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

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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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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ON SOME ENRICHMENTS OF reproducing kernel particle method

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INTERNATIONAL JOURNAL OF COMPUTATIONAL methodS 2004年第3期1卷 519-533页

作者： Zhou, J. X. Wang, X. M. Zhang, Z. Q. Zhang, L. Xi An Jiao Tong Univ Sch Civil Engn & Mech Xian 710049 Peoples R China

In this paper efforts are made to enrich reproducing kernel particle method (RKPM). Firstly, the RKPM shape functions are expressed explicitly in terms of kernel function moments. This avoids numerical matrix inversions and solutions of linear algebraic equations which are involved in classical RKPM, and thus makes RKPM more accurate, faster and more efficient. Then, a recently developed truly meshless body integration technique is introduced into RKPM. It is based on a partition of unity by a set of overlapping patches covering the domain and eliminates background cells completely. We borrow an idea from computational geometry and propose a sweeping-line method to determine the quadrature points inside the domain. The method is robust and effective even for domain with complicated shapes. The truly meshless integration technique in combination with this sweeping-line method make implementation of RKPM quite simple, smart and especially very advantageous when nodes are irregularly scattered. Numerical results presented herein demonstrate that these enrichments make RKPM more efficient, versatile and particularly truly meshless.

关键词： Meshless method reproducing kernel particle method partition of unity

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Numerical aspects of a real-time sub-structuring technique in structural dynamics

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INTERNATIONAL JOURNAL FOR NUMERICAL methodS IN ENGINEERING 2007年第11期72卷 1261-1313页

作者： Sajeeb, R. Roy, D. Manohar, C. S. Indian Inst Sci Dept Civil Engn Struct Lab Bangalore 560012 Karnataka India

A time domain coupling technique, involving combined computational and experimental modelling, for vibration analysis of structures built-up of linear/non-linear substructures is developed. The study permits, in principle, one or more of the substructures to be modelled experimentally with measurements being made only on the interfacial degrees of freedom. The numerical and experimental substructures are allowed to communicate in real time within the present framework. The proposed strategy involves a two-stage scheme: the first is iterative in nature and is implemented at the initial stages of the solution in a non-real-time format;the second is non-iterative, employs an extrapolation scheme and proceeds in real time. Issues on time delays during communications between different substructures are discussed. An explicit integration procedure is shown to lead to solutions with high accuracy while retaining path sensitivity to initial conditions. The stability of the integration scheme is also discussed and a method for numerically dissipating the temporal growth of high-frequency errors is presented. For systems with non-linear substructures, the integration procedure is based on a multi-step transversal linearization method;and, to account for time delays, we employ a multi-step extrapolation scheme based on the reproducing kernel particle method. Numerical illustrations on a few low-dimensional vibrating structures are presented and these examples are fashioned after problems of seismic qualification testing of engineering structures using real-time substructure testing techniques. Copyright (c) 2007 John Wiley & Sons, Ltd.

关键词： real-time sub-structuring numerical and experimental sub-structures extrapolations reproducing kernel particle method explicit and implicit time-stepping schemes

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Development of parallel 3D RKPM meshless bulk forming simulation system

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ADVANCES IN ENGINEERING SOFTWARE 2007年第2期38卷 87-101页

作者： Wang, H. Li, Guangyao Han, X. Zhong, Zhi Hua Hunan Univ Coll Mech & Automot Engn State Key Lab Adv Technol Vehicle Design & Manufa Changsha Peoples R China

A parallel computational implementation of modern meshless system is presented for explicit for 3D bulk forming simulation problems. The system is implemented by reproducing kernel particle method. Aspects of a coarse grain parallel paradigm-domain decompose method-are detailed for a Lagrangian formulation using model partitioning. Integration cells are uniquely assigned on each process element and particles are overlap in boundary zones. Partitioning scheme multilevel recursive spectrum bisection approach is applied. The parallel contact search algorithm is also presented. Explicit message passing interface statements are used for all communication among partitions on different processors. The parallel 3D system is developed and implemented into 3D bulk metal forming problems, and the simulation results demonstrated the efficiency of the developed parallel reproducing kernel particle method system. (C) 2006 Elsevier Ltd. All rights reserved.

关键词： parallel reproducing kernel particle method bulk forming domain decomposition method partition message passing interface speedup

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Immersed finite element method for fluid-structure interactions

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JOURNAL OF FLUIDS AND STRUCTURES 2007年第6期23卷 839-857页

作者： Zhang, L. T. Gay, M. Rensselaer Polytech Inst Dept Mech Aerosp & Nucl Engn Troy NY 12180 USA

In this paper, we present a detailed derivation of the numerical method, Immersed Finite Element method (IFEM), for the solution of fluid-structure interaction problems. This method is developed based on the Immersed Boundary (IB) method that was initiated by Peskin, with additional capabilities in handling nonuniform and independent meshes and applying arbitrary boundary conditions on both fluid and solid domains. A higher order interpolation function is adopted from one of the mesh-free methods, the reproducing kernel particle method (RKPM), which relieves the uniformity constraint of fluid meshes. Two 2-D example problems are presented to illustrate the capabilities of the algorithm. The accuracy in the numerical analysis demonstrates that the IFEM algorithm is a reliable and robust numerical approach to solve fluid and deformable solid interactions. (c) 2007 Published by Elsevier Ltd.

