基于时域自适应精细算法的插值型无单元Galerkin法及其在弹性动力学中的应用

将插值型无单元Galerkin法与时域自适应精细算法相结合,提出一种求解弹性动力学问题的方法。通过时域分段展开,将时空耦合的初边值问题转换为一系列的空间边值问题,进而采用加权残值法推导递推形式的插值型无单元Galerkin法求解方程。该方法不仅能方便地直接施加本质边界条件,并且可以避免时间步长较大造成的精度损失。数值算例给出的结果验证了该方法的有效性。     

一维无迭代自适应网格技术在Burgers方程中的应用

基于变分原理的自适应网格技术需用迭代法求解,非常耗时.故提出了一种新的自适应网格技术,根据权函数曲线下面积相等的原则,重新分布格点位置,一步到位,无需迭代,大大提高了自适应网格技术的效率.利用具有解析解的一维Burgers方程对这一新技术进行验证,发现它能根据不同的权函数确定不同的格点分布;对同一种权函数,当其随时间变化时,对应的格点分布也相应变化.研究表明:该技术能根据问题的求解,在解的大梯度区自动加密网格,从而可计算出激波.     

一种基于预估-修正策略的非定常流网格自适应加密/稀疏方法

给出一种非定常流动数值模拟的网格自适应处理方法.在"求解流动方程-自适应调整网格"的流程中,引入预估-修正步.根据自适应周期内每个时间步上的流场预估解,计算单元上的事后误差估算值.建立考虑解演变的网格自适应指示器,并进行多层次单元加密-稀疏的动态网格自适应处理.在自适应网格上重新计算流场.每个自适应周期中,流动演变区域的网格获得加密;而前一个周期中的特征现象已离开区域的网格被稀疏.应用边界非协调的当地DFD(Domain-Free Discretization)方法求解流动方程.为验证网格自适应处理方法,针对静止圆柱和自推进游鱼的流动进行了数值实验.     

求解非线性偏微分方程的自适应小波精细积分法

以Burgers方程为例,提出了一种求解偏微分方程的自适应多层插值小波配置法,通过引入一种具有插值特性的拟Shannon小波并利用插值小波理论构造了多层自适应插值小波算子,从而在空间实现了偏微分方程的自适应离散.另外,精细时程积分方法和外推法的引入不但有助于提高求解速度和数值结果的精度,而且使时间积分步长的选取具有了自适应性.     

奇异微分方程边值问题的数值解法

本文给出求解奇异微分方程边值问题的正则化方法。解在奇点邻域内展开成级数形式,在余下区间上推导出正则边值问题,应用差分方法求解,并给出收敛结果和数值算例。     

自适应间断有限元方法求解三维欧拉方程

将龙格库塔间断有限元方法(RDDG)与自适应方法相结合,求解三维欧拉方程.区域剖分采用非结构四面体网格,依据数值解的变化采用自适应技术对网格进行局部加密或粗化,减少总体网格数目,提高计算效率.给出四种自适应策略并分析不同自适应策略的优缺点.数值算例表明方法的有效性.     

一类非线性偏微分方程初边值问题的逐层优化算法

针对非线性偏微分方程初边值问题,基于差分法和动态设计变量优化算法原理,以时间计算层上离散节点的未知函数值为设计变量,以离散节点的差分方程组构造程式化的目标函数,提出了离散节点处未知函数值的逐层高精度优化算法.编制通用程序求解具体典型算例.并通过与解析解对比,表明了求解方法的正确性和有效性,为广泛的工程应用提供条件.     

基于保角映射的镜像法的应用

将镜像法与保角变换法相结合,求解复杂形状边界的二维边值问题,得出电场分布规律,并通过Matlab数据处理软件的数值计算功能进行数值模拟,绘制出其电场线与等势线(面)分布图,验证了所得结论的正确性.此方法拓展了镜像法的应用范围,为求解边界形状复杂的二维静电场的边值问题提供了一种新的思路和途径.     

