Table Of ContentSolid 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
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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
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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
