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1.
Jiun‐Shyan Chen Cheng‐Tang Wu Sangpil Yoon Yang You 《International journal for numerical methods in engineering》2001,50(2):435-466
Domain integration by Gauss quadrature in the Galerkin mesh‐free methods adds considerable complexity to solution procedures. Direct nodal integration, on the other hand, leads to a numerical instability due to under integration and vanishing derivatives of shape functions at the nodes. A strain smoothing stabilization for nodal integration is proposed to eliminate spatial instability in nodal integration. For convergence, an integration constraint (IC) is introduced as a necessary condition for a linear exactness in the mesh‐free Galerkin approximation. The gradient matrix of strain smoothing is shown to satisfy IC using a divergence theorem. No numerical control parameter is involved in the proposed strain smoothing stabilization. The numerical results show that the accuracy and convergent rates in the mesh‐free method with a direct nodal integration are improved considerably by the proposed stabilized conforming nodal integration method. It is also demonstrated that the Gauss integration method fails to meet IC in mesh‐free discretization. For this reason the proposed method provides even better accuracy than Gauss integration for Galerkin mesh‐free method as presented in several numerical examples. Copyright © 2001 John Wiley & Sons, Ltd. 相似文献
2.
Michael Hillman Jiun‐Shyan Chen 《International journal for numerical methods in engineering》2016,107(7):603-630
Convergent and stable domain integration that is also computationally efficient remains a challenge for Galerkin meshfree methods. High order quadrature can achieve stability and optimal convergence, but it is prohibitively expensive for practical use. On the other hand, low order quadrature consumes much less CPU but can yield non‐convergent, unstable solutions. In this work, an accelerated, convergent, and stable nodal integration is developed for the reproducing kernel particle method. A stabilization scheme for nodal integration is proposed based on implicit gradients of the strains at the nodes that offers a computational cost similar to direct nodal integration. The method is also formulated in a variationally consistent manner, so that optimal convergence is achieved. A significant efficiency enhancement over a comparable stable and convergent nodal integration scheme is demonstrated in a complexity analysis and in CPU time studies. A stability analysis is also given, and several examples are provided to demonstrate the effectiveness of the proposed method for both linear and nonlinear problems. Copyright © 2015 John Wiley & Sons, Ltd. 相似文献
3.
Thomas‐Peter Fries Ted Belytschko 《International journal for numerical methods in engineering》2008,74(7):1067-1087
Stress‐point integration provides significant reductions in the computational effort of mesh‐free Galerkin methods by using fewer integration points, and thus facilitates the use of mesh‐free methods in applications where full integration would be prohibitively expensive. The influence of stress‐point integration on the convergence and stability properties of mesh‐free methods is studied. It is shown by numerical examples that for regular nodal arrangements, good rates of convergence can be achieved. For non‐uniform nodal arrangements, stress‐point integration is associated with a mild instability which is manifested by small oscillations. Addition of stabilization improves the rates of convergence significantly. The stability properties are investigated by an eigenvalue study of the Laplace operator. It is found that the eigenvalues of the stress‐point quadrature models are between those of full integration and nodal integration. Stabilized stress‐point integration is proposed in order to improve convergence and stability properties. Copyright © 2007 John Wiley & Sons, Ltd. 相似文献
4.
Do Wan Kim Yongsik Kim 《International journal for numerical methods in engineering》2003,56(10):1445-1464
A pseudo‐spectral point collocation meshfree method is proposed. We apply a scheme of approximating derivatives based on the moving least‐square reproducing kernel approximations. Using approximated derivatives, we propose a new point collocation method. Unlike other meshfree methods that require direct calculation of derivatives for shape functions, with the proposed scheme, approximated derivatives are obtained in the process of calculating the shape function itself without further cost. Moreover, the scheme does not require the regularity of the window function, which ensures the regularity of shape functions. In this paper, we show the reproducing property and the convergence of interpolation for approximated derivatives of shape functions. As numerical examples of the proposed scheme, Poisson and Stokes problems are considered in various situations including the case of randomly generated node sets. In short, the proposed scheme is efficient and accurate even for complicated geometry such as the flow past a cylinder. Copyright © 2003 John Wiley & Sons, Ltd. 相似文献
5.
