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COMPUTER methodS IN APPLIED MECHANICS AND ENGINEERING 2004年第21-22期193卷 2051-2067页

作者： Zhang, L Gerstenberger, A Wang, XD Liu, WK Northwestern Univ Inst Technol Dept Mech Engn Evanston IL 60208 USA Polytech Univ MetroTech Ctr 6 Dept Mech Engn Brooklyn NY 11201 USA

In this paper, the immersed finite element method (IFEM) is proposed for the solution Of Complex fluid and deformable structure interaction problems encountered in many physical models. In IFEM, a Lagrangian solid mesh moves on top of a background Eulerian fluid mesh which spans over the entire computational domain. Hence, the mesh generation is greatly simplified. Moreover, both fluid and solid domains are modeled with the finite element methods and the continuity between the fluid and solid sub-domains are enforced via the interpolation of the velocities and the distribution of the forces with the reproducing kernel particle method (RKPM) delta function. In comparison with the immersed boundary (1B) method, the higher-ordered RKPM delta function enables the fluid domain to have non-uniform spatial meshes with arbitrary geometries and boundary conditions. The use Of Such kernel functions may eventually open doors to multi-scale and multi-resolution modelings of complex fluid-structure interaction problems. Rigid and deformable spheres dropping in channels are simulated to demonstrate the unique capabilities of the proposed method. The results compare well with the experimental data. To the authors' knowledge, these are the first solutions that deal with particulate flows with very flexible solids. (C) 2004 Elsevier B.V. All rights reserved.

关键词： immersed finite element method reproducing kernel particle method fluid-structure interaction particulate flow immersed boundary method

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reproducing kernel hierarchical partition of unity, part I - Formulation and theory

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INTERNATIONAL JOURNAL FOR NUMERICAL methodS IN ENGINEERING 1999年第3期45卷 251-288页

作者： Li, SF Liu, WK Northwestern Univ Dept Engn Mech Evanston IL 60208 USA

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 focal region, because the discrete wavelet kernels form a 'partition of nullity'. These newly developed kernels can be used as the multiscale basis to solve partial differential equations in numerical computation as a p-type refinement. Copyright (C) 1999 John Wiley & Sons, Ltd.

关键词： meshless hierarchical partition of unity wavelet methods moving least-squares interpolant reproducing kernel particle method synchronized convergence

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Meshfree truncated hierarchical refinement for isogeometric analysis

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COMPUTATIONAL MECHANICS 2018年第6期62卷 1583-1597页

作者： Atri, H. R. Shojaee, S. Shahid Bahonar Univ Kerman Dept Civil Engn Kerman Iran

In this paper truncated hierarchical B-spline (THB-spline) is coupled with reproducing kernel particle method (RKPM) to blend advantages of the isogeometric analysis and meshfree methods. Since under certain conditions, the isogeometric B-spline and NURBS basis functions are exactly represented by reproducing kernel meshfree shape functions, recursive process of producing isogeometric bases can be omitted. More importantly, a seamless link between meshfree methods and isogeometric analysis can be easily defined which provide an authentic meshfree approach to refine the model locally in isogeometric analysis. This procedure can be accomplished using truncated hierarchical B-splines to construct new bases and adaptively refine them. It is also shown that the THB-RKPM method can provide efficient approximation schemes for numerical simulations and represent a promising performance in adaptive refinement of partial differential equations via isogeometric analysis. The proposed approach for adaptive locally refinement is presented in detail and its effectiveness is investigated through well-known benchmark examples.

关键词： Truncated hierarchical B-spline Meshfree reproducing kernel particle method NURBS Refinement

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Explicit 3-D RKPM shape functions in terms of kernel function moments for accelerated computation

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COMPUTER methodS IN APPLIED MECHANICS AND ENGINEERING 2005年第9-11期194卷 1027-1035页

作者： Zhou, JX Wang, XM Zhang, ZQ Zhang, L Xian Jiaotong Univ Sch Civil Engn & Mech Xian 710049 Peoples R China

The construction of meshless shape functions is more time-consuming than evaluation of FEM shape functions. Therefore. it is of great importance to take measures to speed up the computation of meshless shape functions. 3-D meshless shape functions and their derivatives are. in the context of reproducing kernel particle method (RKPM). expressed explicitly in terms of kernel function moments for the very first time. This avoids solutions of linear algebraic equations and numerical inversions encountered in standard RKPM implementation. thus speeds up computation of meshless shape functions. A numerical test is performed in a hexahedral domain with the mere purpose of comparing the computation time for shape functions construction between the standard RKPM implementation and the enhanced procedure. Then two 3-D elastostatics;numerical examples are presented. which demonstrate that the proposed unique treatment of RKPM shape functions is especially effective. (C) 2004 Elsevier B.V. All rights reserved.

