Light-scattering model for aerosol particles with irregular shapes and inhomogeneous compositions using a parallelized pseudo-spectral time-domain technique

To improve the modeling accuracy of radiative transfer,the scattering properties of aerosol particles with irregular shapes and inhomogeneous compositions should be simulated accurately.To this end,a light-scattering model for nonspherical particles is established based on the pseudo-spectral time domain(PSTD)technique.In this model,the perfectly matched layer with auxiliary differential equation(ADE-PML),an excellent absorption boundary condition(ABC)in the finite difference time domain generalized for the PSTD,and the weighted total field/scattered field(TF/SF)technique is employed to introduce the incident light into 3 D computational domain.To improve computational efficiency,the model is further parallelized using the Open MP technique.The modeling accuracy of the PSTD scheme is validated against Lorenz–Mie,Aden–Kerker,T-matrix theory and DDA for spheres,inhomogeneous particles and nonspherical particles,and the influence of the spatial resolution and thickness of ADE-PML on the modeling accuracy is discussed as well.Finally,the parallel computational efficiency of the model is also analyzed.The results show that an excellent agreement is achieved between the results of PSTD and well-tested scattering models,where the simulation errors of extinction efficiencies are generally smaller than 1%,indicating the high accuracy of our model.Despite its low spatial resolution,reliable modeling precision can still be achieved by using the PSTD technique,especially for large particles.To suppress the electromagnetic wave reflected by the absorption layers,a six-layer ADE-PML should be set in the computational domain at least.

Chebyshev Pseudo-Spectral Method for Solving Fractional Advection-Dispersion Equation

Fractional differential equations have recently been applied in various areas of engineering, science, finance, applied mathematics, bio-engineering and others. However, many researchers remain unaware of this field. In this paper, an efficient numerical method for solving the fractional Advection-dispersion equation (ADE) is considered. The fractional derivative is described in the Caputo sense. The method is based on Chebyshev approximations. The properties of Chebyshev polynomials are used to reduce ADE to a system of ordinary differential equations, which are solved using the finite difference method (FDM). Moreover, the convergence analysis and an upper bound of the error for the derived formula are given. Numerical solutions of ADE are presented and the results are compared with the exact solution.

Comparison of finite difference and pseudo-spectral methods in forward modelling based on metal ore model of random media

With more applications of seismic exploration in metal ore exploration,forward modelling of seismic wave has become more important in metal ore. Finite difference method and pseudo-spectral method are two important methods of wave-field simulation. Results of previous studies show that both methods have distinct advantages and disadvantages: Finite difference method has high precision but its dispersion is serious; pseudospectral method considers both computational efficiency and precision but has less precision than finite-difference. The authors consider the complex structural characteristics of the metal ore,furthermore add random media in order to simulate the complex effects produced by metal ore for wave field. First,the study introduced the theories of random media and two forward modelling methods. Second,it compared the simulation results of two methods on fault model. Then the authors established a complex metal ore model,added random media and compared computational efficiency and precision. As a result,it is found that finite difference method is better than pseudo-spectral method in precision and boundary treatment,but the computational efficiency of pseudospectral method is slightly higher than the finite difference method.

An Accurate Numerical Solution for the Modified Equal Width Wave Equation Using the Fourier Pseudo-Spectral Method

In this study, the numerical solution for the Modified Equal Width Wave (MEW) equation is presented using Fourier spectral method that use to discretize the space variable and Leap-frog method scheme for time dependence. Test problems including the single soliton wave motion, interaction of two solitary waves and interaction of three solitary waves will use to validate the proposed method. The three invariants of the motion are evaluated to determine the conservation properties of the generated scheme. Finally, a Maxwellian initial condition pulse is then studied. The L2 and L∞ error norms are computed to study the accuracy and the simplicity of the presented method.

Solving a Nonlinear Multi-Order Fractional Differential Equation Using Legendre Pseudo-Spectral Method

In this paper, we apply the Legendre spectral-collocation method to obtain approximate solutions of nonlinear multi-order fractional differential equations (M-FDEs). The fractional derivative is described in the Caputo sense. The study is conducted through illustrative example to demonstrate the validity and applicability of the presented method. The results reveal that the proposed method is very effective and simple. Moreover, only a small number of shifted Legendre polynomials are needed to obtain a satisfactory result.

