General boundary element method for non-linear heat transfer problems governed by hyperbolic heat conduction equation

The general Boundary Element Method (BEM) for strongly non-linear problems proposed by Liao (1995) is further applied to solve a two-dimensional unsteady non-linear heat transfer problem in the time domain, governed by the hyperbolic heat conduction equation (HHCE) with the temperature-dependent thermal conductivity coefficients which are different in the x and y directions. This paper confirms that the general BEM can be used to solve even those non-linear unsteady heat transfer problems whose governing equations do not contain any linear terms in spatial domain.     

Transfer matrix solutions for three-dimensional consolidation of a multi-layered soil with compressible constituents

In this paper, transfer matrix solutions for three-dimensional consolidation of a multi-layered soil considering the compressibility of pore fluid are presented. The derivation of the solutions starts with the fundamental differential equations of Biot’s three-dimensional consolidation theory, takes into account the compressibility of pore fluid in the Cartesian coordinate system, and introduces the extended displacement functions. The relationship of displacements, stresses, excess pore water pressure, and flux between the ground surface (z = 0) and an arbitrary depth z is established for Biot’s three-dimensional consolidation problem of a finite soil layer with compressible pore fluid by taking the Laplace transform with respect to t and the double Fourier transform with respect to x and y, respectively. Based on this relationship of the transfer matrix, the continuity between layers, and the boundary conditions, the solutions for Biot’s three-dimensional consolidation problem of a multi-layered soil with compressible constituents in a Laplace-Fourier transform domain is obtained. The final solutions in the physical domain are obtained by inverting the Laplace-Fourier transforms. Numerical analysis is carried out by using a corresponding program based on the solutions developed in this study. This analysis demonstrates that the compressibility of pore fluid has a remarkable effect on the process of consolidation.     

An iterative defect‐correction type meshless method for acoustics

Accurate numerical simulation of acoustic wave propagation is still an open problem, particularly for medium frequencies. We have thus formulated a new numerical method better suited to the acoustical problem: the element‐free Galerkin method (EFGM) improved by appropriate basis functions computed by a defect correction approach. One of the EFGM advantages is that the shape functions are customizable. Indeed, we can construct the basis of the approximation with terms that are suited to the problem which has to be solved. Acoustical problems, in cavities Ω with boundary T, are governed by the Helmholtz equation completed with appropriate boundary conditions. As the pressure p(x,y) is a complex variable, it can always be expressed as a function of cosθ(x,y) and sinθ(x,y) where θ(x,y) is the phase of the wave in each point (x,y). If the exact distribution θ(x,y) of the phase is known and if a meshless basis {1, cosθ(x,y), sinθ (x,y) } is used, then the exact solution of the acoustic problem can be obtained. Obviously, in real‐life cases, the distribution of the phase is unknown. The aim of our work is to resolve, as a first step, the acoustic problem by using a polynomial basis to obtain a first approximation of the pressure field p(x,y). As a second step, from p(x,y) we compute the distribution of the phase θ(x,y) and we introduce it in the meshless basis in order to compute a second approximated pressure field p(x,y). From p(x,y), a new distribution of the phase is computed in order to obtain a third approximated pressure field and so on until a convergence criterion, concerning the pressure or the phase, is obtained. So, an iterative defect‐correction type meshless method has been developed to compute the pressure field in Ω. This work will show the efficiency of this meshless method in terms of accuracy and in terms of computational time. We will also compare the performance of this method with the classical finite element method. Copyright © 2003 John Wiley & Sons, Ltd.     

Effect of ellipse orientation on the thermoelastic behaviour of skew laminated composite plate with elliptical cutout

An effort is made to study the thermoelastic behaviour of a cross-ply laminated composite skew plate with elliptical cutout subjected to pressure and non-linearly varying temperature loading in the present analysis. Orientation of the elliptical cut out is varied from 0° to 180° with respect to horizontal at an interval of 30° in the anti clockwise direction is considered for the present analysis. A three-dimensional heat conduction analysis in fibre reinforced composite laminates has been simulated by finite element method to get realistic temperature in the laminate under different thermal boundary conditions. A finite element method, which works on the basis of three-dimensional theory of elasticity, is employed to evaluate the stresses and deformations. The effect of orientation due to pressure loading on the stresses and transverse deflection is observed to be insignificant. The magnitudes of the in-plane normal stresses, σ _x and σ _y, for temperature loading are greatly affected by ellipse orientation and are observed to be minimum at the ellipse orientation of 0° and 90°, respectively. The in-plane and inter-laminar shear stresses are observed to be minimum at the ellipse orientation of 90°.     

