Time domain modeling of damping using anelastic displacement fields and fractional calculus

A fractional derivative model of linear viscoelasticity based on the decomposition of the displacement field into an anelastic part and elastic part is developed. The evolution equation for the anelastic part is then a differential equation of fractional order in time. By using a fractional order evolution equation for the anelastic strain the present model becomes very flexible for describing the weak frequency dependence of damping characteristics. To illustrate the modeling capability, the model parameters are fit to available frequency domain data for a high damping polymer. By studying the relaxation modulus and the relaxation spectrum the material parameters of the present viscoelastic model are given physical meaning. The use of this viscoelastic model in structural modeling is discussed and the corresponding finite element equations are outlined, including the treatment of boundary conditions. The anelastic displacement field is mathematically coupled to the total displacement field through a convolution integral with a kernel of Mittag–Leffler function type. Finally a time step algorithm for solving the finite element equations are developed and some numerical examples are presented.     

无限长圆柱体中广义电磁热弹耦合问题研究

基于L-S广义热弹性理论,研究了处于磁场中无限长理想圆柱导体在边界受热冲击作用时的电磁热弹耦合问题.建立了广义电磁热弹耦合的有限元方程,为避免积分变换方法求解带来的精度丟失.采用将有限元方程直接在时间域求解的方法,得到了圆柱体中的温度、位移、应力、感应磁场和感应电场的分布规律,反映了热的波动性及电磁热弹的耦合效应.结果表明,将有限元方程直接在时间域求解,可以获得各物理量的准确分布.得到温度在热波波前处的阶跃,准确地反应热波的波动效应.     

A new finite element scheme using the Lagrangian framework for simulation of viscoelastic fluid flows

A new numerical scheme for simulation of viscoelastic fluid flows was designed, making use of finite element algorithms generally regarded as advantageous for tackling the problem. This includes the Lagrangian approach for the solution of viscoelastic constitutive equation using the co-deformational frame of reference with a possibility of analytically solving the equation along the particles trajectories, which in turn allowed eluding the solution of any system of linear equations for the stress. Then, the full ellipticity of the momentum conservation equation was utilised thanks to a possibility of accurate determination of the stress tensor independently of the velocity field at the current stage of computation. The needed independent stress was calculated at each time step on the basis of the past deformation history, which in turn was determined on the basis of the past velocity fields, all incorporated into a modified Euler time stepping algorithm. Owing to explicit inclusion of the full viscous term from the viscoelastic model into the momentum conservation equation, no stress splitting was necessary. The trajectory feet tracking was done accurately using a semi-analytic solution of the displacement gradient evolution equation and a weak formulation of the kinematics equation, the latter at the expense of solving an extra symmetric system of linear equations.The error expressed in the form of the Sobolev norms was determined using a comparison with available analytical solution for UCM fluid in the transient regime or numerically obtained steady-state stress values for the PTT fluid in Couette flow. The implementation of the PTT fluid model was done by modifying the relative displacement gradient tensor so that a new convective frame was defined.The stability of the algorithm was assessed using the well-known benchmark problem of a sphere sedimenting in a tube with viscoelastic fluid. The stable numerical results were obtained at high Weissenberg numbers, with the limit of convergence Wi=6.6, exceeding any previously reported values. The robustness of the code was proven by simulation of the Weissenberg effect (the rod-climbing phenomenon) with the use of PTT fluid.     

New truncated expansion method and soliton-like solution of variable coefficient KdV-MKdV equation with three arbitrary functions

The truncated expansion method for finding explicit and exact soliton-like solution of variable coefficient nonlinear evolution equation was described. The crucial idea of the method was first the assumption that coefficients of the truncated expansion formal solution are functions of time satisfying a set of algebraic equations, and then a set of ordinary different equations of undetermined functions that can be easily integrated were obtained. The simplicity and effectiveness of the method by application to a general variable coefficient KdV-MKdV equation with three arbitrary functions of time is illustrated.     

气体动理学格式研究进展

介绍了近年来气体动理学格式(gas-kinetic scheme, GKS, 亦简称BGK 格式) 的主要研究进展, 重点是高阶精度动理学格式及适合从连续流到稀薄流全流域的统一动理学格式. 通过对速度分布函数的高阶展开和对初值的高阶重构, 构造了时间和空间均为三阶精度的气体动理学格式. 研究表明, 相比于传统的基于Riemann 解的高阶格式, 新格式不仅考虑了网格单元界面上物理量的高阶重构, 而且在初始场的演化阶段耦合了流体的对流和黏性扩散, 也能够保证解的高阶精度. 该研究为高精度计算流体力学(computatial uiddymamics, CFD) 格式的建立提供了一条新的途径. 通过分子离散速度空间直接求解Boltzmann 模型方程,在每个时间步长内将宏观量的更新和微观气体分布函数的更新紧密地耦合在一起, 建立了适合任意Knudsen(kn) 数的统一格式, 相比于已有的直接离散格式具有更高的求解效率. 最后, 本文还讨论了合理的物理模型对数值方法的重要性. 气体动理学方法的良好性能来自于Boltzmann 模型方程对计算网格单元界面上初始间断的时间演化的准确描述. 气体自由运动与碰撞过程的耦合是十分必要的. 通过分析数值激波层内的耗散机制,我们认识到采用Euler 方程的精确Riemann 解作为现代可压缩CFD 方法的基础具有根本的缺陷, 高马赫数下的激波失稳现象不可避免. 气体动理学格式为构造数值激波结构提供了一个重要的可供参考的物理机制.      

