桁架结构几何非线性问题的力密度法

桁架结构大都具有较强的几何非线;浊,受荷载后易出现较大的脯变形。提出一种基于力密度的针对桁架结构几何大变形问题的解法;引入杆件单元的力密度矩阵,推导出相应非线性方程的Jacobi矩阵;与有限单元法集成求解的思想相同,采用力密度矩阵建立结构变形后整体的精确非线性平衡方程。研究结果表明：应用Newton-Raphson迭代法求解,采用适当的迭代收敛精度可得到较精确的桁架结构位移解。     

一种新的求解对流扩散边值问题的无网格方法

基于配点法和楔形基函数,提出了一种新的求解对流扩散边值问题的无网格方法。通过一维和二维的问题验证了该数值方法的可行性;并根据数值算例和分析,可以看到该数值方法能达到满意的收敛效果。该数值方法的隐格式形式能够有效地消除对流占优问题的数值振荡现象,是一种真正的无网格方法。     

Improved L-P method for solving strongly nonlinear problems

ＩｎｔｒｏｄｕｃｔｉｏｎＴｈｅｓｔｒｏｎｇｌｙｎｏｎｌｉｎｅａｒｏｓｃｉｌｌａｔｉｏｎｐｒｏｂｌｅｍ¨ｕ ω20 ｕ =εｆ(ｕ , ｕ ,¨ｕ) , (1 )ｗｈｅｒｅｔｈｅｓｙｍｂｏｌ“·”ｄｅｎｏｔｅｓｔｈｅｄｉｆｆｅｒｅｎｔｉａｔｉｏｎｗｉｔｈｒｅｓｐｅｃｔｔｏｔｈｅｉｎｄｅｐｅｎｄｅｎｔｖａｒｉａｂｌｅｔａｎｄω0ａｎｄεａｒｅａｌｌｔｈｅａｒｂｉｔｒａｒｙｃｏｎｓｔａｎｔｓ,ｈａｖｅｉｎｔｈｅｒｅｃｅｎｔｙｅａｒｓｓｅｖｅｒａｌｉｍｐｒｏｖｅｄＬ＿Ｐｍｅｔｈｏｄｓｗｈｅｎεｉｓｎｏｔｓｍａｌｌ (ｓｔｒｏ…     

两点边值问题的小波配点法

根据多分辨分析,提出用任意连续的尺度函数构造区间上的插值基函数,形成以尺度函数为基础的求解两点边值问题的小波配点法.该方法中,尺度函数不受紧支撑、插值等性质的限制,计算复杂度小,数值解收敛性由多分辨分析理论保证.同时,给出边值条件的积分处理方法,能够方便地处理任意边界条件,当尺度函数不具有高阶导数时,该方法也能有效使用.数值算例表明,该方法是一个高效、高精度的算法.     

二维弹性新型快速多极虚边界元的最小二乘法

在虚边界元最小二乘法的方程求解中采用新型的快速多极展开和广义极小残值法,提出了一种二维弹性新型快速多极虚边界元最小二乘法的求解思想。基于二维弹性问题原有的快速多极虚边界元最小二乘法的展开格式,通过引入对角化的概念,以更新展开传递格式;相对于原有快速多极算法,该方法可进一步提高计算效率且仍能保证具有较高的计算精度。数值算例说明了该方法的可行性、计算效率、计算精度均较高。     

Singular boundary method for 2D dynamic poroelastic problems

This paper extends a strong-form meshless boundary collocation method, named the singular boundary method (SBM), for the solution of dynamic poroelastic problems in the frequency domain, which is governed by Biot equations in the form of mixed displacement–pressure formulation. The solutions to problems are represented by using the fundamental solutions of the governing equations in the SBM formulations. To isolate the singularities of the fundamental solutions, the SBM uses the concept of the origin intensity factors to allow the source points to be placed on the physical boundary coinciding with collocation points, which avoids the auxiliary boundary issue of the method of fundamental solutions (MFS). Combining with the origin intensity factors of Laplace and plane strain elastostatic problems, this study derives the SBM formulations for poroelastic problems. Five examples for 2D poroelastic problems are examined to demonstrate the efficiency and accuracy of the present method. In particular, we test the SBM to the multiply connected domain problem, the multilayer problem and the poroelastic problem with corner stress singularities, which are all under varied ranges of frequencies.     

