粘弹性力学的对应原理及其数值反演方法

积分变换是处理粘弹性混合边值问题的重要数学工具,积分变换的应用使粘弹性混合边值问题在象空间与相应弹性混合边值问题对应起来,从而使粘弹性混合边值问题的求解可以继承和借鉴弹性问题的求解方法,再利用积分反演方法就可求得时间域粘弹性边值问题的解．本文结合国内外的研究成果,就粘弹性力学中存在的各种对应原理及数值反演方法进行了归类和总结．结合在求解粘弹性边值问题中的应用,对各类方法的特点进行了评述,并指出存在的问题及发展新的数值方法的研究重点．      

多层正交异性矩形薄板弯曲振动稳定解析解

构造了带有补充项的双重正弦傅里叶级数通解来求解各种边界条件的多层正交各向异性矩形薄板的弯曲、振动和稳定问题.将坐标轴取在中性面上,求出用挠度表示的应力表达式,然后由横截面上每单位宽度的应力合成板的内力;再将层合板的内力代入板的平衡方程中得到板的控制方程,将多层板的物理参数折算为等价的单层板物理参数;最后联立控制方程与边界条件,求得未知量的系数并代入本文的通解中.本文的通解不需要叠加即可求解各种边界条件的板的弯曲、振动和稳定问题;现有的对于单层板的研究都可以用本文的方法拓展到多层板领域;对于复杂边界条件的板,也可以使用该通解分析.     

Non-element method of solving 2D boundary problems defined on polygonal domains modeled by Navier equation

The paper presents a non-element method of solving boundary problems defined on polygonal domains modeled by corner points. To solve these problems a parametric integral equation system (PIES) is used. The system is characterized by a separation of the approximation of boundary geometry from the approximation of boundary functions. This feature makes it possible to effectively investigate the convergence of the obtained solutions with no need of performing the approximation of boundary geometry. The testing examples included confirm high accuracy of the solutions.     

求解二维弹性问题的径向基差分法

常规的配点型无网格法在求解弹性力学问题中,存在求解精度差和纽曼边界条件处理等局限．为解决这一问题,通过利用流体力学中基于径向基构造的差分格式(RBF-FD),来求解弹性力学平面问题．同时,为了进一步提高求解精度,对纽曼边界条件采用Hermite插值进行处理．数值算例表明,该方法具备良好的收敛性,并有着较高的精度,可有效解决传统配点型无网格法精度差的问题．同时,也表明该方法可以应用于弹性力学问题的求解．     

Factorization method in the geometric inverse problem of static elasticity

The factorization method, which has previously been used to solve inverse scattering problems, is generalized to geometric inverse problems of static elasticity. We prove that finitely many defects (cavities, cracks, and inclusions) in an isotropic linearly elastic body can be determined uniquely if the operator that takes the forces applied to the body outer boundary to the outer boundary displacements due to these forces is known.     

Development of a three-dimensional solver based on the local domain-free discretization and immersed boundary method and its application for incompressible flow problems

Recently, the author and two other coauthors have proposed a two-dimensional hybrid local domain-free discretization and immersed boundary method (LDFD-IBM), which can be used to solve the flow problem with complex geometries. In this paper, the LDFD-IBM is extended to solve a three-dimensional unsteady incompressible flow with the complex computational domain. The technical issues related to the implementation of the LDFD-IBM in three-dimensional problems are discussed in detail, particularly for the discretization of Navier-Stokes equations, mesh strategies for a three-dimensional flow, and the fast algorithm on the identification of the status of mesh nodes (ie, to identify if the mesh node is located in the solid domain, in the fluid domain, or near the immersed boundary). Numerical tests show that the LDFD-IBM can accurately solve three-dimensional incompressible problems with ease.     

The boundary value problems for thin coats

We formulate and solve three boundary value problems for the coats of finite and infinite domains in the plane.     