Pseudo-Spectral Method for Space Fractional Diffusion Equation

This paper presents a numerical scheme for space fractional diffusion equations (SFDEs) based on pseudo-spectral method. In this approach, using the Guass-Lobatto nodes, the unknown function is approximated by orthogonal polynomials or interpolation polynomials. Then, by using pseudo-spectral method, the SFDE is reduced to a system of ordinary differential equations for time variable t. The high order Runge-Kutta scheme can be used to solve the system. So, a high order numerical scheme is derived. Numerical examples illustrate that the results obtained by this method agree well with the analytical solutions.

Calculation of Turbulent Flow with Pseudo-spectral Method

This paper deals with the numerical simulation of incompressible turbulent boundary flow of a flat plate with the pseudo-spectral matrix method. In order to appear more than 10 nodes in the turbulent base-stratum and transition of 43×43 computational grids,a coordinate transformation is put up from physical panel to computational panel. Several zero turbulent models are computed comparatively. The results are credible when comparing with the previous methods.

Pseudo-spectral Approximations for a Class of the Kdv-Burgers Type Equation

In this paper, the pseudo-spectral approximations for a class of the Kdv-Burgers type equation is presented. Convergence and stability of the approximation have been proved by Sobolev's inequalities and the bounded extensive method of the nonlinear function. Finally, the numerical examples are proposed.

A New Framework of Convergence Analysis for Solving the General Nonlinear Schrodinger Equation using the Fourier Pseudo-Spectral Method in Two Dimensions

This paper aims to build a new framework of convergence analysis of conservative Fourier pseudo-spectral method for the general nonlinear Schr¨odinger equation in two dimensions,which is not restricted that the nonlinear term is mere cubic.The new framework of convergence analysis consists of two steps.In the first step,by truncating the nonlinear term into a global Lipschitz function,an alternative numerical method is proposed and proved in a rigorous way to be convergent in the discrete L2 norm;followed in the second step,the maximum bound of the numerical solution of the alternative numerical method is obtained by using a lifting technique,as implies that the two numerical methods are the same one.Under our framework of convergence analysis,with neither any restriction on the grid ratio nor any requirement of the small initial value,we establish the error estimate of the proposed conservative Fourier pseudo-spectral method,while previous work requires the certain restriction for the focusing case.The error bound is proved to be of O(h^(r)+t^(2))with grid size h and time step t.In fact,the framework can be used to prove the unconditional convergence of many other Fourier pseudo-spectral methods for solving the nonlinear Schr¨odinger-type equations.Numerical results are conducted to indicate the accuracy and efficiency of the proposed method,and investigate the effect of the nonlinear term and initial data on the blow-up solution.

Flight strategy optimization for high-altitude long-endurance solar-powered aircraft based on Gauss pseudo-spectral method

Solar-powered aircraft have attracted great attention owing to their potential for longendurance flight and wide application prospects.Due to the particularity of energy system,flight strategy optimization is a significant way to enhance the flight performance for solar-powered aircraft.In this study,a flight strategy optimization model for high-altitude long-endurance solar-powered aircraft was proposed.This model consists of three-dimensional kinematic model,aerodynamic model,energy collection model,energy store model and energy loss model.To solve the nonlinear optimal control problem with process constraints and terminal constraints,Gauss pseudo-spectral method was employed to discretize the state equations and constraint equations.Then a typical mission flying from given initial point to given final point within a time interval was considered.Results indicate that proper changes of the attitude angle contribute to increasing the energy gained by photovoltaic cells.Utilization of gravitational potential energy can partly take the role of battery pack.Integrating these two measures,the optimized flight strategy can improve the final state of charge compared with current constant-altitude constant-velocity strategy.The optimized strategy brings more profits on condition of lower sunlight intensity and shorter daytime.

