Axially symmetric thermal stress of an external circular crack under general thermal conditions

Summary The paper presents a solution for the linear thermoelastic problem of determining axisymmetric stress and displacement fields in an isotropic elastic solid of infinite extent weakened by an external circular crack under general mechanical loadings and general thermal conditions. The mechanical loadings and thermal conditions applied on the crack faces are axisymmetric, being non-symmetric about the crack plane. In similar lines of [7], equations of equilibrium of an elastic solid conducting heat have been solved using Hankel transforms and Abel operators of the first kind. Expressions for stress, displacement, temperature and heat flux functions are obtained in terms of Abel transforms of the first kind of the jumps of stress, displacement, temperature and heat flux at the crack plane. Two types of thermal conditions, that is, general surface temperatures and general heat flux on faces of the crack are considered. In both the cases, closed form solutions have been obtained for the unknown functions solving Abel type of integral equations. Explicit expressions for stresses, displacements, temperature fields, stress intensity factors have been obtained. Two special cases of thermal conditions in which: (i) crack faces are subjected to constant non-symmetric temperatures over a circular ring area, (ii) crack faces are subjected to constant non-symmetric heat flux over a circular ring area, have been considered. In some special cases, results have been compared with those from the literature.     

Optimal control applied to water flow using second order adjoint method

This paper presents an optimal control applied to water flow using the first and second order adjoint equations. The gradient of the performance function with respect to control variables is analytically obtained by the first order adjoint equation. It is not necessary to compute the Hessian matrix directly using the second order adjoint equation. Two numerical studies have been performed to show the adaptability of the present method. The performance of the second order adjoint method is compared with that of the weighted gradient method, Broyden–Fletcher–Goldfarb–Shanno method and Lanczos method. The precise forms of the adjoint equations and the gradient to use for the minimisation algorithm are derived. The computation by the Lanczos method is shown as superior to those of the other methods discussed in this paper. The message passing interface library is used for the communication of parallel computing.     

Non-symmetric CG-like schemes and the finite element solution of the advection–dispersion equation

Seven leading iterative methods for non-symmetric linear systems (GMRES, BCG, QMR, CGS, Bi-CGSTAB, TFQMR and CGNR) are compared in the specific context of solving the advection–dispersion equation by a classic approach: The space derivatives are approximated by linear finite elements while an implicit scheme is used to integrate the time derivatives. Convergence formulas that predict the behaviour of the iterative methods as a function of the discretization parameters are developed and validated by experiments. It is shown that all methods converge nicely when the coefficent matrix of the linear system is close to normal and the finite element approximation of the advection–dispersion equation yields accurate results.     

小孔节流气体静压润滑的离散化和计算收敛

通过在雷诺方程式中增加一流量项,以避免求解雷诺方程时,为满足流量条件而求解压力梯度所带来的噪声和误差,并将不同坐标系内的雷诺方程式变换成为相同形式,以简化数值计算;对加权余量法和变分求极值法这2种将微分方程离散化的方法进行了分析;探讨了小气膜时出现发散的原因并提出了应对措施。     

An immersed boundary finite difference method for LES of flow around bluff shapes

A three‐dimensional numerical model using large eddy simulation (LES) technique and incorporating the immersed boundary (IMB) concept has been developed to compute flow around bluff shapes. A fractional step finite differences method with rectilinear non‐uniform collocated grid is employed to solve the governing equations. Bluff shapes are treated in the IMB method by introducing artificial force terms into the momentum equations. Second‐order accurate interpolation schemes for all sorts of grid points adjacent to the immersed boundary have been developed to determine the velocities and pressure at these points. To enforce continuity, the methods of imposition of pressure boundary condition and addition of mass source/sink terms are tested. It has been found that imposing suitable pressure boundary condition (zero normal gradient) can effectively reproduce the correct pressure distribution and enforce mass conservation around a bluff shape. The present model has been verified and applied to simulate flow around bluff shapes: (1) a square cylinder and (2) the Tsing Ma suspension bridge deck section model. Complex flow phenomena such as flow separation and vortex shedding are reproduced and the drag coefficient, lift coefficient, and pressure coefficient are calculated and analyzed. Good agreement between the numerical results and the experimental data are obtained. The model is proven to be an efficient tool for flow simulation around bluff bodies in time varying flows. Copyright © 2004 John Wiley & Sons, Ltd.     

