ebook img

Precursors of Isogeometric Analysis: Finite Elements, Boundary Elements, and Collocation Methods PDF

602 Pages·2019·19.674 MB·English
Save to my drive
Quick download
Download
Most books are stored in the elastic cloud where traffic is expensive. For this reason, we have a limit on daily download.

Preview Precursors of Isogeometric Analysis: Finite Elements, Boundary Elements, and Collocation Methods

Solid Mechanics and Its Applications Christopher G. Provatidis Precursors of Isogeometric Analysis Finite Elements, Boundary Elements, and Collocation Methods Solid Mechanics and Its Applications Volume 256 Series editors J. R. Barber, Ann Arbor, USA Anders Klarbring, Linköping, Sweden Founding editor G. M. L. Gladwell, Waterloo, ON, Canada Aims and Scope of the Series Thefundamentalquestionsarisinginmechanicsare:Why?,How?,andHowmuch? The aim of this series is to provide lucid accounts written by authoritative researchersgivingvisionandinsightinansweringthesequestionsonthesubjectof mechanics as it relates to solids. The scope of the series covers the entire spectrum of solid mechanics. Thus it includes the foundation of mechanics; variational formulations; computational mechanics; statics, kinematics and dynamics of rigid and elastic bodies: vibrations of solids and structures; dynamical systems and chaos; the theories of elasticity, plasticity and viscoelasticity; composite materials; rods, beams, shells and membranes; structural control and stability; soils, rocks and geomechanics; fracture; tribology; experimental mechanics; biomechanics and machine design. Themedianlevelofpresentationistothefirstyeargraduatestudent.Sometexts aremonographs defining thecurrentstateofthe field; othersareaccessibletofinal year undergraduates; but essentially the emphasis is on readability and clarity. More information about this series at http://www.springer.com/series/6557 Christopher G. Provatidis Precursors of Isogeometric Analysis Finite Elements, Boundary Elements, and Collocation Methods 123 Christopher G.Provatidis Schoolof MechanicalEngineering National Technical University of Athens Athens, Greece ISSN 0925-0042 ISSN 2214-7764 (electronic) Solid MechanicsandIts Applications ISBN978-3-030-03888-5 ISBN978-3-030-03889-2 (eBook) https://doi.org/10.1007/978-3-030-03889-2 LibraryofCongressControlNumber:2018960750 ©SpringerNatureSwitzerlandAG2019 Thisworkissubjecttocopyright.AllrightsarereservedbythePublisher,whetherthewholeorpart of the material is concerned, specifically the rights of translation, reprinting, reuse of illustrations, recitation, broadcasting, reproduction on microfilms or in any other physical way, and transmission orinformationstorageandretrieval,electronicadaptation,computersoftware,orbysimilarordissimilar methodologynowknownorhereafterdeveloped. The use of general descriptive names, registered names, trademarks, service marks, etc. in this publicationdoesnotimply,evenintheabsenceofaspecificstatement,thatsuchnamesareexemptfrom therelevantprotectivelawsandregulationsandthereforefreeforgeneraluse. The publisher, the authors and the editors are safe to assume that the advice and information in this book are believed to be true and accurate at the date of publication. Neither the publisher nor the authorsortheeditorsgiveawarranty,expressorimplied,withrespecttothematerialcontainedhereinor for any errors or omissions that may have been made. The publisher remains neutral with regard to jurisdictionalclaimsinpublishedmapsandinstitutionalaffiliations. ThisSpringerimprintispublishedbytheregisteredcompanySpringerNatureSwitzerlandAG Theregisteredcompanyaddressis:Gewerbestrasse11,6330Cham,Switzerland Dedicated to my Parents: Gabriel and Elisabeth and my Family: Eleni, Gabriel, Athanasios, Paul Preface This book was written with the purpose to be a self-contained text which thor- oughly summarizes the attempts toward “CAD/CAE integration” so as to shorten thegapbetweenComputer-AidedEngineering(CAE)andComputer-AidedDesign (CAD)communities.Inordertomakethistopicfullyclear,besidesthetheory,alot ofbenchmarktests,rangedbetweenpotentialproblemsandelasticityproblems,are solved in an instructive way and are thoroughly commented. Havingcontributedwithmorethan60scientificpapersonthistopicsince1980s, Ifelttheneedtocommunicatemyoverallfindingsonanideathatfirstappearedin 1970s in industrial research centers (such as General Motors) and continues till nowadays in the framework of the contemporary nonuniform rational B-splines (NURBS)-based isogeometric analysis (IGA). Although our overall attempt was substantiallydevelopedwellbefore thedateonwhichIGAappeared,therealityof thisnumerous communityhasdeterminedthetitle ofthebook,i.e.,“Precursorsof Isogeometric Analysis” with emphasis given inengineering analysis. Summarizing most of that useful knowledge associated with the CAGD interpolations which preceded the NURBS interpolation and formed the basis for the development of relevant CAD-based macroelements, it will become clear that the global approxi- mation involved in all CAD-based methods (including the NURBS-based IGA) is the actual cause for the high accuracy of the numerical solution. Close contact with young researchers has given me the impression that a lot of basic knowledge has not been fully absorbed, and this book highly contributes on thecompletionofthedialoguebetweenfiniteelementanalysis(FEA)andtheCAD communities. The book deals with three computer methods, i.e., the Galerkin–Ritz (finite element method: FEM), the Global Collocation as well as the Boundary Element Method (BEM) using CAD-based macroelements (with global approxi- mation). This is the reason why this book refers to “CAD/CAE” instead of only “CAD/FEA” integration. Theextensivenumericalexamplesofthisbookrefertobothstaticanddynamic analysis. Except of Laplace and Poisson equation, particular attention was paid on the eigenvalue extraction of acoustic cavities for which closed-form analytical solutions