非齐次弹性力学问题双互易边界元方法研究北大核心CSCD

基于弹性力学边界元方法理论,将边界元法与双互易法结合,采用指数型基函数对非齐次项进行插值得到双互易边界积分方程.将边界积分方程离散为代数方程组,利用已知边界条件和方程特解求解方程组,得出域内位移和边界面力.指数型基函数的形状参数是由插值点最近距离的最小值决定,采用这种形状参数变化方案,分析径向基函数(RBF)插值精度以及插值稳定性.再次将指数型基函数应用到双互易边界元法中,分析双互易边界元方法下计算精度及稳定性,验证了指数型插值函数作为双互易边界元方法的径向基函数解决弹性力学域内体力项问题的有效性.     

基于厚板拉伸振动精确化方程求解动应力集中问题

过去,对拉伸平板考虑应力集中的工程设计多借鉴弹性力学平面问题分析求解结果,例如弹性力学Kirsch问题的解或弹性动力学平面问题的解．基于厚板拉伸振动精确化方程,对含圆孔平板中弹性波散射与动应力集中问题进行了研究．研究结果表明:1) 两种模型得到的开孔附近的应力是不同的;2) 当入射波波数变大或者说入射波频率变高时,动应力集中系数最大值趋于单位1．含孔平板拉伸振动的动应力集中系数最大值达到3.30,以及基于弹性动力学平面问题模型得到的结果为2.77．对数值计算结果做了分析讨论, 可以看到,当孔径厚度比是a/h=0.10,基于平板拉伸振动精确化方程得到的动应力集中系数可以达到最大值,超出基于弹性动力学平面问题所得到结果的19%．分析方法和数值计算结果可望能在工程平板结构的动力学分析和强度设计中得到应用．     

边界节点法计算二维瞬态热传导问题

采用边界节点法(BKM)结合双重互易法(DRM)求解二维瞬态热传导问题.采用差分格式处理时间变量,可将原瞬态热传导方程转化为一系列非齐次修正的Helmholtz方程.随后,方程的解可分为特解和齐次解两部分计算,引入双重互易法在区域内部配点求解方程的特解,采用边界节点法仅需边界配点求解方程的齐次解.给出的数值算例显示该方法计算精度高,适用性好,具有很好的稳定性和收敛性,适合求解瞬态热传导问题.     

Pasternak地基上简支板振动问题的准格林函数方法

提出一种新的数值方法——准格林函数方法．以Pasternak地基上简支多边形薄板的振动问题为例,详细阐明了准格林函数方法的思想．即利用问题的基本解和边界方程构造一个准格林函数,这个函数满足了问题的齐次边界条件,采用格林公式将Pasternak地基上薄板自由振动问题的振型控制微分方程化为两个耦合的第二类Fredholm积分方程．边界方程有多种选择,在选定一种边界方程的基础上,可以通过建立一个新的边界方程来表示问题的边界,以克服积分核的奇异性,最后由积分方程的离散化方程组有非平凡解的条件,求得固有频率．数值方法表明,该方法具有较高的精度．     

各向异性双晶和三晶体晶界附近应力场分析

采用率相关晶体滑移有限元程序对不同取向晶粒构成的双晶体和三晶体在晶界和三晶交点附近的应力集中特性进行了计算分析．双晶体的数值结果表明,不同取向晶粒的晶界附近应力场具有较大的应力梯度,存在应力集中现象;三晶体由于晶界之间的相互作用使得三晶交点可能造成应力集中地,也可能不造成应力集中,晶界附近的应力结构与双晶体晶界附近的应力结构亦不相同,这主要取决于三个晶粒的晶体取向．对双晶体和三晶体的分析说明,不同取向的晶粒具有不同的变形规律．因此研究金属材料的损伤、断裂问题至少需要采用晶体滑移理论从细观的角度分析不同晶粒之间的相互作用．     

关于薄板的无网格局部边界积分方程方法中的友解

无网格局部边界积分方程方法是最近发展起来的一种新的数值方法,这种方法综合了伽辽金有限元、边界元和无单元伽辽金法的优点,是一种具有广阔应用前景的、真正的无网格方法．把无网格局部边界积分方程方法应用于求解薄板问题,给出了薄板无网格局部边界积分方程方法所需要的友解及其全部公式．     

地下结构物在爆炸冲击波作用下的动力分析

提出了一种求解任意形地下结构物在爆炸冲击波作用下的动应力集中问题的半解析方法．爆炸冲击波以平面SH波的形式入射,并用Fourier变换方法将其转换到频域,不同形状地下结构物的导纳函数由复变函数和保角映射的方法求得．利用Fourier逆变换,进一步合成得到时域中的地下结构的动力响应,最后,对正方形、三角形及马蹄形孔洞附近的动应力集中系数作了数值计算,并给出了具体结果．     

