周期性结构热动力时间-空间多尺度分析

研究一种时间-空间多尺度渐近均匀化分析方法，模拟不同的极端热和动力载荷下微尺度多 相周期性结构中热动力响应问题，并建立一个广义的波动函数场控制方程描述热动力响应. 通过引入一个放大空间尺度和两个缩小时间尺度，在不同时间尺度上获得由空间非均匀性引 起的波动效应和非局部效应. 根据高阶均匀化理论在空间和时间上进行均匀化，获得高阶非 局部函数场波动方程. 并进一步用C0连续修正了高阶非局部函数场波动方程的有限元近 似解，使问题的求解避免了对有限元离散的C1连续性要求. 并与经典的空间均匀化方法 相比较，指出了经典的空间均匀化方法的局限性，进一步以一维非傅立叶热传导和热动力问 题为例，讨论了各种情况下方法的正确性与有效性.     

Multiple spatial and temporal scales method for numerical simulation of non-classical heat conduction problems: one dimensional case

A multiple spatial and temporal scales method is studied to simulate numerically the phenomenon of non-Fourier heat conduction in periodic heterogeneous materials. The model developed is based on the higher-order homogenization theory with multiple spatial and temporal scales in one dimensional case. The amplified spatial scale and the reduced temporal scale are introduced respectively to account for the fluctuations of non-Fourier heat conduction due to material heterogeneity and non-local effect of the homogenized solution. By separating the governing equations into various scales, the different orders of homogenized non-Fourier heat conduction equations are obtained. The reduced time dependence is thus eliminated and the fourth-order governing differential equations are derived. To avoid the necessity of C¹ continuous finite element implementation, a C⁰ continuous mixed finite element approximation scheme is put forward. Numerical results are shown to demonstrate the efficiency and validity of the proposed method.     

Evolving Microstructure and Homogenization

In this article we formulate a mathematical model for the temporally evolving microstructure generated by phase changes and study the homogenization of this model. The investigations are partially formal, since we do not prove existence or convergence of solutions of the microstructure model to solutions of the homogenized problem. To model the microstructure, the sharp interface approach is used. The evolution of the interface is governed by an everywhere defined distribution partial differential equation for the characteristic function of one of the phases. This avoids the disadvantage commonly associated with this approach of an evolution equation only defined on the interface. To derive the homogenized problem, a family of solutions of the microstructure problem depending on the fast variable is introduced. The homogenized problem obtained contains a history functional, which is defined by the solution of an initial-boundary value problem in the representative volume element. In the special case of a temporally fixed microstructure the homogenized problem is reduced to an evolution equation to a monotone operator. Received March 15, 2000     

Multiscale modeling of a fluid saturated medium with double porosity: Relevance to the compact bone

In this paper, we develop a model of a homogenized fluid-saturated deformable porous medium. To account for the double porosity the Biot model is considered at the mesoscale with a scale-dependent permeability in compartments representing the second-level porosity. This model is treated by the homogenization procedure based on the asymptotic analysis of periodic “microstructure”. When passing to the limit, auxiliary microscopic problems are introduced, which provide the corrector basis functions that are needed to compute the effective material parameters. The macroscopic problem describes the deformation-induced Darcy flow in the primary porosities whereas the microflow in the double porosity is responsible for the fading memory effects via the macroscopic poro-visco-elastic constitutive law. For the homogenization procedure, we use the periodic unfolding method. We discuss also the stress and flow recovery at multiple scales characterizing the heterogeneous material. The model is proposed as a theoretical basis to describe compact bone behavior on multiple scales.     

Goal‐oriented space–time adaptivity in the finite element Galerkin method for the computation of nonstationary incompressible flow

This paper presents a general strategy for designing adaptive space–time finite element discretizations of the nonstationary Navier–Stokes equations. The underlying framework is that of the dual weighted residual method for goal‐oriented a posteriori error estimation and automatic mesh adaptation. In this approach, the error in the approximation of certain quantities of physical interest, such as the drag coefficient, is estimated in terms of local residuals of the computed solution multiplied by sensitivity factors, which are obtained by numerically solving an associated dual problem. In the resulting local error indicators, the effects of spatial and temporal discretization are separated, which allows for the simultaneous adjustment of time step and spatial mesh size. The efficiency of the proposed method for the construction of economical meshes and the quantitative assessment of the error is illustrated by several test examples. Copyright © 2012 John Wiley & Sons, Ltd.     

