters space. The implementation procedure of the proposed method is elaborated. Some details of the computation including the selection of the parameters are discussed. The rationality and effectiveness of the proposed method is verified by some examples. Some features of the numerical results are observed.     

On generalized logistic equations with a non-homogeneous differential operator

We consider a generalized logistic equation of superdiffusive type, driven by a non-homogeneous nonlinear differential operator, which incorporates the p-Laplacian, the (p, q)-differential operator and the generalized p-mean curvature differential operator. Using variational methods coupled with truncation and comparison techniques, we prove a bifurcation-type theorem describing the dependence of positive solutions on the parameter λ > 0.     

On the heat-flow distribution from and the temperature profile on an equally heated rectangle,calculated by series development

A method is proposed to approximate the solutions of a certain class of differential equations, linear or nonlinear, in two or three, dimensions, provided that the boundary conditions are given on a rectangle or a parallelepiped, respectively. For other boundary shapes the co-ordinate system must be transformed to meet that requirement. As in illustration, an example is given for the solution of Poisson's equation in two dimensions with a constant heat source, giving the temperatures on the rectangle together with the heat-flow distribution along its edges. The basis of the method is a Taylor-series development around one point; the result is given in terms of as many partial derivatives in that point as is desired. A similar method has already been described by Small² for the ‘heat equation’ ?θ/?t = ?²θ/?x²². Compared with finite difference and finite element solutions, these methods have the advantage that the solution is continuous, whereas first and second derivatives such as heat fluxes are available at hardly any effort. The results are compared with those of the exact solution. Even if the size of the determinant is limited to 4 × 4, the accuracy is already better than 98.98 per cent. When more effort is spent to solve a system of 10 equations, the accuracy is better than 99–95 per cent.     

A CCC‐r chart for high‐yield processes

The cumulative count of a conforming (CCC) chart is used to monitor high‐quality processes and is based on the number of items inspected until observing r non‐conforming ones. This charting technique is known as a CCC‐r chart. The function of the CCC‐r chart is the sensitive detection of an upward shift in the fraction defectives of the process, p. As r gets larger, the CCC‐r chart becomes more sensitive to small changes of upward shift in p. However, since many observations are required to obtain a plotting point on the chart, the cost is fairly high. For this trade‐off problem it is necessary to determine the optimal number of non‐conforming items observed before a point is plotted, the sampling (inspection) interval, and the lower control limit for the chart. In this paper a simplified optimal design method is proposed. For illustrative purposes, some numerical results for the optimal design parameter values are provided. The expected profits per cycle obtained using the proposed optimal design method are compared with those obtained using other misspecified parameter values. The effects of changing these parameters on the profit function are shown graphically. Copyright © 2001 John Wiley & Sons, Ltd.     

Spectral element methods for nonlinear spatio-temporal dynamics of an Euler-Bernoulli beam

Spectral element methods are high order accurate methods which have been successfully utilized for solving ordinary and partial differential equations. In this paper the space-time spectral element (STSE) method is employed to solve a simply supported modified Euler-Bernoulli nonlinear beam undergoing forced lateral vibrations. This system was chosen for analysis due to the availability of a reference solution of the form of a forced Duffing's equation. Two formulations were examined: i) a generalized Galerkin method with Hermitian polynomials as interpolants both in spatial and temporal discretization (HHSE), ii) a mixed discontinuous Galerkin formulation with Hermitian cubic polynomials as interpolants for spatial discretization and Lagrangian spectral polynomials as interpolants for temporal discretization (HLSE). The first method revealed severe stability problems while the second method exhibited unconditional stability and was selected for detailed analysis. The spatialh-convergence rate of the HLSE method is of order α=p _s+1 (wherep _s is the spatial polynomial order). Temporalp-convergence of the HLSE method is exponential and theh-convergence rate based on the end points (the points corresponding to the final time of each element) is of order 2p _T−1 ≤α≤2p _T+1 (wherep _T is the temporal polynomial order). Due to the high accuracy of the HLSE method, good results were achieved for the cases considered using a relatively large spatial grid size (4 elements for first mode solutions) and a large integration time step (1/4 of the system period for first mode solutions, withp _T=3). All the first mode solution features were detected including the onset of the first period doubling bifurcation, the onset of chaos and the return to periodic motion. Two examples of second mode excitation produced homogeneous second mode and coupled first and second mode periodic solutions. Consequently, the STSE method is shown to be an accurate numerical method for simulation of nonlinear spatio-temporal dynamical systems exhibiting chaotic response.     

Cosserat point element (<Emphasis Type="Italic">CPE</Emphasis>) for finite deformation of orthotropic elastic materials

An eight node brick Cosserat point element (CPE) has been developed for the numerical solution of three-dimensional problems of hyperelastic nonlinear orthotropic elastic materials. In the Cosserat approach, a strain energy function for the CPE is proposed which satisfies restrictions due to a nonlinear form of the patch test. Part of the strain energy of the CPE is characterized by a three-dimensional strain energy function that depends on physically based nonlinear orthotropic invariants. Special attention has been focused on developing closed form expressions for constitutive coefficients in another part of the strain energy that characterizes the response to inhomogeneous deformations appropriate for orthotropic material response. A number of example problems are presented which demonstrate that the CPE is a robust user friendly element for finite deformations of orthotropic elastic materials, which does not exhibit unphysical locking or hourglassing for thin structures or nearly incompressible materials.     

