Numerical Analysis for a Nonlocal Parabolic Problem

This article is devoted to the study of the finite element approximation for a nonlocal nonlinear parabolic problem. Using a linearised Crank-Nicolson Galerkin finite element method for a nonlinear reaction-diffusion equation, we establish the convergence and error bound for the fully discrete scheme. Moreover, important results on exponential decay and vanishing of the solutions in finite time are presented. Finally, some numerical simulations are presented to illustrate our theoretical analysis.     

等压固结条件下湘江饱和砂土动力特性研究

在大量的动三轴试验基础上,研究了等压固结条件下湘江饱和砂土振动孔隙水压力、应力应变滞回圈以及应力路径的发展变化规律。研究结果表明:孔压发展变化过程与土体的剪胀、剪缩密切相关;孔压与轴向动应变之间的变化关系显示饱和砂土具有明显的各向异性;从孔压与滞回圈面积累积之和的关系看,孔压的增长伴随着能量的损失;对于围压较大的情形,用ExpAssoc函数进行拟合效果较好。隧振次增加应力应变滞回圈也在发生变化,第一阶段滞回圈呈倒置的帽子形,第二阶段呈菱角形状。在振动初始阶段,应力路径基本上呈倾斜的直线变化;随孔压的增加,加载和卸载两条应力路径逐渐分开,呈纺锤形。     

BEM analysis of Biot’s model for dynamic compaction of concrete in successive placement

Dynamic compaction process is analyzed by applying the boundary element method (BEM) based on Biot’s two-phase model of solid and liquid. In order to analyze mixed concrete, concrete properties at early ages are determined experimentally. Applying these properties, porepressure distribution under dynamic compaction is calculated. A feasibility of the analysis is confirmed by the compaction test. Varying the intervals between the first placement and the second, experimentally jointed beams of concrete are made and tested under bending. Analyzing BEM models of two layers, pore-pressure distributions under vibration are analyzed, taking into account the effect of placing intervals. It is demonstrated that pore-pressure distribution may not drastically change until 3 to 4 hours after placement, suggesting that the underlying layer is still responsive to vibration. Distributions of pore pressures analyzed are compared with the strengths of beams. Thus good correlation between the bending strengths of jointed beams and generation of pore pressures at the joint is confirmed. It is confirmed that low pore-pressure around the joint could result in the decrease in the bending strength, sometimes leading to nucleation of cold joints.     

Superconvergence of Fully Discrete Finite Elements for Parabolic Control Problems with Integral Constraints

A quadratic optimal control problem governed by parabolic equations with integral constraints is considered. A fully discrete finite element scheme is constructed for the optimal control problem, with finite elements for the spatial but the backward Euler method for the time discretisation. Some superconvergence results of the control, the state and the adjoint state are proved. Some numerical examples are performed to confirm theoretical results.     

Compression of time‐generated matrices in two‐dimensional time‐domain elastodynamic BEM analysis

This paper describes a new scheme to improve the efficiency of time‐domain BEM algorithms. The discussion is focused on the two‐dimensional elastodynamic formulation, however, the ideas presented apply equally to any step‐by‐step convolution based algorithm whose kernels decay with time increase. The algorithm presented interpolates the time‐domain matrices generated along the time‐stepping process, for time‐steps sufficiently far from the current time. Two interpolation procedures are considered here (a large number of alternative approaches is possible): Chebyshev–Lagrange polynomials and linear. A criterion to indicate the discrete time at which interpolation should start is proposed. Two numerical examples and conclusions are presented at the end of the paper. Copyright © 2004 John Wiley & Sons, Ltd.     

Ray-transfer-matrix model for accurate pulsed cavity ring-down measurement in the mismatching case

A theoretical model based on the ray-transfer matrix is developed for the pulsed cavity ring-down (CRD) technique to numerically investigate the influence of the geometric parameters of the pulsed-CRD arrangement on the CRD signal. By fitting the spatial distribution of the pulsed laser beam to that of the TEM(00) cavity mode, the geometric parameters are optimized to obtain perfect matching between the laser beam and the ring-down cavity. It is indicated by the numerical simulations that as long as the laser power exiting the ring-down cavity is fully collected, a single exponential-decay signal, identical to the perfectly-matched CRD signal, is obtained in the mismatching case to determine accurately the cavity decay time. Intensity fluctuations appear in the mismatched CRD signal if the laser power exiting the ring-down cavity is not fully collected. Both the conventional exponential decay fitting approach and a linear fitting procedure are employed to analyze these mismatched CRD signals and the latter is recommended to make an accurate pulsed-CRD measurement.     