Higher-order accurate implicit time integration schemes for transport problems

The present paper is concerned with the numerical solution of transient transport problems by means of spatial and temporal discretization methods. The generalized initial boundary value problem of various nonlinear transport phenomena like heat transfer or mass transport is discretized in space by p-finite elements. After finite element discretization, the resulting first-order semidiscrete balance has to be solved with respect to time. Next to the classical generalized-α integration method predicated on the Newmark approach and the evaluation at a generalized midpoint also implicit Runge–Kutta time integration schemes, are presented. Both families of finite difference-based integration schemes are derived for general first-order problems. In contrast to the above-mentioned algorithms, temporal discontinuous and continuous Galerkin methods evaluate the balance equation not at a selected time instant within the timestep, but in an integral sense over the whole time step interval. Therefore, the underlying semidiscrete balance and the continuity of the primary variables are weakly formulated within time steps and between time steps, respectively. Continuous Galerkin methods are obtained by the strong enforcement of the continuity condition as special cases. The introduction of a natural time coordinate allows for the application of standard higher-order temporal shape functions of the p-Lagrange type and the well-known Gau?–Legendre quadrature of associated time integrals. It is shown that arbitrary order accurate integration schemes can be developed within the framework of the proposed temporal p-Galerkin methods. Selected benchmark analyses of calcium diffusion demonstrate the properties of all three methods with respect to non-smooth initial or boundary conditions. Furthermore, the robustness of the present time integration schemes is also demonstrated for the highly nonlinear reaction–diffusion problem of calcium leaching, including the pronounced changes of the reaction term and non-smooth changes of Dirichlet boundary conditions of calcium dissolution.     

时域自适应算法求解弹性地基薄板的动力问题

为更精确地求解弹性地基薄板的动力响应,发展了一种分段时域自适应算法,通过变量在离散时段内的展开,将时空耦合的初边值问题转化为一系列递推的基于有限元(FEM)的空间问题求解,通过自适应计算保持稳定的计算精度。数值算例表明:本文解与解析解相比最大相对误差不超过3.59%;当步长较大时四阶Runge-Kutta法和Newmark法均失效,本文所提算法仍可得到满意的计算结果。     

On the Barenblatt Model for Non-Equilibrium Two Phase Flow in Porous Media

We prove global existence and uniqueness of solutions to a quasilinear Goursat problem, which was proposed by G. I. Barenblatt to describe non-equilibrium two phase fluid flow in permeable porous media. When the equilibrium relaxation time tends to zero, the solution is shown to converge to the entropy solution of the corresponding initial-boundary value problem for the classical Buckley-Leverett equation.     

PEGASE: A NAVIER–STOKES SOLVER FOR DIRECT NUMERICAL SIMULATION OF INCOMPRESSIBLE FLOWS

A hybrid conservative finite difference/finite element scheme is proposed for the solution of the unsteady incompressible Navier–Stokes equations. Using velocity–pressure variables on a non-staggeredgrid system, the solution is obtained with a projection method basedon the resolution of a pressure Poisson equation. The new proposed scheme is derived from the finite element spatial discretization using the Galerkin method with piecewise bilinear polynomial basis functions defined on quadrilateral elements. It is applied to the pressure gradient term and to the non-linear convection term as in the so-called group finite element method. It ensures strong coupling between spatial directions, inhibiting the development of oscillations during long-term computations, as demonstrated by the validation studies. Two- and three-dimensional unsteady separated flows with open boundaries have been simulated with the proposed method using Cartesian uniform mesh grids. Several examples of calculations on the backward-facing step configuration are reported and the results obtained are compared with those given by other methods. © 1997 by John Wiley & Sons, Ltd. Int. j. numer. methods fluids 24: 833–861, 1997.     

A simple and efficient BEM implementation of quasistatic linear visco-elasticity

A simple yet efficient procedure to solve quasistatic problems of special linear visco-elastic solids at small strains with equal rheological response in all tensorial components, utilizing boundary element method (BEM), is introduced. This procedure is based on the implicit discretisation in time (the so-called Rothe method) combined with a simple “algebraic” transformation of variables, leading to a numerically stable procedure (proved explicitly by discrete energy estimates), which can be easily implemented in a BEM code to solve initial-boundary value visco-elastic problems by using the Kelvin elastostatic fundamental solution only. It is worth mentioning that no inverse Laplace transform is required here. The formulation is straightforward for both 2D and 3D problems involving unilateral frictionless contact. Although the focus is to the simplest Kelvin–Voigt rheology, a generalization to Maxwell, Boltzmann, Jeffreys, and Burgers rheologies is proposed, discussed, and implemented in the BEM code too. A few 2D and 3D initial-boundary value problems, one of them with unilateral frictionless contact, are solved numerically.     