Three-dimensional finite element analysis of cracked thick plates in bending

By using the finite element technique, stress intensity factors have been obtained for finite rectangular plates and the results have been given for various h/a, W/a and L/W ratios. By using a three-dimensional isoparametric element, the problem has been considered as a three-dimensional one and the variation of stress intensity factor across the plate thickness has been found to be nonlinear.     

A residual a posteriori error estimate for partition of unity finite elements for three-dimensional transient heat diffusion problems using multiple global enrichment functions

In this article, a study of residual based a posteriori error estimation is presented for the partition of unity finite element method (PUFEM) for three-dimensional (3D) transient heat diffusion problems. The proposed error estimate is independent of the heuristically selected enrichment functions and provides a useful and reliable upper bound for the discretization errors of the PUFEM solutions. Numerical results show that the presented error estimate efficiently captures the effect of h-refinement and q-refinement on the performance of PUFEM solutions. It also efficiently reflects the effect of ill-conditioning of the stiffness matrix that is typically experienced in the partition of unity based finite element methods. For a problem with a known exact solution, the error estimate is shown to capture the same solution trends as obtained by the classical L₂ norm error. For problems with no known analytical solutions, the proposed estimate is shown to be used as a reliable and efficient tool to predict the numerical errors in the PUFEM solutions of 3D transient heat diffusion problems.     

Three-dimensional flow with heat transfer of a viscoelastic fluid over a stretching surface with a magnetic field

The present research study deals with the steady flow and heat transfer of a viscoelastic fluid over a stretching surface in two lateral directions with a magnetic field applied normal to the surface. The fluid far away from the surface is ambient and the motion in the flow field is caused by stretching surface in two directions. This result is a three-dimensional flow instead of two-dimensional as considered by many authors. Self-similar solutions are obtained numerically. For some particular cases, closed form analytical solutions are also obtained. The numerical calculations show that the skin friction coefficients in x- and y-directions and the heat transfer coefficient decrease with the increasing elastic parameter, but they increase with the stretching parameter. The heat transfer coefficient for the constant heat flux case is higher than that of the constant wall temperature case.     

A finite element method for the study of solidification processes in the presence of natural convection

A new numerical technique has been developed for the analysis of two-dimensional transient solidification processes in the presence of time-dependent natural convection in the melt. The method can cope with irregular, transient morphologies of the solid—liquid interface using a new Galerkin formulation for the energy balance on the solid—liquid interface. The finite element solution to the Galerkin formulation yields the displacement of individual nodes on the solid—liquid interface. The displacement of the nodes is expressed by uncoupled components in the x and y directions. The fluid flow problem was solved using a ‘penalty’ formulation. Numerical experiments were performed for Rayleigh numbers as high as 10⁶ to demonstrate the method and to indicate the effect of natural convection on the solid—liquid interface morphology.     

Nonlinear superposition of unidirectional and plane flows of viscoelastic Lodge fluids

Summary. The paper deals with steady fully developed flows through straight channels of constant cross-section under the influence of an axial pressure gradient and under the action of a moving wall. Those flow fields are characterized by Cartesian velocity components of the type u(x,y), v(x,y) and w(x,y) and thus may be considered to be the superposition of a longitudinal and a transverse plane part. They possess some striking properties, not only in case of a Newtonian fluid, but also for quasi-linear viscoelastic fluids with single-integral constitutive equations. We find a one-sided coupling between the axial component w and the transverse components u and v. Accordingly, the transverse flow does not depend on the axial boundary conditions and can be treated first as a two-dimensional problem. Afterwards, a linear integro-differential boundary value problem concerning w(x,y) results, from which some superposition theorems follow, especially with respect to integral quantities like the axial volume flux and the power input. This has some general implications regarding the pumping characteristics of the combined pressure-drag flow, and that at arbitrary values of the Reynolds number and of the Deborah number.     

