A stable implicit nodal integration-based particle finite element method(N-PFEM)for modelling saturated soil dynamics

In this study,we present a novel nodal integration-based particle finite element method(N-PFEM)designed for the dynamic analysis of saturated soils.Our approach incorporates the nodal integration technique into a generalised Hellinger-Reissner(HR)variational principle,creating an implicit PFEM formulation.To mitigate the volumetric locking issue in low-order elements,we employ a node-based strain smoothing technique.By discretising field variables at the centre of smoothing cells,we achieve nodal integration over cells,eliminating the need for sophisticated mapping operations after re-meshing in the PFEM.We express the discretised governing equations as a min-max optimisation problem,which is further reformulated as a standard second-order cone programming(SOCP)problem.Stresses,pore water pressure,and displacements are simultaneously determined using the advanced primal-dual interior point method.Consequently,our numerical model offers improved accuracy for stresses and pore water pressure compared to the displacement-based PFEM formulation.Numerical experiments demonstrate that the N-PFEM efficiently captures both transient and long-term hydro-mechanical behaviour of saturated soils with high accuracy,obviating the need for stabilisation or regularisation techniques commonly employed in other nodal integration-based PFEM approaches.This work holds significant implications for the development of robust and accurate numerical tools for studying saturated soil dynamics.

SECOND-ORDER ACCURATE DIFFERENCE METHOD FOR THE SINGULARLY PERTURBED PROBLEM OF FOURTH-ORDER ORDINARY DIFFERENTIAL EQUATIONS

In this paper, we construct a uniform second-order difference scheme for a class of boundary value problems of fourth-order ordinary differential equations. Finally, a numerical example is given.

The Numerical Accuracy Analysis of Asymptotic Homogenization Method and Multiscale Finite Element Method for Periodic Composite Materials

In this paper,we discuss the numerical accuracy of asymptotic homogenization method(AHM)and multiscale finite element method(MsFEM)for periodic composite materials.Through numerical calculation of the model problems for four kinds of typical periodic composite materials,the main factors to determine the accuracy of first-order AHM and second-order AHM are found,and the physical interpretation of these factors is given.Furthermore,the way to recover multiscale solutions of first-order AHM and MsFEM is theoretically analyzed,and it is found that first-order AHM and MsFEM provide similar multiscale solutions under some assumptions.Finally,numerical experiments verify that MsFEM is essentially a first-order multiscale method for periodic composite materials.

Numerical Experiments Using MATLAB: Superconvergence of Nonconforming Finite Element Approximation for Second-Order Elliptic Problems

The superconvergence in the finite element method is a phenomenon in which the fi-nite element approximation converges to the exact solution at a rate higher than the optimal order error estimate. Wang proposed and analyzed superconvergence of the conforming finite element method by L2-projections. However, since the conforming finite element method (CFEM) requires a strong continuity, it is not easy to construct such finite elements for the complex partial differential equations. Thus, the nonconforming finite element method (NCFEM) is more appealing computationally due to better stability and flexibility properties compared to CFEM. The objective of this paper is to establish a general superconvergence result for the nonconforming finite element approximations for second-order elliptic problems by L2-projection methods by applying the idea presented in Wang. MATLAB codes are published at https://github.com/annaleeharris/Superconvergence-NCFEM for anyone to use and to study. The results of numerical experiments show great promise for the robustness, reliability, flexibility and accuracy of superconvergence in NCFEM by L2- projections.

The use of the node-based smoothed finite element method to estimate static and seismic bearing capacities of shallow strip footings

The node-based smoothed finite element method(NS-FEM)is shortly presented for calculations of the static and seismic bearing capacities of shallow strip footings.A series of computations has been performed to assess variations in seismic bearing capacity factors with both horizontal and vertical seismic accelerations.Numerical results obtained agree very well with those using the slip-line method,revealing that the magnitude of the seismic bearing capacity is highly dependent upon the combinations of various directions of both components of the seismic acceleration.An upward vertical seismic acceleration reduces the seismic bearing capacity compared to the downward vertical seismic acceleration in calculations.In addition,particular emphasis is placed on a separate estimation of the effects of soil and superstructure inertia on each seismic bearing capacity component.While the effect of inertia forces arising in the soil on the seismic bearing capacity is non-trivial,and the superstructure inertia is the major contributor to reductions in the seismic bearing capacity.Both tables and charts are given for practical application to the seismic design of the foundations.

