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该文基于插值型移动最小二乘法,将无网格局部Petrov-Galerkin(MLPG)法用于二维耦合热弹性动力学问题的求解。修正的Fourier热传导方程和弹性动力控制方程通过加权余量法来离散,Heaviside分段函数作为局部弱形式的权函数,从而得到描述热耦合问题的二阶常微分方程组。然后利用微分代数方法,温度和位移作为辅助变量,将上述二阶常微分方程组转换成常微分代数系统,采用Newmark逐步积分法进行求解。该方法无需Laplace变换可直接得到温度场和位移场数值结果,同时插值型移动最小二乘法构造的形函数由于满足Kroneckerdelta特性,因此能直接施加本质边界条件。最后通过两个数值算例来验证该方法的有效性。 相似文献
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IMLS方形影响域法 总被引:1,自引:0,他引:1
展示了一些最新无网格法的研究进展,给出了一种新型无网格法-IMLS方形域无网格法。该法中未知变量的近似采用IMLS技术,局部影响域形状采用方形几何形态。这些技术的具体实施展现了节点布置和数值积分的无网格特点,并自然满足Dirichlet边界条件。该方法可以容易推广到求解非线性问题以及非均匀介质的力学问题。此外,还计算了两个弹性力学平面问题的例子,所得计算结果证明:该方法是一种具有收敛快、精度高、简便有效的通用方法,在工程中具有广阔的应用前景。 相似文献
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用一种修正的无网格局部Petrov-Galerkin方法求解了不可压超弹性材料平面应力问题。在建立求解方程过程中,采用径向基函数耦合多项式构造近似函数,并以Heaviside分段函数作为加权函数简化了刚度矩阵的域积分,引入平面应力假设避免了材料不可压引起的数值求解困难。数值算例表明:该文方法求解不可压超弹性材料平面应力问题具有稳定性好、精度高的特点。 相似文献
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利用基于滑动Kriging插值的无网格局部Petrov-Galerkin (MLPG) 法来求解二维结构动力问题,Heaviside分段函数作为局部弱形式的权函数并采用精细积分法来离散时间域。基于滑动Kriging插值构造的形函数满足Kronecker Delta性质,因此可以直接施加本质边界条件。刚度矩阵形成过程中只涉及到边界积分,而没有涉及到区域积分和奇异积分。计算结果表明:基于滑动Kriging插值的MLPG法具有模拟简单、计算精度高等优点。 相似文献
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对扩展有限元方法(XFEM)的发展及其与其他数值方法的联系进行了综述。该文主要包括以下内容:首先对无网格法的发展背景和历程进行了介绍,并从近似位移场构造的角度对众多的无网格法进行了比较分析;从单位分解理论的形式出发,阐述了XFEM的特点及其与传统有限元法、无网格方法的联系;归纳了关于XFEM的应用研究及其自身理论发展的主要研究方向,并对XFEM的发展进行了展望。 相似文献
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数值求解对流占优的高阶非线性偏微分方程存在近似高阶导数和抑制数值振荡两方面的困难.本文采用容易近似高阶导数的无单元Galerkin方法,并借鉴迎风稳定化方法的思想,建立了基于偏心支持域的迎风无单元Galerkin方法.为保证无单元Galerkin方法在近似高阶导数时形函数满足一致性条件,本文在构造形函数时采用了一种定义在局部坐标中的平移多项式基函数.数值结果表明,使用平移多项式基函数的迎风无单元Galerkin方法在求解对流占优的高阶非线性偏微分方程时,具有精度高、稳定性好和实施简单的优点. 相似文献
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We discuss the symmetric smoothed particle hydrodynamics (SSPH) method for generating basis functions for a meshless method.
It admits a larger class of kernel functions than some other methods, including the smoothed particle hydrodynamics (SPH),
the modified smoothed particle hydrodynamics (MSPH), the reproducing kernel particle method (RKPM), and the moving least squares
(MLS) methods. For finding kernel estimates of derivatives of a function, the SSPH method does not use derivatives of the
kernel function while other methods do, instead the SSPH method uses basis functions different from those employed to approximate
the function. It is shown that the SSPH method and the RKPM give the same value of the kernel estimate of a function but give
different values of kernel estimates of derivatives of the function. Results computed for a sine function defined on a one-dimensional
domain reveal that the L
2, the H
1 and the H
2 error norms of the kernel estimates of a function computed with the SSPH method are less than those found with the MSPH method.
