Extended Complex tanh-Function Method and Exact Solutions to （2＋1）-Dimensional Hirota Equation

在这篇论文，一个新扩大复杂 tanh 功能方法为构造旅行波浪，非旅行的波浪，和系数函数被介绍“非线性的方程的象 soliton 一样解决方案。这个方法比复杂 tanh 功能方法更强大[混乱， Solitons 和分数维图形 20 (2004 ) 1037 ] 。丰富的新答案(2+1 ) 维的 Hirota 方程被使用这个方法和符号的计算系统枫树获得。

FRACTURE CALCULATION OF BENDING PLATES BY BOUNDARY COLLOCATION METHOD

Fracture of Kirchhoff plates is analyzed by the theory of complex variables and boundary collocation method. The deflections, moments and shearing forces of the plates are assumed to be the functions of complex variables. The functions can satisfy a series of basic equations and governing conditions, such as the equilibrium equations in the domain, the boundary conditions on the crack surfaces and stress singularity at the crack tips. Thus, it is only necessary to consider the boundary conditions on the external boundaries of the plate, which can be approximately satisfied by the collocation method and least square technique. Different boundary conditions and loading cases of the cracked plates are analyzed and calculated. Compared to other methods, the numerical examples show that the present method has many advantages such as good accuracy and less computer time. This is an effective semi_analytical and semi_numerical method.

围缘扁钢加强的开圆孔无限平板应力解析算法

为了研究围缘扁钢加强的开圆孔无限平板应力解析计算方法,近似认为围缘扁钢区域的平均应力处于广义平面应力状态。运用复变函数论的方法,将应力函数表示为两个待定的解析函数;列出围缘扁钢的边界条件、围缘扁钢与平板之间的应力连接条件和位移连接条件,求出应力函数,根据求出的应力函数便可求出应力。为了描述围缘扁钢减缓孔口应力集中的效果,提出围缘扁钢加强系数的定义。算例计算表明:本文提出的解析算法计算结果与有限元法计算结果吻合良好,围缘扁钢加强的开圆孔无限平板应力普遍小于开圆孔无限平板应力,围缘扁钢加强效果明显;相同体积的加强构件,围缘扁钢的加强效果优于环形加厚板。

磁电弹性材料中含有带四条裂纹的正方形孔的反平面断裂问题

针对反平面剪切力作用下,磁电弹性材料中含有带四条裂纹的正方形孔口的断裂问题进行了深入地探索。基于线弹性断裂理论和Riemann-Schwarz解析延拓定理,利用复变函数方法和留数定理,通过构造合适的数值保角映射函数,将解析函数边值问题转化为柯西积分方程组进行解答,获得了磁电非渗透边界条件下裂纹尖端断裂力学参数的显式表达式。通过与已有结果的对比,验证了方法的有效性。利用数值算例描述了孔洞尺寸、四条裂纹长度和机–电–磁载荷等因素对裂纹扩展参数的影响规律。结果表明:水平右裂纹对孔口裂纹扩展影响更显著;垂直裂纹影响水平裂纹的扩展趋势;在磁电非渗透边界条件下,随着机械载荷的增大,裂纹尖端的应力强度因子变大,而电载荷和磁载荷对裂纹的扩展与机械载荷的大小和方向密切相关。该分析方法为求解复杂多连通域的智能复合材料问题提供了一条有效途径,研究成果为含裂纹的复合材料或结构的优化设计提供了科学依据。

非圆形隧洞衬砌支护下应力与位移的新解法

以往在利用平面弹性复变函数方法进行衬砌支护作用下非圆形隧洞的力学分析时,一般将洞室外域保角映射到象平面的单位圆外域,并认为可以利用同一个映射函数将衬砌截面映射到象平面的圆环域内。当衬砌厚度很薄时,这种方法是可行的,而当衬砌相对较厚时,利用同一映射函数所得到的衬砌截面形状发生严重失真。为解决此问题,本文引入两套映射函数,分别将洞室外域映射到象平面的单位圆的圆外域,衬砌截面映射到圆环域。这种方法可以适应于不同的衬砌厚度,利用这种方式获得的衬砌截面,在衬砌较厚时仍能保证衬砌厚度相对均匀。在求解解析函数过程中不再采用传统的幂级数解法,而是使用边界配点法,将两套映射函数结合,获得了该问题的应力解和位移解。本文以马蹄形隧洞为例,求解结果与ANSYS软件所得数值解吻合很好,分析了衬砌厚度对隧洞应力和位移的影响。根据本文解答可以方便、快捷地进行相近条件下隧道的初步设计。