UNIFORM ERROR BOUNDS OF A CONSERVATIVE COMPACT FINITE DIFFERENCE METHOD FOR THE QUANTUM ZAKHAROV SYSTEM IN THE SUBSONIC LIMIT REGIME

In this paper,we consider a uniformly accurate compact finite difference method to solve the quantum Zakharov system(QZS)with a dimensionless parameter 0<ε≤1,which is inversely proportional to the acoustic speed.In the subsonic limit regime,i.e.,when 0<ε?1,the solution of QZS propagates rapidly oscillatory initial layers in time,and this brings significant difficulties in devising numerical algorithm and establishing their error estimates,especially as 0<ε?1.The solvability,the mass and energy conservation laws of the scheme are also discussed.Based on the cut-off technique and energy method,we rigorously analyze two independent error estimates for the well-prepared and ill-prepared initial data,respectively,which are uniform in both time and space forε∈(0,1]and optimal at the fourth order in space.Numerical results are reported to verify the error behavior.

Extremum of second-order directional derivatives

In this paper, the extremum of second-order directional derivatives, i.e. the gradient of first-order derivatives is discussed. Given second-order directional derivatives in three nonparallel directions, or given second-order directional derivatives and mixed directional derivatives in two nonparallel directions, the formulae for the extremum of second-order directional derivatives are derived, and the directions corresponding to maximum and minimum are perpendicular to each other.

SINE TRANSFORM PRECONDITIONERS FOR SECOND-ORDER PARTIAL DIFFERENTIAL EQUATIONS

In this paper, we are concerned with the numerical solution of second-order partial differential equations. We analyse the use of the Sine Transform precondilioners for the solution of linear systems arising from the discretization of p.d.e. via the preconditioned conjugate gradient method. For the second-order partial differential equations with Dirichlel boundary conditions, we prove that the condition number of the preconditioned system is O(1) while the condition number of the original system is O(m 2) Here m is the number of interior gridpoints in each direction. Such condition number produces a linear convergence rale.

供水管网大口径管道爆管事件形成机理与精细模拟

近年来城市供水管网爆管事件频发对城市的运行安全造成直接影响。结合2017年上海“11·16”溧阳路四平路爆管事件,系统介绍了本次爆管事件概况,同时,利用本次爆管事件现场调查数据,采用3D激光扫描技术和三维数值建模方法,对本次爆管管道进行了建模分析,研究了供水管道在存在初期裂缝情况下,裂缝末端应力集中与管内水压力和裂缝长度发展变化影响关系,采用动力分析方法模拟了管道从裂缝发展到管体破坏形成爆管的物理全过程。结果表明:在供水管网管道早期裂缝末端,由于应力集中管体裂缝继续发展,且随管道水压增大而增大,其增速与裂缝长度密切相关;管道发生开裂后其临界破坏水压随早期裂缝长度的增加而降低,结合本次爆管事件周边获得的3个测点实测水压分析可知,爆管管道早期裂缝断面圆心角应大于26°;管道爆管破坏过程模拟显示,管道残片随裂缝开展向管顶转动,直至形成贯通裂缝完全脱离管道,脱落管片在内水压及外部荷载作用下会发生剧烈的转动和向外弹射,从而形成爆管并产生极大的瞬间破坏作用。

On the superconvergence of a WG method for the elliptic problem with variable coefficients

This article extends a recently developed superconvergence result for weak Galerkin(WG)approximations for modeling partial differential equations from constant coefficients to variable coefficients.This superconvergence features a rate that is two orders higher than the optimal-order error estimates in the usual energy and L^(2)norms.The extension from constant to variable coefficients for the modeling equations is highly non-trivial.The underlying technical analysis is based on a sequence of projections and decompositions.Numerical results confirm the superconvergence theory for second-order elliptic problems with variable coefficients.

