Heat transfer analysis of composite slabs using meshless element Free Galerkin method

This paper deals with three dimensional heat transfer analysis of composite slabs using meshless element free Galerkin method. The element free Galerkin method (EFG) method utilizes moving least square (MLS) approximants to approximate the unknown function of temperature Tx). These approximants are constructed by using a weight function, a basis function and a set of coefficients that depends on position. Penalty and Lagrange multiplier techniques have been used to enforce the essential boundary conditions. MATLAB codes have been developed to obtain the EFG results. Two new basis functions namely trigonometric and polynomial have been proposed. A comparison has been made among the results obtained using existing (linear) and proposed (trigonometric and polynomial) basis functions for three dimensional heat transfer in composite slabs. The effect of penalty parameter on EFG results has also been discussed. The results obtained by EFG method are compared with those obtained by finite element method     

New complex Fourier shape functions for the analysis of two-dimensional potential problems using boundary element method

In this paper, boundary element analysis for two-dimensional potential problems is investigated. In this study, the boundary element method (BEM) is reconsidered by proposing new shape functions to approximate the potentials and fluxes. These new shape functions, called complex Fourier shape function, are derived from complex Fourier radial basis function (RBF) in the form of exp(iωr). The proposed shape functions may easily satisfy various functions such as trigonometric, exponential, and polynomial functions. In order to illustrate the validity and accuracy of the present study, several numerical examples are examined and compared to the results of analytical and with those obtained by classic real Lagrange shape functions. Compared to the classic real Lagrange shape functions, the proposed complex Fourier shape functions show much more accurate results.     

A local Heaviside weighted meshless method for two-dimensional solids using radial basis functions

 A meshless method is developed for the stress analysis of two-dimensional solids, based on a local weighted residual method with the Heaviside step function as the weighting function over a local subdomain. Trial functions are constructed using radial basis functions (RBF). The present method is a truly meshless method based only on a number of randomly located nodes. No domain integration is needed, no element matrix assembly is required and no special treatment is needed to impose the essential boundary conditions. Effects of the sizes of local subdomain and interpolation domain on the performance of the present method are investigated. The behaviour of shape parameters of multiquadrics (MQ) has been systematically studied. Example problems in elastostatics are presented and compared with closed-form solutions and show that the proposed method is highly accurate and possesses no numerical difficulties. Received: 10 November 2002 / Accepted: 5 March 2003     

Critical evaluation of spline,exponential and Gaussian test functions for elastostatic analysis of metal parts using meshless method

The smoothness of shape functions computed using moving least-square approximation is affected by the choice of trial function and order of basis function. This paper presents elastostatic analysis using Meshless Local Petrov Galerkin method (MLPG) with three types of test functions, i.e., Gaussian, exponential and spline. The numerical results for three case studies, i.e., cantilever, plate with a hole and pressurized thick cylinder, are presented. Computational results indicate that the performance of Gaussian test function is the best, followed by exponential and spline functions, but the performance of exponential test function can be improved by optimizing its scaling parameters.     

Interpolation functions in control volume finite element method

 The main contribution of this paper is the study of interpolation functions in control volume finite element method used in equal order and applied to an incompressible two-dimensional fluid flow. Especially, the exponential interpolation function expressed in the elemental local coordinate system is compared to the classic linear interpolation function expressed in the global coordinate system. A quantitative comparison is achieved by the application of these two schemes to four flows that we know the analytical solutions. These flows are classified in two groups: flows with privileged direction and flows without. The two interpolation functions are applied to a triangular element of the domain then; a direct comparison of the results given by each interpolation function to the exact value is easily realized. The two functions are also compared when used to solve the discretized equations over the entire domain. Stability of the numerical process and accuracy of solutions are compared. Received: 20 October 2002 / Accepted: 2 December 2002     

A novel meshless approach – Local Kriging (LoKriging) method with two-dimensional structural analysis

This paper develops a novel meshless approach, called Local Kriging (LoKriging) method, which is based on the local weak form of the partial differential governing equations and employs the Kriging interpolation to construct the meshless shape functions. Since the shape functions constructed by this interpolation have the delta function property based on the randomly distributed points, the essential boundary conditions can be implemented easily. The local weak form of the partial differential governing equations is obtained by the weighted residual method within the simple local quadrature domain. The spline function with high continuity is used as the weight function. The presently developed LoKriging method is a truly meshless method, as it does not require the mesh, either for the construction of the shape functions, or for the integration of the local weak form. Several numerical examples of two-dimensional static structural analysis are presented to illustrate the performance of the present LoKriging method. They show that the LoKriging method is highly efficient for the implementation and highly accurate for the computation.     

