Interaction between elliptical and ellipsoidal inclusions under bending stress fields

Summary  This paper deals with interaction problems of elliptical and ellipsoidal inclusions under bending, using singular integral equations of the body force method. The problems are formulated as a system of singular integral equations with Cauchy-type or logarithmic-type singularities, where unknown functions are densities of body forces distributed in the x,y and r,θ,z directions in infinite bodies having the same elastic constants as those of the matrix and inclusions. In order to satisfy the boundary conditions along the elliptical and the ellipsoidal boundaries, the unknown functions are approximated by a linear combination of fundamental density functions and polynomials. The present method is found to yield the exact solutions for a single elliptical or spherical inclusion under a bending stress field. It yields rapidly converging numerical results for interface stresses in the interaction of inclusions. Received 9 September 1999; accepted for publication 15 January 2000     

Material sensitivity analysis in homogenization of linear elastic composites

Summary  The main goal of the paper is to present theoretical aspects and the finite element method (FEM) implementation of the sensitivity analysis in homogenization of composite materials with linear elastic components, using effective modules approach. The deterministic sensitivity analysis of effective material properties is presented in a general form for an n-components periodic composite, and is illustrated by the examples of 1D as well as of 2D heterogeneous structures. The results of the sensitivity analysis presented in the paper confirm the usefulness of the homogenization method in computational analysis of composite materials the method may be applied to computational optimization of engineering composites, to the shape-sensitivity studies and, after some probabilistic extensions, to stochastic sensitivity analysis of random composites. Received 10 November 2000; accepted for publication 24 April 2001     

Variational principles in elasticity with nonlinear stress-strain relation

Since 1979, a series of papers have been published concerning the variational principles and generalized variational principles in elasticity such as [1] (1979), [6] (1980), [2,3] (1983) and [4,5] (1984). All these papers deal with the elastic body with linear stress-strain relations. In 1985, a book was published on generalized variational principles dealing with some nonlinear elastic body, but never going into detailed discussion. This paper discusses particularly variational principles and generalized variational principles for elastic body with nonlinear stress-strain relations. In these discussions, we find many interesting problems worth while to pay some attention. At the same time, these discussions are also instructive for linear elastic problems. When the strain is small, the high order terms may be neglected, the results of this paper may be simplified to the well-known principles in ordinary elasticity problems.     

Nonsimilar boundary-layer flow over a stationary permeable plate in a rotating fluid with a magnetic field

Summary  The nonsimilar boundary-layer flow and heat transfer over a stationary permeable surface in a rotating fluid in the presence of magnetic field, mass transfer and free stream velocity are studied. The parabolic partial differential equations governing the flow have been solved numerically by using a difference–differential method. For small streamwise distance, these partial differential equations are also solved by a perturbation technique with Shanks transformation. For uniform mass transfer, analytical solutions are obtained. The surface skin friction coefficients and the Nusselt number increase with the magnetic field, suction and streamwise distance from the leading edge of the plate except the skin friction coefficient in the y-direction which decreases with the increasing magnetic field. Received 4 December 2001; accepted for publication 24 September 2002     

Homotopy perturbation method to obtain new solitary solutions with compact support for Boussinesq‐like B(2n, 2n) equations with fully nonlinear dispersion

The aim of this paper is to obtain new solitary solutions with compact support for Boussinesq‐like B(2n, 2n) equations with fully nonlinear dispersion using the homotopy perturbation method (HPM). The special case B(2, 2) is chosen to illustrate the concrete scheme of the HPM in B(2n, 2n) equations. General formulas for the solutions of B(2n, 2n) equations are established. Copyright © 2009 John Wiley & Sons, Ltd.     

Forced response of quadratic nonlinear oscillator: comparison of various approaches

The primary resonances of a quadratic nonlinear system under weak and strong external excitations are investigated with the emphasis on the comparison of different analytical approximate approaches. The forced vibration of snap-through mechanism is treated as a quadratic nonlinear oscillator. The Lindstedt-Poincaré method, the multiple-scale method, the averaging method, and the harmonic balance method are used to determine the amplitude-frequency response relationships of the steady-state responses. It is demonstrated that the zeroth-order harmonic components should be accounted in the application of the harmonic balance method. The analytical approximations are compared with the numerical integrations in terms of the frequency response curves and the phase portraits. Supported by the numerical results, the harmonic balance method predicts that the quadratic nonlinearity bends the frequency response curves to the left. If the excitation amplitude is a second-order small quantity of the bookkeeping parameter, the steady-state responses predicted by the second-order approximation of the LindstedtPoincaré method and the multiple-scale method agree qualitatively with the numerical results. It is demonstrated that the quadratic nonlinear system implies softening type nonlinearity for any quadratic nonlinear coefficients.     