关键词： meshless method reproducing kernel particle method shape functions moments

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Meshless numerical solution for nonlocal integral differentiation equation with application in peridynamics

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ENGINEERING ANALYSIS WITH BOUNDARY ELEMENTS 2022年第0期144卷 569-582页

作者： Yao, Wu -Wen Zhou, Xiao-Ping Chongqing Univ Sch Civil Engn Chongqing 400045 Peoples R China

The mid-point quadrature rule is often used to solve the nonlocal integral differentiation equations, in which a central material point is used to calculate the integral of the subdomain in the meshless geometry. However, it can only achieve the linear approximation and the first order convergence rate, and it is only accurate enough when the entire cell overlaps with the neighborhood of a material point. In this paper, to achieve the higher precision and the higher rate of convergence speed, a novel coupling method of reproducing kernel particle method and Gaussian-Legendre quadrature scheme is proposed to solve the nonlocal integral differentiation equations in the meshless geometry, in which the mid-point quadrature scheme is replaced with Gauss quad-rature scheme to calculate the integral of the subdomain in the meshless geometry, the reproducing kernel particle method is employed to construct the shape functions for material points in the nonlocal model, and to approximate the value of the Gaussian quadrature node. Moreover, the meshless solution for the peridynamic equation is derived detailedly based on the proposed method. Compared with the traditional meshless solution for the nonlocal equations, the proposed solution has higher precision and higher rate of convergence speed in different problems. The numerical results demonstrate improved results in both mathematics and mechanical problems using the proposed method.

关键词： Nonlocal integral differentiation equations Meshless numerical solution reproducing kernel particle method Gaussian-Legendre quadrature Peridynamics

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The meshless method for a two-dimensional parabolic problem with a source parameter

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APPLIED MATHEMATICS AND COMPUTATION 2008年第2期202卷 730-737页

作者： Cheng, Rongjun Ge, Hongxia Zhejiang Univ Ningbo Inst Technol Dept Fundamental Course Ningbo 315100 Zhejiang Peoples R China Anhui Normal Univ Dept Math Wuhu 241000 Peoples R China Ningbo Univ Fac Sci Ningbo 315211 Peoples R China

In this paper, the reproducing kernel particle method (RKPM) is used for finding the solution of a two-dimensional parabolic inverse problem with a source control parameter, and the corresponding discrete equations are obtained. Comparing with the numerical methods based on mesh, the reproducing kernel particle method only needs the scattered nodes instead of meshing the domain of the problem. The reproducing kernel particle method is an efficient mesh free technique for the numerical solution of partial differential equations. The result of numerical example is presented. (C) 2008 Elsevier Inc. All rights reserved.

关键词： reproducing kernel particle method meshless method parabolic equation overspecification inverse problem

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Free vibrations of elastically embedded stocky single-walled carbon nanotubes acted upon by a longitudinally varying magnetic field

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MECCANICA 2015年第12期50卷 3041-3067页

作者： Kiani, Keivan KN Toosi Univ Technol Dept Civil Engn Tehran Iran

The mechanical properties of nano-scaled composites reinforced by carbon nanotubes are enhanced by application of appropriate magnetic fields, however, little is known on the free dynamic response of the magnetically affected stocky single-walled carbon nanotubes (SWCNTs) with elastic supports. Using nonlocal Rayleigh, Timoshenko, and higher-order beam theory, the equations of free transverse vibration of elastically embedded SWCNTs subjected to a longitudinally varying magnetic field are obtained. Since finding an analytical solution to the equations of motion is a very difficult task, an efficient meshless method is proposed. The frequencies of the magnetically affected stocky SWCNTs are evaluated for different boundary conditions. The convergence checks of the proposed numerical models are carried out. In a special case, the obtained results are also compared with those of assumed mode method, and a reasonably good agreement is achieved. Subsequently, the roles of the slenderness ratio of the SWCNT, small-scale parameter, strength of the magnetic field, lateral and rotational interactions of the SWCNT with its surrounding medium on the fundamental frequency are addressed in some detail. The capabilities of the proposed models in capturing the frequencies of the magnetically affected nanostructure are also comprehensively investigated.