A two-dimensional finite element method for simulating the constitutive response and microstructure of polycrystals during high temperature plastic deformation

We describe a finite element method designed to model the mechanisms that cause superplastic deformation. Our computations account for grain boundary sliding, grain boundary diffusion, grain boundary migration, and surface diffusion, as well as thermally activated dislocation creep within the grains themselves. Front tracking and adaptive mesh generation are used to follow changes in the grain structure. The method is used to solve representative boundary value problems to illustrate its capabilities.     

A numerical solution of infiltration

A fully implicit finite difference scheme is used to evaluate one-dimensional infiltration. The method makes it possible to solve a general infiltration problem; nonhomogeneous soil profile and saturated-unsaturated seepage can be treated. To solve special problems of hydrology and soil physics, several types of boundary conditions are formulated and numerically expressed. The type of boundary conditions may vary in time depending on the values of the unknown function. High accuracy of solution is emphasized. Several applications of this method are presented.     

Strong-form framework for solving boundary value problems with geometric nonlinearity

In this paper, we present a strong-form framework for solving the boundary value problems with geometric nonlinearity, in which an incremental theory is developed for the problem based on the Newton-Raphson scheme. Conventionally, the finite element methods (FEMs) or weak-form based meshfree methods have often been adopted to solve geometric nonlinear problems. However, issues, such as the mesh dependency, the numerical integration, and the boundary imposition, make these approaches computationally inefficient. Recently, strong-form collocation methods have been called on to solve the boundary value problems. The feasibility of the collocation method with the nodal discretization such as the radial basis collocation method (RBCM) motivates the present study. Due to the limited application to the nonlinear analysis in a strong form, we formulate the equation of equilibrium, along with the boundary conditions, in an incremental-iterative sense using the RBCM. The efficacy of the proposed framework is numerically demonstrated with the solution of two benchmark problems involving the geometric nonlinearity. Compared with the conventional weak-form formulation, the proposed framework is advantageous as no quadrature rule is needed in constructing the governing equation, and no mesh limitation exists with the deformed geometry in the incremental-iterative process.     

A review of boundary and finite element methods in fracture mechanics

Reviewed in this work are the methods of finite and boundary element as applied to solve fracture mechanics problems. The former requires the discretization of the interior of the domain while the latter involves computing an integral equation over the boundary of the domain. Applications of these methods are made to two-dimensional elastic crack problems. Efficiency and accuracy of different approaches are discussed and compared by examples. The boundary element procedure employing special Green's functions for the plane crack problem is shown to be superior. The correlation between the hybrid element formulations and boundary element regions embedded into a finite element model is also given.     

夹层板的复变边界元解法

本文用全纯函数表示微分方程△f(x,y)-λ(~2)f(x,y)=0的一般解,粮据全纯函数的Bekya积分表示法,建立了复数域内的边界积分方程并针对各种边界条件下Reissner型夹层板、Hoff型夹层板进行了数值求解。     

Boundary integral equations for 2D elasticity and its application in discrete dislocation simulation in finite body: 2. Numerical implementation

In the previous paper by Yu and Diab (2013), several sets of boundary integral equations are derived for general anisotropic materials and corresponding equations for materials with different classes of symmetry are deduced. The work presented herein implements two sets of boundary element schemes to numerically solve the stress field. The integration on the element that has the singular point of the kernel is bounded and can be evaluated analytically. Four benchmark elastic problems are solved numerically to show the advantage of the two schemes over the conventional boundary element formulation in eliminating the boundary layer effect. The one with the weaker singularity has better convergence and gives more accurate results. The presented formulation also provides a direct approach to solve for stress field in a finite solid body in the presence of dislocations. Combined with discrete dislocations dynamics, boundary value problems with dislocations in finite bodies can be solved. Two examples, bending of a single crystal beam and pure shearing of a polycrystalline solid, are simulated by discrete dislocation dynamics using the scheme that has the weaker singularity. The comparisons with the published results using the well-established superposition technique validate the proposed formulation and show its quick convergence.     

