Table Of ContentCISM International Centre for Mechanical Sciences 566
Courses and Lectures
Jörg Schröder
Peter Wriggers Editors
Advanced
Finite
Element
Technologies
International Centre
for Mechanical Sciences
CISM International Centre for Mechanical
Sciences
Courses and Lectures
Volume 566
Series editors
The Rectors
Friedrich Pfeiffer, Munich, Germany
Franz G. Rammerstorfer, Vienna, Austria
Elisabeth Guazzelli, Marseille, France
The Secretary General
Bernhard Schrefler, Padua, Italy
Executive Editor
Paolo Serafini, Udine, Italy
Theseriespresentslecturenotes,monographs,editedworksandproceedingsinthe
field of Mechanics, Engineering, Computer Science and Applied Mathematics.
Purpose of the series is to make known in the international scientific and technical
community results obtained in some of the activities organized by CISM, the
International Centre for Mechanical Sciences.
More information about this series at http://www.springer.com/series/76
ö ö
J rg Schr der Peter Wriggers
(cid:129)
Editors
Advanced Finite Element
Technologies
123
Editors
Jörg Schröder PeterWriggers
Institut für Mechanik Institut für Kontinuumsmechanik
UniversitätDuisburg-Essen LeibnizUniversität Hannover
Essen Hannover
Germany Germany
ISSN 0254-1971 ISSN 2309-3706 (electronic)
CISMInternational Centre for MechanicalSciences
ISBN978-3-319-31923-0 ISBN978-3-319-31925-4 (eBook)
DOI 10.1007/978-3-319-31925-4
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Preface
Advanced Finite Element Technologies are essential for the solution of almost all
problems in computational mechanics. One of the great attractions of the finite
element method is its enormous range of applicability, which varies from classical
subjects like mechanical, aerospace, automotive, and civil engineering, to new
scientificdisciplineslikeinformationtechnology,appliedphysics,orbiomechanics.
Due to the substantial developments in several fields, as for instance materials
science, production methods or optimization processes, many engineering and
mathematical approaches for novel finite elements were developed during the last
decades. The growing demand for reliable, accurate, and highly efficient finite
elementsparticularlyinthefieldofnonlinearitieshasledtoanumberofinteresting
finite element formulations.
The CISM course on “Advanced Finite Element Technologies”, held in Udine
from October 6 to 10, 2014, was addressed to master students, doctoral students,
postdocs, and experienced researchers in engineering, applied mathematics, and
materials science who wished to broaden their knowledge in e.g. advanced mixed
Galerkin and least-squares FEM, discontinuous Galerkin methods as well as the
related mathematical analysis.
It is our pleasure to thank the lecturers of the CISM course: Ferdinando
Auricchio (Pavia, Italy), Antonio Huerta (Barcelona, Spain), Daya Reddy (Cape
Town, South Africa), Gerhard Starke (Essen, Germany), as well as the additional
contributorstotheseCISMlecturenotesAdrienLefieux(Atlanta,USA),Benjamin
Müller (Essen, Germany), Alessandro Reali (München, Germany), Alexander
Schwarz (Essen, Germany), Ruben Sevilla (Swansea, Wales), and Karl Steeger
(Essen,Germany).Wefurthermorethankthe55participantsfrom13countrieswho
madethecourseasuccess.Finally,weextendourthankstotheRectors,theBoard,
and the staff of CISM for the excellent support and kind help.