奇异边界法模拟二维弹性力学问题

奇异边界法是一种新的边界型无网格数值离散方法.该方法使用基本解作为插值基函数,在继承传统边界型方法优点的同时,不需要费时费力的网格划分和奇异积分,数学简单,编程容易,是一个真正的无网格方法.为避免配置点与插值源点重合时带来的基本解源点奇异性,该方法提出了源点强度因子的概念,从而将边界型强格式方法的核心归结为求解源点强度因子.论文首次将该方法应用于求解平面弹性力学问题.数值算例表明,本文算法稳定,效率高,并可达到很高的计算精度.     

INTERPOLATION PERTURBATION METHOD FOR SOLVING THE BOUNDARY LAYER TYPE PROBLEMS

ＹｕａｎＹｉｗｕ（袁镒吾）（ＲｅｃｅｉｖｅｄＯｃｔ．２，１９９４；ＣｏｍｍｕｎｉｃａｔｅｄｂｙＣｈｉｅｎＷｅｉｚａｎｇ）ＩＮＴＥＲＰＯＬＡＴＩＯＮＰＥＲＴＵＲＢＡＴＩＯＮＭＥＴＨＯＤＦＯＲＳＯＬＶＩＮＧＴＨＥＢＯＵＮＤＡＲＹＬＡＹＥＲＴＹＰＥＰＲＯＢ...     

不规则区域平面弹性问题的正则区域重心插值配点法

将不规则区域嵌入到规则的矩形区域,在矩形区域上将弹性平面问题的控制方程采用重心Lagrange插值离散,得到控制方程矩阵形式的离散表达式。在边界节点上利用重心插值离散边界条件,规则区域采用置换法施加边界条件,不规则区域采用附加法施加边界条件,得到求解平面弹性问题的过约束线性代数方程组,采用最小二乘法进行求解,得到整个规则区域上的位移数值解。利用重心插值计算得到不规则区域内任意节点的位移值,计算精度可到10^-14以上。数值算例验证了所建立方法的有效性和计算精度。     

Wavelet multiresolution interpolation Galerkin method for nonlinear boundary value problems with localized steep gradients

The wavelet multiresolution interpolation for continuous functions defined on a finite interval is developed in this study by using a simple alternative of transformation matrix. The wavelet multiresolution interpolation Galerkin method that applies this interpolation to represent the unknown function and nonlinear terms independently is proposed to solve the boundary value problems with the mixed Dirichlet-Robin boundary conditions and various nonlinearities, including transcendental ones, in which the discretization process is as simple as that in solving linear problems, and only common two-term connection coefficients are needed. All matrices are independent of unknown node values and lead to high efficiency in the calculation of the residual and Jacobian matrices needed in Newton’s method, which does not require numerical integration in the resulting nonlinear discrete system. The validity of the proposed method is examined through several nonlinear problems with interior or boundary layers. The results demonstrate that the proposed wavelet method shows excellent accuracy and stability against nonuniform grids, and high resolution of localized steep gradients can be achieved by using local refined multiresolution grids. In addition, Newton’s method converges rapidly in solving the nonlinear discrete system created by the proposed wavelet method, including the initial guess far from real solutions.

     

几何非线性分析的无网格伽辽金算法

利用几何非线性的应变-位移关系,在小应变假设的条件下,推导出二维几何非线性问题中的无网格伽辽金法的计算格式。由于无网格方法中的形函数不具备Kronecker delta性质,文中采用罚方法来实现本质边界条件。数值实例表明,无网格伽辽金法在处理几何非线性问题时,具有较高的计算精度,是一种有效的数值计算方法。     

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.     

一种准零刚度系统隔振性能的几何参数优化分析

提出了一种采用菱形连杆组件作为负刚度机构的准零刚度隔振器(下文简称菱形准零刚度隔振器).通过静力学分析方法,建立了菱形准零刚度隔振器数学模型,并与其他调节变量较少的隔振器模型进行了对比;以被测量曲线在隔振器平衡位置处的曲率作为评价参数,研究了负刚度机构几何参数对系统刚度、阻尼非线性的影响,推导了利用几何参数进行隔振优化...     

On pressure boundary conditions for thermoconvective problems

Solving numerically hydrodynamical problems of incompressible fluids raises the question of handling first order derivatives (those of pressure) in a closed container and determining its boundary conditions. A way to avoid the first point is to derive a Poisson equation for pressure, although the problem of taking the right boundary conditions still remains. To remove this problem another formulation of the problem has been used consisting of projecting the master equations into the space of divergence‐free velocity fields, so pressure is eliminated from the equations. This technique raises the order of the differential equations and additional boundary conditions may be required. High‐order derivatives are sometimes troublesome, specially in cylindrical coordinates due to the singularity at the origin, so for these problems a low order formulation is very convenient. We research several pressure boundary conditions for the primitive variables formulation of thermoconvective problems. In particular we study the Marangoni instability of an infinite fluid layer and we show that the numerical results with a Chebyshev collocation method are highly correspondent to the exact ones. These ideas have been applied to linear stability analysis of the Bénard–Marangoni (BM) problem in cylindrical geometry and the results obtained have been very accurate. Copyright © 2002 John Wiley & Sons, Ltd.     