基于复变函数理论的无中导连拱隧道应力与位移解析解

无中导连拱隧道由于后行洞衬砌搭接于先行洞衬砌拱腰之上,围岩与衬砌受力情况复杂,开展相应的围岩与衬砌应力和位移分析具有重要的工程意义。基于复变函数理论和Schwarz交替法,提出了深埋条件下考虑衬砌的无中导连拱隧道围岩与衬砌应力和变形的计算方法。针对无中导连拱隧道两洞具有共用边界的特点,在利用柯西积分法进行求解时,将单洞衬砌与围岩接触的边界积分条件转化为仅存在单洞时的完整边界积分与共用边界积分之差,对于Schwarz交替法中求解“附加面力”的过程亦进行类似处理。给出共用边界应力与位移的计算方法。通过算例,将上述解析结果和数值仿真结果进行对比,证明了方法的可行性和准确性。

水声MIMO-OFDM通信中的空频迭代信道估计与均衡

在MIMO-OFDM水声通信系统中,由于信道间的相互干扰和水声信道严重时延扩展产生的频率选择性衰落,系统的通信误码率较高。针对这一问题,研究了空频编码的MIMO-OFDM通信,提出空频迭代信道估计与均衡(Spatial Frequency Iterative Channel Estimation and Equalization,SFICEE)方法。该方法通过载波间的空频正交性进行各收发阵元对的信道估计,并通过空频均衡获得符号初始估计,迭代更新信道估计,而后通过符号后验软信息反馈进行迭代空频软均衡。仿真结果表明,当误码率为10^(-3)时,文中所提出的SFICEE方法经过二次迭代与STBC方法相比具有4.8 d B的性能增益,相对于SFBC方法有2.8 d B的性能提升。当输入信噪比相同时,文中所提出方法的星座图更加收敛,可以更好地降低水下通信系统的误码率。

充分发挥高等数学教育在应用型人才培养中的作用——以“复变函数与积分变换”课程为例

为了应对本科院校培养应用型人才方向和社会对应用型人才要求的变化,以“复变函数与积分变换”课程为例,首先,在教学方法上,采用启发式、引导式、类比式教学法和形象直观教学法,注重知识的实际应用,培养学生应用能力;其次,在教学模式上,注重以兴趣为导向的教学模式,以案例驱动教学为导向的教学模式,以分层教学为导向的教学模式,强化数学知识的应用,为培养应用型人才服务;再次,在教学手段上,利用现代化教育技术、数学软件,鼓励学生参加数学实践活动,培养应用式思维;最后,优化课程教学内容,调整、完善本课程知识体系,强化应用性环节,通过改革考核方法,加强应用能力考核机制,训练和培养学生的应用水平,充分发挥高等数学教育在应用型人才培养的作用。

Two kinds of contact problems in three-dimensional icosahedral quasicrystals

Two kinds of contact problems, i.e., the frictional contact problem and the adhesive contact problem, in three-dimensional （3D） icosahedral quasicrystals are dis- cussed by a complex variable function method. For the frictional contact problem, the contact stress exhibits power singularities at the edge of the contact zone. For the adhe- sive contact problem, the contact stress exhibits oscillatory singularities at the edge of the contact zone. The numerical examples show that for the two kinds of contact problems, the contact stress exhibits singularities, and reaches the maximum value at the edge of the contact zone. The phonon-phason coupling constant has a significant effect on the contact stress intensity, while has little impact on the contact stress distribution regu- lation. The results are consistent with those of the classical elastic materials when the phonon-phason coupling constant is 0. For the adhesive contact problem, the indentation force has positive correlation with the contact displacement, but the phonon-phason cou- pling constant impact is barely perceptible. The validity of the conclusions is verified.