Uniqueness of weak solutions to dynamic problems in the elasticity theory with boundary conditions of Winkler and inertial types

A uniqueness theorem for the weak solution of an initial-boundary value problem in the anisotropic elasticity theory with the boundary conditions that “do not conserve” energy, namely, with the impedance and inertial type conditions is proved. The chosen method of proof does not require the positive definiteness of the elastic constant tensor (the case that may arise when solving the problems by the homogenization method for composite materials), but it requires to take the energy variation law as a postulate.     

Dynamical problems of continuous media with random boundary data

A spatial correlation method is formulated for linear dynamical problems in continuum mechanics with random boundary data. The essential feature of the method is the formulation of a nonstochastic mixed initial-boundary value problem for the (matrix) spatial correlation function of the (vector) state variable. Whenever the Green's function of the (stochastic) problem can not be obtained in terms of known functions, a numerical solution of the meansquare response and other second order response statistics by the spatial correlation method is several hundred folds more efficient than any other available method. Further improvements in the computational efficiency of the method for a steady state stationary response process are also noted.     

Multiscale block spectral solution for unsteady flows

Many problems of interest are characterized by 2 distinctive and disparate scales and a huge multiplicity of similar small‐scale elements. The corresponding scale‐dependent solvability manifests itself in the high gradient flow around each element needing a fine mesh locally and the similar flow patterns among all elements globally. In a block spectral approach making use of the scale‐dependent solvability, the global domain is decomposed into a large number of similar small blocks. The mesh‐pointwise block spectra will establish the block‐block variation, for which only a small set of blocks need to be solved with a fine mesh resolution. The solution can then be very efficiently obtained by coupling the local fine mesh solution and the global coarse mesh solution through a block spectral mapping. Previously, the block spectral method has only been developed for steady flows. The present work extends the methodology to unsteady flows of short temporal and spatial scales (eg, those due to self‐excited unsteady vortices and turbulence disturbances). A source term–based approach is adopted to facilitate a two‐way coupling in terms of time‐averaged flow solutions. The global coarse base mesh solution provides an appropriate environment and boundary condition to the local fine mesh blocks, while the local fine mesh solution provides the source terms (propagated through the block spectral mapping) to the global coarse mesh domain. The computational method will be presented with several numerical examples and sensitivity studies. The results consistently demonstrate the validity and potential of the proposed approach.     

Development of a class of multiple time‐stepping schemes for convection–diffusion equations in two dimensions

In this paper we present a class of semi‐discretization finite difference schemes for solving the transient convection–diffusion equation in two dimensions. The distinct feature of these scheme developments is to transform the unsteady convection–diffusion (CD) equation to the inhomogeneous steady convection–diffusion‐reaction (CDR) equation after using different time‐stepping schemes for the time derivative term. For the sake of saving memory, the alternating direction implicit scheme of Peaceman and Rachford is employed so that all calculations can be carried out within the one‐dimensional framework. For the sake of increasing accuracy, the exact solution for the one‐dimensional CDR equation is employed in the development of each scheme. Therefore, the numerical error is attributed primarily to the temporal approximation for the one‐dimensional problem. Development of the proposed time‐stepping schemes is rooted in the Taylor series expansion. All higher‐order time derivatives are replaced with spatial derivatives through use of the model differential equation under investigation. Spatial derivatives with orders higher than two are not taken into account for retaining the linear production term in the convection–diffusion‐reaction differential system. The proposed schemes with second, third and fourth temporal accuracy orders have been theoretically explored by conducting Fourier and dispersion analyses and numerically validated by solving three test problems with analytic solutions. Copyright © 2006 John Wiley & Sons, Ltd.     