Toshio Nagashima 《International journal for numerical methods in engineering》1999,46(3):341-385
The meshless method is expected to become an effective procedure for realizing a CAD/CAE seamless system for analyses ranging from modelling to computation, because time‐consuming mesh generation processes are not required. In the present study, a new meshless approach, referred to as the Node‐By‐Node Meshless method is proposed, in which only nodal data is utilized to discretize the governing equations, which are derived using either the principle of virtual work or the Galerkin method. In this method, three key methodologies are utilized: (i) nodal integration using stabilization terms, (ii) interpolation by the Moving Least‐Squares Method, and (iii) a node‐by‐node iterative solver. This paper presents the formulation of the proposed method along with numerical results obtained for two‐dimensional elastostatic and eigenvalue problems. Copyright © 1999 John Wiley & Sons, Ltd. 相似文献
6.
Amit Shaw Shravan Bendapudi D. Roy 《International journal for numerical methods in engineering》2008,73(10):1434-1467
An error‐reproducing and interpolating kernel method (ERIKM), which is a novel and improved form of the error‐reproducing kernel method (ERKM) with the nodal interpolation property, is proposed. The ERKM is a non‐uniform rational B‐splines (NURBS)‐based mesh‐free approximation scheme recently proposed by Shaw and Roy (Comput. Mech. 2007; 40 (1):127–148). The ERKM is based on an initial approximation of the target function and its derivatives by NURBS basis functions. The errors in the NURBS approximation and its derivatives are then 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 obtained in the first step. In the ERKM, the interpolating property at the boundary is achieved by repeating the knot (open knot vector). However, for most problems of practical interest, employing NURBS with open knots is not possible because of the complex geometry of the domain, and consequently ERKM shape functions turn out to be non‐interpolating. In ERIKM, the error functions are obtained through localized Kriging based on a minimization of the squared variance of the estimate with the reproduction property as a constraint. Interpolating error functions so obtained are then added to the NURBS approximant. While enriching the ERKM with the interpolation property, the ERIKM naturally possesses all the desirable features of the ERKM, such as insensitivity to the support size and ability to reproduce sharp layers. The proposed ERIKM is finally applied to obtain strong and weak solutions for a class of linear and non‐linear boundary value problems of engineering interest. These illustrations help to bring out the relative numerical advantages and accuracy of the new method to some extent. Copyright © 2007 John Wiley & Sons, Ltd. 相似文献
7.
8.
K. M. Liew Yumin Cheng S. Kitipornchai 《International journal for numerical methods in engineering》2005,64(12):1610-1627
In this paper, we present a direct meshless method of boundary integral equation (BIE), known as the boundary element‐free method (BEFM), for two‐dimensional (2D) elastodynamic problems that combines the BIE method for 2D elastodynamics in the Laplace‐transformed domain and the improved moving least‐squares (IMLS) approximation. The formulae for the BEFM for 2D elastodynamic problems are given, and the numerical procedures are also shown. The BEFM is a direct numerical method, in which the basic unknown quantities are the real solutions of the nodal variables, and the boundary conditions can be implemented directly and easily that leads to a greater computational precision. For the purpose of demonstration, some selected numerical examples are solved using the BEFM. Copyright © 2005 John Wiley & Sons, Ltd. 相似文献
9.
A new formulation of the element‐free Galerkin (EFG) method is presented in this paper. EFG has been extensively popularized in the literature in recent years due to its flexibility and high convergence rate in solving boundary value problems. However, accurate imposition of essential boundary conditions in the EFG method often presents difficulties because the Kronecker delta property, which is satisfied by finite element shape functions, does not necessarily hold for the EFG shape function. The proposed new formulation of EFG eliminates this shortcoming through the moving kriging (MK) interpolation. Two major properties of the MK interpolation: the Kronecker delta property (?I( s J)=δIJ) and the consistency property (∑In?I( x )=1 and ∑In?I( x )xIi=xi) are proved. Some preliminary numerical results are given. Copyright © 2002 John Wiley & Sons, Ltd. 相似文献
10.
G. R. Liu G. Y. Zhang 《International journal for numerical methods in engineering》2008,74(7):1128-1161
It is well known that the displacement‐based fully compatible finite element method (FEM) provides a lower bound in energy norm for the exact solution to elasticity problems. It is, however, much more difficult to bound the solution from above for general problems in elasticity, and it has been a dream of many decades to find a systematical way to obtain such an upper bound. This paper presents a very important and unique property of the linearly conforming point interpolation method (LC‐PIM): it provides a general means to obtain an upper bound solution in energy norm for elasticity problems. This paper conducts first a thorough theoretical study on the LC‐PIM: we derive its weak form based on variational principles, study a number of properties of the LC‐PIM, and prove that LC‐PIM is variationally consistent and that it produces upper bound solutions. We then demonstrate these properties through intensive numerical studies with many examples of 1D, 2D, and 3D problems. Using the LC‐PIM together with the FEM, we now have a systematical way to numerically obtain both upper and lower bounds of the exact solution to elasticity problems, as shown in these example problems. Copyright © 2007 John Wiley & Sons, Ltd. 相似文献
11.
