H 1 space-time discontinuous finite element method for convection-diffusion equations

An H~1 space-time discontinuous Galerkin (STDG) scheme for convectiondiffusion equations in one spatial dimension is constructed and analyzed. This method is formulated by combining the H~1 Galerkin method and the space-time discontinuous finite element method that is discontinuous in time and continuous in space. The existence and the uniqueness of the approximate solution are proved. The convergence of the scheme is analyzed by using the techniques in the finite difference and finite element methods. An optimal a-priori error estimate in the L~∞ (H~1 ) norm is derived. The numerical exper- iments are presented to verify the theoretical results.     

On the asymptotic solutions for a class of second order differential equations with slowly varying coefficients

In this paper we study the asymptotic expansions of the solutions for a class of secondorder ordinary differential equations with slowly varying coefficients.The defect of theknown works on these problems is noted,and the results in[1—4]are improved andextended by means of the modified method of multiple scales.     

Asymptotic solutions of mathieu equation with damping

This paper first reduces the motion equation of a collapsible tube to the Mathieu equation with damping. Then the stability charts correcting the accuracy to each order are obtained with the method of asymptotic expansions. The accuracy of the results obtained with the average variational method is shown. And some phenomena observed in the experiment are also explained.     

A reconstructed edge-based smoothed DSG element based on global coordinates for analysis of Reissner–Mindlin plates

A reconstructed edge-based smoothed triangular element, which is incorporated with the discrete shear gap(DSG) method, is formulated based on the global coordinate for analysis of Reissner–Mindlin plates. A symbolic integration combined with the smoothing technique is implemented to calculate the smoothed finite element matrices,which is integrated along the boundaries of each smoothing cell. Numerical results show that the proposed element is free from shear locking, and its results are in good agreement with the exact solutions, even for very thin plates with extremely distorted elements. The proposed element gives more accurate results than the original DSG element without smoothing, and it can be taken as an alternative element for analysis of Reissner–Mindlin plates. The prominent feature of the present element is that the integration scheme is unified in the smoothed form for all of the finite element matrices.     

Superconvergence analysis of bi-k-degree rectangular elements for two-dimensional time-dependent Schr?dinger equation

Superconvergence has been studied for long, and many different numerical methods have been analyzed. This paper is concerned with the problem of superconvergence for a two-dimensional time-dependent linear Schr?dinger equation with the finite element method. The error estimate and superconvergence property with order O(h~(k+1))in the H~1 norm are given by using the elliptic projection operator in the semi-discrete scheme. The global superconvergence is derived by the interpolation post-processing technique. The superconvergence result with order O(h~(k+1)+ τ~2) in the H~1 norm can be obtained in the Crank-Nicolson fully discrete scheme.     

ASYMPTOTIC EXPANSIONS OF ZEROS FOR KRAWTCHOUK POLYNOMIALS WITH ERROR BOUNDS

Krawtchouk polynomials are frequently applied in modern physics. Based on the results which were educed by Li and Wong, the asymptotic expansions of Krawtchouk polynomials are improved by using Airy function, and uniform asymptotic expansions are got. Furthermore, the asymptotic expansions of the zeros for Krawtchouk polynomials are again deduced by using the property of the zeros of Airy function, and their corresponding error bounds are discussed. The obtained results give the asymptotic property of Krawtchouk polynomials with their zeros, which are better than the results educed by Li and Wong.     

ON SINGULAR PERTURBATION METHOD OF PERTURBED BIFURCATION PROBLEMS

In this paper, the general mathematical principle is over-allexplained and a new general technique is presented in order to calculate uniformly asymptotic expansions of solutions of the perturbed bifurcation problem (1.6) in the vicinity of y=O. λ=O,δ=0, by means of singular perturbation method. Simultaneously, Newton's polygon is generalized.Finally, the calculating results of two examples are given.     

AN INVESTIGATION ON A NEW INTEGRAL TRANSFORM-BOUNDARY ELEMENT METHOD FOR ELASTODYNAMICS

The numerical methods of Fourier eigen transform FET and its inversion are discussedand applied to the boundary element method for elastodynamics. The program for solving elastody-namic problems with the boundary element method is developed and some examples are given. Fromthe numerical results of the examples, we know the method can increase the computing speed 5～10times and the accuracy is guaranteed.     

Nonconforming local projection stabilization for generalized Oseen equations

A new method of nonconforming local projection stabilization for the gen- eralized Oseen equations is proposed by a nonconforming inf-sup stable element pair for approximating the velocity and the pressure. The method has several attractive features. It adds a local projection term only on the sub-scale （H ≥ h）. The stabilized term is simple compared with the residual-free bubble element method. The method can handle the influence of strong convection. The numerical results agree with the theoretical expectations very well.     