Jörg Schröder
Peter Wriggers
v
Contents
Functional Analysis, Boundary Value Problems and Finite
Elements . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1
Batmanathan Dayanand Reddy
Discretization Methods for Solids Undergoing Finite
Deformations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 17
Peter Wriggers
Three-Field Mixed Finite Element Methods in Elasticity. . . . . . . . . . . . 53
Batmanathan Dayanand Reddy
Stress-Based Finite Element Methods in Linear and Nonlinear
Solid Mechanics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 69
Benjamin Müller and Gerhard Starke
Tutorial on Hybridizable Discontinuous Galerkin (HDG) for
Second-Order Elliptic Problems. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 105
Ruben Sevilla and Antonio Huerta
Least-Squares Mixed Finite Element Formulations for Isotropic and
Anisotropic Elasticity at Small and Large Strains. . . . . . . . . . . . . . . . . 131
Jörg Schröder, Alexander Schwarz and Karl Steeger
Theoretical and Numerical Elastoplasticity. . . . . . . . . . . . . . . . . . . . . . 177
Batmanathan Dayanand Reddy
On the Use of Anisotropic Triangles with Mixed Finite Elements:
Application to an “Immersed” Approach for Incompressible
Flow Problems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 195
Ferdinando Auricchio, Adrien Lefieux and Alessandro Reali
vii
Functional Analysis, Boundary Value
Problems and Finite Elements
BatmanathanDayanandReddy
Abstract Thischapterpresents,first,anoverviewofthemathematicaltoolsrequired
to undertake studies of the well-posedness of linear boundary value problems and
theirapproximationsbyfiniteelements.Intheremainderofthiswork,thesetoolsare
usedtoexaminetheexistenceanduniquenessofsolutionstoweakboundaryvalue
problems, and convergence of finite element approximations. The emphasis is on
second-order partial differential equations, with the governing equations for linear
elasticitybeingthekeymodelproblem.
1 Introduction
Wewillbeconcernedwithboundaryvalueproblemsthatariseinsolidmechanics.
Thesetypicallytaketheformofasinglepartialdifferentialequation(PDE)or,often,a
systemofPDEs.Inapplicationssuchasplasticityandcontact,theproblemcomprises
asetofequationsaswellasinequalities.
Thepurposeofthischapteristopresentanoverviewofmathematicalfundamen-
tals that are essential to qualitative studies of boundary value problems as well as
theirapproximationsbythefiniteelementmethod.Theaimofaqualitativestudyis
togleaninformationaboutaproblemanditssolutionintheabsenceofaclosed-form
solution,whichisgenerallythecaseincomplexproblemsofcontinuummechanics.
Whenapproximateapproachessuchasthefiniteelementmethodareused,suchstud-
iesareabletotellusaboutthequalityofanapproximationanditsrateofconvergence
totheactualsolution,againintheabsenceofknowledgeabouttheexactsolution.
Wefocusinthischapteronlinearproblems,inwhichthegoverningequationtakes
theform Au = f.Here Aisalinearoperatorandtheright-handside f isgiven.A
simpleexampleistheproblemofanEuler–Bernoullibeam.Thegoverningequation
isafourth-orderdifferentialequation
B
B.D.Reddy( )
CentreforResearchinComputationalandAppliedMechanics,
UniversityofCapeTown,Rondebosch,SouthAfrica
e-mail:daya.reddy@uct.ac.za
©CISMInternationalCentreforMechanicalSciences2016 1
J.SchröderandP.Wriggers(eds.),AdvancedFiniteElementTechnologies,
CISMInternationalCentreforMechanicalSciences566,
DOI10.1007/978-3-319-31925-4_1
2 B.D.Reddy
d4u
Au = EI = f (1)
dx4
inwhichu denotes thetransversedisplacement, f theappliedloading,and E and
I are, respectively, Young’s modulus and the second moment of area of the beam
cross-section. From a qualitative point of view, we wish to know: (a) whether the
problemhasasolution,thatis,whether f ∈ R(A),therangeof A;and(b)whether
thatsolutionisunique,thatis,whetherthenullspaceof A,denoted N(A),consists
of only the zero element. These terms have yet to be defined. It will be seen that
the answer to the two questions will depend to some extent on the specification
of the boundary conditions. The range of A comprises all continuous functions.