Nonlinear oscillations of laminated plates using an accurate four-node rectangular shear flexible material finite element

The objective of the present paper is to investigate the large amplitude vibratory behaviour of unsymmetrically laminated plates. For this purpose, an efficient and accurate four-node shear flexible rectangular material finite element (MFE) with six degrees of freedom per node (three displacements (u, v, w) along thex, y andz axes, two rotations (θ _x and θ _y ) abouty andx axes and twist (θ _xy )) is developed. The element assumes bi-cubic polynomial distribution with sixteen generalized undetermined coefficients for the transverse displacement. The fields for section rotations θ _x and θ _y , and in-plane displacementsu andv are derived using moment-shear equilibrium and in-plane equilibrium equations of composite strips along thex- andy-axes. The displacement field so derived not only depends on the element coordinates but is a function of extensional, bending-extensional coupling, bending and transverse shear stiffness as well. The element stiffness and mass matrices are computed numerically by employing 3×3 Gauss-Legendre product rules. The element is found to be free ofshear locking and does not exhibit any spurious modes. In order to compute the nonlinear frequencies, linear mode shape corresponding to the fundamental frequency is assumed as the spatial distribution and nonlinear finite element equations are reduced to a single nonlinear second-order differential equation. This equation is solved by employing thedirect numerical integration method. A series of numerical examples are solved to demonstrate the efficacy of the proposed element.     

A procedure for advection and diffusion in thin cavities

In this paper a finite element formulation is proposed for the calculation of advection and diffusion in a thin cavity. For these kinds of systems, very high aspect ratio elements are necessary for cost-effective simulation. Locally, element dimensions, say in x and y, are comparable, whereas the dimension in the transverse direction z is orders of magnitude smaller than those for x and y. In our formulation, the three-dimensional basis functions for interpolation are constructed as a tensor product of the basis functions that span the lateral (x, y) plane of an element and those that span the transverse direction. Unknowns along the transverse direction are solved implicitly, in a line-by-line fashion, using the tridiagonal matrix algorithm, while the out-of-line unknowns are treated either explicitly or semi-implicitly. Several applications to material processing are discussed, as are the computationally-intensive components of three variations of our basic procedure.This work was supported by a Grant from Technalysis Incorporated, Indianapolis, Indiana, USA. This support is gratefully acknowledged.     

Convergence,accuracy and stability of finite element approximations of a class of non-linear hyperbolic equations

Numerical stability criteria and rates of convergence are derived for finite element approximations of the non- linear wave equation u_tt?F(u_x) = f(x, t), where F(u_x) possesses properties generally encountered in non-linear elasticity. Piecewise linear finite element approximations in x and central difference approximations in t are studied.     

A Microscopic Model of Superconductor Stability

The ratio J _Crit[T(x,y,t)]/J _Crit[T(x,y,t ₀)] of critical current densities (t ₀ indicating start of a disturbance) integrated over sample cross section serves to calculate the “stability function”, Φ(t), to predict under which conditions zero-loss transport current is possible. Critical current density and stability function are correlated with (conventional) timescale, t, in the superconductor (the “phonon aspect”). However, the stability problem is not simply restricted to coupled conduction/radiation heat transfer. It is questionable whether decay of electron pairs and subsequent recombination of excited electron states to a new dynamic equilibrium (the “electron aspect” under a disturbance) proceeds on the same timescale. A sequential model has been defined to calculate lifetimes of the excited electron states. These are estimated from analogy to the nucleon–nucleon, pion-mediated Yukawa interaction, from an aspect of the Racah-problem (expansion of an antisymmetric N-particle wave function from a N?1 parent state) and from the uncertainty principle, all in dependence of the local (transient) temperature field; with these approximations, the sequential model accounts for the retarded electron–phonon interaction. The numerical analysis is applied to NbTi and YBaCuO filaments in a standard matrix. As a result, the difference between both timescales can be significant, in particular near the phase transition: in the NbTi filament, a minimum distance of at least 60 μm (in this example) from the location of a disturbance should be observed for reliable stability analysis. This difference could have consequences also for safe operation of a resistive fault current limiter.     