Anti-plane analysis of semi-infinite crack in piezoelectric strip

Using the complex variable function method and the technique of the conformal mapping, the fracture problem of a semi-infinite crack in a piezoelectric strip is studied under the anti-plane shear stress and the in-plane electric load. The analytic solutions of the field intensity factors and the mechanical strain energy release rate are presented under the assumption that the surface of the crack is electrically impermeable. When the height of the strip tends to infinity, the analytic solutions of an infinitely large piezoelectric solid with a semi-infinite crack are obtained. Moreover, the present results can be reduced to the well-known solutions for a purely elastic material in the absence of the electric loading. In addition, numerical examples are given to show the influences of the loaded crack length, the height of the strip, and the applied mechanical/electric loads on the mechanical strain energy release rate.

Exact solutions of two semi-infinite collinear cracks in piezoelectric strip

Using the complex variable function method and the conformal mapping technique, the fracture problem of two semi-infinite collinear cracks in a piezoelectric strip is studied under the anti-plane shear stress and the in-plane electric load on the partial crack surface. Analytic solutions of the field intensity factors and the mechanical strain energy release rate are derived under the assumption that the surfaces of the crack are electrically impermeable. The results can be reduced to the well-known solutions for a purely elastic material in the absence of an electric load. Moreover, when the distance between the two crack tips tends to infinity, analytic solutions of a semi-infinite crack in a piezoelectric strip can be obtained. Numerical examples are given to show the influence of the loaded crack length, the height of the strip, the distance between the two crack tips, and the applied mechanical/electric loads on the mechanical strain energy release rate. It is shown that the material is easier to fail when the distance between two crack tips becomes shorter, and the mechanical/electric loads have greater influence on the propagation of the left crack than those of the right one.

Implicit Shape Reconstruction of Unorganized Points Using PDE-Based Deformable 3D Manifolds

In this work we consider the problem of shape reconstruction from an unorganized data set which has many important applications in medical imaging, scientific computing, reverse engineering and geometric modelling. The reconstructed surface is obtained by continuously deforming an initial surface following the Partial Differential Equation (PDE)-based diffusion model derived by a minimal volume-like variational formulation. The evolution is driven both by the distance from the data set and by the curvature analytically computed by it. The distance function is computed by implicit local interpolants defined in terms of radial basis functions. Space discretization of the PDE model is obtained by finite co-volume schemes and semi-implicit approach is used in time/scale. The use of a level set method for the numerical computation of the surface reconstruction allows us to handle complex geometry and even changing topology,without the need of user-interaction. Numerical examples demonstrate the ability of the proposed method to produce high quality reconstructions. Moreover, we show the effectiveness of the new approach to solve hole filling problems and Boolean operations between different data sets.

Analytic solutions to problem of elliptic hole with two straight cracks in one-dimensional hexagonal quasicrystals

By means of the complex variable function method and the technique of conformal mapping, the anti-plane shear problem of an elliptic hole with two straight cracks in one-dimensional hexagonal quasicrystals is investigated. The solution of the stress intensity factor （SIF） for mode III problem has been found. Under the condition of limitation, both the known results and the SIF solution at the crack tip of a circular hole with two straight cracks and cross crack in one-dimensional hexagonal quasicrystals can be obtained.

On a Problem of an Infinite Plate with a Curvilinear Hole inside the Unit Circle

In this work, we used the complex variable methods to derive the Goursat functions for the first and second fundamental problem of an infinite plate with a curvilinear hole C. The hole is mapped in the domain inside a unit circle by means of the rational mapping function. Many special cases are discussed and established of these functions. Also, many applications and examples are considered. The results indicate that the infinite plate with a curvilinear hole inside the unit circle is very pronounced.

