Computing Streamfunction and Velocity Potential in a Limited Domain of Arbitrary Shape.Part II:Numerical Methods and Test Experiments

Built on the integral formulas in Part I,numerical methods are developed for computing velocity potential and streamfunction in a limited domain.When there is no inner boundary（around a data hole） inside the domain,the total solution is the sum of the internally and externally induced parts.For the internally induced part,three numerical schemes（grid-staggering,local-nesting and piecewise continuous integration） are designed to deal with the singularity of the Green＇s function encountered in numerical calculations.For the externally induced part,by setting the velocity potential（or streamfunction） component to zero,the other component of the solution can be computed in two ways：（1） Solve for the density function from its boundary integral equation and then construct the solution from the boundary integral of the density function.（2） Use the Cauchy integral to construct the solution directly.The boundary integral can be discretized on a uniform grid along the boundary.By using local-nesting（or piecewise continuous integration）,the scheme is refined to enhance the discretization accuracy of the boundary integral around each corner point（or along the entire boundary）.When the domain is not free of data holes,the total solution contains a data-hole-induced part,and the Cauchy integral method is extended to construct the externally induced solution with irregular external and internal boundaries.An automated algorithm is designed to facilitate the integrations along the irregular external and internal boundaries.Numerical experiments are performed to evaluate the accuracy and efficiency of each scheme relative to others.

Anti-plane deformations around arbitrary-shaped canyons on a wedge-shape half-space:moment method solutions

The wave propagation behavior in an elastic wedge-shaped medium with an arbitrary shaped cylindrical canyon at its vertex has been studied.Numerical computation of the wave displacement field is carried out on and near the canyon surfaces using weighted-residuals(moment method).The wave displacement fields are computed by the residual method for the cases of elliptic,circular,rounded-rectangular and flat-elliptic canyons,The analysis demonstrates that the resulting surface displacement depends,as in similar previous analyses,on several factors including,but not limited,to the angle of the wedge,the geometry of the vertex,the frequencies of the incident waves,the angles of incidence,and the material properties of the media.The analysis provides intriguing results that help to explain geophysical observations regarding the amplification of seismic energy as a function of site conditions.

A new analytical-numerical method for calculating interacting stresses of a multi-hole problem under both remote and arbitrary surface stresses

Based on the elementary solutions and new integral equations,a new analytical-numerical method is proposed to calculate the interacting stresses of multiple circular holes in an infinite elastic plate under both remote stresses and arbitrarily distributed stresses applied to the circular boundaries.The validity of this new analytical-numerical method is verified by the analytical solution of the bi-harmonic stress function method,the numerical solution of the finite element method,and the analytical-numerical solutions of the series expansion and Laurent series methods.Some numerical examples are presented to investigate the effects of the hole geometry parameters(radii and relative positions)and loading conditions(remote stresses and surface stresses)on the interacting tangential stresses and interacting stress concentration factors(SCFs).The results show that whether the interference effect is shielding(k<1)or amplifying(k>1)depends on the relative orientation of holes(α)and remote stresses(σ^∞x,σ^∞y).When the maximum principal stress is aligned with the connecting line of two-hole centers andσ^∞y<0.5σ^∞x,the plate containing two circular holes has greater stability than that containing one circular hole,and the smaller circular hole has greater stability than the bigger one.This new method not only has a simple formulation and high accuracy,but also has an advantage of wide applications over common analytical methods and analytical-numerical methods in calculating the interacting stresses of a multi-hole problem under both remote and arbitrary surface stresses.

A simplified two-dimensional boundary element method with arbitrary uniform mean flow

To reduce computational costs, an improved form of the frequency domain boundary element method（BEM） is proposed for two-dimensional radiation and propagation acoustic problems in a subsonic uniform flow with arbitrary orientation. The boundary integral equation（BIE） representation solves the two-dimensional convected Helmholtz equation（CHE） and its fundamental solution, which must satisfy a new Sommerfeld radiation condition（SRC） in the physical space. In order to facilitate conventional formulations, the variables of the advanced form are expressed only in terms of the acoustic pressure as well as its normal and tangential derivatives, and their multiplication operators are based on the convected Green＇s kernel and its modified derivative. The proposed approach significantly reduces the CPU times of classical computational codes for modeling acoustic domains with arbitrary mean flow. It is validated by a comparison with the analytical solutions for the sound radiation problems of monopole,dipole and quadrupole sources in the presence of a subsonic uniform flow with arbitrary orientation.