Theory and application of numerical simulation method of capillary force enhanced oil production

A kind of second-order implicit upwind fractional step finite difference methods are presented for the numerical simulation of coupled systems for enhanced(chemical)oil production with capillary force in the porous media.Some techniques,e.g.,the calculus of variations,the energy analysis method,the commutativity of the products of difference operators,the decomposition of high-order difference operators,and the theory of a priori estimate,are introduced.An optimal order error estimate in the l~2 norm is derived.The method is successfully used in the numerical simulation of the enhanced oil production in actual oilfields.The simulation results are satisfactory and interesting.     

P 1-nonconforming triangular finite element method for elliptic and parabolic interface problems

The lowest order P₁-nonconforming triangular finite element method (FEM) for elliptic and parabolic interface problems is investigated. Under some reasonable regularity assumptions on the exact solutions, the optimal order error estimates are obtained in the broken energy norm. Finally, some numerical results are provided to verify the theoretical analysis.     

ANALYSIS OF 3-D NOTCHED/CRACKED STRUCTURES BY USING EXTENDED BOUNDARY ELEMENT METHOD 1)

在线弹性理论中,三维 V 形切口/裂纹结构尖端区域存在多重应力奇异性,常规数值方法不易求解. 本文提出和建立了三维扩展边界元法 (XBEM),用于分析三维线弹性 V 形切口/裂纹结构完整的位移和应力场. 先将三维线弹性 V 形切口/裂纹结构分为尖端小扇形柱和挖去小扇形柱后的外围结构. 尖端小扇形柱内的位移函数采用自尖端径向距离 $r$ 的渐近级数展开式表达,其中尖端区域的应力奇异指数、位移和应力特征角函数通过插值矩阵法获得. 而级数展开式各项的幅值系数作为基本未知量. 挖去扇形域后的外围结构采用常规边界元法分析. 两者方程联立求解可获得三维 V 形切口/裂纹结构完整的位移和应力场,包括切口/裂纹尖端区域精细的应力场. 扩展边界元法具有半解析法特征,适用于一般三维 V 形切口/裂纹结构完整位移场和应力场的分析,其解可精细描述从尖端区域到整体结构区域的完整应力场. 作者研制了三维扩展边界元法程序,文中给出了两个算例,通过计算结果分析,表明了扩展边界元法求解三维 V 形切口/裂纹结构完整应力场的准确性和有效性.     

Investigation of Coulomb force effects on ethylene glycol based nanofluid laminar flow in a porous enclosure

Forced convection heat transfer of ethylene glycol based nanofluid with Fe_3O_4 inside a porous medium is studied using the electric field. The control volume based finite element method(CVFEM) is selected for numerical simulation. The impact of the radiation parameter(R_d), the supplied voltage(?φ), the volume fraction of nanofluid(?), the Darcy number(Da), and the Reynolds number(Re) on nanofluid treatment is demonstrated. Results prove that thermal radiation increases the temperature gradient near the positive electrode. Distortion of isotherms increases with the enhance of the Darcy number and the Coulomb force.     

A new stabilized method for quasi-Newtonian flows

For a generalized quasi-Newtonian flow, a new stabilized method focused on the low-order velocity-pressure pairs, （bi）linear/（bi）linear and （bi）linear/constant element, is presented. The pressure projection stabilized method is extended from Stokes problems to quasi-Newtonian flow problems. The theoretical framework developed here yields an estimate bound, which measures error in the approximate velocity in the W 1,r（Ω） norm and that of the pressure in the L r＇ （Ω） （1/r ＋ 1/r＇ = 1）. The power law model and the Carreau model are special ones of the quasi-Newtonian flow problem discussed in this paper. Moreover, a residual-based posterior bound is given. Numerical experiments are presented to confirm the theoretical results.     