Multisymplectic Fourier pseudo-spectral integrators for Klein-Gordon-Schrdinger equations

A multisymplectic Fourier pseudo-spectral scheme, which exactly preserves the discrete multisym- plectic conservation law, is presented to solve the Klein-Gordon-SchrSdinger equations. The scheme is of spectral accuracy in space and of second order in time. The scheme preserves the discrete multisymplectic conservation law and the charge conservation law. Moreover, the residuals of some other conservation laws are derived for the geometric numerical integrator. Extensive numerical simulations illustrate the numerical behavior of the multisymplectic scheme, and demonstrate the correctness of the theoretical analysis.

AN EXPLICIT PSEUDO-SPECTRAL SCHEME WHIT ALMOST UNCONDITIONAL STABILITY FOR THE CAHN-HILLIARD EQUATION

Proposes an explicit fully discrete three-level pseudo-spectral scheme with unconditional stability for the Cahn-Hilliard equation. Equations for pseudo-spectral scheme; Analysis of linear stability of critical points.

Modeling of Borehole Radar for Well Logging Using Pseudo-spectral Time Domain Algorithm

In this article, numerical modeling of borehole radar for well logging in time domain is developed using pseudo-spectral time domain algorithm in axisymmetric cylindrical coordinate for proximate true formation model. The conductivity and relative permittivity logging curves are obtained from the data of borehole radar for well logging. Since the relative permittivity logging curve is not affected by salinity of formation water, borehole radar for well logging has obvious advantages as compared with conventional electrical logging. The borehole radar for well logging is a one-transmitter and two-receiver logging tool. The conductivity and relative permittivity logging curves are obtained successfully by measuring the amplitude radio and the time difference of pulse waveform from two receivers. The calculated conductivity and relative permittivity logging curves are close to the true value of surrounding formation, which tests the usability and reliability of borehole radar for well logging. The numerical modeling of borehole radar for well logging laid the important foundation for researching its logging tool.

VTI介质修正声学近似qP波波动方程与模拟

各向异性介质中纵横波通常耦合在一起传播,纵横波解耦是地震波传播理论研究的重要内容。经典的声学近似通过设置垂向qSV波速度V_(S0)为0来解耦qP波场,但是存在退化qSV波等问题。本研究将垂向qSV波速度V_(S0)考虑成波数k_(x)、k_(z)和各向异性参数ε、δ的函数,对声学近似进行修正,推导了VTI介质修正声学近似qP波频散关系和波动方程。VTI介质修正声学近似qP波波动方程包含椭圆项和非椭圆项两部分,因此采用混合有限差分/伪谱算法进行求解,即采用有限差分法求解椭圆项、伪谱法求解非椭圆项。频散关系分析和数值示例表明,基于修正声学近似的qP波波动方程不包含退化qSV波,是纯qP波方程,与弹性波方程模拟结果吻合较好,且具有较高精度;该方程在ε≥δ和ε<δ的VTI介质中均是稳定的,并且精度高于声学近似模拟结果。

广义概率密度演化方程的Chebyshev拟谱法

概率密度演化方法(probability density evolution equation,PDEM)为非线性随机结构的动力响应分析提供了新的途径.通过PDEM获得结构响应概率密度函数(probability density function,PDF)的关键步骤是求解广义概率密度演化方程(generalized probability density evolution equation,GDEE).对于GDEE的求解通常采用有限差分法,然而,由于GDEE是初始条件间断的变系数一阶双曲偏微分方程,通过有限差分法求解GDEE可能会面临网格敏感性问题、数值色散和数值耗散现象.文章从全局逼近的角度出发,基于Chebyshev拟谱法为GDEE构造了全局插值格式,解决了数值色散、数值耗散以及网格敏感性问题.考虑GDEE的系数在每个时间步长均为常数,推导了GDEE在每一个时间步长内时域上的序列矩阵指数解.由于序列矩阵指数解形式上是解析的,从而很好地克服了数值稳定性问题.两个数值算例表明,通过Chebyshev拟谱法结合时域的序列矩阵指数解求解GDEE得到的结果与精确解以及Monte Carlo模拟的结果非常吻合,且数值耗散和数值色散现象几乎可以忽略.此外,拟谱法具有高效的收敛性且序列矩阵指数解不受CFL (Courant-Friedrichs-Lewy)条件的限制,因此该方法具有良好的数值稳定性和计算效率.