基于保角映射的磁像法的应用

将磁像法与保角映射相结合,求解边界形状复杂的二维静磁场的边值问题,得出磁场的分布规律,并通过MATLAB数据处理软件的数值计算功能进行数值模拟,绘制出其磁感线分布图,验证了所得结论的正确性.此方法拓展了磁像法的应用范围,为求解边界形状复杂的二维静磁场的边值问题提供了一种新的思路和途径.     

基于保角映射的格林函数法及其应用

将格林函数法与保角映射相结合,求解复杂形状边界的二维静电场边值问题,得出电场分布规律,并通过MATLAB数据处理软件的数值计算功能进行数值模拟,绘制出其电场线与等势线(面)分布图,验证了所得结论的正确性.此方法拓展了格林函数法的应用范围,为求解边界形状复杂的二维静电场的边值问题提供了一种新的思路和途径.     

Criterion of the incubation time in the problems of pulsed fracture and electric breakdown

The problems of a pulsed strength of continuum media are considered in terms of the structural-time approach that is based on the concept of the incubation fracture time. This approach makes it possible to describe phenomena that arise under high-velocity external effects. A limiting condition that determines the instant of rupture or breakdown is proposed on the basis of the structural-time approach. A way to interpret and to determine the incubation time is proposed. A phenomenological model of an electric breakdown of solid dielectrics is formulated. Examples are considered where the structural-time approach is applied to problems of spall fracture, crack initiation, and a pulsed breakdown of dielectrics. A procedure for describing the time dependence of the electric strength (volt-second characteristic) is described in detail. The results of the calculations are found to be in good agreement with experimental data.     

High-performance modeling acoustic and elastic waves using the parallel Dichotomy Algorithm

A high-performance parallel algorithm is proposed for modeling the propagation of acoustic and elastic waves in inhomogeneous media. An initial boundary-value problem is replaced by a series of boundary-value problems for a constant elliptic operator and different right-hand sides via the integral Laguerre transform. It is proposed to solve difference equations by the conjugate gradient method for acoustic equations and by the GMRES(k) method for modeling elastic waves. A preconditioning operator was the Laplace operator that is inverted using the variable separation method. The novelty of the proposed algorithm is using the Dichotomy Algorithm [26], which was designed for solving a series of tridiagonal systems of linear equations, in the context of the preconditioning operator inversion. Via considering analytical solutions, it is shown that modeling wave processes for long instants of time requires high-resolution meshes. The proposed parallel fine-mesh algorithm enabled to solve real application seismic problems in acceptable time and with high accuracy. By solving model problems, it is demonstrated that the considered parallel algorithm possesses high performance and efficiency over a wide range of the number of processors (from 2 to 8192).     

Exactly solvable time-dependent models in quantum mechanics and their applications

Methods for obtaining exact and approximate solutions of the evolution of quantum-mechanical problems are discussed. The cyclic evolution of quantum systems described by time-periodic Hamiltonians is analyzed. A class of time-periodic Hamiltonians is constructed in the close analytical form. The corresponding cyclic solutions are calculated. Time-dependent Hamiltonians are generated whose expectation values calculated with cyclic solutions are time independent. It is shown that the expectation values of the spin projection calculated with the same cyclic solutions, as well as the probability density of finding a particle at a given space-time point, are also time independent. Therefore, the approach can be used to simulate quantum dynamic potential wells with the particle localization effect. Nonadiabatic geometric phases are expressed in terms of the cyclic solutions. Exactly solvable time-dependent problems are used to construct a universal set of gates for quantum computers. A method for obtaining entanglement operators is discussed.     

Molecular dynamics simulation of reverse micelles: Standing problems after 25 years of research

The main problems of molecular dynamics simulation of reverse micelles are considered. The most important problem is that, in a simulation, a micelle can exist for an infinite time, while in a real chemical experiment, it exists for finite time or not formed at all. A criterion of rejection of such low-probable micelles for well-studied surface-active compounds has been proposed.     