Whereas the L
2 and the H
2 norms of the error in the estimates computed with the SSPH method are less than those with the RKPM, the H
1 norm of the error in the RKPM estimate is slightly less than that found with the SSPH method. The error norms for a sample
problem computed with six kernel functions show that their rates of convergence with an increase in the number of uniformly
distributed particles are the same and their magnitudes are determined by two coefficients related to the decay rate of the
kernel function. The revised super Gauss function has the smallest error norm and is recommended as a kernel function in the
SSPH method. We use the revised super Gauss kernel function to find the displacement field in a linear elastic rectangular
plate with a circular hole at its centroid and subjected to tensile loads on two opposite edges. Results given by the SSPH
and the MSPH methods agree very well with the analytic solution of the problem. However, results computed with the SSPH method
have smaller error norms than those obtained from the MSPH method indicating that the former will give a better solution than
the latter. The SSPH method is also applied to study wave propagation in a linear elastic bar. 相似文献
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主要工作是推导了一种基于新的形函数形式的无网格方法。由于目前的无网格方法中,紧支域的大小和形状不同直接影响整体场函数的分布,有时会造成计算结果的严重失真。为了弱化紧支域尺寸的选取对计算结果的影响,以节点约束的形式构造一种形函数对位移场进行近似。一维位移场的计算实例表明方法能有效的弱化覆盖尺寸对计算结果的影响。一个简单的三维数值算例也表明利用该形函数构造的应变场进行无网格方法计算是可行的。方法的缺点是弱化核函数覆盖尺寸的影响是以增大计算量为代价的,需进一步改进。 相似文献
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SSPH basis functions for meshless methods, and comparison of solutions with strong and weak formulations 总被引:1,自引:0,他引:1
We propose a new and simple technique called the Symmetric Smoothed Particle Hydrodynamics (SSPH) method to construct basis
functions for meshless methods that use only locations of particles. These basis functions are found to be similar to those
in the Finite Element Method (FEM) except that the basis for the derivatives of a function need not be obtained by differentiating
those for the function. Of course, the basis for the derivatives of a function can be obtained by differentiating the basis
for the function as in the FEM and meshless methods. These basis functions are used to numerically solve two plane stress/strain
elasto-static problems by using either the collocation method or a weak formulation of the problem defined over a subregion
of the region occupied by the body; the latter is usually called the Meshless Local Petrov–Galerkin (MLPG) method. For the
two boundary-value problems studied, it is found that the weak formulation in which the basis for the first order derivatives
of the trial solution are derived directly in the SSPH method (i.e., not obtained by differentiating the basis function for
the trial solution) gives the least value of the L2-error norm in the displacements while the collocation method employing the strong formulation of the boundary-value problem
has the largest value of the L2-error norm. The numerical solution using the weak formulation requires more CPU time than the solution with the strong formulation
of the problem. We have also computed the L2-error norm of displacements by varying the number of particles, the number of Gauss points used to numerically evaluate domain
integrals appearing in the weak formulation of the problem, the radius of the compact support of the kernel function used
to generate the SSPH basis, the order of complete monomials employed for constructing the SSPH basis, and boundary conditions
used at a point on a corner of the rectangular problem domain. It is recommended that for solving two-dimensional elasto-static
problems, the MLPG formulation in which derivatives of the trial solution are found without differentiating the SSPH basis
function be adopted. 相似文献
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H. CHICKERMANE H. C. GEA 《International journal for numerical methods in engineering》1996,39(5):829-846
A new method for solving structural optimization problems using a local function approximation algorithm is proposed. This new algorithm, called the Generalized Convex Approximation (GCA), uses the design sensitivity information from the current and previous design points to generate a sequence of convex, separable subproblems. The paper contains the derivation of the parameters associated with the approximation and the formulation of the approximated problem. Numerical results from standard test problems solved using this method are presented. It is observed that this algorithm generates local approximations which lead to faster convergence for structural optimization problems. 相似文献