基于切比雪夫拟谱法的移动荷载作用下简支梁动力响应计算

采用切比雪夫拟谱法求解移动荷载作用下简支梁的动力响应问题,该方法通过重新选取插值点的方式并借助转换矩阵对边界条件进行施加,对集中荷载采用高斯函数法进行正则化处理。将本文方法计算结果与解析解和有限元解进行比较,结果表明,在划分较少单元数量时,采用本文方法计算结果与解析解吻合良好,求解效率高于有限元方法。采用本文方法对Dirac函数的正则化参数σ_(n)进行敏感性分析,结果表明,当在±3σ_(n)范围内布置配点数6~10个时,计算结果能取得较好的精度。

考虑云团衰减作用的激光制导炮弹弹道规划

针对多云天气条件下,云团对激光信号产生较大衰减作用从而影响末制导性能这一问题,提出一种考虑云团对激光衰减作用的滑翔增程激光制导炮弹弹道规划方法。经过合理假设建立云团近似模型,在此基础上设计云团规避—控制能量最优复合规避目标函数(Compound Evasion Objective Function,CEOF)。根据激光导引头工作特性和制导炮弹的飞行特性建立弹道规划模型,采用Radau伪谱法将弹道规划问题转化为非线性规划问题(Non-linear Programming,NLP)。调用NLP求解器SNOPT求解。选取多组云团算例进行仿真分析,仿真结果验证所提弹道规划方法的有效性。研究结果表明,与传统弹道规划方法进行仿真对比,新的弹道规划方法以增加较少动能损耗、控制能量消耗和攻击时长为代价,大幅减少了云团对激光的衰减作用,验证了其优越性。

面向突防的滑翔制导炮弹弹道规划方法

针对滑翔制导炮弹在不可避免的威胁区域内选择突防方案的问题,从量化威胁值的角度建立了敌方防御手段的数学模型,基于模型设计了全程综合威胁值最低的规划指标,提出考虑目标防御威胁的弹道规划方法。为实现滑翔制导炮弹全程飞行过程中初始弹道倾角、偏角、火箭点火时刻、滑翔启控时刻等各参数的最佳匹配,建立了多阶段全弹道规划模型,并采用hp自适应伪谱法将最优控制问题转换为非线性规划问题求解。通过仿真验证了在该指标下滑翔制导炮弹对目标防御的规避效果,分析了影响有效性的因素。与传统弹道规划方法进行对比,证明了所提方法的优越性。

制导炮弹协同方案弹道快速规划

为求解复杂的多弹多阶段协同弹道规划问题,提出一种增广集中式-协同弹道规划算法(AC-CTPM)。首先根据制导炮弹飞行过程中各阶段特性建立5阶段弹道规划模型,之后将n_p发制导炮弹的5阶段弹道规划问题组合扩展为较为复杂的5n_p阶段最优控制问题,然后采用多阶段Radau伪谱法将无限维最优控制问题(OCP)离散为有限维非线性规划问题(NLP),最后调用成熟的NLP求解器SNOPT求解。为提高对复杂的5n_p阶段最优控制问题的求解效率,在AC-CTPM中提出一种将2D标称弹道转换为3D预测方案弹道的协同弹道规划问题初始值获取方法(IGVAM),每一发制导炮弹分别以各自发射点为原点建立新地面坐标系,在新地面坐标系内快速规划出2D方案弹道,之后通过扩展和坐标转换将新坐标系中规划的2D方案弹道转换成原地面坐标系中的3D方案弹道,这些3D方案弹道组合构成协同弹道规划问题的初始预测。采用AC-CTPM算法对单炮多发、多炮齐射同时弹着任务场景进行仿真求解,获得了满足炮弹自身约束以及协同约束的协同方案弹道,验证了AC-CTPM算法的有效性。与分布式协同弹道规划算法(D-CTPM)以及传统集中式协同弹道规划算法(TC-CTPM)进行仿真对比,结果表明:AC-CTPM算法规划的协同方案弹道的目标函数平均比TC-CTPM算法优5.07%,比D-CTPM算法优32.98%,而AC-CTPM算法的求解耗时却比TC-CTPM算法减少86.48%,比D-CTPM减少82.36%,验证了AC-CTPM算法的优越性。