Simulation of sheet metal extrusion processes with Arbitrary Lagrangian-Eulerian method

An Arbitrary Lagrangian-Eulerian(ALE) method was employed to simulate the sheet metal extrusion process,aiming at avoiding mesh distortion and improving the computational accuracy.The method was implemented based on MSC/MARC by using a fractional step method,i.e.a Lagrangian step followed by an Euler step.The Lagrangian step was a pure updated Lagrangian calculation and the Euler step was performed using mesh smoothing and remapping scheme.Due to the extreme distortion of deformation domain,it was almost impossible to complete the whole simulation with only one mesh topology.Therefore,global remeshing combined with the ALE method was used in the simulation work.Based on the numerical model of the process,some deformation features of the sheet metal extrusion process,such as distribution of localized equivalent plastic strain,and shrinkage cavity,were revealed.Furthermore,the differences between conventional extrusion and sheet metal extrusion process were also analyzed.

A hybrid subcell-remapping algorithm for staggered multi-material arbitrary Lagrangian-Eulerian methods

A new flux-based hybrid subcell-remapping algorithm for staggered multimaterial arbitrary Lagrangian-Eulerian (MMALE) methods is presented. This new method is an effective generalization of the original subcell-remapping method to the multi-material regime (LOUBERE, R. and SHASHKOV,M. A subcell remapping method on staggered polygonal grids for arbitrary-Lagrangian-Eulerian methods. Journal of Computational Physics, 209, 105–138 (2005)). A complete remapping procedure of all fluid quantities is described detailedly in this paper. In the pure material regions, remapping of mass and internal energy is performed by using the original subcell-remapping method. In the regions near the material interfaces, remapping of mass and internal energy is performed with the intersection-based fluxes where intersections are performed between the swept regions and pure material polygons in the Lagrangian mesh, and an approximate approach is then introduced for constructing the subcell mass fluxes. In remapping of the subcell momentum, the mass fluxes are used to construct the momentum fluxes by multiplying a reconstructed velocity in the swept region. The nodal velocity is then conservatively recovered. Some numerical examples simulated in the full MMALE regime and several purely cyclic remapping examples are presented to prove the properties of the remapping method.

Methods for Derivation of Density Matrix of Arbitrary Multi-Mode Gaussian States from Its Phase Space Representation

We present a method for derivation of the density matrix of an arbitrary multi-mode continuous variable Gaussian entangled state from its phase space representation.An explicit computer algorithm is given to reconstruct the density matrix from Gaussian covariance matrix and quadrature average values.As an example,we apply our method to the derivation of three-mode symmetric continuous variable entangled state.Our method can be used to analyze the entanglement and correlation in continuous variable quantum network with multi-mode quantum entanglement states.

Simulations of the Electron Spectrum of Quantum Wires in <i>n</i>-Si of Arbitrarily Doping Profile by Thomas-Fermi Method

Electron spectrum in doped n-Si quantum wires is calculated by the Thomas-Fermi (TF) method under finite temperatures. The many-body exchange corrections are taken into account. The doping profile is arbitrary. At the first stage, the electron potential energy is calculated from a simple two-dimensional equation. The effective iteration scheme is proposed there that is valid for multidimensional problems. Then the energy levels and wave functions of this quantum well are simulated from the Schr&#246;dinger equations. The expansion by the full set of eigenfunctions of the linear harmonic oscillator is used. The quantum mechanical perturbation theory can be utilized to compute the energy levels. Generally, the perturbation theory for degenerate energy levels should be used.

Single-Step Arbitrary Lagrangian-Eulerian Discontinuous Galerkin Method for 1-D Euler Equations

We propose an explicit,single-step discontinuous Galerkin method on moving grids using the arbitrary Lagrangian-Eulerian approach for one-dimensional Euler equations.The grid is moved with the local fluid velocity modified by some smoothing,which is found to con-siderably reduce the numerical dissipation introduced by Riemann solvers.The scheme preserves constant states for any mesh motion and we also study its positivity preservation property.Local grid refinement and coarsening are performed to maintain the mesh qual-ity and avoid the appearance of very small or large cells.Second,higher order methods are developed and several test cases are provided to demonstrate the accuracy of the proposed scheme.