Perspective on post-menopausal osteoporosis: establishing an interdisciplinary understanding of the sequence of events from the molecular level to whole bone fractures

Current drug treatments for post-menopausal osteoporosis cannot eliminate bone fractures, possibly because the mechanisms responsible for bone loss are not fully understood. Although research within various disciplines has significantly advanced the state of knowledge, fundamental findings are not widely understood between different disciplines. For that reason, this paper presents noteworthy experimental findings from discrete disciplines focusing on post-menopausal osteoporosis. These studies have established that, in addition to bone loss, significant changes in bone micro-architecture, tissue composition and micro-damage occur. Cellular processes and molecular signalling pathways governing pathological bone resorption have been identified to a certain extent. Ongoing studies endeavour to determine how such changes are initiated at the onset of oestrogen deficiency. It emerges that, because of the discrete nature of previous research studies, the sequence of events that lead to bone fracture is not fully understood. In this paper, two sequences of multi-scale changes are proposed and the experimental challenges that need to be overcome to fully define this sequence are outlined. Future studies must comprehensively characterize the time sequence of molecular-, cellular- and tissue-level changes to attain a coherent understanding of the events that ultimately lead to bone fracture and inform the future development of treatments for post-menopausal osteoporosis.     

A Time Splitting Space Spectral Element Method for the Cahn-Hilliard Equation

We propose and analyse a class of fully discrete schemes for the Cahn-Hilliard equation with Neumann boundary conditions. The schemes combine large-time step splitting methods in time and spectral element methods in space. We are particularly interested in analysing a class of methods that split the original Cahn-Hilliard equation into lower order equations. These lower order equations are simpler and less computationally expensive to treat. For the first-order splitting scheme, the stability and convergence properties are investigated based on an energy method. It is proven that both semi-discrete and fully discrete solutions satisfy the energy dissipation and mass conservation properties hidden in the associated continuous problem. A rigorous error estimate, together with numerical confirmation, is provided. Although not yet rigorously proven, higher-order schemes are also constructed and tested by a series of numerical examples. Finally, the proposed schemes are applied to the phase field simulation in a complex domain, and some interesting simulation results are obtained.     

A Fully Discrete Spectral Method for Fisher’s Equation on the Whole Line

A generalised Hermite spectral method for Fisher’s equation in genetics with different asymptotic solution behaviour at infinities is proposed, involving a fully discrete scheme using a second order finite difference approximation in the time. The convergence and stability of the scheme are analysed, and some numerical results demonstrate its efficiency and substantiate our theoretical analysis.     

Assumed strain finite element methods for conserving temporal integrations in non‐linear solid dynamics

This paper presents a new class of assumed strain finite elements to use in combination with general energy‐momentum‐conserving time‐stepping algorithms so that these conservation properties in time are preserved by the fully discretized system in space and time. The case of interest corresponds to nearly incompressible material responses, in the fully non‐linear finite strain elastic and elastoplastic ranges. The new elements consider the classical scaling of the deformation gradient with an assumed Jacobian (its determinant) defined locally through a weighted averaging procedure at the element level. The key aspect of the newly proposed formulation is the definition of the associated linearized strain operator or B‐bar operator. The developments presented here start by identifying the conditions that this discrete operator must satisfy for the fully discrete system in time and space to inherit exactly the conservation laws of linear and angular momenta, and the conservation/dissipation law of energy for elastic and inelastic problems, respectively. Care is also taken of the preservation of the relative equilibria and the corresponding group motions associated with the momentum conservation laws, and characterized by purely rotational and translational motions superimposed to the equilibrium deformed configuration. With these developments at hand, a new general B‐bar operator is introduced that satisfies these conditions. The new operator not only accounts for the spatial interpolations (e.g. bilinear displacements with piece‐wise constant volume) but also depends on the discrete structure of the equations in time. The aforementioned conservation/dissipation properties of energy and momenta are then proven to hold rigorously for the final numerical schemes, unconditionally of the time step size and the material model (elastic or elastoplastic). Different finite elements are considered in this framework, including quadrilateral and triangular elements for plane problems and brick elements for three‐dimensional problems. Several representative numerical simulations are presented involving, in particular, the use of energy‐dissipating momentum‐conserving time‐stepping schemes recently developed by the author and co‐workers for general finite strain elastoplasticity in order to illustrate the properties of the new finite elements, including these conservation/dissipation properties in time and their locking‐free response in the quasi‐incompressible case. Copyright © 2007 John Wiley & Sons, Ltd.     