Numerical simulation of axisymmetric unsteady incompressible flow by a vorticity-velocity method

A new numerical method for solving the axisymmetric unsteady incompressible Navier-Stokes equations using vorticity-velocity variables and a staggered grid is presented. The solution is advanced in time with an explicit two-stage Runge-Kutta method. At each stage a vector Poisson equation for velocity is solved. Some important aspects of staggering of the variable location, divergence-free correction to the velocity field by means of a suitably chosen scalar potential and numerical treatment of the vorticity boundary condition are examined. The axisymmetric spherical Couette flow between two concentric differentially rotating spheres is computed as an initial value problem. Comparison of the computational results using a staggered grid with those using a non-staggered grid shows that the staggered grid is superior to the non-staggered grid. The computed scenario of the transition from zero-vortex to two-vortex flow at moderate Reynolds number agrees with that simulated using a pseudospectral method, thus validating the temporal accuracy of our method.     

Numerical analysis of a quasistatic piezoelectric problem with damage

The quasistatic evolution of the mechanical state of a piezoelectric body with damage is numerically studied in this paper. Both damage and piezoelectric effects are included into the model. The variational formulation leads to a coupled system composed of two linear variational equations for the displacement field and the electric potential, and a nonlinear parabolic variational equation for the damage field. The existence of a unique weak solution is stated. Then, a fully discrete scheme is introduced by using a finite element method to approximate the spatial variable and an Euler scheme to discretize the time derivatives. Error estimates are derived on the approximate solutions, from which the linear convergence of the algorithm is deduced under suitable regularity conditions. Finally, a two-dimensional example is presented to demonstrate the behaviour of the solution. To cite this article: J.R. Fernández et al., C. R. Mecanique 336 (2008).     

Verification of deforming polarized structure computation by using a closed-form solution

Various engineering systems exploit the conversion between electromagnetic and mechanical work. It is important to compute this coupling accurately, and we present a method for solving the governing equations simultaneously (at once) without a staggering scheme. We briefly present the theory for coupling the elecgoverning equations as well as the variational formulation that leads to the weak form. This weak form is nonlinear and couples various fields. In order to solve the weak form, we use the finite element method in space and the finite difference method in time for the discretization of the computational domain. Numerical problems are circumvented by selecting the field equations carefully, and the weak form is assembled using standard shape functions. In order to examine the accuracy of the method, for the case of a linear elastic material under small deformations, we present and use an analytic solution. Comparison of the computation to the closed-form solution shows that the computational approach is reliable and models the jump of the electromagnetic fields across the interface between two different materials.     

On Nonlinear Evolution Approach for Convective Flow during Alloy Solidification

We consider the problem of nonlinear buoyant flow in a horizontal mushy layer during alloy solidification. We study the nonlinear evolution of such flow based on a recently developed realistic model for the mushy layer. The evolution approach is based on a Landau type equation for the amplitude of the secondary nonlinear solution, which is derived in this article. Using both analytical and computational methods, we calculate the solution to the evolution equation for both subcritical and supercritical regimes. We find, in particular, that for a passive mush, where the permeability is constant, and supercritical regime, the primary solution is linearly unstable to the secondary solution which becomes a steady stable solution for sufficiently large time, while the secondary solution decays to zero for the subcritical regime. On the other hand, for a realistic reactive mush, where the permeability is variable, the secondary flow can break down in a finite time for either supercritical or subcritical regime, which indicates existence of some kind of bursting behavior. These results are then compared to the corresponding ones based on the weakly nonlinear theory.     

