Advanced Finite Element Technologies PDF
Preview Advanced Finite Element Technologies
CISM 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 LibraryofCongressControlNumber:2016936423 ©CISMInternationalCentreforMechanicalSciences2016 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 foranyerrorsoromissionsthatmayhavebeenmade. Printedonacid-freepaper ThisSpringerimprintispublishedbySpringerNature TheregisteredcompanyisSpringerInternationalPublishingAGSwitzerland 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:[email protected] ©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)
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