Error Estimates for Advanced Galerkin Methods

505 Pages·2019·22.575 MB·English

by  Marcus Olavi Rüter

Lecture Notes in Applied and Computational Mechanics 88
Marcus Olavi Rüter
Error Estimates
for Advanced
Galerkin
Methods
Lecture Notes in Applied and Computational
Mechanics
Volume 88
Series Editors
Peter Wriggers, Institut für Baumechanik und Numerische Mechanik, Leibniz
Universität Hannover, Hannover, Niedersachsen, Germany
Peter Eberhard, InstituteofEngineering andComputational Mechanics, University
of Stuttgart, Stuttgart, Germany
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ü
Marcus Olavi R ter
Error Estimates for Advanced
Galerkin Methods
123
Marcus OlaviRüter
Department ofCivil andEnvironmental Engineering
University of California, LosAngeles
USA
ISSN 1613-7736 ISSN 1860-0816 (electronic)
Lecture Notesin AppliedandComputational Mechanics
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Preface
The finite element method (FEM) enjoys tremendous popularity for solving (initial)
boundary value problems that arise in the broad field of Computational Mechanics.
However,certainshortcomingscannotbedenied;forexampleitsinabilitytodealwith
arbitrary crack-propagation problems, which goes along with being tied to a mesh,
whoseconstructioncanbeatediousundertaking.Thepasttwodecadeshavewitnessed
thedevelopmentofnovelapproachesthataimtoremedyseveralshortcomingsofthe
finiteelementmethod.Sincetheseapproachesevolvedfromthefiniteelementmethod,
which is the archetypal (mesh-based) Galerkin method, they can be considered
advancedGalerkinmethods.Theextendedfiniteelementmethod(XFEM)isonesuch
advanced Galerkin method in the sense of a mesh-based method, which allows for
arbitrarycrackpropagationthroughtheelements.Galerkinmeshfreemethods,suchas
theelement-freeGalerkin(EFG)andreproducingkernelparticlemethods(RKPM),go
onestepfurtherandintroducenewtypesofshapefunctionswithhighregularityintothe
Galerkin method. As its name implies, the scattered particles in combination with
arbitrary support sizes of the associated shape functions naturally bypass the con-
structionofamesh.TheadvancedGalerkinmethodspresentedinthismonographare
appliedtothefiniteandlinearizedhyperelasticityproblemswithafocusonthelatter
theory. Since crack propagation problems play a central role throughout this mono-
graph,theframeworksofbothNewtonianandEshelbianmechanicsarediscussed.
Inmostcases,theshapefunctionsandthenumericalintegrationschemesusedin
Galerkin mesh-based and meshfree methods are not able to reproduce the exact
solution to the problem at hand. As a consequence, errors are introduced into the
method.ToensurereliabilityandtoimprovetheaccuracyoftheGalerkinsolution,
these errors need to be controlled. Central aspects of this monograph therefore
concern discussions of the sources of error in terms of both shape functions and
(classical and modern) numerical integration schemes and the control of error in
terms of a posteriori error estimation procedures.
Even though the development of error estimation procedures and the construc-
tion of meshfree shape functions seem at first sight not to share any similarities, it
turns out that the well-known least-squares method can be applied to both. To be
more precise, the least-squares method can be used to find the “best” solution to a
v
vi Preface
linear system of equations that has no solution. Such linear systems appear in
various error estimation procedures that are built upon a postprocessing of the
Galerkin solution. On the other hand, the same concept can be used to derive
meshfree shape functions that lead to the “best” global approximants rather than
interpolants, as is the case in mesh-based methods. Their approximate character,
however, causes difficulties in imposing Dirichlet boundary conditions, which is a
generalprobleminGalerkinmeshfreemethods.It istherefore demonstratedinthis
monograph how the least-squares method can be modified to allow for the
(restricted)impositionofDirichletboundaryconditions.Theseexplanationsareone
example that pave the way to more sophisticated methods, such as the moving
least-squares and the related reproducing kernel particle methods, rather than pro-
vidingpracticallyusefulmethodsontheirownfollowingoneofthemainobjectives
of this monograph that consists of providing deeper insight into various methods
and concepts for a better understanding of the numerical simulation process based
on Galerkin methods. Along this line, this monograph also contains various
appendices, which should help the reader understand the theories presented in this
monograph in the best case without resorting to additional literature.
The basis for this monograph is my PhD and particularly my habilitation thesis
thatIwrotein2016andthatincludespartsofmyworkcarriedoutbetween2005and
2016 at Aalto University, Finland, the Leibniz Universität Hannover (LUH),
Germany, and the University of California, Los Angeles (UCLA), USA. As such,
thismonographisintendedtoassistgraduatestudents,researchers,andpractitioners,
who aim to explore advanced Galerkin methods and a posteriori error estimation
proceduresrelatedtothesemethods.BecauseofthemanyadditionalsectionsthatI
included afterwards and that provide a basic understanding of the subject matter,
several sections can also be well understood by undergraduate students. This is
particularly the case because some sections are based on my experience as an
instructor of various undergraduate and graduate classes held at the Civil and
EnvironmentalEngineering (CEE) DepartmentattheUniversity ofCalifornia,Los
Angeles. These classes include “Applied Numerical Computing and Modeling in
Civil and Environmental Engineering”, “Elementary Structural Mechanics”, and
“Finite ElementAnalysis ofStructures”.