AN OVER-PENALIZED WEAK GALERKIN METHOD FOR SECOND-ORDER ELLIPTIC PROBLEMS

The weak Galerkin （WG） finite element method was first introduced by Wang and Ye for solving second order elliptic equations, with the use of weak functions and their weak gradients. The basis function spaces depend on different combinations of polynomial spaces in the interior subdomains and edges of elements, which makes the WG methods flexible and robust in many applications. Different from the definition of jump in discontinuous Galerkin （DG） methods, we can define a new weaker jump from weak functions defined on edges. Those functions have double values on the interior edges shared by two elements rather than a limit of functions defined in an element tending to its edge. Naturally, the weak jump comes from the difference between two weak flmctions defined on the same edge. We introduce an over-penalized weak Galerkin （OPWG） method, which has two sets of edge-wise and element-wise shape functions, and adds a penalty term to control weak jumps on the interior edges. Furthermore, optimal a priori error estimates in H1 and L2 norms are established for the finite element （Pk（K）, Pk（e）, RTk（K））. In addition, some numerical experiments are given to validate theoretical results, and an incomplete LU decomposition has been used as a preconditioner to reduce iterations from the GMRES, CO, and BICGSTAB iterative methods.

对流扩散方程的变步长摄动有限差分格式

摄动有限差分(PFD)方法是构造高精度差分格式的一种新方法。变步长摄动有限差分方法是等步长摄动有限差分方法的发展和推广。对需要局部加密网格的计算问题,变步长PFD格式不需要对自变量进行数学变换,且和等步长PFD格式一样,具有如下的共同特点:从变步长一阶迎风格式出发,通过把非微商项(对流系数和源项)作变步长摄动展开,展开幂级数系数通过消去摄动格式修正微分方程的截断误差项求出,由此获得高精度变步长PFD格式。该格式在一、二和三维情况下分别仅使用三、五和七个基点,且具有迎风性。文中利用变步长PFD格式对对流扩散反应模型方程,变系数方程及Burgers方程等进行了数值模拟,并与一阶迎风和二阶中心格式及其问题的精确解作了比较。数值试验表明,与一阶迎风和二阶中心格式相比,变步长PFD格式具有精度高,稳定性与收敛性好的特点。变步长PFD格式与等步长PFD格式相比,变步长PFD解在薄边界层型区域的分辨率得到了明显的提高。

用有限区域风速场准确求解流函数和速度势场的方法

流函数和速度势是气象业务和研究中常用于表述风速的一组变量。用有限区域风速场,使用有限差分方法求解得到的流函数和速度势场重建初始风速场,由于受区域边界的限制往往有明显的偏差。虽然有许多求解方法的研究,但是,至今仍尚未见到一种真正准确的求解计算方案。首先,介绍用Arakawa A网格和D网格分布的有限区域风速场求解流函数和速度势场的一般有限差分计算方法,探讨用它们的解重建风速场产生误差的原因。然后,针对这些原因,对给定的有限区域,通过线性外推初始风速场,扩展求解计算区域,使用协调、一致的有限差分格式方案,准确计算求解区域的边界有旋风速、散度风速和速度势的定解边界条件,以及恰当选择流函数、速度势、涡度和散度等变量的分布网格,设计了用上述两种网格分布的风速场准确求解流函数、速度势场的方案,并对其正确性加以证明,它们可以推广应用于其他Arakawa网格。用实际资料试验同样显示,方案避免了重建风速场误差的出现,与初始风速场相比,全场风速最大偏差精度达到10-12m/s或以上,在计算机精度造成的计算误差影响范围内。本文的研究很好解决了长期以来用有限区域风速场、使用有限差分方法无法准确求解流函数和速度势场的问题。

基于阻抗边界法的变压器杂散损耗计算与分析

为研究变压器结构件中的杂散损耗问题,基于Problem 21基准族中的P21-B模型进行了详细实验研究和仿真分析.采用测量与仿真结合法对不同激励电流导磁钢板中的杂散损耗进行了测量分析,并且针对现有测量方法的不足,引入温度系数因子对其修正,得到了更为准确的杂散损耗测量结果;同时,借助工程电磁场数值计算软件MagNet,分别采用传统有限元法和有限元与阻抗边界结合法对该模型进行仿真计算.通过对比分析,发现相较于传统有限元法,有限元与阻抗边界结合法在得到较为准确的损耗结果的同时,还能减少计算成本,节省计算资源,更适合工程应用,所得结论和结果有助于通过优化设计提高变压器的性能指标.