关键词： immersed finite element method fluid-structure interactions Immersed boundary method reproducing kernel particle method

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On the utilization of the reproducing kernel particle method for the numerical simulation of plane strain rolling

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INTERNATIONAL JOURNAL OF MACHINE TOOLS & MANUFACTURE 2003年第1期43卷 89-102页

作者： Shangwu, X Liu, WK Cao, J Rodrigues, JMC Martins, PAF Inst Super Tecn Seccao Tecnol Mecan P-1049001 Lisbon Portugal Northwestern Univ Dept Mech Engn Evanston IL 60208 USA Inst Super Tecn Dept Engn Mecan P-1049001 Lisbon Portugal

This paper draws from fundamental research on the reproducing kernel particle method (RKPM) to the development of an innovative numerical approach for analyzing rolling under plane strain conditions. The approach is based on the flow formulation for slightly compressible rigid-plastic materials and a detailed description of the method and its numerical implementation is presented with the objective of making clear the fundamental differences to the well-established finite element method for slightly compressible rigid-plastic materials. Special emphasis is placed on the construction of shape functions and their derivatives, enforcement of the essential boundary conditions and treatment of frictional effects, along the contact interface between the workpiece and the roll. The effectiveness of the proposed approach is discussed by comparing theoretical predictions with experimental data found in the literature. (C) 2002 Published by Elsevier Science Ltd.

关键词： plane strain rolling reproducing kernel particle method slightly compressible rigid-plastic materials

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Enriched reproducing kernel particle Approximation for simulating problems involving moving interfaces

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Lecture Notes in Computational Science and Engineering 2007年 57卷 149-164页

作者： Joyot, Pierre Trunzler, Jean Chinesta, Fransisco LIPSI ESTIA Technopole Izarbel 64210 Bidart France LMSP 151 Bd. de l'Hôpital 75013 Paris France

ISBN: (纸本)9783540462149

In this paper we propose a new approximation technique within the context of meshless methods able to reproduce functions with discontinuous derivatives. This approach involves some concepts of the reproducing kernel particle method (RKPM), which have been extended in order to reproduce functions with discontinuous derivatives. This strategy will be referred as Enriched reproducing kernel particle Approximation (E-RKPA). The accuracy of the proposed technique will be compared with standard RKP approximations (which only reproduces polynomials).

关键词： Discontinuous derivatives Enriched approximation Meshless methods reproducing kernel particle method

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Immersed finite element method and its applications to biological systems

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COMPUTER methodS IN APPLIED MECHANICS AND ENGINEERING 2006年第13-16期195卷 1722-1749页

作者： Liu, WK Liu, YL Farrell, D Zhang, L Wang, XS Fukui, Y Patankar, N Zhang, YJ Bajaj, C Lee, J Hong, JH Chen, XY Hsu, HY Northwestern Univ Dept Mech Engn Evanston IL 60208 USA Tulane Univ Dept Mech Engn New Orleans LA 70118 USA New Jersey Inst Technol Dept Math Sci Newark NJ 07102 USA Northwestern Univ Feinberg Sch Med Dept Cell & Mol Biol Chicago IL 60611 USA Univ Texas Dept Comp Sci Inst Computat Engn & Sci Austin TX 78712 USA Univ Texas Dept Comp Sci Dept Comp Sci Austin TX 78712 USA Seoul Natl Univ Sch Mech & Aerosp Engn Seoul 151742 South Korea

This paper summarizes the newly developed immersed finite element method (IFEM) and its applications to the modeling of biological systems. This work was inspired by the pioneering work of Professor T.J.R. Hughes in solving fluid-structure interaction problems. In IFEM, a Lagrangian solid mesh moves on top of a background Eulerian fluid mesh which spans the entire computational domain. Hence, mesh generation is greatly simplified. Moreover, both fluid and solid domains are modeled with the finite element method and the continuity between the fluid and solid subdomains is enforced via the interpolation of the velocities and the distribution of the forces with the reproducing kernel particle method (RKPM) delta function. The proposed method is used to study the fluid-structure interaction problems encountered in human cardiovascular systems. Currently, the heart modeling is being constructed and the deployment process of an angioplasty stent has been simulated. Some preliminary results on monocyte and platelet deposition are presented. Blood rheology, in particular, the shear-rate dependent de-aggregation of red blood cell (RBC) clusters and the transport of deformable cells, are modeled. Furthermore, IFEM is combined with electrokinetics to study the mechanisms of nano/bio filament assembly for the understanding of cell motility. (c) 2005 Elsevier B.V. All rights reserved.

关键词： immersed finite element method reproducing kernel particle method fluid-structure interaction cytoskeletal dynamics red blood cell aggregation thrombosis cardiovascular system micro-circulation naiio-electro-mechanical-sensors cell motility surgical corrective procedures

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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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