CVBEM and FVM Computational Model Comparison for Solving Ideal Fluid Flow in a 90-Degree Bend

While finite volume methodologies (FVM) have predominated in fluid flow computations, many flow problems, including groundwater models, would benefit from the use of boundary methods, such as the Complex Variable Boundary Element Method (CVBEM). However, to date, there has been no reporting of a comparison of computational results between the FVM and the CVBEM in the assessment of flow field characteristics. In this work, the CVBEM is used to develop a flow field vector outcome of ideal fluid flow in a 90-degree bend which is then compared to the computational results from a finite volume model of the same situation. The focus of the modelling comparison in the current work is flow field trajectory vectors of the fluid flow, with respect to vector magnitude and direction. Such a comparison is necessary to validate the development of flow field vectors from the CVBEM and is of interest to many engineering flow problems, specifically groundwater modelling. Comparison of the CVBEM and FVM flow field trajectory vectors for the target problem of ideal flow in a 90-degree bend shows good agreement between the considered methodologies.

压电材料平面Ⅰ型裂纹模拟仿真分析

文中利用复变函数知识,基于压电材料第二类本构方程推导了压电材料平面Ⅰ型裂纹尖端的应力场和电位移场及机械应变能释放率的解析解,采用虚拟闭合裂纹法和Abaqus软件对压电材料平面型裂纹进行模拟仿真分析,根据仿真结果得到的机械应变释放数值解,讨论了裂纹参数与施加力电载荷对裂纹扩展的影响。将仿真模拟的数值解与解析解相对比,数值解结果与解析解符合程度较好,验证了虚拟裂纹闭合法与有限元软件结合方法的有效性。

The complex variable reproducing kernel particle method for elasto-plasticity problems

On the basis of reproducing kernel particle method(RKPM),using complex variable theory,the complex variable reproducing kernel particle method(CVRKPM) is discussed in this paper.The advantage of the CVRKPM is that the correction function of a two-dimensional problem is formed with one-dimensional basis function when the shape function is formed.Then the CVRKPM is applied to solve two-dimensional elasto-plasticity problems.The Galerkin weak form is employed to obtain the discretized system equation,the penalty method is used to apply the essential boundary conditions.And then,the CVRKPM for two-dimensional elasto-plasticity problems is formed,the corresponding formulae are obtained,and the Newton-Raphson method is used in the numerical implementation.Three numerical examples are given to show that this method in this paper is effective for elasto-plasticity analysis.

浅埋隧道围岩应力场的计算复变函数求解法

对于受地表边界和地面荷载影响的浅埋隧道的围岩应力场,由于在数学处理上存在一定的困难,很难用解析解来进行分析,而通常采用边界元或有限元的数值方法来解答。为了求解浅埋隧道的应力场,采用边界配点来确定边界条件,同时用保角映射将一个含圆孔的半无限空间区域映射为圆环域,然后把这个区域内的解析函数展开成Laurent级数的形式,利用Muskhelishvili的复变函数理论和最小二乘法来确定解析函数的各项系数,从而求得浅埋隧道围岩压力的半数值、半解析解,最后通过算例给出了围岩应力的分布情况。计算结果表明,该方法计算精度高、计算量小,具有应用价值。

基于土层位移差的地下结构抗震反应位移法分析

采用平面弹性理论复变函数方法,推导具有小孔无限介质在无限远处受剪应力的水平位移,得出有小孔与无小孔的位移关系,应用在基于土层位移差的反应位移法土层位移求解。为验证该方法的适用性,设某地质条件下典型圆形盾构隧道与矩形地下车站为基准模型,用不同方法求解土层位移,用反应位移法计算土体刚度变化的结构内力。结果表明,基于土层位移差的反应位移法适用于埋深较大的地下结构,不适用硬土情况;对地基弹簧刚度求解精度要求较高。

深埋双隧洞开挖的解析延拓法求解

将隧洞所处的地下空间转换为平面应变情况下的弹性半空间,考虑两隧洞开挖次序的影响,利用解析延拓法及Schwarz交替法求解出弹性半空间内2个任意形状任意尺寸的双隧洞在任意相对位置下其半空间内任意一点的应力解和位移解的隐式表达式。将双圆形隧洞作为特例,考虑到隧洞埋深与孔径相比比较大,从而不考虑重力梯度的影响,将重力化为作用在边界上的均布外荷载,考虑2个隧洞衬砌法向反力的影响,求解出弹性半空间内圆形双隧洞任意一点处的应力值和位移值解析解的具体显式表达式。最后利用二维有限元分析对理论推导的两圆形孔洞解析解进行验证,表明解的可靠性。