I would like to express my gratitude to the Series Editors, i.e. the Profs. Peter
WriggersandPeterEberhard,forinvitingmetocontributetothisdistinguishedseries
from Springer. Along this line, I would also like to thank Pierpaolo Riva and
Arunkumar Raviselvam from Springer for a good collaboration. Since this mono-
graphreliesonmyhabilitationthesis,Iwouldliketoexpressmyappreciationtothe
reviewers,i.e.theProfs.Jiun-Shyan(JS)Chen,UdoNackenhorst,JörgSchröder,and
thelateProf.ErwinStein,thechairmanofthecommitteeProf.RaimundRolfes,and
allothercommitteemembers.IamparticularlygratefultothelateProf.ErwinStein
and Prof. Jiun-Shyan (JS) Chen, who helped shape my academic career in myriad
waysandwhointroducedmetotheinterestingworldsofaposteriorierrorestimation
procedures and Galerkin meshfree methods. It goes without saying that the work
detailed in this monograph would not have been possible without the fruitful dis-
cussions, teaching possibilities, and joint works with my former supervisors and
Preface vii
esteemed colleagues, the Profs. and Drs. Ulrich Brink, Sheng-Wei Chi, Tymofiy
Gerasimov,MichaelHillman,SergeyKorotov,FredrikLarsson,StephanOhnimus,
Kenneth Runesson, Rolf Stenberg, and Ertugrul Taciroglu. I would further like to
thanktheGermanResearchFoundation(DFG),SandiaNationalLaboratories(SNL),
andtheUSArmyEngineerResearchandDevelopmentCenter(ERDC)forproviding
fundingofmyworkandGrantM.Galloway,B.S.andDr.HaoyanWeiforhelpingme
findmistakesandforprovidingvaluablesuggestionsonthestyleofthismonograph.
Myacknowledgmentswouldnotbecompletewithoutthankingmyparentsforpro-
vidingthemostpeacefulandinspirationalworkenvironmentontheFinnishlakeside
thatIcanthinkof.
I would like to dedicate this monograph to my former supervisor, mentor, and
friend—one of the most prolific and reputable pioneers of Computational
Mechanics—Prof.ErwinStein,whopassedawayattheageof87onDecember19,
2018.Atthattime,ErwinSteinhadstartedtowritetheforewordtothismonograph,
which he wanted to provide before he unexpectedly passed away. It had been his
unfinished dream for a long time to write a monograph about his point of view on
error-controlled adaptive finite element methods applied to various engineering
problemsthathewaspassionateabout.In2016,ashisplansbecamemoreprecise,
weaimedtowritethismonographtogetheruntilmyhabilitation(andtheemanating
present monograph) thwarted these plans. He told me during one of our many
conversations: “Marcus, you have to write the monograph now by yourself. I am
confident that you will succeed to write it in our characteristic engineering style.”
Erwin Stein was truly one of a kind. He was an avid scientific polyglot, who was
inexorably committed to theoretical and practical engineering, mathematics, phi-
losophy, and particularly engineering history, always following Leibniz’s motto
“theoria cum praxis”. Not only were his vast wine and fishing expertise second to
none, his fine character, boundless energy, broad general knowledge, and pros-
perousscientificcareerwereallreflectedinthemanyaccoladeshereceived,e.g.the
Gauss-Newton,Ritz-Galerkin,andZienkiewiczmedals,tomentionafew.Whenhe
wasonceinvitedtowriteanobituary,hetoldmenotonlytowritetheusualpraise
but also to mention the rough edges and flaws of a person that belong to their
character. Using his advice in good faith, I would like to mention his assertive,
direct,andcharismaticpersonality,whichdidnotmakediscussionswithhimeasy,
buthewasalwaysfairandhadhisheartintherightplace.DearErwin,Iamgrateful
forthetimeweshareddiscussingresearch,foryourfriendship,foryourinspiration
and support in various ways, and for the vast contributions that you left for the
scientific community and beyond. I remember the bust of Gottfried Wilhelm
Leibniz, whom you admired, in your garden. You placed the bust proudly and
purposely so that you could look up at and to Leibniz. Like you looked up to
Leibniz, I will always look up to you.