关键词： Single-walled carbon nanotube Longitudinally varying magnetic field Transverse vibration Nonlocal beam theories reproducing kernel particle method

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A variational multiscale stabilized and locking-free meshfree formulation for Reissner-Mindlin plate problems

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COMPUTATIONAL MECHANICS 2022年第1期69卷 59-93页

作者： Huang, Tsung-Hui Natl Tsing Hua Univ Dept Power Mech Engn Hsinchu Taiwan

In this study, a variational multiscale stabilized locking-free meshfree formulation is introduced for modeling Reissner-Mindlin plate problems under arbitrary plate thickness. Under this framework, the plate quantities are decoupled into coarse-scale and fine-scale components in the variational equations, where the fine-scale solution represents a correction to the residual of the coarse-scale equations that can be solved by an effective collocation-type approach with an approximation method meeting locking free conditions. The substitution of fine-scale solutions in the coarse-scale system leads to a residual-based Galerkin formulation. In the proposed framework, the reproducing kernel approximation, as well as the smoothed gradient and divergence, are adopted to ensure the bending exactness in the Galerkin formulation. The multiscale approach is also beneficial for problems exhibiting localized phenomena. The effectiveness of the proposed method is tested by solving a series of numerical examples and compared with classical methods.

关键词： reproducing kernel particle method Stabilized nodal integration Variational multiscale method Shear locking Reissner-Mindlin plate

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A combined meshfree/finite strip method for analysis of plates with perforations and cracks

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THIN-WALLED STRUCTURES 2017年第Feb.期111卷 113-125页

作者： Khezri, M. Abbasi, M. Rasmussen, K. J. R. Univ Sydney Sch Civil Engn Sydney NSW 2006 Australia

This work presents a coupling of the finite strip method (FSM) and the meshless reproducing kernel particle method (RKPM), which is inspired by the complementary features of both methods. A straightforward and effective method is developed to accurately integrate these methods, and the obtained coupled technique is utilised for analysing two-dimensional linear elastic plates weakened by arbitrarily shaped perforations or cracks. The RKPM is used in the regions surrounding the cracks, i.e. parts of the domain with high gradients and irregular boundaries, whereas the FSM is adopted in the rest of the domain. The sub-domains of these methods are conjoined over transition zones by utilising a modified collocation approach which ensures the linear consistency of the approximation field. A series of numerical examples and studies of the obtained convergence rates are presented to demonstrate the accuracy and effectiveness of the developed coupled method.

关键词： Finite strip method Meshless reproducing kernel particle method Combined method Perforated plates

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Boundary locking induced by penalty enforcement of essential boundary conditions in Mesh-free methods

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COMPUTER methodS IN APPLIED MECHANICS AND ENGINEERING 2008年第13-16期197卷 1167-1183页

作者： Cho, Jin Yeon Song, You Me Choi, Yun Hyuk Inha Univ Dept Aerosp Engn Inchon 402751 South Korea

Mesh-free methods continue to have problems in enforcing the essential boundary conditions because of the absence of the Kronecker delta property in mesh-free shape functions, constructed from moving least squares method, reproducing kernel particle method, and others. In this paper, this problem in dealing with the essential boundary conditions is examined. Boundary locking, induced by the penalty enforcement of the essential boundary conditions in mesh-free methods, is first explored, and the reason for obtaining an over-constrained solution is explained theoretically. To justify these observations, various numerical examples are also presented. Additionally, based on further theoretical investigations, a remedy to circumvent this problem of the over-constraint phenomena, caused by the penalty enforcement of essential boundary conditions, is proposed. (c) 2007 Elsevier B.V. All rights reserved.

关键词： boundary locking Penalty method Essential boundary conditions Mesh-free method over-constraint phenomena moving least squares method reproducing kernel particle method

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