The hybrid element-free Galerkin method for three-dimensional wave propagation problems

This paper presents a hybrid element-free Galerkin (HEFG) method for solving wave propagation problems. By introducing the dimension split method, the three-dimensional wave propagation problems are transformed into a series of two-dimensional ones in other one-dimensional directions. The two-dimensional problems are solved using the improved element-free Galerkin (IEFG) method, and the finite difference method is used in the one-dimensional splitting direction and the time space. Then, the formulas of the HEFG method for three-dimensional wave propagation problems are obtained. Numerical examples are selected to show the effectiveness and the advantage of the HEFG method. The convergence and error analysis of the HEFG method are discussed according to the numerical results under different splitting directions, weight functions, node distributions, scale parameters of the influence domain, penalty factors, and time steps. The numerical results are given to show the convergence and advantages of the HEFG method over the IEFG method. Comparing with the IEFG method, the HEFG method has greater computational precision and speed for three-dimensional wave propagation problems.     

圆锥壳的渐进分布传递函数解

本文给出一种求解圆锥薄壳线弹性变形的渐进传递函数方法。壳体的三个位移函数首先沿环向展开为Fourier级数,由此得到解耦的偏微分方程,它包括一个空间变量和一个时间变量。对时间变量进行Laplace变换后进一步将其简化为含复参数s的常微分方程,它的系数是坐标的函数。引入小参数ε＝L／r0sinα,用摄动方法得到一组常微分方程,它可以用渐进分布传递函数方法求解。将各子锥段的解进行综合,构造出了由多段锥壳构成的组合壳体的传递函数解。文中给出了数值算例并与有限元的结果进行了比较。     

Boundary collocation method for acoustic eigenanalysis of three-dimensional cavities using radial basis function

 In this paper, a theoretical formulation based on the collocation method is presented for the eigenanalysis of arbitrarily shaped acoustic cavities. This article can be seen as the extension of non-dimensional influence function (NDIF) method proposed by Kang et al. (1999, 2000a) extending from two-dimensional to three-dimensional case. Unlike the conventional collocation techniques in the literature, approximate functions used in this paper are two-point functions of which the argument is only the distance between the two points. Based on this radial basis expansion, the acoustic field can be represented more exactly. The field solution is obtained through the linear superposition of radial basis function, and boundary conditions can be applied at the discrete points. The influence matrix is symmetric regardless of the boundary shape of the cavity, and the calculated eigenvalues rapidly converge to the exact values by using only a few boundary nodes. Moreover, the method results in true and spurious boundary modes, which can be obtained from the right and left unitary vectors of singular value decomposition, respectively. By employing the updating term and document of singular value decomposition (SVD), the true and spurious eigensolutions can be sorted out, respectively. The validity of the proposed method are illustrated through several numerical examples. Received: 29 August 2001 / Accepted: 27 June 2002 Financial support from the National Science Council under Grant No. NSC-90-2211-E-019-006 for National Taiwan Ocean University is gratefully acknowledged.     

A radial basis function approach in the meshless local Petrov-Galerkin method for Euler-Bernoulli beam problems

A meshless local Petrov-Galerkin (MLPG) method that uses radial basis functions rather than generalized moving least squares (GMLS) interpolations to develop the trial functions in the study of Euler-Bernoulli beam problems is presented. The use of radial basis functions (RBF) in meshless methods is demonstrated for C¹ problems for the first time. This interpolation choice yields a computationally simpler method as fewer matrix inversions and multiplications are required than when GMLS interpolations are used. Test functions are chosen as simple weight functions as in the conventional MLPG method. Patch tests, mixed boundary value problems, and problems with complex loading conditions are considered. The radial basis MLPG method yields accurate results for deflections, slopes, moments, and shear forces, and the accuracy of these results is better than that obtained using the conventional MLPG method.Lockheed Martin Space Operations     

热弹性动力学耦合问题的插值型移动最小二乘无网格法研究

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。     

Calculation of stress intensity factors for functionally graded materials by using the weight functions derived by the virtual crack extension technique