Los Angeles, USA Marcus Olavi Rüter
March 2019
Contents
1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1
1.1 Computational Validation and Verification Strategies . . . . . . . . . . 1
1.2 Advanced Galerkin Methods. . . . . . . . . . . . . . . . . . . . . . . . . . . . 5
1.3 A Posteriori Error Estimates . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7
1.4 On Duality in Computational Mechanics . . . . . . . . . . . . . . . . . . . 9
1.5 Organization of the Monograph. . . . . . . . . . . . . . . . . . . . . . . . . . 10
References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11
2 Newtonian and Eshelbian Mechanics . . . . . . . . . . . . . . . . . . . . . . . . 15
2.1 Kinematics. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 16
2.1.1 The Deformation of an Elastic Body. . . . . . . . . . . . . . . . 16
2.1.2 Strain Measures . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 20
2.2 Balance Equations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 22
2.2.1 Conservation of Mass. . . . . . . . . . . . . . . . . . . . . . . . . . . 22
2.2.2 The Concept of Mechanical Stress . . . . . . . . . . . . . . . . . 23
2.2.3 Stress Measures . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 23
2.2.4 Conservation of Physical and Pseudo Momenta. . . . . . . . 25
2.2.5 Conservation of Moments of Physical and Pseudo
Momenta. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 27
2.2.6 Conservation of Energy . . . . . . . . . . . . . . . . . . . . . . . . . 29
2.2.7 The Entropy Inequality. . . . . . . . . . . . . . . . . . . . . . . . . . 30
2.2.8 The Clausius-Planck Inequality. . . . . . . . . . . . . . . . . . . . 31
2.2.9 Thermodynamic Potentials . . . . . . . . . . . . . . . . . . . . . . . 32
2.3 Constitutive Equations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33
2.3.1 The Specific Strain-energy Function . . . . . . . . . . . . . . . . 33
2.3.2 Compressible Hyperelastic Materials. . . . . . . . . . . . . . . . 35
2.3.3 (Nearly) Incompressible Hyperelastic Materials . . . . . . . . 41
2.3.4 On the Choice of the Specific Strain-energy Function . . . 48
References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 52
ix
x Contents
3 Boundary Value Problems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 55
3.1 Compressible Elasticity. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 56
3.1.1 Finite Hyperelasticity . . . . . . . . . . . . . . . . . . . . . . . . . . . 56
3.1.2 Linearized Elasticity. . . . . . . . . . . . . . . . . . . . . . . . . . . . 60
3.1.3 The Poisson Problem . . . . . . . . . . . . . . . . . . . . . . . . . . . 64
3.1.4 Uniaxial Deformations . . . . . . . . . . . . . . . . . . . . . . . . . . 67
3.2 (Nearly) Incompressible Elasticity . . . . . . . . . . . . . . . . . . . . . . . . 69
3.2.1 Finite Hyperelasticity . . . . . . . . . . . . . . . . . . . . . . . . . . . 69
3.2.2 Linearized Elasticity. . . . . . . . . . . . . . . . . . . . . . . . . . . . 71
References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 73
4 Galerkin Methods. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 75
4.1 Galerkin Weak Forms. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 76
4.1.1 Weighted Residual Methods . . . . . . . . . . . . . . . . . . . . . . 76
4.1.2 Finite-dimensional Test and Solution Spaces . . . . . . . . . . 79
4.1.3 From Continuous to Discrete Problems . . . . . . . . . . . . . . 80
4.2 The Finite Element Method. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 84
4.2.1 Preliminaries . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 84
4.2.2 Lagrangian Interpolants . . . . . . . . . . . . . . . . . . . . . . . . . 85
4.2.3 The Isoparametric Concept . . . . . . . . . . . . . . . . . . . . . . . 88
4.2.4 The Conventional Qk- and the Mixed
Qk-Pk(cid:1)1-elements. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 90
4.2.5 The Mixed RT -P -element. . . . . . . . . . . . . . . . . . . . . . . 91
0 0
4.3 The Extended Finite Element Method . . . . . . . . . . . . . . . . . . . . . 94
4.3.1 The Extended Finite Element Interpolant. . . . . . . . . . . . . 95
4.3.2 An Alternative Set of Crack Tip Enrichment
Functions. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 98
4.4 The Element-free Galerkin and Reproducing Kernel
Particle Methods. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 99
4.4.1 Least-squares Approximants . . . . . . . . . . . . . . . . . . . . . . 101
4.4.2 Moving Least-squares Approximants. . . . . . . . . . . . . . . . 112
4.4.3 Reproducing Kernel Approximants . . . . . . . . . . . . . . . . . 128
4.4.4 Differentiation of the MLS and RK Shape Functions . . . . 133
4.4.5 Nitsche’s Method. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 139
References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 143
5 Numerical Integration. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 149
5.1 Gauss Quadrature. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 151
5.1.1 The Integration Scheme . . . . . . . . . . . . . . . . . . . . . . . . . 151
5.1.2 The Linear Integration Constraint . . . . . . . . . . . . . . . . . . 153
5.2 Stabilized Conforming Nodal Integration . . . . . . . . . . . . . . . . . . . 154
5.2.1 The Enhanced Assumed Strain Method. . . . . . . . . . . . . . 155
5.2.2 On the Construction of the Enhanced Strains. . . . . . . . . . 158
5.2.3 Stability and Consistency . . . . . . . . . . . . . . . . . . . . . . . . 160