Investigation of diffusion in p-version ‘LSFE’ and ‘STLSFE’ formulations

This paper deals with investigation of diffusion for p-version least squares finite element formulation (LSFEF) and p-version space-time coupled least squares finite element formulation (STLSFEF) for steady-state and transient problems. Convection dominated flows result in hyperbolic system of equations which leads to ill-conditioned matrices when using Galerkin formulation. Various techniques (SUPG, SUPG-with discontinuity capturing operator etc.) have been devised to overcome the difficulties arising primarily due to hyperbolic terms and sharp gradients. In this paper, it is demonstrated that when using p-version STLSFEF or LSFEF, no such difficulties are encountered in formulation as well as in the solution procedure. Almost all numerical processes suffer from numerical diffusion to some extent, however, it is demonstrated in this paper that in p-version STLSFE and LSFE formulations numerical diffusion can be completely eliminated by mesh refinement and p-level increase and the formulations are free of inherent diffusion. Several model problems are considered with dominant convective terms to investigate diffusion in p-version LSFEF and STLSFEF. Two dimensional convection-diffusion problems are used as steady state representative cases. One dimensional transient problems considered in this paper include pure advection, convection-diffusion and Burgers' equation. Numerical results are also compared with exact solutions and those reported in the literature.     

An accurate low order plate bending element with thickness change and enhanced strains

A three-dimensional variational formulation is used to obtain a plate bending element which includes the thickness change of the plate. The nodal degrees of freedom for the four-node element are the deflection w, the rotations θ _x and θ _y , and the thickness change H. Bilinear functions of the in-plane coordinates ξ and η are used for the approximation of the deflection, the rotations and the thickness change. Integration in thickness direction is performed analytically. One key feature of the element is that the three-dimensional constitutive equations for the six stresses have not to be modified. Using eight enhanced strain terms, a well performing plate bending element is obtained. Received 14 September 2000     

Sampling Methods for Censuses and Surveys (4th Ed.)

Functional responses are encountered when units are observed over time. Although the whole function itself is not observed, a sufficiently large number of evaluations, as is common with modern recording equipment, are assumed to be available. Functional regression analysis relates the smooth functional response, y(t), to known covariates, x, by a linear combination of parameter functions, β(t), which are to be estimated. The model takes the standard form, y(t) = x ^T β(t) + ?(t). This approach provides an alternative to standard longitudinal data methods used in the biological sciences, where less and noisier data necessitate parametric modeling. The methodology is illustrated by an application in ergonomics.     

A new 3D finite element for adaptive h-refinement in 1-irregular meshes

A new finite element, viable for use in the three-dimensional simulation of transient physical processes with sharply varying solutions, is presented. The element is intended to function in adaptive h-refinement schemes as a versatile transition between regions of different refinement levels, ensuring interelement continuity by constructing a piecewise linear solution at the element boundaries, and retaining all degrees of freedom in the solution phase. Construction of the element shape functions is described, and a numerical example is presented which illustrates the advantages of using such an element in an adaptive refinement problem. The new element can be used in moving-front problems, such as those found in reservoir engineering and groundwater flow applications.     

丝网印版变形对印刷位置精度的建模及分析

目的 研究丝网印版变形对印刷位置精度的影响,建立关于丝网印版变形的理论计算模型和有限元模型。方法 根据实际调研,构建丝网印版三维模型,分析丝网印版可能的变形及其对印刷位置精度的影响,建立关于网距、图文位置、预张力等参数的理论计算模型;将丝网印版合理简化后,构建有限元模型并计算分析。结果 随网距的增大,x、y2个方向上的位置误差均增大,且增大趋势近似二次曲线;图文位置对x向位置误差的影响趋势近似三次曲线,但与y向位置误差之间存在线性正相关关系,在丝网印版中心位置处的误差接近于0;预张力与位置误差之间存在线性负相关关系。结论 丝网印版变形的理论计算模型与有限元模型的分析计算结果基本吻合,该模型能较好地预测各参数对位置精度的影响,为控制印刷位置精度的参数选择提供了指导。     

Blended infinite elements for paraboblic boundary value problems

A common method for numerically approximating two-point parabolic boundary value problems of the form u_t = L[u]+f(u) defined of the semi-infinite strip S = [0, 1]×[0, ∞] is to first discretize the spatial operator in the differential equation and then solve for the time evolution. Such an approach typically involves solving a system of algebriaic equations at a sequence of time steps. In this paper we take a different approach and subdivide S into a collection of semi-infinite substrips S_i = [x_i, x_i+1]×[0, ∞], and use blending function techniques to derive finite parameter functions e_i(x, t) defined on S_i. Spectral matching methods are used in deriving e_i to ensure that (u ? e_i) can be made small on S_i. Galerkin's method, with associated integration sover the entire space-time domain S, is then used to generate approximations to u(x, t) based upon the so defined infinite element (e_i, S_i). Approximations are hence found for all (x, t) in S by solving one well structed system of algebraic equations. We apply the method to several linear and non-linear problms.