Poiseuille-Bénard流的出口边界条件及其谱元法计算

研究二维矩形管道中底部加热的不可压缩Ｐｏｉｓｅｕｉｌｌｅ－Ｂｅｎａｒｄ流的谱元法数值计算问题．讨论各种不同的出口边界条件的处理及其对谱元法数值模拟的影响．通过干扰区、干扰幅度和计算时间的比较,确定比较理想的出口边界条件．     

Acoustic simulation using a novel approach for reducing dispersion error

It is well‐known that the traditional finite element method (FEM) fails to provide accurate results to the Helmholtz equation with the increase of wave number because of the ‘pollution error’ caused by numerical dispersion. In order to overcome this deficiency, a gradient‐weighted finite element method (GW‐FEM) that combines Shepard interpolation and linear shape functions is proposed in this work. Three‐node triangular and four‐node tetrahedral elements that can be generated automatically are first used to discretize the problem domain in 2D and 3D spaces, respectively. For each independent element, a compacted support domain is then formed based on the element itself and its adjacent elements sharing common edges (or faces). With the aid of Shepard interpolation, a weighted acoustic gradient field is then formulated, which will be further used to construct the discretized system equations through the generalized Galerkin weak form. Numerical examples demonstrate that the present algorithm can significantly reduces the dispersion error in computational acoustics. Copyright © 2016 John Wiley & Sons, Ltd.     

The integration methods of Vacco dynamics equation of nonlinear nonholonomic system

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Fully coupled flow-induced vibration of structures under small deformation with GMRES method

Lagrangian-Eulerian formulations based on a generalized variational principle of fluid-solid coupling dynamics are established to describe flow-induced vibration of a structure under small deformation in an incompressible viscous fluid flow. The spatial discretization of the formulations is based on the multi-linear interpolating functions by using the finite element method for both the fluid and solid structures. The generalized trapezoidal rule is used to obtain apparently non-symmetric linear equations in an incremental form for the variables of the flow and vibration. The nonlinear convective term and time factors are contained in the non-symmetric coefficient matrix of the equations. The generalized minimum residual （GMRES） method is used to solve the incremental equations. A new stable algorithm of GMRES-Hughes-Newmark is developed to deal with the flow-induced vibration with dynamical fluid-structure interaction in complex geometries. Good agreement between the simulations and laboratory measurements of the pressure and blade vibration accelerations in a hydro turbine passage was obtained, indicating that the GiViRES-Hughes-Newmark algorithm presented in this paper is suitable for dealing with the flow-induced vibration of structures under small deformation.     

The moment‐guided Monte Carlo method

In this work we propose a new approach for the numerical simulation of kinetic equations through Monte Carlo schemes. We introduce a new technique that permits to reduce the variance of particle methods through a matching with a set of suitable macroscopic moment equations. In order to guarantee that the moment equations provide the correct solutions, they are coupled to the kinetic equation through a nonequilibrium term. The basic idea, on which the method relies, consists in guiding the particle positions and velocities through moment equations so that the concurrent solution of the moment and kinetic models furnishes the same macroscopic quantities. Copyright © 2010 John Wiley & Sons, Ltd.     

Integration of linear and some model non-linear equations of motion of incompressible fluids

We suggest a new exact method that allows one to construct solutions to a wide class of linear and some model non-linear hydrodynamic-type systems. The method is based on splitting a system into a few simpler equations; two different representations of solutions (non-symmetric and symmetric) are given. We derive formulas that connect solutions to linear three-dimensional stationary and non-stationary systems (corresponding to different models of incompressible fluids in the absence of mass forces) with solutions to two independent equations, one of which being the Laplace equation and the other following from the equation of motion for any velocity component at zero pressure. To illustrate the potentials of the method, we consider the Stokes equations, describing slow flows of viscous incompressible fluids, as well as linearized equations corresponding to Maxwell's and some other viscoelastic models. We also suggest and analyze a differential-difference fluid model with a constant relaxation time. We give examples of integrable non-linear hydrodynamic-type systems. The results obtained can be suitable for the integration of linear hydrodynamic equations and for testing numerical methods designed to solve non-linear equations of continuum mechanics.     