Suppose,forexample,thattheload f isconstantandthattheboundaryconditions
areu(0)=u(cid:3)(cid:3)(L)=u(cid:3)(cid:3)(cid:3)(L)=0,forabeamoflength L:thatis,zerodisplacement
atoneendandzeromomentandshearforceattheother.Thisgivesthesolution
(cid:2) (cid:3)
f 1 1 1
u(x)= x4− Lx3+ L2x2+Cx . (2)
EI 24 6 4
ThereremainsoneconstantCtobefound,whichisasitshouldbebecausethereisa
furtherboundaryconditionthatneedstobespecifiedatx =0.Ifweassumethatthe
outstandingboundaryconditionisu(cid:3)(0)=0,sothattheendx =0isclamped,then
wefindthatC =0.Thenullspaceof Acomprisesallsolutionsusuchthat Au =0,
andheretheonlysuchsolutionisu =0,sothat N(A)={0}.Thesolution(2)with
C =0isthusunique.
Ontheotherhand,assumethattheboundaryconditionisu(cid:3)(cid:3)(0),sothatwehave
zero moment at the end x = 0. Physically it is clear that the system cannot be in
equilibrium. This is confirmed by the fact that C is now undetermined: N(A) =
{u |u(x) =Cx}whichgivesaninfinitenumberoffurthersolutionscorresponding
tothebeamrotatingabouttheendx =0.
The above simple example illustrates for a somewhat obvious case the kind of
informationthatcanbeobtainedbyseekingqualitativeinformationaboutthesolu-
tion.Weshallformalizethisprocessinthefollowingsections.
A further example of a boundary value problem is the system of equations for
equilibriumofisotropiclinearelasticbodies.Thegoverningequationsoftheproblem
areasfollows:
Equilibrium: −divσ = f, (3a)
Hooke’slaw: σ =Cε(u)=λdiv u+2με(u), (3b)
Strain-displacement: ε(u)= 1(∇u+[∇u]T). (3c)
2
HereλandμaretheLaméconstants,σ isthestressandε(u)isthestraintensor.
SubstitutionintheequilibriumequationyieldstheLaméequation
Au:=−div [Cε(u)]=−(λ+μ)∇ div u−μ∇2u= f (4)
FunctionalAnalysis,BoundaryValueProblemsandFiniteElements 3
Fig.1 Deformationofan
elasticbar
foragivenbodyforce f.Tothissystemofequationsasetofboundaryconditions
mustbeadded.ForthedomainshowninFig.1,thedisplacementu=0ontheend
x =0whilethetractiont =σnisprescribedovertherestoftheboundary.
APDEoforder2mrequiresmboundaryconditionsateachpointontheboundary,
with derivatives of order no greater than 2m −1. Thus the Euler–Bernoulli beam
equation is an equation of the fourth order and this requires two boundary condi-
tionsateachendofthebeam.TheelasticityproblemleadstoasystemofPDEsof
second order, and so requires one vector-valued boundary condition at each point
ontheboundary.Forasecond-orderPDEorsystemofPDEsaboundarycondition
involvingthedisplacementiscalledaDirichletcondition,whilethatinvolvingthe
first derivatives of the displacement, through the traction for example, is called a
Neumannboundarycondition.
1.1 WeakFormulations
Questionsaroundtheexistenceofsolutionsandtheiruniquenessmaybeeffectively
approachednotonlybystudyingtheoriginalPDEandboundaryconditions,butalso
byreformulatingtheprobleminaweakorvariationalform.Wetakethegoverning
equationsasamodelproblemforequilibriumofalinearelasticbodyassetoutin
(4) and use these to show how the weak formulation is constructed. Suppose that
thebodyoccupiesthedomain(cid:2)⊂Rd (d =2,3)withboundary(cid:3)comprisingtwo
nonoverlappingparts(cid:3) and(cid:3) .Supposefurtherthattheboundaryconditionsare
u t
u=0 on (cid:3) , t =σn= t on (cid:3) . (5)
u t
We start by introducing a test function v, which is a function smooth enough to
bedifferentiated,andwhichsatisfiesthehomogeneousDirichletboundarycondition
v =0on(cid:3) .Next,taketheinnerproductofbothsidesofEq.(4)withvandintegrate
u
overthedomain(cid:2):
(cid:4) (cid:4)
− div Cε(u)·vdx = f ·vdx. (6)
(cid:2) (cid:2)