The periodic cracks of an infinite anisotropic media for plane skew-symmetric loadings

This paper attempts to solve the periodic crack problems of infinitive anisotropic media for plane skew-symmetric loadings by means of the method of complex function. The problems are now reduced to the determination of two complex functions that must satisfy certain boundary conditions. In this paper, the stresses, the displacements and the boundary conditions are assumed to be periodic, and further, the stresses are assumed to be bounded at infinity. The solutions are expressed in closed forms.     

An FMM for waveguide problems of 2-D Helmholtz’ equation and its application to eigenvalue problems

This paper discusses an FMM for solving waveguide problems and associated eigenvalue problems for Helmholtz’ equation in a two dimensional infinite strip with homogeneous Neumann boundary condition on the sides. Layer potentials with Green’s function for this problem are evaluated efficiently with the help of the method of images and FMM. We apply FMM to solve some boundary value problems in waveguides and associated resonance frequency problems using the Sakurai–Sugiura projection method after discussing the required analytic continuation of the solutions to complex frequencies. Some numerical examples show the accuracy and the efficiency of the proposed method.     

边界元法在求解复合材料力学问题中的应用

本文讨论了边界元法在求解复合材料的微观力学、宏观力学和结构力学问题中的应用,并指出边界元法用于分析复合材料及其结构的力学问题的优点和局限性。      

非均质问题中的无网格边界单元法

介绍了一种不需要内部网格计算非均匀介质问题的边界元算法.该算法是建立在一种能将任何区域积分转换成边界积分的径向积分转换法基础上,首先用对应各向同性问题的基本解来建立以正规化位移表示的非均质问题的积分方程,然后用径向积分转换法将出现在积分方程中的区域积分转换成边界积分,从而形成不需要使用内部网格来计算区域积分的纯边界元算法.与其它无网格法相比,此方法需要很少的内部点,有些问题甚至不需要内部点都能得到满意的结果,因此,可以计算大型的三维非均匀介质工程问题.由于此方法继承了边界元和无网格算法的优点,因而具有广阔的发展前景.     

A meshless numerical method for solving slow mixed convections in containers with discontinuous boundary data

This paper describes the formulations of the method of fundamental solutions (MFS), which is a famous meshless numerical method representing a sought solution by a series of fundamental solutions to solve slow mixed convections in containers with discontinuous boundary data. In the derivations, the fundamental solutions were obtained by using the Hörmander operator decomposition technique. All the velocities, temperatures, pressures, stresses and thermal fluxes corresponding to the fundamental solutions were addressed explicitly in tensor forms. Although the MFS is highly accurate for smooth boundary data, its convergence becomes poor when it is applied to problems with discontinuous boundary data. To compensate for this drawback, we enriched the MFS by adding the local discontinuous solutions to the series of fundamental solutions. This enriched MFS was applied to solve the benchmark problems of a lid‐driven cavity and natural convection in rectangular containers. In addition, the numerical solutions were compared with the analytical solutions. Then, the meshless numerical method was further utilized to solve mixed convections in a triangular cavity and a cavity with a cosine‐shaped bottom. These numerical results demonstrated the applicability of the enriched MFS to two‐dimensional mixed convections in containers with discontinuous boundary data. Copyright © 2010 John Wiley & Sons, Ltd.     

Somigliana's method applied to plane problems of elastic half-spaces

In this paper Somigliana's method has been utilized to solve the two-dimensional boundary value problems of elastic half-spaces. We have considered traction and displacement problem, shear problem, contact and crack problems and the solutions, thus derived, have been compared with those derived from dislocation considerations.     

压电介质平面问题的基本解

应用复变函数的方法,对于压电介质平面问题,分析导出了无限介质或半无限介质受任意 集中载荷作用时的复势函数基本解;这些结果可作为边界元法的基本解,以求解具有复杂边界压电体的平面问题。