Computing axisymmetric,laminar internal flows

A simple computational scheme is developed to compute laminar flows inside axisymmetric ducts. It is based on the Keller box method where the equations are approximated at the centre of the downstream face of each computational box. The coupling between the pressure gradient and the velocities for internal flow has been observed to introduce stability problems for the Keller box method that are not present for external, boundary layer flow problems. The difference scheme for the velocities is coupled to an iterative scheme to solve for the pressure gradient at each axial step. Example results for developing flow in a pipe and in a 2° conical diffuser are presented.     

Semi-implicit two-level predictor–corrector methods for non-linearly coupled,hydrostatic, barotropic/baroclinic flows

The unsteady shallow-water equations for barotropic/baroclinic (free-surface/density-stratified) flows with non-linear coupling of density transport and momentum are solved using a family of two-time-level, semi-implicit predictor–corrector methods (PC2). The PC2 methods are a general family that includes the popular TRIM method for hydrostatic flows. PC2 is characterised by four ‘θ’ parameters controlling the time ‘n’ and ‘n + 1’ weighting of (1) free surface gradient, (2) predictor step, (3) baroclinic gradient and (4) density temporal interpolation. Stability of the non-linear coupling between momentum and density transport for PC2 is examined in the inviscid limit. Central difference and quadratic (QUICK) spatial interpolation for density are compared. Second-order temporal accuracy for both barotropic and baroclinic flows is simultaneously obtained with appropriate θ parameters, which has previously been shown to be impractical for TRIM. The 2nd-order PC2 method has near-neutral non-linear stability (slightly positive amplification factor) where linear theory predicts exactly neutral stability. QUICK is shown to be preferable to central difference spatial discretisation to reduce the amplification factor. Adjusting the baroclinic weighting or adding small artificial viscosities can stabilise the model for non-linear internal wave simulations.     

拟谱-微分求积混合方法求解一类双曲电报方程

拟谱方法和微分求积法是两类重要的无网格法,二者都已在科学和工程计算中获得了广泛应用。采用拉格朗日插值多项式作为二者的试函数,且采用同一种网格点分布,指出了在空间域上,微分求积法是拟谱方法的一种特殊形式。在此基础上,结合二者各自的特点,提出了拟谱-微分求积混合方法用于求解一类双曲电报方程。理论分析和数值测试表明,新方法在空间域上具有谱精度收敛性,在时间域上是A-稳定的,比较适合于求解多维电报方程。     

ARC-length method for differential equations

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Using a segregated finite element scheme to solve the incompressible Navier-Stokes equations

In this paper, a segregated finite element scheme for the solution of the incompressible Navier-Stokes equations is proposed which is simpler in form than previously reported formulations. A pressure correction equation is derived from the momentum and continuity equations, and equal-order interpolation is used for both the velocity components and pressure. Algorithms such as this have been known to lead to checkerboard pressure oscillations; however, the pressure correction equation of this scheme should not produce these oscillations. The method is applied to several laminar flow situations, and details of the methods used to achieve converged solutions are given.     

不可压粘流N－S方程的边界积分解法

对原变量的Ｎ－Ｓ方程进行一阶时间离散，采用共轭梯度法解除压强－速度的耦合．对所得的一系列Ｌａｐｌａｃｅ方程、Ｐｏｓｓｉｏｎ方程和Ｈｅｌｍｈｏｔｚ方程均进行边界积分法求解，首次得到了粘性Ｎ－Ｓ方程的边界积分表示式．圆柱的定常、非定常尾迹计算结果表明了本文方法的有效性．     

Phase field approach with anisotropic interface energy and interface stresses: Large strain formulation