应力偶对孔洞附近应力集中的影响

将求解无限弹性平面中孔洞附近应力集中问题的复变函数方法,推广到微极弹性介质的应力集中问题上去,在复平面上给出了二维微极弹性理论应力集中问题的一般解,它可由解析函数与“域函数”构造出来,并利用保角映射的方法来满足非圆孔洞的边界条件。在此基础上建立了求解微极弹性理论中应力集中问题的一般求解方法。最后,对圆形孔洞附近的应力集中系数作了数值计算,并给出了具体结果。     

第二类混合变分不等式的MRM-边界元分析

本文以弹性力学中的摩擦问题为背景,采用多重互易方法(MRM方法),边界元方法,将摩擦问题中的第二类混合变分不等式化解为MRM-边界混合变分不等式,给出了MRM-边界混合变分不等式解的存在唯—性,通过引入变换将原MRM-边界混合变分不等式化解为标准的凸极值问题,采用正则化方法处理后,给出了MRM-边界混合变分不等式的迭代分解方法。文末给出了数值算例。     

短纤维复合材料的本征应变边界积分方程计算模型

提出了短纤维复合材料的本征应变边界积分方程计算模型,并采用新发展的边界点法进行了求解．模型依据Eshelby等效夹杂物的概念并借助Eshelby张量,采用迭代方法来计算基体中各种性能短纤维的本征应变,其中所需的Eshelby张量不难通过解析或数值方法获得．由于未知量只出现在边界上,与已有的有限元和边界元模型相比,提出的计算模型可极大地减小异质体问题的求解规模,提高计算效率．通过数值算例计算了代表性体积单元上各种短纤维复合材料的整体弹性性能,验证了计算模型和求解方法的正确性和有效性．     

A subregion DRBEM formulation for the dynamic analysis of two-dimensional cracks

The dual reciprocity boundary element method employing the step by step time integration technique is developed to analyse two-dimensional dynamic crack problems. In this method the equation of motion is expressed in boundary integral form using elastostatic fundamental solutions. In order to transform the domain integral into an equivalent boundary integral, a general radial basis function is used for the derivation of the particular solutions. The dual reciprocity boundary element method is combined with an efficient subregion boundary element method to overcome the difficulty of a singular system of algebraic equations in crack problems. Dynamic stress intensity factors are calculated using the discontinuous quarter-point elements. Several examples are presented to show the formulation details and to demonstrate the computational efficiency of the method.     

A symplectic analytical wave propagation model for damping and steady state forced vibration of orthotropic composite plate structure

An analytical wave propagation model is proposed in this paper for damping and steady state forced vibration of orthotropic composite plate structure by using the symplectic method. By solving an eigen-problem derived in the symplectic dual system of free bending vibration of orthotropic rectangular thin plates, the wave shape of plate is obtained in symplectic analytical form for any combination of simple boundary conditions along the plate edges. And then the specific damping capacity of wave mode is obtained symplectic analytically by using the strain energy theory. The steady state forced vibration of built-up plates structure is calculated by combining the wave propagation model and the finite element method. The vibration of the uniform plate domain of the built-up plates structure is described using symplectic analytical waves and the connector with discontinuous geometry or material is modeled using finite elements. In the numerical examples, the specific damping capacity of orthotropic rectangular thin plate with three different combinations of boundary condition is first calculated and analyzed. Comparisons of the present method results with respect to the results from the finite element method and from the Rayleigh–Ritz method validate the effectiveness of the present method. The relationship between the specific damping capacity of wave mode and that of modal mode is expounded. At last, the damped steady state forced vibration of a two plates system with a connector is calculated using the hybrid solution technique. The availability of the symplectic analytical wave propagation model is further validated by comparing the forced response from the present method with the results obtained using the finite element method.     

Asymptotic analysis of three-dimensional dynamic equations of a thin plate : PMM vol. 38, n 6, 1974, pp. 1072–1078

Two-dimensional dynamic equations of thin plate vibrations are obtained from the three-dimensional dynamic equations of elasticity theory on the basis of an asymptotic method [1 – 3], Such an approach permits establishing the limits of applicability of the two-dimensional dynamic equations and the corresponding boundary and initial conditions, and indicating the means of obtaining refined results.The question of the construction of an inner state of stress of a thin plate under dynamic conditions is examined herein. The possibility of considering states of stress with distinct variability in time and in the coordinates and with a distinct relationship between the displacement intensities, is taken into account.     

