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Methods of Fundamental Solutions in Solid Mechanics presents the fundamentals of continuum mechanics, the foundational concepts of the MFS, and methodologies and applications to various engineering problems. Eight chapters give an overview of meshless methods, the mechanics of solids and structures, the basics of fundamental solutions and radical basis functions, meshless analysis for thin beam bending, thin plate bending, two-dimensional elastic, plane piezoelectric problems, and heat transfer in heterogeneous media. The book presents a working knowledge of the MFS that is aimed at solving real-world engineering problems through an understanding of the physical and mathematical characteristics of the MFS and its applications. Explains foundational concepts for the method of fundamental solutions (MFS) for the advanced numerical analysis of solid mechanics and heat transfer Extends the application of the MFS for use with complex problems Considers the majority of engineering problems, including beam bending, plate bending, elasticity, piezoelectricity and heat transfer Gives detailed solution procedures for engineering problems Offers a practical guide, complete with engineering examples, for the application of the MFS to real-world physical and engineering challenges
This book mainly focuses on the major area: Computational Mechanics. Computational mechanics is widely used in nanomechanics and micromechanics, continuum mechanics, and many other mechanical systems. The main focus throughout this book will be to address methods concerning the field of continuum mechanics. Continuum mechanics studies bodies at the macroscopic level by developing continuum models with a homogenized microstructure. The two traditional areas of application are solid and thermal-fluid mechanics.Over the past century, energy and variational principles have become popular methods when obtaining approximate solutions to practical problems in applied mechanics. In addition, these methods enable engineers to carry out more effective simulations. In fact, most simulation and computation software are based upon concepts from energy and variational approaches.This book combines the essential ideas and methods behind current energy applications and variational theory in theoretical, applied mechanics. The emphasis is on understanding physical and computational applications of variational methodology rather than on rigorous mathematical formalism.Although there are some excellent books for engineering analysis using variational techniques to solve engineering problems, in this manuscript, we intend to guide the reader through the classical topics of energy and variational principles through the fundamental concepts to the extent of a first-year graduate student. What makes this book distinct from all others is that students usually grasp abstract and complex formulations through problem-solving, which is the major strength of this book.This book is intended to provide a theoretical and practical foundation for approximations to differential equations, including the finite element method. The target audience is first-year graduate students who have had little exposure to energy and variational principles. Practicing engineers will also benefit from the approach of this manuscript as they will be able to learn the theoretical aspects of typical approximation methods such as the finite element methods, basically, by their own. Thus, we can assure that this book will fill up a void in the personal library of many engineers who are trying to, or planning, to these methods in their next analysis.
This monograph focuses on the numerical methods needed in the context of developing a reliable simulation tool to promote the use of renewable energy. One very promising source of energy is the heat stored in the Earth’s crust, which is harnessed by so-called geothermal facilities. Scientists from fields like geology, geo-engineering, geophysics and especially geomathematics are called upon to help make geothermics a reliable and safe energy production method. One of the challenges they face involves modeling the mechanical stresses at work in a reservoir. The aim of this thesis is to develop a numerical solution scheme by means of which the fluid pressure and rock stresses in a geothermal reservoir can be determined prior to well drilling and during production. For this purpose, the method should (i) include poroelastic effects, (ii) provide a means of including thermoelastic effects, (iii) be inexpensive in terms of memory and computational power, and (iv) be flexible with regard to the locations of data points. After introducing the basic equations and their relations to more familiar ones (the heat equation, Stokes equations, Cauchy-Navier equation), the “method of fundamental solutions” and its potential value concerning our task are discussed. Based on the properties of the fundamental solutions, theoretical results are established and numerical examples of stress field simulations are presented to assess the method’s performance. The first-ever 3D graphics calculated for these topics, which neither requiring meshing of the domain nor involving a time-stepping scheme, make this a pioneering volume.
This book is the 3rd edition of an introduction to modern computational mechanics based on the finite element method. This third edition is largely extended, adding many new examples to let the reader understand the principles better by performing calculations by hand, as well as numerical example to practice the finite element approach to engineering problems. The new edition comes together with a set of digital flash cards with questions and answers that improve learning success. Featuring over 100 more pages, the new edition will help students succeed in mechanics courses by showing them how to apply the fundamental knowledge they gained in the first years of their engineering education to more advanced topics. In order to deepen readers’ understanding of the equations and theories discussed, each chapter also includes supplementary problems. These problems start with fundamental knowledge questions on the theory presented in the respective chapter, followed by calculation problems. In total, over 80 such calculation problems are provided, along with brief solutions for each. Test your knowledge with questions and answers about the book in the Springer Nature Flashcards app.
A self-contained and systematic development of an aspect of analysis which deals with the theory of fundamental solutions for differential operators, and their applications to boundary value problems of mathematical physics, applied mathematics, and engineering, with the related computational aspects.
This book is an introduction to computational mechanics, proceeding from basic computational tools to advanced computational procedures and applications. Emphasis is placed on the numerical techniques and how they form the bases for algorithms. Numerous worked examples in structural mechanics, heat transfer, fluid flow, and biomechanics are given with the numerical codes to illustrate how the methods are applied. A concluding section addresses advanced applications in such areas as finite volume methods and biomechanics.
This book explores the numerical algorithms underpinning modern finite element based computational mechanics software. It covers all the major numerical methods that are used in computational mechanics. It reviews the basic concepts in linear algebra and advanced matrix theory, before covering solution of systems of equations, symmetric eigenvalue solution methods, and direct integration of discrete dynamic equations of motion, illustrated with numerical examples. This book suits a graduate course in mechanics based disciplines, and will help software developers in computational mechanics. Increased understanding of the underlying numerical methods will also help practicing engineers to use the computational mechanics software more effectively.
Methods of mathematical modelling applied in contemporary computational mechanics can be divided into purely numerical and analytical-numerical procedures. In this book, the first part is a general presentation of the boundary collocation approach and its numerous variants and in the second part the method is applied to many engineering problems.
During the last two decades the boundary element method has experienced a remarkable evolution. Contemporary concepts and techniques leading to the advancements of capabilities and understanding of the mathematical and computational aspects of the method in mechanics are presented. The special emphasis on theoretical and numerical issues, as well as new formulations and approaches for special and important fields of solid and fluid mechanics are considered. Several important and new mathematical aspects are presented: singularity and hypersingular formulations, regularity, errors and error estimators, adaptive methods, Galerkin formulations, coupling of BEM-FEM and non-deterministic (stochastic and fuzzy) BEM formulations. Novel developments and applications of the boundary element method in various fields of mechanics of solids and fluids are considered: heat conduction, diffusion and radiation, non-linear problems, dynamics and time-depending problems, fracture mechanics, thermoelasticity and poroelasticity, aerodynamics and acoustics, contact problems, biomechanics, optimization and sensitivity analysis problems, ill posed and inverse problems, and identification problems.