Arbitrary Lagrangian‑Eulerian Discontinuous Galerkin Methods for KdV Type Equations

In this paper,several arbitrary Lagrangian-Eulerian discontinuous Galerkin(ALE-DG)methods are presented for Korteweg-de Vries(KdV)type equations on moving meshes.Based on the L^(2) conservation law of KdV equations,we adopt the conservative and dissipative numerical fuxes for the nonlinear convection and linear dispersive terms,respectively.Thus,one conservative and three dissipative ALE-DG schemes are proposed for the equations.The invariant preserving property for the conservative scheme and the corresponding dissipative properties for the other three dissipative schemes are all presented and proved in this paper.In addition,the L^(2)-norm error estimates are also proved for two schemes,whose numerical fuxes for the linear dispersive term are both dissipative type.More precisely,when choosing the approximation space with the piecewise kth degree polynomials,the error estimate provides the kth order of convergence rate in L^(2)-norm for the scheme with the conservative numerical fuxes applied for the nonlinear convection term.Furthermore,the(k+1∕2)th order of accuracy can be proved for the ALE-DG scheme with dissipative numerical fuxes applied for the convection term.Moreover,a Hamiltonian conservative ALE-DG scheme is also presented based on the conservation of the Hamiltonian for KdV equations.Numerical examples are shown to demonstrate the accuracy and capability of the moving mesh ALE-DG methods and compare with stationary DG methods.

Negative Norm Estimates for Arbitrary Lagrangian-Eulerian Discontinuous Galerkin Method for Nonlinear Hyperbolic Equations

In this paper,we present the negative norm estimates for the arbitrary Lagrangian-Eulerian discontinuous Galerkin(ALE-DG)method solving nonlinear hyperbolic equations with smooth solutions.The smoothness-increasing accuracy-conserving(SIAC)filter is a post-processing technique to enhance the accuracy of the discontinuous Galerkin(DG)solutions.This work is the essential step to extend the SIAC filter to the moving mesh for nonlinear problems.By the post-processing theory,the negative norm estimates are vital to get the superconvergence error estimates of the solutions after post-processing in the L2 norm.Although the SIAC filter has been extended to nonuniform mesh,the analysis of fil-tered solutions on the nonuniform mesh is complicated.We prove superconvergence error estimates in the negative norm for the ALE-DG method on moving meshes.The main dif-ficulties of the analysis are the terms in the ALE-DG scheme brought by the grid velocity field,and the time-dependent function space.The mapping from time-dependent cells to reference cells is very crucial in the proof.The numerical results also confirm the theoreti-cal proof.

THE SOLUTION OF A CRACK EMANATING FROM AN ARBITRARY HOLE BY BOUNDARY COLLOCATION METHOD

In this paper a group of stress functions has been proposed for the calculation of a crack emanating from a hole with different shape (including circular, elliptical, rectangular, or rhombic hole) by boundary collocation method. The calculation results show that they coincide very well with the existing solutions by other methods for a circular or elliptical hole with a crack in an infinite plate. At the smae time, a series of results for different holes in a finite plate has also been obtained in this paper. The proposed functions and calculation procedure can be used for a plate of a crack emanating from an arbitrary hole.

BOUNDARY ELEMENT METHOD FOR ARBITRARY ELASTIC THIN SHELL

In this paper,a boundary element scheme for arbitrary elastic thin shells is elaborated,Based on BEM of 3D linear elasticity and Kirchhoff's hypothesis,boundary integral equations for shells are deduced. As a result,only Kelvin's solution is used,the difficulty,in finding a fundamental solution of arbitrary shells is successfully avoided.

收放伞绳对十字型伞静态气动特性影响的数值模拟研究

为研究十字型降落伞作为主动控制装置时,其可控性与气动特性的规律,建立了不同伞绳收缩比δ为0、10.0%、12.5%、15.0%、20.0%、25.0%时十字型伞无限质量充气的有限元模型,基于任意拉格朗日-欧拉方法进行数值模拟,分析了十字型降落伞收缩伞绳,引起伞面几何构型不对称对气动特性的影响.结果表明:不同攻角为0°、5°、10°时,十字型降落伞的平均阻力系数随伞绳收缩比δ的增大而减小;十字型伞的瞬时升力峰值随着伞绳收缩比δ的增加而增大,方向始终由伞绳收缩一侧指向另一侧,具有较强的时变特性.

下卧软弱夹层地基ALE法强夯加固数值分析

随着高能级强夯快速发展,软弱夹层对地基加固影响问题凸显,其吸能缓冲作用机理尚不清晰,夯击参数选取经验性强。基于现场试验和任意拉格朗日-欧拉(arbitrary Lagrange-Euler, ALE)法强夯加固仿真开展地基应力波传播过程研究,探讨下卧软弱夹层对加固效果影响机理。对夯击能级、夹层埋深、夹层厚度等因素进行参数研究,改进强夯参数设计流程。结果表明:软弱夹层对夯击应力波向下传播过程有迟滞作用,地基应力场在此深度内发生断层;当首击地基加固深度为1/2软弱夹层埋深时,可兼顾能级控制与施工效率需要;当软弱夹层埋深大于10 m或厚度超过2 m时加固难度较大,需配合其他工法共同加固处理。

