Central difference solutions of the kinematic model of settling of polydisperse suspensions and three-dimensional particle-scale simulations

The extension of Kynch's kinematic theory of sedimentation of monodisperse suspensions to polydisperse mixtures leads to a nonlinear system of conservation laws for the volume fractions of each species. In this paper, we show that a second-order central (Riemann-solver-free) scheme for the solution of systems of conservation laws can be employed as an efficient tool for the simulation of the settling and the separation of polydisperse suspensions. This is demonstrated by comparison with a published experimental study of the settling of a bidisperse suspension. In addition, we compare the prediction of the one-dimensional kinematic sedimentation model with a three-dimensional particle-scale simulation.     

抛物型方程移动界面的并行差分格式

本文构造了抛物型方程的带移动界面的一般并行差分格式,并证明了其稳定性。许多简单实用的差分格式都能从中导出。理论分析和数值试验检验了算法的稳定性,精度和并行性。     

拟线性双重退化抛物方程的自相似解

本文研究一类带有非线性梯度项的拟线性双重退化抛物方程的自相似奇性解的存在性及其分类。该类方程包含著名的Hamilton-Jacobi方程,牛顿流和非牛顿流渗流方程。本文通过标准的皮卡迭代方法,得到了该方程的自相似奇性解存在的充分条件,同时给出了相应奇性解的分类。     

一类半线性抛物方程的差分格式及收敛性和稳定性

本文讨论一类半线性抛物方程初边值问题的线性化二层隐式差分格式,证明了差分格式的可解性、收敛性和无条件稳定性,并指出了在离散L2模和L∞模意义下的收敛阶数为O(h2 τ2)。数值例子验证了理论分析结果。     

Computational algorithms for solving the coefficient inverse problem for parabolic equations

Among inverse problems for partial differential equations, we distinguish coefficient inverse problems, which are associated with the identification of coefficients and/or the right-hand side of an equation using some additional information. When considering time-dependent problems, the identification of the coefficient dependences on space and on time is usually separated into individual problems. In some cases, we have linear inverse problems (e.g. identification problems for the right-hand side of an equation); this situation essentially simplify their study. This work deals with the problem of determining in a multidimensional parabolic equation the lower coefficient that depends on time only. To solve numerically a non-linear inverse problem, linearized approximations in time are constructed using standard finite difference approximations in space. The computational algorithm is based on a special decomposition, where the transition to a new time level is implemented via solving two standard elliptic problems.     

Alternative time-marching schemes for elastodynamic analysis with the domain boundary element method formulation

This work presents alternative time-marching schemes for performing elastodynamic analysis by the Boundary Element Method. The use of the static fundamental solution and the maintenance of the domain integral associated to the accelerations characterize the formulation employed in this work. It is called D-BEM, D meaning domain. Time response is obtained by employing step-by-step time-marching procedures similar to those adopted in the Finite Element Method. Among all integration procedures, Houbolt scheme became the most popular used to march in time with D-BEM formulation, in spite of the presence of a high numerical damping. In order to improve the integration, this work presents alternative schemes that can be used to perform elastodynamic analysis by the BEM with a better damping control. In order to verify the accuracy of the proposed scheme, three examples are presented and discussed at the end of this work.     

An unconditionally stable alternating segment difference scheme of eight points for the dispersive equation

A group of new Saul'yev‐type asymmetric difference schemes to approach the dispersive equation are given here. On the basis of these schemes, an alternating difference scheme with intrinsic parallelism for solving the dispersive equation is constructed. The scheme is unconditionally stable. Numerical experiments show that the method has high accuracy. Copyright © 2006 John Wiley & Sons, Ltd.     

对称正则长波方程的拟紧致守恒差分格式

本文就对称正则长波方程的初边值问题进行了数值研究,提出了一个三层线性拟紧致差分格式,该格式具有较高精度且合理模拟了初边值问题的守恒性质。文章在先验估计基础上运用能量分析方法分析了格式的稳定性及二阶收敛性。数值结果验证了格式的有效性。     

On the robustness and efficiency of integration algorithms for a 3D finite strain phenomenological SMA constitutive model

Most devices based on shape memory alloys experience large rotations and moderate or finite strains. This motivates the development of finite‐strain constitutive models together with the appropriate computational counterparts. To this end, in the present paper a three‐dimensional finite‐strain phenomenological constitutive model is investigated and a robust and efficient integration algorithm is proposed. Properly defining the variables, extensively used regularization schemes are avoided and a nucleation–completion criterion is defined. Moreover, introducing a logarithmic mapping, a new form of time‐discrete equations is proposed. The solution algorithm as well as a suitable initial guess for the resultant nonlinear equations are also deeply discussed. Extensive numerical tests are performed to show robustness as well as efficiency of the proposed integration algorithm. Implementation of the integration algorithm within a user‐defined subroutine UMAT in the commercial nonlinear finite element software ABAQUS/Standard makes also possible the solution of a variety of boundary value problems. The obtained results show the efficiency and robustness of the proposed approach and confirm the improved efficiency (in terms of solution CPU time) when a nucleation–completion criterion is used instead of regularization schemes, as well as when a logarithmic mapping is used for the time‐discrete evolution equation instead of an exponential mapping. Copyright © 2010 John Wiley & Sons, Ltd.     