On the a priori and a posteriori error analysis of a two‐fold saddle‐point approach for nonlinear incompressible elasticity

In this paper, we reconsider the a priori and a posteriori error analysis of a new mixed finite element method for nonlinear incompressible elasticity with mixed boundary conditions. The approach, being based only on the fact that the resulting variational formulation becomes a two‐fold saddle‐point operator equation, simplifies the analysis and improves the results provided recently in a previous work. Thus, a well‐known generalization of the classical Babu?ka–Brezzi theory is applied to show the well‐posedness of the continuous and discrete formulations, and to derive the corresponding a priori error estimate. In particular, enriched PEERS subspaces are required for the solvability and stability of the associated Galerkin scheme. In addition, we use the Ritz projection operator to obtain a new reliable and quasi‐efficient a posteriori error estimate. Finally, several numerical results illustrating the good performance of the associated adaptive algorithm are presented. Copyright © 2006 John Wiley & Sons, Ltd.     

基于有限质点法的结构动力非线性行为分析

有限质点法是以向量力学为基础的崭新的结构分析方法,该文将其应用到结构动力非线性行为分析中。该方法将结构离散为质点群,采用牛顿定律描述质点的运动,通过对质点行为的模拟和分析计算结构行为。该文介绍了有限质点法的基本思想,提出了该方法分析结构“动”与“静”,“线性”与“非线性”问题的独特思路。以杆系结构为例,推导了该方法求解结构动力反应,及几何、材料非线性问题的基本公式。通过三个数值算例,验证了该方法在结构动力非线性行为分析中的正确性和适用性。有限质点法在处理动力非线性问题时无需迭代求解和特殊修正,与传统方法相比在结构复杂行为分析中有明显的优势。     

Schwindungsgleichung: Bestimmung der Koeffizienten aus Sinterversuchen – mit Hilfe mehrfacher Regression

Sintering Equation: Determination of its coefficients by experiments – using multiple Regression Sintering is a method for volume-compression (or volume-contraction) of powdered or grained material applying high temperature (less than the melting point of the material). Mäkipirtti tried in ([2]) to find an equation which describes the process of sintering by its main parameters sintering time, sintering temperature and volume contracting. Such equation is called a sintering equation. It also contains some coefficients which characterise the behaviour of the material during the process of sintering. These coefficients have to be determined by experiments. Here we show that some linear regressions will produce wrong coefficients, but multiple regression results in an useful sintering equation.     

Convergence of the fixed point theorem with random parameters

Solving stochastic non‐linear dynamical problems represents a formidable task which, in many cases, can be achieved solely through numerical simulation techniques. This is true for high dimensional as well as low dimensional problems. One method to deal with the non‐linearity is to use the fixed point theorem which gives the convergence conditions of the iterative scheme towards the equilibrium point of the equation. In this paper we look at the particular case where the equilibrium equation depends on a random variable. This case arises for instance in the study of coupled non‐linear dynamical systems when structural uncertainties are introduced in the dynamical systems. We give almost sure and L ^p convergence conditions for the simulation iterative scheme. Copyright © 2003 John Wiley & Sons, Ltd.     

Indirect T‐Trefftz and F‐Trefftz methods for solving boundary value problem of Poisson equation

Abstract

The Poisson equation can be solved by first finding a particular solution and then solving the resulting Laplace equation. In this paper, a computational procedure based on the Trefftz method is developed to solve the Poisson equation for two‐dimensional domains. The radial basis function approach is used to find an approximate particular solution for the Poisson equation. Then, two kinds of Trefftz methods, the T‐Trefftz method and F‐Trefftz method, are adopted to solve the resulting Laplace equation. In order to deal with the possible ill‐posed behaviors existing in the Trefftz methods, the truncated singular value decomposition method and L‐curve concept are both employed. The Poisson equation of the type, ?² u = f(x, u), in which x is the position and u is the dependent variable, is solved by the iterative procedure. Numerical examples are provided to show the validity of the proposed numerical methods and some interesting phenomena are carefully discussed while solving the Helmholtz equation as a Poisson equation. It is concluded that the F‐Trefftz method can deal with a multiply connected domain with genus p(p > 1) while the T‐Trefftz method can only deal with a multiply connected domain with genus 1 if the domain partition technique is not adopted.     