Comparative study of honeycomb optimization using Kriging and radial basis function

Structural optimization for crashworthiness criteria is of particular significance especially at early stage of design. The comparative study of Kriging and radial basis function network (RBFN) was performed in order to improve the crashworthiness effects of honeycomb. Improving the crashworthiness characteristic of honeycomb was achieved using LS-OPT® and domain reduction strategy. This optimization is performed on the basis of validated numerical simulation to establish the approximated model to illustrate the relationship between the responses and design variables. The results showed that Kriging meta-model is excelled in accuracy, robustness and efficiency compared to radial basis function (RBF) and crashworthiness characteristic of honeycomb is improved by 4%.     

RBF collocation method and stability analysis for phononic crystals

The mesh-free radial basis function (RBF) collocation method is explored to calculate band structures of periodic composite structures. The inverse multi-quadric (MQ), Gaussian, and MQ RBFs are used to test the stability of the RBF collocation method in periodic structures. Much useful information is obtained. Due to the merits of the RBF collocation method, the general form in this paper can easily be applied in the high dimensional problems analysis. The stability is fully discussed with different RBFs. The choice of the shape parameter and the effects of the knot number are presented.     

A Cartesian grid technique based on one‐dimensional integrated radial basis function networks for natural convection in concentric annuli

This paper reports a radial basis function (RBF)‐based Cartesian grid technique for the simulation of two‐dimensional buoyancy‐driven flow in concentric annuli. The continuity and momentum equations are represented in the equivalent stream function formulation that reduces the number of equations from three to one, but involves higher‐order derivatives. The present technique uses a Cartesian grid to discretize the problem domain. Along a grid line, one‐dimensional integrated RBF networks (1D‐IRBFNs) are employed to represent the field variables. The capability of 1D‐IRBFNs to handle unstructured points with accuracy is exploited to describe non‐rectangular boundaries in a Cartesian grid, while the method's ability to avoid the reduction of convergence rate caused by differentiation is instrumental in improving the quality of the approximation of higher‐order derivatives. The method is applied to simulate thermally driven flows in annuli between two circular cylinders and between an outer square cylinder and an inner circular cylinder. High Rayleigh number solutions are achieved and they are in good agreement with previously published numerical data. Copyright © 2007 John Wiley & Sons, Ltd.     

Augmenting the immersed boundary method with Radial Basis Functions (RBFs) for the modeling of platelets in hemodynamic flows

We present a new computational method by extending the immersed boundary (IB) method with a geometric model based on parametric radial basis function (RBF) interpolation of the Lagrangian structures. Our specific motivation is the modeling of platelets in hemodynamic flows, although we anticipate that our method will be useful in other applications involving surface elasticity. The efficacy of our new RBF‐IB method is shown through a series of numerical experiments. Specifically, we test the convergence of our method and compare our method with the traditional IB method in terms of computational cost, maximum stable time‐step size, and volume loss. We conclude that the RBF‐IB method has advantages over the traditional IB method and is well‐suited for modeling of platelets in hemodynamic flows. Copyright © 2015 John Wiley & Sons, Ltd.     

A simultaneous space-time wavelet method for nonlinear initial boundary value problems

A high-precision and space-time fully decoupled numerical method is developed for a class of nonlinear initial boundary value problems. It is established based on a proposed Coiflet-based approximation scheme with an adjustable high order for the functions over a bounded interval, which allows the expansion coefficients to be explicitly expressed by the function values at a series of single points. When the solution method is used, the nonlinear initial boundary value problems are first spatially discretized into a series of nonlinear initial value problems by combining the proposed wavelet approximation and the conventional Galerkin method, and a novel high-order step-by-step time integrating approach is then developed for the resulting nonlinear initial value problems with the same function approximation scheme based on the wavelet theory. The solution method is shown to have the N th-order accuracy, as long as the Coiflet with [0, 3 N-1]compact support is adopted, where N can be any positive even number. Typical examples in mechanics are considered to justify the accuracy and efficiency of the method.