Shaofan Li Wing Kam Liu 《International journal for numerical methods in engineering》1999,45(3):251-288
This work is concerned with developing the hierarchical basis for meshless methods. A reproducing kernel hierarchical partition of unity is proposed in the framework of continuous representation as well as its discretized counterpart. To form such hierarchical partition, a class of basic wavelet functions are introduced. Based upon the built‐in consistency conditions, the differential consistency conditions for the hierarchical kernel functions are derived. It serves as an indispensable instrument in establishing the interpolation error estimate, which is theoretically proven and numerically validated. For a special interpolant with different combinations of the hierarchical kernels, a synchronized convergence effect may be observed. Being different from the conventional Legendre function based p‐type hierarchical basis, the new hierarchical basis is an intrinsic pseudo‐spectral basis, which can remain as a partition of unity in a local region, because the discrete wavelet kernels form a ‘partition of nullity’. These newly developed kernels can be used as the multi‐scale basis to solve partial differential equations in numerical computation as a p‐type refinement. Copyright © 1999 John Wiley & Sons, Ltd. 相似文献
12.
Edouard Yreux Jiun‐Shyan Chen 《International journal for numerical methods in engineering》2017,109(7):1045-1064
Reproducing kernel particle method (RKPM) has been applied to many large deformation problems. RKPM relies on polynomial reproducing conditions to yield desired accuracy and convergence properties but requires appropriate kernel support coverage of neighboring nodes to ensure kernel stability. This kernel stability condition is difficult to achieve for problems with large particle motion such as the fragment‐impact processes that commonly exist in extreme events. A new reproducing kernel formulation with ‘quasi‐linear’ reproducing conditions is introduced. In this approach, the first‐order polynomial reproducing conditions are approximately enforced to yield a nonsingular moment matrix. With proper error control of the first‐order completeness, nearly second‐order convergence rate in L2 norm can be achieved while maintaining kernel stability. The effectiveness of this quasi‐linear RKPM formulation is demonstrated by modeling several extremely large deformation and fragment‐impact problems. Copyright © 2016 John Wiley & Sons, Ltd. 相似文献
13.
Abstract: The present work refers to the errors imposed by the recently introduced random‐grid mesh‐free full field strain measurement method. Excluding systematic errors of the digital camera, the method itself is not an error‐free procedure. A possible cause of errors could be the misplacement of the spot‐centre (centroid) with regard of the spot boundaries. Another cause of errors is the limited order of approximation in the field function. A third one, emerges from the so‐called ‘sub‐pixel effect’. This kind of error is difficult to trace, so direct comparison between the results of the method and exact solutions is required. In the present work, proper analytical or numerical derivations of those errors are presented and reasonable upper limits are estimated. Finally, numerical and experimental examples are presented to demonstrate the accuracy of the method. 相似文献
14.
Shaofan Li Wing Kam Liu 《International journal for numerical methods in engineering》2000,48(9):1285-1309
In this paper, a comprehensive account on using mesh‐free methods to simulate strain localization in inelastic solids is presented. Using an explicit displacement‐based formulation in mesh‐free computations, high‐resolution shear‐band formations are obtained in both two‐dimensional (2‐D) and three‐dimensional (3‐D) simulations without recourse to any mixed formulation, discontinuous/incompatible element or special mesh design. The numerical solutions obtained here are insensitive to the orientation of the particle distributions if the local particle distribution is quasi‐uniform, which, to a large extent, relieves the mesh alignment sensitivity that finite element methods suffer. Moreover, a simple h‐adaptivity procedure is implemented in the explicit calculation, and by utilizing a mesh‐free hierarchical partition of unity a spectral (wavelet) adaptivity procedure is developed to seek high‐resolution shear‐band formations. Moreover, the phenomenon of multiple shear band and mode switching are observed in numerical computations with a relatively coarse particle distribution in contrast to the costly fine‐scale finite element simulations. Copyright © 2000 John Wiley & Sons, Ltd. 相似文献
15.