The weight function method provides a powerful approach for calculating the stress intensity factors for a homogeneous cracked body subjected to mechanical loadings. In this paper, the basic equations of weight function method for mode I and mixed mode crack problems in a two-dimensional functionally graded crack system are derived based on the Betti’s reciprocal theorem. The weight functions derived by the virtual crack extension technique are further used to calculate the stress intensity factors of functionally graded materials (FGMs). The practicability and accuracy of this proposed method has been confirmed by the comparison with theoretical or numerical solutions available in the literatures. On account that the repeated extractions of the distributions of normal stress and shear stress in the uncracked component along the prospective crack line under different loadings can be avoided using the method presented in this paper, this method can be potentially used for optimal design for FGMs under multiple-load cases.     

Extended finite element method and fast marching method for three-dimensional fatigue crack propagation

A numerical technique for planar three-dimensional fatigue crack growth simulations is proposed. The new technique couples the extended finite element method (X-FEM) to the fast marching method (FMM). In the X-FEM, a discontinuous function and the two-dimensional asymptotic crack-tip displacement fields are added to the finite element approximation to account for the crack using the notion of partition of unity. This enables the domain to be modeled by finite elements with no explicit meshing of the crack surfaces. The initial crack geometry is represented by level set functions, and subsequently signed distance functions are used to compute the enrichment functions that appear in the displacement-based finite element approximation. The FMM in conjunction with the Paris crack growth law is used to advance the crack front. Stress intensity factors for planar three-dimensional cracks are computed, and fatigue crack growth simulations for planar cracks are presented. Good agreement between the numerical results and theory is realized.     

A NURBS-based Error Reproducing Kernel Method with Applications in Solid Mechanics

A novel method for derivation of mesh-free shape functions is proposed. The first step in the method is to approximate a function and its derivatives through non-uniform-rational-B-spline (NURBS) basis functions. However since NURBS functions neither reproduce polynomials of degree higher than one nor interpolate the control points (also referred to as grid or nodal points), the approximated function leads to uncontrolled errors over the domain including the nodal points. Accordingly the error function in the NURBS approximation and its derivatives are reproduced via a family of non-NURBS basis functions. The non-NURBS basis functions are constructed using a polynomial reproduction condition and added to the NURBS approximation of the function obtained in the first step. Since any desired order of continuity in the approximation can be achieved through NURBS, the proposed error reproducing kernel method (ERKM) can even approximate functions with discontinuous derivatives. Moreover, thanks to the variation diminishing property of NURBS, it has advantages in representing sharp layers without the so-called Gibbs‘ or Runge’s phenomena. Since derivatives are reproduced within polynomial spaces of appropriately reduced dimensions, differentiability requirements of the kernel functions are avoided. Any compactly supported continuous function, monotonically decreasing on either side of its maximum, may be used as the weight function (unlike other mesh free approximations). As it turns out, a target function is mainly approximated via NURBS and error functions are just supposed to add corrections, whose magnitudes are typically an order less than those of the NURBS components. The proposed method is observed to be nearly insensitive to the support size of the weight function. The proposed method is next applied to some linear and nonlinear boundary value problems of typical interest in solid mechanics. Some of these results are compared with those obtained via the standard form of RKPM. In the process, the relative numerical advantages and accuracy of the new method are brought out to an extent.     

Determination of approximate point load weight functions for embedded elliptical cracks

This paper presents the application of weight function method for the calculation of stress intensity factors in embedded elliptical cracks under complex two-dimensional loading conditions. A new general mathematical form of point load weight function is proposed based on the properties of weight functions and the available weight functions for two-dimensional cracks. The existence of this general weight function form has simplified the determination of point load weight functions significantly. For an embedded elliptical crack of any aspect ratio, the unknown parameters in the general form can be determined from one reference stress intensity factor solution. This method was used to derive the weight functions for embedded elliptical cracks in an infinite body and in a semi-infinite body. The derived weight functions are then validated against available stress intensity factor solutions for several linear and non-linear stress distributions. The derived weight functions are particularly useful for the fatigue crack growth analysis of planer embedded cracks subjected to fluctuating non-linear stress fields resulting from surface treatment (shot peening), stress concentration or welding (residual stress).     