A study on the multi-freedom broadband impedance model for time-domain simulations

The purpose of this paper is to construct a general broadband impedance model, which is suited for predicting acoustic propagation problems in time domain.A multi-freedom broadband impedance model for sound propagation over impedance surfaces is proposed and the corresponding time domain impedance boundary condition is presented.Basing on the extended Helmholtz resonator,the multi-freedom impedance model is constructed through combing with a sum of rational functions in the form of general complex-conjugate pole-residue pairs and it is proved that the impedance model is well posed.The impedance boundary condition can be implemented into a computational aeroacoustics solver by a recursive convolution technique, which results in a fast and computationally efficient algorithm.The two dimensional and three dimensional benchmark problems are selected to validate the accuracy of the proposed impedance model and time domain simulations.The numerical results are in good agreement with the reference solutions.It is demonstrated that the proposed impedance model can be used to describe the broadband characteristics of acoustic liners,and the corresponding time domain impedance boundary condition is viable and accurate for the prediction of sound propagation over broadband impedance surfaces.     

Effective and flexible analysis for propagation in time varying waveguides

The equivalence between the propagation of dispersive modal fields in two-dimensional waveguides, and plane waves in a one-dimensional plasma is presented. Exploitation of this equivalence allows a time domain variant of the effective index approach to be used to model dispersive waveguides problems very efficiently. A time domain integral equation is developed for this important practical case and the stability of a computer algorithm based upon it is improved by means of both a semi-implicit formulation and the use of a modified space–time mesh.     

A new combined stable and dispersion relation preserving compact scheme for non-periodic problems

A new compact scheme is presented for computing wave propagation problems and Navier–Stokes equation. A combined compact difference scheme is developed for non-periodic problems (called NCCD henceforth) that simultaneously evaluates first and second derivatives, improving an existing combined compact difference (CCD) scheme. Following the methodologies in Sengupta et al. [T.K. Sengupta, S.K. Sircar, A. Dipankar, High accuracy schemes for DNS and acoustics, J. Sci. Comput. 26 (2) (2006) 151–193], stability and dispersion relation preservation (DRP) property analysis is performed here for general CCD schemes for the first time, emphasizing their utility in uni- and bi-directional wave propagation problems – that is relevant to acoustic wave propagation problems. We highlight: (a) specific points in parameter space those give rise to least phase and dispersion errors for non-periodic wave problems; (b) the solution error of CCD/NCCD schemes in solving Stommel Ocean model (an elliptic p.d.e.) and (c) the effectiveness of the NCCD scheme in solving Navier–Stokes equation for the benchmark lid-driven cavity problem at high Reynolds numbers, showing that the present method is capable of providing very accurate solution using far fewer points as compared to existing solutions in the literature.     

二维抛物型问题的Strang型交替分段区域分裂格式

给出求解二维抛物型方程的Strang型的交替分段区域分裂格式。交替分段思想可以将区域分为一些不重叠的子区域，Strang型算子分裂技巧通过将高维问题的求解分解为几个低维问题的求解来降低其求解的复杂度。方法是无条件稳定的，理论分析了截断误差。数值算例说明格式的有效性及时空的二阶精度.     

A mathematical model for pattern formation in biological systems

A system of reaction-diffusion equations with applications in biological problems is presented and studied theoretically and numerically. Solutions demonstrating the following types of behaviour have been obtained: (i) time constant, spatially inhomogeneous, (ii) time periodic, spatially homogeneous, and (iii) time periodic, travelling wave-like. Some applications in embryological problems are discussed.     

Some general problems of fast electrons transport

A generalization is given of the segments method in the form of a multistep method with generalized time for computing the transport of fast particles. The integral equation for a flow with generalized time in the phase space of variables is written under the assumption that the flow cuts the generalized time surface at right angles. The Green's function for the differential flow operator is the kernel of the integral equation. It is also shown that such an integral equation which can be obtained from a nonstationary kinetic equation provides a uniform consistent algorithm for solving either nonstationary or stationary problems. Examples of Green's functions are given for an operator of differential flow of fast electrons.Translated from Izvestiya Vysshikh Uchebnykh Zavedenii, Fizika, No. 8, pp. 110–114, August, 1974.The author would like to express his thanks to A. A. Vorob'ev and B. A. Kononov for their encouragement, to A. P. Yalovets for discussing the work with him, and to A. M. Kol'chuzhkin for going through the text.