M-integral analysis for microcrack-damaged anisotropic composite materials

Summary  This paper presents an M-integral analysis for the microcracked anisotropic composite materials. By using an elementary solution derived for a single finite crack subjected to a concentrated force on crack faces, the problem of strong interacting, arbitrarily oriented and located microcracks in an anisotropic composite materials is reduced to a system of Fredholm integral equations. The crack-tip fracture parameters, such as the stress intensity factors, are evaluated from a numerical solution of the system of integral equations. Its dependence on the coordinate system, calculation, and physical interpretation of the M-integral are discussed in the interaction problem. Finally, a numerical example of the damage evaluation by the M-integral analysis is given. Received 24 September 1999; accepted for publication 8 February 2000     

A spectral‐element discontinuous Galerkin thermal lattice Boltzmann method for conjugate heat transfer applications

We present a spectral‐element discontinuous Galerkin thermal lattice Boltzmann method for fluid–solid conjugate heat transfer applications. Using the discrete Boltzmann equation, we propose a numerical scheme for conjugate heat transfer applications on unstructured, non‐uniform grids. We employ a double‐distribution thermal lattice Boltzmann model to resolve flows with variable Prandtl (Pr) number. Based upon its finite element heritage, the spectral‐element discontinuous Galerkin discretization provides an effective means to model and investigate thermal transport in applications with complex geometries. Our solutions are represented by the tensor product basis of the one‐dimensional Legendre–Lagrange interpolation polynomials. A high‐order discretization is employed on body‐conforming hexahedral elements with Gauss–Lobatto–Legendre quadrature nodes. Thermal and hydrodynamic bounce‐back boundary conditions are imposed via the numerical flux formulation that arises because of the discontinuous Galerkin approach. As a result, our scheme does not require tedious extrapolation at the boundaries, which may cause loss of mass conservation. We compare solutions of the proposed scheme with an analytical solution for a solid–solid conjugate heat transfer problem in a 2D annulus and illustrate the capture of temperature continuities across interfaces for conductivity ratio γ > 1. We also investigate the effect of Reynolds (Re) and Grashof (Gr) number on the conjugate heat transfer between a heat‐generating solid and a surrounding fluid. Steady‐state results are presented for Re = 5?40 and Gr = 10⁵?10⁶. In each case, we discuss the effect of Re and Gr on the heat flux (i.e. Nusselt number Nu) at the fluid–solid interface. Our results are validated against previous studies that employ finite‐difference and continuous spectral‐element methods to solve the Navier–Stokes equations. Copyright © 2016 John Wiley & Sons, Ltd.     

A new general-purpose least-squares finite element model for steady incompressible low-viscosity laminar flow using isoparametric C1-continuous Hermite elements

This paper deals with a critical evaluation of various finite element models for low-viscosity laminar incompressible flow in geometrically complex domains. These models use Galerkin weighted residuals UVP, continuous penalty, discrete penalty and least-squares procedures. The model evaluations are based on the use of appropriate tensor product Lagrange and simplex quadratic triangular elements and a newly developed isoparametric Hermite element. All of the described models produce very accurate results for horizontal flows. In vertical flow domains, however, two different cases can be recognized. Downward flows, i.e. when the gravitational force is in the direction of the flow, usually do not present any special problem. In contrast, laminar flow of low-viscosity Newtonian fluids where the gravitational force is acting in the direction opposite to the flow presents a difficult case. We show that only by using the least-squares method in conjunction with C¹-continuous Hermite elements can this type of laminar flow be modelled accurately. The problem of smooth isoparametric mapping of C¹ Hermite elements, which is necessary in dealing with geometrically complicated domains, is tackled by means of an auxiliary optimization procedure. We conclude that the least-squares method in combination with isoparmetric Hermite elements offers a new general-purpose modelling technique which can accurately simulate all types of low-viscosity incompressible laminar flow in complex domains.     