需求受价格影响的市场环境下生产-分销系统动态仿真

利用系统动力学方法对需求受价格影响的市场环境下的生产-分销系统的动态性进行了仿真研究。论文设置了3种不同的需求-价格函数关系,分别为下降趋势的线性关系、非线性幂函数关系和非线性指数关系。通过仿真比较了不同需求函数的牛鞭效应。研究结果表明,需求与价格的关系为下降趋势的线性关系时的牛鞭效应最大,而需求与价格的关系为非线性幂函数关系和非线性指数关系时的牛鞭效应接近。研究结论对改善具有价格敏感性的产品生产与分销策略,提高企业这种产品的市场竞争力有参考意义。     

A hybrid finite difference and moving least square method for elasticity problems

In this paper, a novel hybrid finite difference and moving least square (MLS) technique is presented for the two-dimensional elasticity problems. A new approach for an indirect evaluation of second order and higher order derivatives of the MLS shape functions at field points is developed. As derivatives are obtained from a local approximation, the proposed method is computationally economical and efficient. The classical central finite difference formulas are used at domain collocation points with finite difference grids for regular boundaries and boundary conditions are represented using a moving least square approximation. For irregular shape problems, a point collocation method (PCM) is applied at points that are close to irregular boundaries. Neither the connectivity of mesh in the domain/boundary or integrations with fundamental/particular solutions is required in this approach. The application of the hybrid method to two-dimensional elastostatic and elastodynamic problems is presented and comparisons are made with the boundary element method and analytical solutions.     

Weight-adaptive isogeometric analysis for solving elastodynamic problems based on space-time discretization approach

In this paper, a novel adaptive isogeometric analysis (IGA) is introduced and its application in the numerical solution of two-dimensional elastodynamic problems based on the space-time discretization (STD) approach is studied. In the STD approach, the time is considered as an additional dimension and is discretized the same as the spatial domain. The weights of control points play the main role in the proposed method. In the conventional IGA, the same set of weights is used in the modeling of geometric and solution spaces. The idea is to define two groups of weights: geometric and solution weights. Geometric weights are known and can be determined based on the position of control points, but the solution weights are considered to be unknown and can be determined using a proper strategy so that the accuracy of the solution is optimized. This strategy is based on the minimization of an error function. The results obtained from the proposed method are compared with those obtained from the conventional IGA.     

基于耦合扩展多尺度有限元方法的功能梯度材料热应力分析

以高效模拟功能梯度材料(FGM)微观非均质性对整体热力学性能的影响为研究目的,通过随机形态描述函数(RMDF)法和体积分数的指数分布建立FGM二维微结构,在此基础上,发展了FGM热应力分析的耦合扩展多尺度有限元方法(CEMsFEM)。该方法基于扩展多尺度有限元方法(EMsFEM)的基本思想,对温度场和位移场构造数值基函数,以把微观非均质材料性质带到宏观响应中。同时为了考虑泊松效应导致的不同方向间的耦合作用,在位移场数值基函数中增加了耦合附加项。通过数值基函数建立宏微观单元信息的映射关系,在宏观尺度求解有效方程,节约计算量。为了更好地考虑微观载荷的影响,把结构的真实响应分解为宏观响应和微观扰动,进一步推导出修正的宏观载荷向量。通过不同体积分数分布的FGM在不同载荷工况下的热应力分析算例验证了本文中方法的正确性和有效性,最后讨论了微结构的尺寸效应对结构热力学响应的影响。     

Cell-based maximum entropy approximants for three-dimensional domains: Application in large strain elastodynamics using the meshless total Lagrangian explicit dynamics method

We present the cell-based maximum entropy (CME) approximants in E³ space by constructing the smooth approximation distance function to polyhedral surfaces. CME is a meshfree approximation method combining the properties of the maximum entropy approximants and the compact support of element-based interpolants. The method is evaluated in problems of large strain elastodynamics for three-dimensional (3D) continua using the well-established meshless total Lagrangian explicit dynamics method. The accuracy and efficiency of the method is assessed in several numerical examples in terms of computational time, accuracy in boundary conditions imposition, and strain energy density error. Due to the smoothness of CME basis functions, the numerical stability in explicit time integration is preserved for large time step. The challenging task of essential boundary condition (EBC) imposition in noninterpolating meshless methods (eg, moving least squares) is eliminated in CME due to the weak Kronecker-delta property. The EBCs are imposed directly, similar to the finite element method. CME is proven a valuable alternative to other meshless and element-based methods for large-scale elastodynamics in 3D. A naive implementation of the CME approximants in E³ is available to download at https://www.mountris.org/software/mlab/cme .