扩展边界元法分析切口和裂纹结构应力场的准确性

在线弹性理论中,切口/裂纹结构尖端区域存在奇异应力场,数值方法不易求解。本文建立的扩展边界元法(XBEM)对围绕尖端区域位移函数采用自尖端径向距离 的渐近级数展开式表达,其级数项的幅值系数作为基本未知量,而外部区域采用常规边界元法离散方程。两者方程联立求解可获得切口和裂纹结构完整的位移和应力场。扩展边界元法具有半解析法特征,适用于一般的切口和裂纹结构应力场分析,其解可精细描述从尖端区域到整体结构区域的应力场。作者研制了扩展边界元法程序,文中给出了两个算例,通过计算结果分析,表明扩展边界元法求解切口和裂纹结构应力场的准确性和有效性。     

三维切口/裂纹结构的扩展边界元法分析

在线弹性理论中,三维 V 形切口/裂纹结构尖端区域存在多重应力奇异性,常规数值方法不易求解. 本文提出和建立了三维扩展边界元法 (XBEM),用于分析三维线弹性 V 形切口/裂纹结构完整的位移和应力场. 先将三维线弹性 V 形切口/裂纹结构分为尖端小扇形柱和挖去小扇形柱后的外围结构. 尖端小扇形柱内的位移函数采用自尖端径向距离 $r$ 的渐近级数展开式表达,其中尖端区域的应力奇异指数、位移和应力特征角函数通过插值矩阵法获得. 而级数展开式各项的幅值系数作为基本未知量. 挖去扇形域后的外围结构采用常规边界元法分析. 两者方程联立求解可获得三维 V 形切口/裂纹结构完整的位移和应力场,包括切口/裂纹尖端区域精细的应力场. 扩展边界元法具有半解析法特征,适用于一般三维 V 形切口/裂纹结构完整位移场和应力场的分析,其解可精细描述从尖端区域到整体结构区域的完整应力场. 作者研制了三维扩展边界元法程序,文中给出了两个算例,通过计算结果分析,表明了扩展边界元法求解三维 V 形切口/裂纹结构完整应力场的准确性和有效性.      

On the radiation properties of an asymmetric cylinder

The radiation properties of an asymmetric cylinder, formed by slicing a circular cylinder with a plane making an angle π/2 − Λ with the cylinder axis, are investigated as a model problem of relevance to noise emission by novel aeroengine intakes. We consider the scattering of incident duct modes and use asymptotic analysis in the limit Λ 1; the unsteady flow then comprises an inner incompressible region around the cylinder rim and an outer region comprising the rest of space, and the radiation in the outer region is determined using the method of matched asymptotic expansions. From this we are able to conclude that the correction to the radiation directivity is O(Λ), whilst the correction to the total integrated sound power reaching infinity is much weaker, and is in fact only O(Λ²).     

Scattering of elastic waves by an arbitrary small imperfection in the surface of a half-space

T , the first of two articles, is concerned with the scattering of elastic waves by arbitrary surface-breaking or near surface defects in an isotropic half-plane. We present an analytical solution, by the method of matched asymptotic expansions, when the parameter , which is the ratio of a typical length scale of the imperfection to the incident radiation's wavelength, is small. The problem is formulated for a general class of small defects, including cracks, surface bumps and inclusions, and for arbitrary incident waves. As a straightforward example of the asymptotic scheme we specialize the defect to a two-dimensional circular void or protrusion, which breaks the free surface, and assume Rayleigh wave excitation ; this inner problem is exactly solvable by conformal mapping methods. The displacement field is found uniformly to leading order in , and the Rayleigh waves which are scattered by the crack are explicitly determined. In the second article we use the method given here to tackle the important problem of an inclined edge-crack. In that work we show that the scattered field can be found to any asymptotic order in a straightforward manner, and in particular the Rayleigh wave coefficients are given to O(²).     

血液-血管耦合特性与脉搏波传播特性的关系

脉搏波既不可简单地理解为可压缩血液流体中的压力纵波，也不可简单地理解为沿固体血管传播的涨缩位移横波，而是超乎普通想象的流-固耦合和纵波-横波耦合的复杂波。从分析耦合本构关系的新途径出发，本文中提出了一个流-固耦合/纵波-横波耦合的串联模型，可为解读“位数形势”中医脉诊提供更丰富的信息。结果表明，脉搏波耦合系统的等效体积压缩模量K_s以及相应的耦合系统脉搏波传播速度c_s主要依赖于两个无量纲参数:血液-血管模量比K_b(p)/E(p)和薄壁血管径厚比D(p)/h₀，它们因人而异、因人的不同脉搏位置而异。文中定量分析了它们对c_s的影响，显示人体的K_b/E值在103数量级，从而c_s值在100～101 m/s数量级，以适应人体生理生化反应。由临床有创测量，证实脉搏体积横波与脉搏压力纵波是相耦合地以相同速度传播；还显示脉搏波是在其波阵面上具有氧合生化反应的“生物波”。此外，还讨论了“脉压放大”现象与非线性本构关系和与血管分叉处加载增强反射之间的关系，并讨论了Lewis关于重搏波形成的假设。