水下中远场爆炸冲击波作用下航行体表面动态响应分析

针对水下中远场爆炸作用下航行体表面动态响应复杂、预测困难的问题,基于任意欧拉-拉格朗日数值算法,考虑自由面对冲击波反射的影响,利用爆炸相似理论细化模型网格,运用有限元软件对水下爆炸模型进行求解。分析冲击波作用下航行体表面动态应力峰值出现的时间和区域,并对其动态响应随当量、爆距及水深的变化规律进行研究。利用数值拟合方法得到航行体表面动态应力方程,并通过决定系数验证其可靠性,在定当量、爆距情况下可直接求解其动态应力。研究结果表明:当量增加,动态应力在相同爆距区间内衰减速率增大;爆距增大,动态应力在相同当量区间内增长速率降低;深度每下降100 m,动态应力增长20 MPa左右。所得研究结果可为中远场水下爆炸作用下航行体安全防护和抗冲击研究提供一定参考。

任意分布ST方法及其在随机场线耦合中的应用

随机测试(ST)方法是一种不确定性量化方法,已有的ST方法还不能处理具有任意分布随机变量的情况。通过Gram-Schmidt正交化算法为任意分布随机变量构建混沌多项式基底,对ST方法进行拓展,使其可用于任意分布随机变量的情况。将拓展后的ST方法用于传输线场线耦合分析中。在场线耦合的计算中,由于辐射场的不确定性,入射波极化角、方位角与仰角存在不确定性,因此传输线的响应也呈现出不确定性,导致随机场线耦合问题。针对输入参数服从任意分布的情况,应用ST方法对传输线场线耦合响应进行不确定性量化。对模型的输出响应进行多项式混沌展开,通过求解展开系数得到模型响应的统计信息。结合单随机变量和多随机变量场线耦合算例,得到传输线电压响应的统计信息。最后对比蒙特卡罗模拟方法,验证了ST方法的正确性和高效性。

基于线段近似法的姿态控制线圈磁场位形研究

基于线段近似方法对几何结构相对复杂的偶极场姿态控制(Tilt-Slide-Rotate coils,TSR)线圈的线圈作用力进行了研究。在偶极场倾斜和偏移情况下,分别通过控制对侧和同侧TSR线圈产生磁力使偶极场线圈恢复平衡位置的情形进行了建模分析。计算TSR线圈和偶极场线圈耦合磁场及分析磁场线结构,并采用具有正弦电流形式的TSR线圈激励磁场模拟环向磁扰动,对扰动磁场在赤道面上的庞加莱截面图的磁场线结构和相位进行分析。发现仅通过一组同侧或对侧TSR线圈控制偶极场线圈使其恢复平衡时该过程具有力学不稳定性,需要多组TSR线圈协同控制。对全部TSR线圈施加伪正弦电流,会对背景偶极磁场产生压缩和拉伸效应,并且控制TSR线圈电流来控制压缩位置相位,该过程验证了多TSR线圈控制偶极场线圈姿态的可行性,但同时会使背景磁场发生漂移及形成开放磁场线,这可能会驱动等离子体不稳定性及导致粒子的损失。

空间任意汇交杆系的变形协调几何方程组及其组合学释义

将平面汇交杆系的变形协调问题拓展到空间任意汇交杆系的情形,运用微分解析法,通过对空间任意4根汇交杆的求解,直观地得到了空间任意n根汇交杆系变形协调几何方程组的一般形式,解决了空间汇交杆系的超静定问题。结论中所有下标的顺序符合组合学中字典序数排列,因此可以用计算机编程去求解超静定空间汇交杆系的变形协调方程。

B样条函数法的任意荷载近似计算方法

为优化B样条函数法数值计算流程,提升解决复杂荷载问题的能力。本文基于B样条函数法,通过与相关文献数值算例结果进行比较分析,研究任意荷载的新型计算方法。论文以解圆柱壳结构模型为数值算例,基于文中提出的计算方法,构建出三组不同的单方向荷载逼近函数,分别求解得到特殊点的位移、弯矩、轴力,以及位移和内力沿坐标轴方向的分布,与精确解(扁壳解法)进行比较研究,分析近似计算方法的精确性与实用性。结果显示,在特殊点处二次插值方法的相对误差均低于1.5%,三次、四次插值方法的相对误差均低于0.6%,沿坐标轴方向的分布与精确解均一致,表明本方法具有精度高、处理复杂荷载简便、易于程序编写、构造函数灵活等特点。本方法用逼近函数替代原函数便于数值积分的想法,为未来B样条函数法处理复杂荷载提供新思路。