The displacement solution of conical shell and its application

In this paper,the displacement solution method of the conical shell is presented.Fromthe differential equations in displacement form of conical shell and by introducing adisplacement function,U(s,θ),the differential equations are changed into an eight-ordersoluble partial differential equation about the displacement function U(s,θ)in which thecoefficients are variable.At the same time,the expressions of the displacement and internalforce components of the shell are also given by the displacement function.As special casesof this paper,the displacement function introduced by V.Z.Vlasov in circular cylindricalshell,the basic equation of the cylindrical shell and that of the circular plate are directlyderived.Under the arbitrary loads and boundary conditions,the general bending problem of theconical shell is reduced to finding the displacement function U(s,θ),and the generalsolution of the governing equation is obtained in generalized hypergeometric function,Forthe axisymmetric bending deformation of the     

On the role of (weak) compressibility for fluid-structure interaction solvers

In the present study, a weakly compressible formulation of the Navier-Stokes equations is developed and examined for the solution of fluid-structure interaction (FSI) problems. Newtonian viscous fluids under isothermal conditions are considered, and the Murnaghan-Tait equation of state is employed for the evaluation of mass density changes with pressure. A pressure-based approach is adopted to handle the low Mach number regime, ie, the pressure is chosen as primary variable, and the divergence-free condition of the velocity field for incompressible flows is replaced by the continuity equation for compressible flows. The approach is then embedded into a partitioned FSI solver based on a Dirichlet-Neumann coupling scheme. It is analytically demonstrated how this formulation alleviates the constraints of the instability condition of the artificial added mass effect, due to the reduction of the maximal eigenvalue of the so-called added mass operator. The numerical performance is examined on a selection of benchmark problems. In comparison to a fully incompressible solver, a significant reduction of the coupling iterations and the computational time and a notable increase in the relaxation parameter evaluated according to Aitken's Δ² method are observed.     

参数激励下非线性受控系统的动力学和稳定性

研究两自由度参数激励系统的非线性动力学与控制问题．利用Lagrange方程建立含反馈控制的参激捅及其驱动机构组成的系统动力学方程，以多尺度方法获得一阶近似控制方程．然后，对系统受一阶摸态参激主共振与一、二阶模态间3：1内共振联合作用下的幅额响应及其稳定性，以及反馈参数对系统稳态行为的影响作了详细分析．结果表明，响应的稳定域位置和大小取决于位移反馈，位移立方反馈改变了系统的非线性程度，速度反馈类似于阻尼，可使系统呈现自激振动特性．     

On Singular Limits of Mean-Field Equations

Mean-field equations arise as steady state versions of convection-diffusion systems where the convective field is determined by solution of a Poisson equation whose right-hand side is affine in the solutions of the convection-diffusion equations. In this paper we consider the repulsive coupling case for a system of two convection-diffusion equations. For general diffusivities we prove the existence of a unique solution of the mean-field equation by a variational analysis of a saddle point problem (usually without coercivity). Also we analyze the small-Debye-length limit and prove convergence to either the so-called charge-neutral case or to a double obstacle problem for the limiting potential depending on the data.     

Dynamic fracture behavior of functionally graded magnetoelectroelastic solids by BIEM

Dynamic anti-plane fracture problem of an exponentially graded linear magnetoelectroelastic plane with a finite impermeable crack subjected to time-harmonic SH-waves is solved. Directions of wave propagation and material inhomogeneity are chosen in an arbitrary way. The fundamental solution for the coupled system of partial differential equations with variable coefficients is derived in a closed form by the hybrid usage of both an appropriate algebraic transformation for the displacement vector and the Radon transform. The formulated boundary-value problem is solved by a nonhypersingular traction boundary integral equation method (BIEM). The collocation method and parabolic approximation for the unknown generalized crack opening displacements are used for the numerical solution of the posed problem. Quarter point elements placed next to the crack-tips ensure properly modeling the singular behavior of the field variables around the crack tip. Fracture parameters as stress intensity factor, electric field intensity factor and magnetic field intensity factor are computed. Intensive simulations reveal the sensitivity of the generalized intensity factors (GIF) at the crack-tips to the material inhomogeneity, characteristics of the incident wave, coupling effects, wave-material and wave-crack interaction phenomena.     

A staggered control volume scheme for unstructured triangular grids

The purpose of this work is to introduce and validate a new staggered control volume method for the simulation of 2D/axisymmetric incompressible flows. The present study introduces a numerical procedure for solving the Navier–Stokes equations using the primitive variable formulation. The proposed method is an extension of the staggered grid methodology to unstructured triangular meshes for a control volume approach which features ease of handling of irregularly shaped domains. Two alternative elements are studied: transported scalars are stored either at the sides of an element or at its vertices, while the pressure is always stored at the centre of an element. Two interpolation functions were investigated for the integration of the momentum equations: a skewed mass-weighted upwind function and a flow-oriented exponential shape function. The momentum equations are solved over the covolume of a side or of a vertex and the pressure–velocity coupling makes use of a localized linear reconstruction of the discontinuous pressure field surrounding an element in order to obtain the pressure gradient terms. The pressure equation is obtained through a discretization of the continuity equation which uses the triangular element itself as the control volume. The method is applied to the simulation of the following test cases: backward-facing step flow, flow over a two-dimensional obstacle and flow in a pipe with sudden contraction of cross-sectional area. All numerical investigations are compared with experimental data from the literature. A grid convergence and error analysis study is also carried out for flow in a driven cavity. Results compared favourably with experimental data and so the new control volume scheme is deemed well suited for the prediction of incompressible flows in complex geometries. © 1997 John Wiley & Sons, Ltd.     