Low Cost Optimization Techniques for Solving the Nonlinear Seismic Reflection Tomography Problem

The nonlinear seismic reflection tomography problem consists on minimizing the function g(p) = T – f(p)² ₂, where p is a vector containing the velocity model parameters and depth position of the reflectors, T contains the travel time of the rays, and f(p) is a highly nonlinear function that depends on the velocity of the subsurfaces. This problem has been solved using the Gauss-Newton method, and reconstruction techniques. These methods require either too many storage locations or too many iterations to converge. We present a different approach. We apply recently developed low storage minimization techniques directly on the function g(p). In particular, we use the recent global version of the spectral gradient method, and recent implementations of the conjugate gradient method. We present encouraging preliminary numerical results that indicate that our new approach clearly outperforms the classical techniques in CPU time and accuracy of the approximate solutions.     

The h–p version of the finite element method for problems with interfaces

Regularities of the solutions of interface problems in two dimensions are described in the frame of the weighted Sobolev spaces and countably normed spaces. Based upon the regularity of solutions the geometric meshes and the distribution of polynomial degrees are properly designed so that the h–p version of the finite element method for interface problems can lead to the exponential rate of convergence. Numerical results on an elliptic equation with interfaces are presented. The optimal mesh factor, optimal degree factors, and optimal layer factors of the geometric mesh in neighbourhoods of singular points having varied intensities are discussed from both theoretical and practical point of view.     

The hp‐d‐adaptive finite cell method for geometrically nonlinear problems of solid mechanics

The finite cell method (FCM) combines the fictitious domain approach with the p‐version of the finite element method and adaptive integration. For problems of linear elasticity, it offers high convergence rates and simple mesh generation, irrespective of the geometric complexity involved. This article presents the integration of the FCM into the framework of nonlinear finite element technology. However, the penalty parameter of the fictitious domain is restricted to a few orders of magnitude in order to maintain local uniqueness of the deformation map. As a consequence of the weak penalization, nonlinear strain measures provoke excessive stress oscillations in the cells cut by geometric boundaries, leading to a low algebraic rate of convergence. Therefore, the FCM approach is complemented by a local overlay of linear hierarchical basis functions in the sense of the hp‐d method, which synergetically uses the h‐adaptivity of the integration scheme. Numerical experiments show that the hp‐d overlay effectively reduces oscillations and permits stronger penalization of the fictitious domain by stabilizing the deformation map. The hp‐d‐adaptive FCM is thus able to restore high convergence rates for the geometrically nonlinear case, while preserving the easy meshing property of the original FCM. Accuracy and performance of the present scheme are demonstrated by several benchmark problems in one, two, and three dimensions and the nonlinear simulation of a complex foam sample. Copyright © 2011 John Wiley & Sons, Ltd.     

A CBS‐type stabilizing algorithm for the consolidation of saturated porous media

The presented method stems from the works by Zienkiewicz and co‐workers for coupled fluid/thermal problems starting from the early 1990s. They propose algorithms to overcome the difficulties connected to the application of the FEM to the area of fluid mechanics, which include the problems of singular behaviour in incompressibility and the problems connected to convective terms. The major step forward was to introduce the concept of characteristic lines (the particle paths in a simple convection situation): for a class of problems with a single scalar variable, the equations in the characteristic co‐ordinates regain self‐adjointness. The procedure is called characteristic based split algorithm (CBS). We use here a CBS‐type procedure for a saturated deformable elastic porous medium, in which the fluid velocity is governed by Darcy's equation (which comes directly from Navier–Stokes ones). The physical–mathematical model is a fully coupled one and is here used to study an incompressible flow inside a continuum with incompressible solid grains. The power of the adopted algorithm is to treat the basic equations in their strong form and to transform a usual ‘u–p’ problem into a ‘u–v–p’ one, where u generally indicates the displacement of the solid matrix and p and v the pressure and velocity of the fluid, respectively. Attention is focused on the expression of Darcy's velocity which is considered as the starting point of the algorithm. The accuracy of the scheme is checked by comparing the present predictions in a typical consolidation test with available analytical and numerical u–p solutions. A good fitting among different results has been obtained. It is further shown that the procedure eliminates the oscillations at the onset of consolidation, typical for many schemes. The FEM code Ed‐Multifield has been used for implementing and testing the procedure. Copyright © 2005 John Wiley & Sons, Ltd.     

Generalized finite element method in structural nonlinear analysis – a <Emphasis Type="Italic">p</Emphasis>-adaptive strategy

This paper is concerned with an extension of the generalized finite element method, GFEM, to nonlinear analysis and to the proposition of a p-adaptive strategy. The p-adaptivity is considered due to the nodal enrichment scheme of the method. Here, such scheme consists of multiplying the partition of unity functions by a set of polynomials. In a first part, the performance of the method in nonlinear analysis of a reinforced concrete beam with progressive damage is presented. The adaptive strategy is then proposed on basis of a control over the approximation error. Aiming to estimate the approximation error, the equilibrated element residual method is adapted to the GFEM and to the nonlinear approach. Then, global and local error measures are defined. A numerical example is presented outlining the effectivity index of the error estimator proposed. Finally, a p-adaptive procedure is described and its good performance is illustrated by a numerical example.The authors gratefully acknowledge the Conselho Nacional de Desenvolvimento Cientìifico e Tecnológico (CNPq) at Brazil.