Recovering the Dirichlet-to-Neumann map in inverse scattering problems using integral equation methods

Consider the reconstruction of Dirichlet-to-Neumann map(D-to-N map) from the far-field patterns of the scattered waves in inverse scattering problems, which is the first step in detecting the obstacle boundary by the probe method using far-field measurements corresponding to all incident plane waves. In principle, this problem can be reduced to solving an integral equation of the second kind with the kernels involving the derivatives of the scattered waves for point sources. Based on the mixed reciprocity principle, we propose two simple and feasible numerical schemes for reconstructing D-to-N map. Compared with the well-known obstacle boundary recovering schemes using the simulation of D-to-N map directly, the proposed schemes give the possible ways to realizing the probe methods using practical far-field data, with the advantage of no numerical differentiation for scattered wave in their implementations. We present some numerical examples for the D-to-N map, showing the validity and stability of our schemes.     

Iterative solution schemes for quadratic DRM‐MD

The aim of this work is to find the most suitable iterative technique to solve linear systems of equations arising from the dual reciprocity method in multidomains (DRM‐MD). In this article, the surface variables of the governing equations and the shape functions of the boundary elements are quadratic functions, and for the dual reciprocity approximation the radial basis function (RBF) used is the augmented thin plate spline. A series of tests are carried out studying efficiency and accuracy of different Krylov iterative solvers, every one assessed with several preconditioners. The results are related to intrinsic properties of the linear systems such as condition numbers, sparsity patterns, singular values, and eigenvalues. Besides, the performance of the selected iterative solvers is studied in relation with the subdivision of the domain, taking into account total number of nodes of the whole mesh, number of internal nodes per subdomain, as well as different aspect ratios of the mesh grids. As a result, performances of different refining schemes are considered. © 2008 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq 2008     

Transient magneto-thermoviscoelastic plane waves in a non-homogeneous anisotropic thick strip subjected to a moving heat source

The aim of this paper is to study the transient wave propagations in a non-homogeneous anisotropic thermoviscoelastic thick strip placed in a constant primary magnetic field and subjected to a moving heat source. The governing equations for temperature and displacement fields are solved by means of a dual reciprocity boundary element method (DRBEM). In the case of plane deformation, a numerical scheme for the implementation of the method is presented and the numerical computations are carried out for the temperature, displacement components and thermal stress components. The validity of DRBEM is examined by considering a magneto-thermo-visco-elastic thick strip occupies a rectangular region and good agreement is obtained with existent results. The results obtained are presented graphically to show the effect of inhomogeneity on the displacement components and thermal stress components. Relevant results of previous investigations are deduced as special cases from this study.     

On a method of Atkinson for evaluating domain integrals in the boundary element method

We have shown how to couple a numerical method due to Atkinson for computing particular solutions to a class of elliptic differential equations with a variety of boundary element methods, which alleviates the problem of domain discretization for solving inhomogeneous equations. When the inhomogeneous term is of the form typically used in the dual reciprocity method, Atkinson's formula is such that only boundary integrals need to be approximated numerically. If this method is coupled with the method of potentials, it results in a computational technique which requires neither boundary nor domain discretization. Some numerical results are given validating our approach.     

Boundary element method for dynamic inelastic analysis

The paper shows formulation and application of the boundary element method (BEM) for dynamic analysis of elastoplastic materials. The initial stress approach is used in the elastoplastic analysis. The mass matrices are computed by the dual reciprocity method (DRBEM). Displacements and stresses are computed by the iterative procedure in each time step. The time dependent problem is solved by the direct integration Houbolt method. The methods presented in the paper are applied to compute displacements and stresses in a crank loaded by dynamic forces. (© 2006 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim)     

Modelling of circumferential waves in cylindrical thermoelastic plates with voids

The propagation of thermoelastic waves along circumferential direction in homogeneous, isotropic, cylindrical curved solid plates with voids has been investigated in the context of linear generalized theory of thermoelasticity. The plate is subjected to stress free or rigidly fixed, thermally insulated or isothermal boundary conditions. Mathematical modeling of the problem for the considered cylindrical curved plate with voids leads to a system of coupled partial differential equations. The model has been simplified by using the Helmholtz decomposition technique and the resulting equations are solved by using the method of separation of variables. The formal solution obtained by using Bessel’s functions with complex arguments is utilized to derive the secular equations which govern the wave motion in the plate with voids. The longitudinal shear motion and axially symmetric shear vibration modes get decoupled from the rest of the motion in contrast to non-axially symmetric plane strain vibrations. These modes remain unaffected due to thermal variations and presence of voids. In order to illustrate theoretical developments, numerical solutions have been carried out for a stress free, thermally insulated or isothermal magnesium plate and are presented graphically. The obtained results are also compared with those available in the literature.