求解一维理想磁流体方程的移动网格熵稳定格式

磁流体方程的数值求解在等离子体物理学、天体物理研究以及流动控制等领域具有重要意义,本文构造了用于求解理想磁流体动力学方程的基于移动网格的熵稳定格式,此方法将Roe型熵稳定格式与自适应移动网格算法结合,空间方向采用熵稳定格式对磁流体动力学方程进行离散,利用变分法构造网格演化方程并通过Gauss-Seidel迭代法对其迭代求解实现网格的自适应分布,在此基础上采用守恒型插值公式实现新旧节点上的量值传递,利用三阶强稳定Runge-Kutta方法将数值解推进到下一时间层。数值实验表明,该算法能有效捕捉解的结构(特别是激波和稀疏波),分辨率高,通用性好,具有强鲁棒性。     

一种抛物线浅拱线性单元

为研究抛物线浅拱在复杂受力状态下的线性内力及位移,考虑拱的压弯耦合作用,基于力法和浅拱假定,对抛物线拱平衡方程、几何方程和物理方程进行了分析;求得了抛物线拱内力方程通解格式,构建了抛物线浅拱位移形函数。利用最小势能原理,构造了抛物线浅拱单元。算例表明,本文构造的抛物线浅拱单元计算的抛物线拱内力与理论解计算结果符合较好,在划分一个单元时最大相对误差仅为4.03%,可用于抛物线拱分析。     

SOLUTIONS FOR CYLINDRICAL CAVITY IN SATURATED THERMOPOROELASTIC MEDIUM

Based on the thermodynamics of irreversible processes, the mass conservation equation and heat energy balance equation are established. The governing equations of thermal consolidation for homogeneous isotropic materials are presented, accounting for the coupling effects of the temperature, stress and displacement fields. The case of a saturated medium with a long cylindrical cavity subjected to a variable thermal loading and a variable hydrostatic pressure （or a variable radial water flux） with time is considered. The analytical solutions are derived in the Laplace transform space. Then, the time domain solutions are obtained by a numerical inversion scheme. The results of a typical example indicate that thermodynamically coupled effects have considerable influences on thermal responses.     

On CFL evolution strategies for implicit upwind methods in linearized Euler equations

In implicit upwind methods for the solution of linearized Euler equations, one of the key issues is to balance large time steps, leading to a fast convergence behavior, and small time steps, needed to sufficiently resolve relevant flow features. A time step is determined by choosing a Courant–Friedrichs–Levy (CFL) number in every iteration. A novel CFL evolution strategy is introduced and compared with two existing strategies. Numerical experiments using the adaptive multiscale finite volume solver QUADFLOW demonstrate that all three CFL evolution strategies have their advantages and disadvantages. A fourth strategy aiming at reducing the residual as much as possible in every time step is also examined. Using automatic differentiation, a sensitivity analysis investigating the influence of the CFL number on the residual is carried out confirming that, today, CFL control is still a difficult and open problem. Copyright © 2008 John Wiley & Sons, Ltd.     

The (9,5) HOC formulation for the transient Navier–Stokes equations in primitive variable

We recently proposed an improved (9,5) higher order compact (HOC) scheme for the unsteady two‐dimensional (2‐D) convection–diffusion equations. Because of using only five points at the current time level in the discretization procedure, the scheme was seen to be computationally more efficient than its predecessors. It was also seen to capture very accurately the solution of the unsteady 2‐D Navier–Stokes (N–S) equations for incompressible viscous flows in the stream function–vorticity (ψ – ω) formulation. In this paper, we extend the scope of the scheme for solving the unsteady incompressible N–S equations based on primitive variable formulation on a collocated grid. The parabolic momentum equations are solved for the velocity field by a time‐marching strategy and the pressure is obtained by discretizing the elliptic pressure Poisson equation by the steady‐state form of the (9,5) scheme with the Neumann boundary conditions. In particular, for pressure, we adopt a strategy on the collocated grid in conjunction with ideas borrowed from the staggered grid approach in finite volume. We first apply this extension to a problem having analytical solution and then to the famous lid‐driven square cavity problem. We also apply our formulation to the backward‐facing step problem to see how the method performs for external flow problems. The results are presented and are compared with established numerical results. This new approach is seen to produce excellent comparison in all the cases. Copyright © 2007 John Wiley & Sons, Ltd.