求解变系数对流扩散反应方程的指数型高精度紧致差分方法

本文给出了一种数值求解变系数对流扩散反应方程的指数型高精度紧致差分方法.我们首先将模型方程变形,借助常系数对流扩散方程的指数型高精度紧致差分格式,采用残量修正法得到变系数对流扩散反应方程的指数型高精度紧致差分格式;并从理论上分析了当Pelect数很大时,本文格式达到四阶计算精度时网格步长的限制条件;离散得到的代数方程组可采用追赶法直接求解.数值实验结果与理论分析完全吻合,表明了本文格式对于边界层问题或大梯度变化的物理量求解问题具有的高精度和鲁棒性的优点.     

基于Allen-Cahn方程图像修复的算子分裂方法（英）

本文提出了一种基于Allen-Cahn方程图像修复的算子分裂方法．其核心思想是利用算子分裂方法将原问题分解为一个线性方程和一个非线性方程,线性方程使用有限差分Crank-Nicolson格式进行离散,非线性方程利用解析方法进行求解,因此时间和空间都能达到二阶精度．由于该方法只作用于图像需要修复的区域,而其余区域的像素值与原始图像的保持一样,可以大大提高计算效率．合成图像和真实图像的数值实验验证了该算法的正确性和有效性．     

充分非线性Dullin-Gottwald-Holm方程的Cauchy问题

本文利用Sobolev不等式、算子半群理论和相关的偏微分方程的知识,分析和研究了充分非线性Dullin-Gottwald-Holm方程的性质。通过对该方程进行适当的变形,验证了该方程满足Kato理论的三个条件,并由此证明了该方程局部解的存在性、唯一性和解对初值的连续依赖性。     

长短波方程的两个守恒型紧致有限差分格式

本文对一类耦合非线性长短波方程组进行了数值研究,提出了两个四阶紧致有限差分格式,并证明新格式在离散意义下保持原问题的两个守恒性质,即总质量守恒和总能量守恒．数值实验表明本文格式在时间和空间方向分别具有二阶和四阶精度,具有良好的稳定性且在离散意义下很好地保持总质量和总能量守恒．     

On the use of a spectrum-based model for linear viscoelastic materials

A new spectrum-based model for describing the behavior of time-dependent materials is presented. In this paper, unlike most prior modeling techniques, the time-dependent response of viscoelastic materials is not expressed through the use of series. Instead, certain criteria have been imposed to select a spectrum function that has the potential of describing a wide range of material behavior. Another consequence of choosing the spectrum function of the type used in this paper is to have a few closed form analytic solutions in the theory of linear viscoelasticity. The Laplace transform technique is used to obtain the necessary formulae for viscoelastic Lame' functions, relaxation and bulk moduli, creep bulk and shear compliance, as well as Poisson's ratio. By using the Elastic–Viscoelastic Correspondence Principle (EVCP), material constants appearing in the proposed model are obtained by comparing the experimental data with the solution of the integral equation for a simple tensile test. The resulting viscoelastic functions describe the material properties which can then be used to express the behavior of a material in other loading configurations. The model's potential is demonstrated and its limitations are discussed.     

Poisson方程差分格式的SOR方法中最优松弛因子的回归分析方法

针对二维Poisson方程各种边值问题的典型差分格式,使用回归分析方法给出了求解这些格式的SOR方法中最优松弛因子的计算公式。统计分析与实际计算表明这些公式具有非常好的计算效果。     

Simultaneous inversion for a diffusion coefficient and a spatially dependent source term in the SFADE

This paper deals with an inverse problem of determining a diffusion coefficient and a spatially dependent source term simultaneously in one-dimensional (1-D) space fractional advection–diffusion equation with final observations using the optimal perturbation regularization algorithm. An implicit finite difference scheme for solving the forward problem is set forth, and a fine estimation to the spectrum radius of the coefficient matrix of the difference scheme is given with which unconditional stability and convergence are proved. The simultaneous inversion problem is transformed to a minimization problem, and existence of solution to the minimum problem is proved by continuity of the input–output mapping. The optimal perturbation algorithm is introduced to solve the inverse problem, and numerical inversions are performed with the source function taking on different forms and the diffusion coefficient taking on different values, respectively. The inversion solutions give good approximations to the exact solutions demonstrating that the optimal perturbation algorithm with the Sigmoid-type regularization parameter is efficient for the simultaneous inversion problem in the space fractional diffusion equation.     

On the impact of the scheme for solving the higher dimensional equation in coupled population balance systems

Population balance systems are models for processes in nature and industry that lead to a coupled system of equations (Navier–Stokes equations, transport equations, etc.) where the equations are defined in domains with different dimensions. This paper will study the impact of using different schemes for solving the three‐dimensional (3D) equation of a precipitation process in a two‐dimensional flow domain. The numerical schemes for the 3D equation are assessed with respect to the median of the volume fraction of the particle size distribution and the computational costs. It turns out that in the case of a structured flow field with small variations in time all schemes give qualitatively the same results. For a highly time‐ dependent flow field, the evolution of the median of the volume fraction differs considerably between first order and higher order schemes. Copyright © 2010 John Wiley & Sons, Ltd.