M-integral analysis for a two-dimensional metal/ceramic bimaterial solid with extending subinterface microcracks

Summary  The problem of the extension of subinterface microcracks in an infinite metal/ceramic bimaterial solid is studied. For the microcrack growth, the values of the M-integral are calculated under the assumption of a self-similar growth. First, the role that the M-integral plays in a metal/ceramic bimaterial solid with growing subinterface cracks is analyzed. It is concluded that an inherent relation exists between the value of the M-integral and the decrease of the effective elastic moduli for a bimaterial solid with growing subinterface microcracks. Second, it is concluded that mutual amplification and shielding effects exist during the microcrack extension, while they are substantially dependent on the increment of the microcrack length as well as the geometry of the microcrack arrangement under given loads. This strong mutual shielding effect of interacting microcracks makes the microcrack extension become increasingly difficult, and may stop the growth of the microcracks even under constant loads. Also, it is concluded that for a certain microcrack growth, the value of the M-integral in metal/ceramic bimaterial solid is always larger than that in homogeneous brittle solid for the same crack configuration. This means that the same microcrack growth in the former case shows lower stability than that in the latter one, due to the existence of a ductile phase. Received 3 May 2001; accepted for publication 27 June 2002 This work was supported by the Chinese National Nature Science Foundation (Grant 19472053) and supported by the Doctorate Foundation of Xi'an Jiaotong University (Grant DFXJU2000-15).     

Plastic surface strain mapping of bent sheets by image correlation

A technique using a single CCD camera, a precision rotation/translation stage, a telecentric zoom lens, and digital image correlation software is described for measuring surface profiles and surface plastic strain distributions of a bent thin sheet. The measurement principles, based on both parallel and pinhole perspective projections, are outlined and the relevant mathematical equations for computing the profiles and displacement fields on a curved surface are presented. The typical optical setup as well as the experimental measurement and digital image correlation analysis procedure are described. The maximum errors in the in-plane and out-of-plane coordinates or displacements are about ±5 and ±25 μm, respectively, and the maximum errors in surface strain mapping are about 0.1% or less based on a series of evaluation tests on flat and curved sample surfaces over a physical field of view of 15.2 × 11.4 mm². As an application example, the shape and surface plastic strain distribution example, the shape and surface plastic strain distributions around a bent apex of a flat 2 mm thick automotive aluminum AA5182-O sheet, which underwent a 90° bend with three bend ratios of 2t, 1t, and 0.6t, are determined using the proposed technique.     

Exact calculation of the flow of a staggered tandem cascade with moving second blade row

Summary  The plane flow around a tandem cascade of flat plates is calculated by means of conformal mapping. The blades of the two rows are perpendicular to each other. The first row is stationary, the second row moves with constant velocity. The conformal mapping will be constructed by a “mapping flow”. The blades of one row are stream lines and those of the other row are potential lines of the flow. By conformal mapping, the physical flow around the tandem cascade of the physical ζ-plane is converted into a flow between infinitely long straight walls in the z-plane, each wall corresponding to one of the blades. The conditions far upstream and far downstream of the cascade are represented by source-vortices. In the z-plane, the boundary conditions may be easily fulfilled by reflection and repetition of the source-vortices, and the flow may be calculated by well-known methods. The physical flow searched for is then obtained by inverse mapping. Received 24 July 2000; accepted for publication 6 December 2000     

Pansystems studies in stability,bifurcation and chaos

This paper divides fixed subsets into three kinds, mainly discusses the existence of II-type fixed subset, connects the investigations infixed subsets with the studies in non-linear problems, such as stability, bifurcation, chaos, etc., and proposes a kind of discrete simulation to Liapunov stability and his second method.     

Two-level stabilized finite element method for Stokes eigenvalue problem

A two-level stabilized finite element method for the Stokes eigenvalue problem based on the local Gauss integration is considered.This method involves solving a Stokes eigenvalue problem on a coarse mesh with mesh size H and a Stokes problem on a fine mesh with mesh size h = O(H 2),which can still maintain the asymptotically optimal accuracy.It provides an approximate solution with the convergence rate of the same order as the usual stabilized finite element solution,which involves solving a Stokes eigenvalue problem on a fine mesh with mesh size h.Hence,the two-level stabilized finite element method can save a large amount of computational time.Moreover,numerical tests confirm the theoretical results of the present method.     