G. Y. Zhang G. R. Liu Y. Y. Wang H. T. Huang Z. H. Zhong G. Y. Li X. Han 《International journal for numerical methods in engineering》2007,72(13):1524-1543
Linearly conforming point interpolation method (LC‐PIM) is formulated for three‐dimensional elasticity problems. In this method, shape functions are generated using point interpolation method by adopting polynomial basis functions and local supporting nodes are selected based on the background cells. The shape functions so constructed have the Kronecker delta functions property and it allows straightforward imposition of point essential boundary conditions. Galerkin weak form is used for creating discretized system equations, and a nodal integration scheme with strain‐smoothing operation is used to perform the numerical integration. The present LC‐PIM can guarantee linear exactness and monotonic convergence for the numerical results. Numerical examples are used to examine the present method in terms of accuracy, convergence, and efficiency. Compared with the finite element method using linear elements, the LC‐PIM can achieve better efficiency, and higher accuracy especially for stresses. Copyright © 2007 John Wiley & Sons, Ltd. 相似文献
16.
Young‐Sam Cho Seyoung Im 《International journal for numerical methods in engineering》2006,65(4):494-516
Two‐dimensional variable‐node elements compatible with quadratic interpolation are developed using the moving least‐square (MLS) approximation. The mapping from the parental domain to the physical element domain is implicitly obtained from MLS approximation, with the shape functions and their derivatives calculated and saved only at the numerical integration points. It is shown that the present MLS‐based variable‐node elements meet the patch test if a sufficiently large number of integration points are employed for numerical integration. The cantilever problem with non‐matching meshes is chosen to check the feasibility of the present MLS‐based variable‐node elements, and the result is compared with that from the lower‐order case compatible with linear interpolation. Copyright © 2005 John Wiley & Sons, Ltd. 相似文献
17.
Sang‐Hoon Park Sung‐Kie Youn 《International journal for numerical methods in engineering》2001,52(9):997-1012
A new efficient meshfree method is presented in which the first‐order least‐squares method is employed instead of the Galerkin's method. In the meshfree methods based on the Galerkin formulation, the source of many difficulties is in the numerical integration. The current method, in this respect, has different characteristics and is expected to remove some of the integration‐related problems. It is demonstrated through numerical examples that the present formulation is highly robust to integration errors. Therefore, numerical integration can be performed with great ease and effectiveness using very simple algorithms. Copyright © 2001 John Wiley & Sons, Ltd. 相似文献
18.
M. de Buhan P. Frey 《International journal for numerical methods in engineering》2011,86(13):1544-1557
In this paper, we consider a non‐linear viscoelastic model with internal variable, thoroughly analyzed by Le Tallecit et al. (Comput. Methods Appl. Mech. Engrg 1993; 109 :233–258). Our aim is to study here the implementation in three dimensions of a generalized version of this model. Computational results will be analyzed to validate our model on toy problems without geometric complexity, for which pseudo‐analytical solutions are known. At the end, we present a three‐dimensional numerical simulation on a mechanical device. Copyright © 2011 John Wiley & Sons, Ltd. 相似文献
19.
Yuzhou Sun K. M Liew 《International journal for numerical methods in engineering》2012,91(11):1184-1198
This paper examines the interaction between coplanar square cracks by combining the moving least‐squares (MLS) approximation and the derived boundary integral equation (BIE). A new traction BIE involving only the Cauchy singular kernels is derived by applying integration by parts to the traditional boundary integral formulation. The new traction BIE can be directly applied to a crack surface and no displacement BIE is necessary because all crack boundary conditions (both upper and lower ones) are incorporated. A boundary element‐free method is then developed by combining the derived BIE and MLS approximation, in which the crack opening displacement is first expressed as the product of weight functions and the characteristic terms, and the unknown weight is approximated with the MLS approximation. The efficiency of the developed method is tested for isotropic and transversely isotropic media. The interaction between two and three coplanar square cracks in isotropic elastic body is numerically studied and the case of any number of coplanar square cracks is deduced and discussed. Copyright © 2012 John Wiley & Sons, Ltd. 相似文献
20.
直线回归的最小面积法 总被引：6，自引：0，他引：6
自变量、因变量的不同选择,用传统的最小二乘法得到的回归方程是不同的。提出直线回归的最小面积法解决这一问题,推导了该方法的计算公式并讨论了该方法的一些性质。 相似文献