Meshless Methods and Their Numerical Properties

Meshless Methods and Their Numerical Properties

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Features

  • Provides a detailed numerical analysis of widely used meshless approximation methods
  • Demonstrates the numerical properties of each method developed by the authors with differential equations
  • Covers the recent development of meshless methods, including contributions by the authors
  • Include several examples of engineering applications, such as the two-dimensional simulation of pH-sensitive hydrogel, to show how to solve multiphysics problems
  • Offers benchmark results for various engineering applications of meshless methods to help compare and verify other numerical methods
  • Supplies template functions, flowcharts, and data structures to guide researchers in implementing meshless methods
  • Contains more than 185 illustrations

Summary

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.

Table of Contents

Introduction

Formulation of Classical Meshless Methods
Introduction
Fundamentals of Meshless Methods
Common Steps of Meshless Method
Classical Meshless Methods
Summary

Recent Development of Meshless Methods
Introduction
Hermite-Cloud Method
Point Weighted Least-Squares Method
Local Kriging (LoKriging) Method
Variation of Local Point Interpolation Method (vLPIM)
Random Differential Quadrature (RDQ) Method
Summary

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
Summary

Stability Analyses
Introduction
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
Summary

Adaptive Analysis
Introduction
Error Recovery Technique in ARDQ Method
Adaptive RDQ Method
Convergence Analysis in ARDQ Method
Summary

Engineering Applications
Introduction
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
Summary

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

Index

Author Bio(s)

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.

Editorial Reviews

"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

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