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该文采用张量积的试函数逼近形式,即u(x,y,z)={X(x)}T[Z(z)]{Y(y)},成功地建立了三维延拓Kantorovich 法的算法方程式,克服了简单试函数逼近形式的迭代不收敛的数值困难。三维Poisson 方程的数值算例显示了该算法的迭代收敛性以及高精度和高效率。 相似文献
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For design problems involving computation-intensive analysis or simulation processes, approximation models are usually introduced to reduce computation lime. Most approximation-based optimization methods make step-by-step improvements to the approximation model by adjusting the limits of the design variables. In this work, a new approximation-based optimization method for computation-intensive design problems - the adaptive response surface method(ARSM), is presented. The ARSM creates quadratic approximation models for the computation-intensive design objective function in a gradually reduced design space. The ARSM was designed to avoid being trapped by local optima and to identify the global design optimum with a modest number of objective function evaluations. Extensive tests on the ARSM as a global optimization scheme using benchmark problems, as well as an industrial design application of the method, are presented. Advantages and limitations of the approach are also discussed 相似文献
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AbstractWithin the framework of Timoshenko beam theory, the buckling of nano sandwich beams is developed. The material properties are assumed to vary arbitrarily in both axial and thickness directions. These types of beams are referred to as bi-directional functionally graded (BDFG) beams. Two types of nano sandwich beams with different material distribution patterns and immovable supports are considered. Since the size effects play a significant role in mechanical behavior of nanostructures, the small-scale effects are captured by Eringen’s nonlocal theory of elasticity. The governing equations are derived using the variational formulation. Symmetric smoothed particle hydrodynamics (SSPH) and the Galerkin method are adopted as numerical solution approaches. As a truly meshless method, the convergence of the SSPH technique mainly depends on the smoothing length value and distribution of particles in the compact support domain of the kernel function. The Revised Super Gauss Function is used as the kernel function and an optimum value for the smoothing length that bears the fastest convergence rate is obtained. The solution methods are verified through benchmark problems found in the literature. Numerical and illustrative results show that various parameters, including the aspect ratio, nonlocal parameter, gradient indexes, and cross-sectional types have significant effects on the buckling responses of BDFG nano sandwich beams. 相似文献
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This paper presents a very efficient method, referred to as the aggregate method, for solving nonlinear minimax optimization problems, which converts a minimax problem to an unconstrained minimization on a differentiable aggregate function. We derive this function by means of Lagrangean duality and the Jaynes' maximum entropy principle. It can be shown that as a parameter in the function tends to infinity, it approaches the maximum function that denotes a uniform approximation to the problem functions and is non-differentiable. Main features of the present method consist of its algorithmic simplicity and computational high efficiency. 相似文献
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C. L. Tsai Y. L. Guan R. C. Batra D. C. Ohanehi J. G. Dillard E. Nicoli D. A. Dillard 《Computational Mechanics》2013,51(1):19-34
We use symmetric smoothed particle hydrodynamics (SSPH) and moving least squares (MLS) basis functions to analyze six linear elastostatics problems by first deriving their Petrov-Galerkin approximations. With SSPH basis functions one can approximate the trial solution and its derivatives by using different basis functions whereas with MLS basis functions the derivatives of the trial solution involve derivatives of the basis functions used to approximate the trial solution. The class of allowable kernel functions for SSPH basis functions includes constant functions which are excluded in MLS basis functions if derivatives of the trial solution are also to be approximated. We compare results for different choices of weight functions, size of the compact support of the weight function, order of complete polynomials, and number of particles in the problem domain. The two basis functions are also used to analyze crack initiation and propagation in plane stress mode-I deformations of a plate made of a linear elastic isotropic and homogeneous material with particular emphasis on the computation of the T-stress. The crack trajectories predicted by using the two basis functions agree well with those found experimentally. 相似文献