Meshless Methods and Their Numerical Properties: 1st Edition (Paperback) book cover

Meshless Methods and Their Numerical Properties

1st Edition

By Hua Li, Shantanu S. Mulay

CRC Press

447 pages | 187 B/W Illus.

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Meshless, or meshfree methods, which overcome many of the limitations of the finite element method, have achieved significant progress in numerical computations of a wide range of engineering problems. A comprehensive introduction to meshless methods, Meshless Methods and Their Numerical Properties gives complete mathematical formulations for the most important and classical methods, as well as several methods recently developed by the authors. This book also offers a rigorous mathematical treatment of their numerical properties—including consistency, convergence, stability, and adaptivity—to help you choose the method that is best suited for your needs.

Get Guidance for Developing and Testing Meshless Methods

Developing a broad framework to study the numerical computational characteristics of meshless methods, the book presents consistency, convergence, stability, and adaptive analyses to offer guidance for developing and testing a particular meshless method. The authors demonstrate the numerical properties by solving several differential equations, which offer a clearer understanding of the concepts. They also explain the difference between the finite element and meshless methods.

Explore Engineering Applications of Meshless Methods

The book examines how meshless methods can be used to solve complex engineering problems with lower computational cost, higher accuracy, easier construction of higher-order shape functions, and easier handling of large deformation and nonlinear problems. The numerical examples include engineering problems such as the CAD design of MEMS devices, nonlinear fluid-structure analysis of near-bed submarine pipelines, and two-dimensional multiphysics simulation of pH-sensitive hydrogels. Appendices supply useful template functions, flowcharts, and data structures to assist you in implementing meshless methods.

Choose the Best Method for a Particular Problem

Providing insight into the special features and intricacies of meshless methods, this is a valuable reference for anyone developing new high-performance numerical methods or working on the modelling and simulation of practical engineering problems. It guides you in comparing and verifying meshless methods so that you can more confidently select the best method to solve a particular problem.


"This monograph is a blast of extensive and detailed mathematical exposition of meshless methods. It would serve as a quick reference guide for any one studying advanced numerical methods. The content has a blend of both physics and computational mathematics."

—Dr. Dominic Chandar, University of Wyoming, Laramie

"This book contains a comprehensive description of meshless approaches and their numerical algorithms. The easy-to-understand text clarifies some of the most advanced techniques for providing detailed mathematical derivation and worked examples where appropriate."

—Qinghua Qin, Australian National University, Canberra

"The authors presented a thorough, balanced, and informative review of the meshless method, which has been one of the most exciting areas in computational mechanics in the past decade. Drawing on their long-time experience and excellent works on meshless method, the authors covered the different aspects of meshless method skillfully and addressed many of the essential and tough issues including stability head on. The book is an excellent reference for scientists and engineers interested in meshless method and, more generally, numerical methods for partial differential equations."

—Rui Qiao, Associate Professor, Department of Mechanical Engineering, Clemson University, USA

Table of Contents


Formulation of Classical Meshless Methods


Fundamentals of Meshless Methods

Common Steps of Meshless Method

Classical Meshless Methods


Recent Development of Meshless Methods


Hermite-Cloud Method

Point Weighted Least-Squares Method

Local Kriging (LoKriging) Method

Variation of Local Point Interpolation Method (vLPIM)

Random Differential Quadrature (RDQ) Method


Convergence and Consistency Analyses

Introduction to Convergence Analysis

Development of Superconvergence Condition

Convergence Analysis

Application of RDQ Method for Solving Fixed-Fixed and Cantilever Microswitches under Nonlinear Electrostatic Loading

Introduction to Consistency Analysis of RDQ Method

Consistency Analysis of Locally Applied DQ Method

Effect of Uniform and Cosine Distributions of Virtual Nodes on Convergence of RDQ Method


Stability Analyses


Stability Analysis of First Order Wave Equation by RDQ Method

Stability Analysis of Transient Heat Conduction Equation

Stability Analysis of the Transverse Beam Deflection Equation


Adaptive Analysis


Error Recovery Technique in ARDQ Method

Adaptive RDQ Method

Convergence Analysis in ARDQ Method


Engineering Applications


Application of Meshless Methods to Microelectromechanical System Problems

Application of Meshless Method in Submarine Engineering

Application of RDQ Method for 2-D Simulation of pH-Sensitive Hydrogel


Appendix A: Derivation of Characteristic Polynomial Φ(Z)

Appendix B: Definition of Reduced Polynomial Φ1(Z)

Appendix C: Derivation of Discretization Equation by Taylor Series

Appendix D: Derivation of Ratio of Successive Amplitude Reduction Values for Fixed-Fixed Beam using Explicit and Implicit Approaches

Appendix E: Source Code Development


About the Authors

Dr. Hua Li is currently an assistant professor at the School of Mechanical and Aerospace Engineering at Nanyang Technological University in Singapore. His research interests include the modeling and simulation of MEMS, focusing on the use of smart hydrogels in BioMEMS applications; the development of advanced numerical methodologies; and the dynamics of high-speed rotating shell structures. He has authored or co-authored several books and book chapters, as well as more than 110 articles published in top international peer-reviewed journals. His research has been extensively funded by agencies and industries and he acted as the principal investigator of a computational BioMEMS project awarded under A*STAR’s Strategic Research Programme in MEMS.

Dr. Shantanu S. Mulay currently works as a postdoctoral associate with Professor Rohan Abeyaratne of Massachusetts Institute of Technology as part of the Singapore–MIT Alliance for Research and Technology (SMART). Before joining Nanyang Technological University (NTU), Dr. Mulay worked in product enhancement of DMU (CATIA workbench) and the development of NISA (FEM product), where he gained exposure to a variety of areas such as the development of CAD translators, computational geometry, and handling user interfaces of FEM products. During his Ph.D. program at NTU, Dr. Mulay worked extensively in the field of computational mechanics and developed a meshless random differential quadrature (RDQ) method.

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