颗粒群碰撞搜索及CFD-DEM耦合分域求解的推进算法研究

在采用计算流体力学-离散元耦合方法(computational fluiddynamics-discrete element method, CFD-DEM)进行固液两相耦合分析时, 颗粒计算时间步的选取直接影响到耦合计算精度和计算效率. 为此, 本文选取每个目标颗粒为研究对象, 引入插值函数计算时间步的运动位移, 构建可变空间搜索网格; 通过筛选可能碰撞颗粒建立搜索列表, 采用逆向搜索方式判断碰撞颗粒, 从而提出一种改进的DEM方法(modified discreteelement method, MDEM). 该算法在颗粒群与流体耦合计算中, 颗粒计算初始时间步选取不受颗粒碰撞时间限制, 通过自动调整和修正实现大步长, 由颗粒和流体耦合条件实时更新流体计算时间步, 使颗粒计算时间步选取过小导致计算效率低、选取过大导致颗粒碰撞漏判的问题得以解决, 为颗粒与流体耦合的数值模拟提供了行之有效的计算方法. 通过两个颗粒和多个颗粒的数值模拟, 得到的颗粒间碰撞力、碰撞位置及次数, 与理论计算结果的相对误差均低于2%, 与传统的DEM碰撞搜索算法相比, 在选取的3种计算时间步均不会影响计算精度, 且有较高的计算效率. 通过多个颗粒与流体的耦合数值模拟, 采用传统的CFD-DEM方法, 只有颗粒计算时间步选取10$^{-6}$ s或更小才能得到精确解, 而采用本文方法取10$^{-4}$ s也能够得到精确解, 避免了颗粒碰撞随时间步增大而出现的漏判问题, 且计算耗时降低了16.7%.      

A design of residual error estimates for a high order BDF‐DGFE method applied to compressible flows

We deal with the numerical solution of the non‐stationary compressible Navier–Stokes equations with the aid of the backward difference formula – discontinuous Galerkin finite element method. This scheme is sufficiently stable, efficient and accurate with respect to the space as well as time coordinates. The nonlinear algebraic systems arising from the backward difference formula – discontinuous Galerkin finite element discretization are solved by an iterative Newton‐like method. The main benefit of this paper are residual error estimates that are able to identify the computational errors following from the space and time discretizations and from the inexact solution of the nonlinear algebraic systems. Thus, we propose an efficient algorithm where the algebraic, spatial and temporal errors are balanced. The computational performance of the proposed method is demonstrated by a list of numerical experiments. Copyright © 2013 John Wiley & Sons, Ltd.     

Homogenization of scalar wave equations with hysteresis

The paper deals with a scalar wave equation of the form where is a Prandtl–Ishlinskii operator and are given functions. This equation describes longitudinal vibrations of an elastoplastic rod. The mass density and the Prandtl–Ishlinskii distribution function are allowed to depend on the space variable x. We prove existence, uniqueness and regularity of solution to a corresponding initial-boundary value problem. The system is then homogenized by considering a sequence of equations of the above type with spatially periodic data and , where the spatial period tends to 0. We identify the homogenized limits and and prove the convergence of solutions to the solution of the homogenized equation. Received June 17, 1999     

A method for solving the factorized vorticity-stream function equations by finite elements

A new finite element method for solving the time-dependent incompressible Navier-Stokes equations with general boundary conditions is presented. The two second-order partial differential equations for the vorticity and the stream function are factorized, apart from the non-linear advection term, by eliminating the coupling due to the double specification on the stream function at (a part of) the boundary. This is achieved by reducing the no-slip boundary conditions to projection integral conditions for the vorticity field and by evaluating the relevant quantities involved according to an extension of the method of Glowinski and Pironneau for the biharmonic problem. Time integration schemes and iterative algorithms are introduced which require the solution only of banded linear systems of symmetric type. The proposed finite element formulation is compared with its finite difference equivalent by means of a few numerical examples. The results obtained using 4-noded bilinear elements provide an illustration of the superiority of the finite element based spatial discretization.     

ON SINGULAR PERTURBATION FOR A NONLINEAR INITIAL-BOUNDARY VALUE PROBLEM (Ⅱ)

In this paper, we consider a singularly perturbed problem of a kind of quasilinear hyperbolic-parabolic equations, subject to initial-boundary value conditions with moving boundary. When certain assumptions are satisfied and e is sufficiently small, the solution of this problem has a generalized asymptotic expansion (in the Van der Corput sense), which takes the sufficiently smooth solution of the reduced problem as the first term, and is uniformly valid in domain Q where the sufficiently smooth solutioh exists. The layer exists in the neighborhood of t=0. This paper is the development of references.     

MIXED FINITE ELEMENT METHODS FOR THE SHALLOW WATER EQUATIONS INCLUDING CURRENT AND SILT SEDIMENTATION (Ⅰ)-THE CONTINUOUS-TIME CASE

An initial-boundary value problem for shallow equation system consisting of water dynamics equations, silt transport equation, the equation of bottom topography change, and of some boundary and initial conditions is studied, the existence of its generalized solution and semidiscrete mixed finite element (MFE) solution was discussed, and the error estimates of the semidiscrete MFE solution was derived. The error estimates are optimal.