exist. Therefore, the accuracy of the proposed single macroelements for vii viii Preface rectangular and circular cavities refers to an analytically known exact value, a fact that is not possible in free vibration analysis of elastic structures (where the exact solution is not accurately known and therefore is approximated). And since due to theHelmholtzdecomposition(u¼gradUþcurlW),thepartialdifferentialequation of elastic waves splits into two wave propagation equations, the first for the scalar potentialUandthesecondforthevectorpotentialW,thestudyofacousticsmaybe a “crash test” for foreseeing the accuracy of any computational method in elasto- dynamics.Thisfactisthephilosophyofthisbook:Westartwithacousticsandthen continue with elastodynamics. In the summer of 2018, my published work on this particular topic included: – 38 papers in conjunction with Global Galerkin–Ritz – 16 papers in conjunction with the Global Collocation method – 5 papers in conjunction with the Global Boundary Element Method. Synthesizing the numerical findings of the abovementioned papers as a whole, the book in hand consists of the following fourteen chapters: Chapter 1 is a summary of earlier activities in CAD/CAE integration and offers the background or a refreshment of basic knowledge to the reader. Chapter 2 refers to the basics of function interpolation/approximation and elements of CAGD. We start from univariate and then extend to multivariate functions. The main idea is that since in a 2D problem the solution Uðx;yÞ rep- resents a surface, then several possible CAGD interpolations can be used to interpolateit. Inother words,CAD surface interpolationsof this surface are useful for further analysis. Following the historical evolution, Coons, Gordon, Barnhill, Bezier, B-splines, and NURBS interpolations are summarized. In the sequence, Chaps. 3–8 refer to the application of global interpolation in conjunction with the Galerkin–Ritz formulation, whereas other methods are developed inChaps. 11 (collocation) and 12(boundary elements). Inmore details: Chapter3referstoCoonsinterpolation,whichischronologicallythefirstonein CAD theory. Large Coons elements are developed and a lot of engineering prob- lems are solved. Chapter4referstoGordontransfiniteinterpolation,whichischronologicallythe second one in CAD theory. Large Gordon elements are developed and a lot of engineering problems are solved. Chapter5referstoBarnhillinterpolationapplicabletotriangularpatches.Large Barnhill elements are developed and some engineering problems are solved. Chapter6referstononrationalBéziercurvesandsurfaces.Theequivalencywith elements of Lagrangian type is pointed out. The shortcoming of these elements in conjunction with conic sections is explained. Chapter 7 refers to B-splines, covering the older truncated power formulation (before Schoenberg 1946) and the “contemporary” Curry–Schoenberg (1966) enriched by de Boor (1972) and Cox (1972) recursive algorithms. One- and two-dimensional problems are solved. Chapter 8 refers to rational Bézier and rational B-splines (NURBS) surfaces. Preface ix Chapter 9 refers to the C1-continuity problems and focuses on plate bending macroelements. Chapter 10 refers to volume subregions (volume blocks) of hexahedral shape, which in this book are called “superbricks.” Chapter 11 refers to the global collocation method in conjunction with CAD-based macroelements. Chapter 12 refers to the Boundary Element Method in conjunction with CAD-based patches. Chapter 13 refers to domain decomposition problems. Chapter14isareviewoftheoverallperformanceofCAD-basedmacroelements in engineering analysis. Hoping that this textbook will substantially contribute in the CAD/CAE dialogue, bridging the older with the contemporary ideas. Athens, Greece Christopher G. Provatidis, Ph.D. August 2018 Professor of Computer Methods at NTUA Contents 1 Initial Attempts on CAD/CAE Integration . . . . . . . . . . . . . . . . . . . 1 1.1 The Conventional Meaning of Integrated CAD/CAE Systems. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1 1.2 The Meaning of CAD/CAE Integration Adopted in This Book . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2 1.3 CAD Interpolation. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3 1.4 CAE Methods. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7 1.4.1 Finite Element Method (FEM). . . . . . . . . . . . . . . . . . 8 1.4.2 Boundary Element Method (BEM) . . . . . . . . . . . . . . 9 1.4.3 Collocation Method . . . . . . . . . . . . . . . . . . . . . . . . . 11 1.5 Modules of CAD/CAE Integration . . . . . . . . . . . . . . . . . . . . . 11 1.5.1 Mesh Generation . . . . . . . . . . . . . . . . . . . . . . . . . . . 11 1.5.2 Transfinite Elements . . . . . . . . . . . . . . . . . . . . . . . . . 13 1.5.3 Later Attempts . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14 1.6 Recapitulation and Some Historical Remarks . . . . . . . . . . . . . 17 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 18 2 Elements of Approximation and Computational Geometry. . . . . . . 25 2.1 Description of Curve’s Geometry as Well as of a Physical Quantity Along It . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 25 2.2 Definitions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 26 2.3 Approximation Using Univariate Functions. . . . . . . . . . . . . . . 26 2.3.1 Taylor Power Series . . . . . . . . . . . . . . . . . . . . . . . . . 26 2.3.2 One-Dimensional Linear Interpolation . . . . . . . . . . . . 27 2.3.3 One-Dimensional (1D) Piecewise Linear Interpolation. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29 2.3.4 One-Dimensional (1D) Lagrange and Power Series Interpolation. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30 2.3.5 One-Dimensional (1D) Hermite Interpolation . . . . . . . 33 xi

See more

The list of books you might like

Most books are stored in the elastic cloud where traffic is expensive. For this reason, we have a limit on daily download.