Frequency estimation via weighted multipoint interpolated DFT

A weighted multipoint interpolated discrete Fourier transform (WMIpDFT) method for high accuracy frequency estimation of a multi-frequency signal component is presented. This method uses the maximum sidelobe decay windows. The relationships to estimate the frequency when an H-term maximum sidelobe decay window (H ges 2) is used for (2J + 1)-point interpolation (1 les J les H + 1) are given. The effectiveness of the method proposed was analysed by means of computer simulations for a multi-frequency signal without noise as well as with quantisation noise.     

Segregated Runge–Kutta methods for the incompressible Navier–Stokes equations

In this work, we propose Runge–Kutta time integration schemes for the incompressible Navier–Stokes equations with two salient properties. First, velocity and pressure computations are segregated at the time integration level, without the need to perform additional fractional step techniques that spoil high orders of accuracy. Second, the proposed methods keep the same order of accuracy for both velocities and pressures. The segregated Runge–Kutta methods are motivated as an implicit–explicit Runge–Kutta time integration of the projected Navier–Stokes system onto the discrete divergence‐free space, and its re‐statement in a velocity–pressure setting using a discrete pressure Poisson equation. We have analysed the preservation of the discrete divergence constraint for segregated Runge–Kutta methods and their relation (in their fully explicit version) with existing half‐explicit methods. We have performed a detailed numerical experimentation for a wide set of schemes (from first to third order), including implicit and IMEX integration of viscous and convective terms, for incompressible laminar and turbulent flows. Further, segregated Runge–Kutta schemes with adaptive time stepping are proposed. Copyright © 2015 John Wiley & Sons, Ltd.     

算法的有界性和稳定性(英文)

非线性Ｇａｌｅｒｋｉｎ算法是长时间范围内求解非线性发展方程的一种新的数值格式。我们在这篇文章里,提供了全离散非线性Ｇａｌｅｒｋｉｎ算法的有界性和稳定性结果。     

A hybrid material-point spheropolygon-element method for solid and granular material interaction

Capturing the interaction between objects that have an extreme difference in Young's modulus or geometrical scale is a highly challenging topic for numerical simulation. One of the fundamental questions is how to build an accurate multiscale method with optimal computational efficiency. In this work, we develop a material-point-spheropolygon discrete element method (MPM-SDEM). Our approach fully couples the material point method (MPM) and the spheropolygon discrete element method (SDEM) through the exchange of contact force information. It combines the advantage of MPM for accurately simulating elastoplastic continuum materials and the high efficiency of DEM for calculating the Newtonian dynamics of discrete near-rigid objects. The MPM-SDEM framework is demonstrated with an explicit time integration scheme. Its accuracy and efficiency are further analyzed against the analytical and experimental data. Results demonstrate this method could accurately capture the contact force and momentum exchange between materials while maintaining favorable computational stability and efficiency. Our framework exhibits great potential in the analysis of multi-scale, multi-physics phenomena.     

Finite elements in space and time for dynamic contact problems

Dynamic obstacle and Signorini problems are discretized by continuous and discontinuous finite elements in space and time. The resulting discrete problems are attributed to a standard implicit time‐stepping scheme through relaxation of impact phenomena and suitable numerical integration. The method can cope with dynamic contact problems, which is shown by an analysis of a model problem. Moreover, numerical examples demonstrate that it is actually able to approximate the solution of dynamic contact problems, which are not fully covered by the theory. Copyright © 2008 John Wiley & Sons, Ltd.     