A thermodynamically consistent, large-strain, multi-phase field approach (with consequent interface stresses) is generalized for the case with anisotropic interface (gradient) energy (e.g. an energy density that depends both on the magnitude and direction of the gradients in the phase fields). Such a generalization, if done in the “usual” manner, yields a theory that can be shown to be manifestly unphysical. These theories consider the gradient energy as anisotropic in the deformed configuration, and, due to this supposition, several fundamental contradictions arise. First, the Cauchy stress tensor is non-symmetric and, consequently, violates the moment of momentum principle, in essence the Herring (thermodynamic) torque is imparting an unphysical angular momentum to the system. In addition, this non-symmetric stress implies a violation of the principle of material objectivity. These problems in the formulation can be resolved by insisting that the gradient energy is an isotropic function of the gradient of the order parameters in the deformed configuration, but depends on the direction of the gradient of the order parameters (is anisotropic) in the undeformed configuration. We find that for a propagating nonequilibrium interface, the structural part of the interfacial Cauchy stress is symmetric and reduces to a biaxial tension with the magnitude equal to the temperature- and orientation-dependent interface energy. Ginzburg–Landau equations for the evolution of the order parameters and temperature evolution equation, as well as the boundary conditions for the order parameters are derived. Small strain simplifications are presented. Remarkably, this anisotropy yields a first order correction in the Ginzburg–Landau equation for small strains, which has been neglected in prior works. The next strain-related term is third order. For concreteness, specific orientation dependencies of the gradient energy coefficients are examined, using published molecular dynamics studies of cubic crystals. In order to consider a fully specified system, a typical sixth order polynomial phase field model is considered. Analytical solutions for the propagating interface and critical nucleus are found, accounting for the influence of the anisotropic gradient energy and elucidating the distribution of components of interface stresses. The orientation-dependence of the nonequilibrium interface energy is first suitably defined and explicitly determined analytically, and the associated width is also found. The developed formalism is applicable to melting/solidification and crystal-amorphous transformation and can be generalized for martensitic and diffusive phase transformations, twinning, fracture, and grain growth, for which interface energy depends on interface orientation of crystals from either side.     

Global convergence of inexact Newton methods for transonic flow

In computational fluid dynamics, non-linear differential equations are essential to represent important effects such as shock waves in transonic flow. Discretized versions of these non-linear equations are solved using iterative methods. In this paper an inexact Newton method using the GMRES algorithm of Saad and Schultz is examined in the context of the full potential equation of aerodynamics. In this setting, reliable and efficient convergence of Newton methods is difficult to achieve. A poor initial solution guess often leads to divergence or very slow convergence. This paper examines several possible solutions to these problems, including a standard local damping strategy for Newton's method and two continuation methods, one of which utilizes interpolation from a coarse grid solution to obtain the initial guess on a finer grid. It is shown that the continuation methods can be used to augment the local damping strategy to achieve convergence for difficult transonic flow problems. These include simple wings with shock waves as well as problems involving engine power effects. These latter cases are modelled using the assumption that each exhaust plume is isentropic but has a different total pressure and/or temperature than the freestream.     

Use of two‐step split‐operator approach for 2D shallow water flow computation

The objective of this paper is to present a methodology of using a two‐step split‐operator approach for solving the shallow water flow equations in terms of an orthogonal curvilinear co‐ordinate system. This approach is in fact one kind of the so‐called fractional step method that has been popularly used for computations of dynamic flow. By following that the momentum equations are decomposed into two portions, the computation procedure involves two steps. The first step (dispersion step) is to compute the provisional velocity in the momentum equation without the pressure gradient. The second step (propagation step) is to correct the provisional velocity by considering a divergence‐free velocity field, including the effect of the pressure gradient. This newly proposed method, other than the conventional split‐operator methods, such as the projection method, considers the effects of pressure gradient and bed friction in the second step. The advantage of this treatment is that it increases flexibility, efficiency and applicability of numerical simulation for various hydraulic problems. Four cases, including back‐water flow, reverse flow, circular basin flow and unsteady flow, have been demonstrated to show the accuracy and practical application of the method. Copyright © 1999 John Wiley & Sons, Ltd.