摄动有限差分方法研究进展

振动有限差分（ＰＦＤ）方法，既离散徽商项也离散非微商项（包括微商系数），在微商用直接差分近似的前提下提高差分格式的精度和分辨率．ＰＦＤ方法包括局部线化微分方程的摄动精确数值解（ＰＥＮＳ）方法和摄动数值解（ＰＮＳ）方法以及考虑非线性近似的摄动高精度差分（ＰＨＤ）方法。论述了这些方法的基本思想、具体技巧、若干方程（对流扩散方程、对流扩散反应方程、双曲方程、抛物方程和ＫｄＶ方程）的ＰＥＮＳ、ＰＮＳ和ＰＨＤ格式，它们的性质及数值实验．并与有关的数值方法作了必要的比较．最后提出值得进一步研究的一些课题．

求解非定常对流扩散方程的高精度差分格式

提出了一种新的数值求解一维非定常对流扩散方程高精度差分格式.利用Fourier分析方法证明了该格式是无条件稳定的,而且适合于大梯度(高雷诺数)问题的数值求解.数值实验结果证明了本文的精确性、稳定性和对高网络雷诺数问题的强适应性.

三维气固圆柱绕流颗粒扩散的直接数值模拟

为了考察颗粒在圆柱尾流中扩散的运动机理,对气固两相圆柱尾迹流动进行了高精度紧致差分方法的三维直接数值模拟.在对气相流场进行高精度模拟的基础上,采用了Lagrangrian方法来追踪颗粒场的运动.比较了不同颗粒Stokes数为0.01、1、10和不同Reynolds数为180和200下颗粒的扩散函数,考察了三维圆柱尾迹流流场中气相场的运动对颗粒相扩散的影响.模拟结果表明:当气相流场由二维特性向三维特性转变时,由于受到气流场展向涡的影响,颗粒的展向扩散函数有相当程度的增大.颗粒在横向上扩散函数随着粒径的增大而增大,而在展向上颗粒扩散函数则随着粒径的增大而减小.

雷电先导下行过程特高压直流线路电晕向流注放电转化的仿真研究

为了定量研究雷暴过程中电晕空间电荷对特高压直流输电线路后续流注放电起始的影响,基于带通量限制器的2阶有限体积方法和Kaptzov假设,建立雷暴过程中特高压直流输电线路电晕空间电荷分布的数值仿真模型。通过开展动态电场下水平导线电晕放电电流的实测与仿真对比研究,验证了模型的准确性。计算分析典型±1100kV特高压直流输电线路雷暴过程中导地线电晕电流时域波形和电晕空间电荷分布特征。得出下行先导趋近过程会显著增加导、地线表面附近约0.5m范围内正离子密度,并使导、地线电晕电流最大值较雷云电场作用时增加6~7个数量级。随着地线表面附近正离子密度增加,电场最大值从地线表面向导线附近空间移动,而导致后续流注产生。所作研究工作可为后续研究电晕空间电荷对特高压直流线路雷电绕击特性的影响奠定基础。

多面开关柜表面暂态对地电压的分布特性

根据暂态对地电压(transient earth voltage,TEV)的基本原理,运用时域有限差分法(finite-difference time-domain,FDTD),建立了三面并排连接的开关柜1:1精确模型,对三面并排开关柜外表面上分布的33个测量点的TEV分布特性进行仿真研究。结果表明,测量点靠近开关柜表面的缝隙处时可以获得较大的TEV幅值,TEV幅值随测量点与局部放电源之间距离的增大而减小。利用自制的TEV局部放电测试系统,对仿真结果进行验证,得到了较好的一致性,证明使用FDTD法仿真开关柜TEV信号分布特性的有效性。

基于精确子域模型的永磁直线同步电机空载磁场解析计算

准确计算永磁直线同步电机(PMLSM)磁场分布是得到电机参数及性能的前提,而建立精确的数学模型则是求解电机磁场问题的关键。鉴于PMLSM传统解析理论在齿槽效应对气隙磁场影响的求解上存在局限性,提出PMLSM精确子域模型以得到精确磁场分布。计及永磁体相对磁导率、电机槽深、所有槽与槽之间对磁场分布的相互影响,采用标量磁位分别建立PMLSM永磁体、气隙、槽子域拉普拉斯方程。根据各子域交界面边界条件,基于傅里叶级数法,列出所有边界条件方程组,并建立端部等效模型,求解得到各子域标量磁位和空载磁通密度分布,进而探讨各个齿槽、端部区域对气隙磁场空间分布的影响。有限元结果证明了所用解析方法的准确性。