颗粒流动模型研究进展

颗粒流动在自然界中和各种工业过程中广泛存在,但人们对于其机理认识的还不深入。描述颗粒运动的模型有很多,连续介质模型应用简单但准确性比较低,离散微粒学模型是近来人们研究的一个热点,以每个颗粒为考察对象,能够更准确地反应颗粒系统的性质。本文介绍了描述颗粒流动的模型,概述了各模型的理论和应用,通过对多种模型的比较可以看到,每个模型都有一定的使用范围,要更准确、更方便地描述颗粒系统的运动,还要进行深入地研究。     

Error estimates of numerical solutions for a cyclic plasticity problem

A cyclic plasticity problem is numerically analyzed in [13], where a sub-optimal order error estimate is shown for a spatially discrete scheme. In this note, we prove an optimal order error estimate for the spatially discrete scheme under the same solution regularity condition. We also derive an error estimate for a fully discrete scheme for solving the plasticity problem.     

Energy–momentum consistent finite element discretization of dynamic finite viscoelasticity

This paper is concerned with energy–momentum consistent time discretizations of dynamic finite viscoelasticity. Energy consistency means that the total energy is conserved or dissipated by the fully discretized system in agreement with the laws of thermodynamics. The discretization is energy–momentum consistent if also momentum maps are conserved when group motions are superimposed to deformations. The performed approximation is based on a three‐field formulation, in which the deformation field, the velocity field and a strain‐like viscous internal variable field are treated as independent quantities. The new non‐linear viscous evolution equation satisfies a non‐negative viscous dissipation not only in the continuous case, but also in the fully discretized system. The initial boundary value problem is discretized by using finite elements in space and time. Thereby, the temporal approximation is performed prior to the spatial approximation in order to preserve the stress objectivity for finite rotation increments (incremental objectivity). Although the present approach makes possible to design schemes of arbitrary order, the focus is on finite elements relying on linear Lagrange polynomials for the sake of clearness. The discrete energy–momentum consistency is based on the collocation property and an enhanced second Piola–Kirchhoff stress tensor. The obtained coupled non‐linear algebraic equations are consistently linearized. The corresponding iterative solution procedure is associated with newly proposed convergence criteria, which take the discrete energy consistency into account. The iterative solution procedure is therefore not complicated by different scalings in the independent variables, since the motion of the element is taken into account for solving the viscous evolution equation. Representative numerical simulations with various boundary conditions show the superior stability of the new time‐integration algorithm in comparison with the ordinary midpoint rule. Both the quasi‐rigid deformations during a free flight, and large deformations arising in a dynamic tensile test are considered. Copyright © 2009 John Wiley & Sons, Ltd.     

Discrete Green's methods and their application to two-dimensional phase unwrapping

A fully self-contained discrete framework with discrete equivalents of Stokes's, Gauss's, and Green's theorems is presented. The formulation is analogous to that of continuous operators, but totally discrete in nature, and the exact relationships derived are shown to hold provided that a set of predefined rules is followed in building discrete contours and domains. The method allows for an analytical rigor that is not guaranteed if one translates the classical continuous formulations onto a discretized approximated framework. We clarify several issues related to the use of discrete operators, which may play a crucial role in specific applications such as the two-dimensional phase-unwrapping problem, chosen as our main application example, and we show that reconstruction on irregular domains and/or in the presence of undersampling and noise is better formulated in the discrete framework than in the continuous domain.     

Decay of two-qubit correlations in the squeezed Bloch channel

The decay of correlations between two qubits under the influence of a squeezed thermal reservoir is investigated by means of the quantum master equation in the Born–Markov approximation. To find the effect of the reservoir squeezing on the two-qubit correlations, concurrence, quantum discord, classical correlation and total correlation are calculated for the X-states. It is found that, except for quantum discord, the reservoir squeezing always suppresses the decay of the correlations during the time evolution. On the other hand, for quantum discord, the reservoir squeezing enhances the decay in the initial and intermediate time regions while it reduces the decay in the long time region.