Numerical solutions of the shallow water equations with discontinuous bed topography

A simple scheme is developed for treatment of vertical bed topography in shallow water flows. The effect of the vertical step on flows is modelled with the shallow water equations including local energy loss terms. The bed elevation is denoted with z_b^‐ for the left and z_b⁺ for the right values at each grid point, hence exactly representing a discontinuity in the bed topography. The surface gradient method (SGM) is generalized to reconstruct water depths at cell interfaces involving a vertical step so that the fluxes at the cell interfaces can accurately be calculated with a Riemann solver. The scheme is verified by predicting a surge crossing a step, a tidal flow over a step and dam‐break flows on wet/dry beds. The results have shown good agreements compared with analytical solutions and available experimental data. The scheme is efficient, robust, and may be used for practical flow calculations. Copyright © 2002 John Wiley & Sons, Ltd.     

The sudden starting of a floating body in deep water

This paper applies Lagrangian method to discuss the sudden starting of a floating body in deep water and the analytical solutions are obtained. It is known from the numerical results that the disturbing domain extends and the dynamic pressure also increases when the breadth of the floating body keeps constant and its depth increases.     

Influence of unsteady wake on a turbulent separation bubble

 An experimental study was made of turbulent separated and reattaching flow over a blunt body, where unsteady wake was generated by a spoked-wheel type of wake generator with cylindrical rods. The influence of unsteady wake was scrutinized by altering the rotation direction (clockwise and counter-clockwise) and the normalized passing frequency (0 ≤ St _H  ≤ 0.20). The Reynolds number based on the cylindrical rod was Re _d =375. A phase-averaging technique was employed to characterize the unsteady wake. The effect of different rotation directions, which gave a significant reduction of x _R , was examined in detail. The wall pressure fluctuations on the blunt body were analyzed in terms of the spectrum and the coherence. Received: 15 January 2001 / Accepted: 17 July 2001     

Numerical studies of slow viscous rotating flow past a sphere. II

The Navier–Stokes equations, which are the governing equations for a steady, viscous, incompressible fluid rotating about the z-axis with angular velocity ω, are linearized using the Oseen approximation. Two parameters, namely the Reynolds number Re = Ua/v and Re_ω = 2ωa²/v (the Reynolds number w.r.t. rotation), enter the linearized equations. These equations are solved by the Peaceman–Rachford ADI method and the resulting algebraic equations are solved by the SOR method. Streamlines are plotted and compared with the Oseen solution for the non-rotating case. The magnitude of the vorticity vector with increasing θ is also plotted.     

Computation of fluid flow with a parallel multigrid solver

A finite volume numerical method for the prediction of fluid flow and heat transfer in simple geometries was parallelized using a domain decomposition approach. The method is implicit, uses a colocated arrangement of variables and is based on the SIMPLE algorithm for pressure-velocity coupling. Discretization is based on second-order central difference approximations. The algebraic equation systems are solved by the ILU method of Stone.¹ To accelerate the convergence, a multigrid technique was used. The efficiency was examined on three different parallel computers for laminar flow in a pipe with an orifice and natural convection in a closed cavity. It is shown that the total efficiency is made up of three major factors: numerical efficiency, parallel efficiency and load-balancing efficiency. The first two factors were thoroughly investigated, and a model for predicting the parallel efficiency on various computers is presented. Test calculations indicate reasonable total efficiency and favourable dependence on grid size and the number of processors.     

Instability of vibrations of a moving two-mass oscillator on a flexibly supported Timoshenko beam

Summary  Transversal vibrations of a uniformly moving two-mass oscillator on a Timoshenko beam of infinite length supported by a viscoelastic foundation are studied. By using integral transforms, the characteristic equation for the oscillator's vibrations is obtained. It is shown that the equation may have a root with a positive real part. The existence of such a root leads to the exponential increase of the amplitude of the oscillator vibrations, i.e. to instability. The reasons for the instability to occur are discussed. By employing the method of D-decomposition, the instability domains are found in the space of the system parameters. Received 30 October 2000; accepted for publication 28 March 2001