paisintroductiontotheextendedfiniteelementmethodxfem

Pais(2010)IntroductionPais(2010)IntroductiontotheExtendedFiniteEltotheExtendedFiniteElementMethod(XFEM)ementMethod(XFEM)Structural&Multidisciplinary Optimization GroupMechanical and Aerospace Engineering DepartmentOverviewBasic ConceptsLevel Set MethodExtended Finite Element Method(XFEM)Basic formulationLevel set representations of cracks,inclusions,and voidsEnrichment functions for cracks,inclusions,and voidsIntegration of enriched elementsApplications to fluidsReanalysis of XFEMMotivationAlgorithmBasic analysis of methodInitial resultsConclusions2Structural&Multidisciplinary Optimization GroupMechanical and Aerospace Engineering DepartmentEnergy Release Rate/SIFG,energy release rate amount of energy released by unit advance of crackIn general a crack will grow in the direction which maximizes GKi,stress intensity factor characterizes the magnitude of amplification of applied stress at crack tipEspecially useful for linear elastic case which predicts r-1/2 singularity at crack tipSingle value gives state of stress around the crack tip2D:KI and KII,3D:KI,KII,and KIII Mode I:OpeningMode II:In-Plane ShearMode III:Out-of-Plane Shear3Structural&Multidisciplinary Optimization GroupMechanical and Aerospace Engineering DepartmentLevel Set MethodThe level set method1,2 is a front tracking method given in terms of the front velocity with respect to timeThe level set function is commonly discretized and interpolation between points is performedAmmenable to use in FE environmentHas been used in shape optimization3 within the FE framework1Osher et al,1988,Fronts propagating with curvature dependent speed,J.Comp.Phys.,79,12-49.2Stolarksa et al,2001,Modeling crack growth by level sets in the XFEM,Int.J.Num.Meth.Eng.,51,943-960.3Wang et al,2004,Structural shape and topology optimization in a level set based framework,Struct.Multi.Opt.,27,1-19.4Structural&Multidisciplinary Optimization GroupMechanical and Aerospace Engineering DepartmentModeling Crack Growth in FEFinite element mesh corresponds to the domain of the given materialUse singular elements at crack tip to represent the asymptotic crack tip displacement field1As crack grows,must recreate mesh around the crack tip,which can be expensive2Creates challenges in tracking time history of points near crack which are being remeshedDisplacement,stress,or strain1Barsoum,1976,On the use of isoparametric finite elements for linear fracture mechanics,Int.J.Num.Meth.Eng.,10,25-37.2Maligno et al,2010,A three-dimensional numerical study of fatigue crack growth using remeshing,Eng.Frac.Mech.,77,94-111.5Structural&Multidisciplinary Optimization GroupMechanical and Aerospace Engineering DepartmenteXtended Finite Element Method(XFEM)Belytschko1Strong DiscontinuitiesCracksDaux2 and Sukumar3Weak DiscontinuitiesInclusionsVoidsDiscontinuous behavior embedded into elements using local enrichment functions,additional nodal DOFsDoes not require mesh to conform to domainNo remeshing needed for evolving discontinuitiesLevel sets4 used to track discontinuitiesCrackInclusionVoid1Belytschko et al,1999,Elastic crack growth in finite elements with minimal remeshing,Int.J.Num.Meth.Eng.,45,601-620.2Daux et al,2000,Arbitrary branched and intersecting cracks with XFEM,Int.J.Num.Meth.Eng.,48,1741-1760.3Sukumar et al,2001,Modeling holes and inclusion by level sets in the XFEM,Comp.Meth.App.Mech.Eng,190,6183-6200.4Osher et al,1988,Fronts propagating with curvature dependent speed,J.Comp.Phys.,79,12-49.6Structural&Multidisciplinary Optimization GroupMechanical and Aerospace Engineering DepartmentGeneral XFEM ApproximationUse lowercase to represent enrichment,upper case for shiftedShift1 enrichment to recover FEM approximation at nodes where enrichment function is nonzeroInterested in enrichment functions representing discontinuitiesSuperimposing continuous and discontinuous approximationsEnrichment FunctionAdditional Spatial DOF at Node ITraditional FEM Approximation1Belytschko et al,2001,Arbitrary discontinuities in finite elements,Int.J.Num.Meth.Eng.,50,993-1013.7Structural&Multidisciplinary Optimization GroupMechanical and Aerospace Engineering DepartmentLevel Set Representation of CracksIntroduced by Stolarska1 as an extension of the work by Sethian2Intersection of two level sets defines the open sectionCan be updated if desired using well known methodsNarrow band3,fast marching method4 decrease computational time1Stolarksa et al,2001,Modeling crack growth by level sets in the XFEM,Int.J.Num.Meth.Eng.,51,943-960.2Osher et al,1988,Fronts propagating with curvature dependent speed,J.Comp.Phys.,79,12-49.3Adalsteinsson et at,1995,A Fast Level Set Method for Propagating Interfaces,J.Comp.Phys.,118,269-277.4Sethian,1996,A Fast Marching Level Set Method for Monotonically Advancing Fronts,Proc.Nat.Acad.Sci.,93,1591-1595.Structural&Multidisciplinary Optimization GroupMechanical and Aerospace Engineering DepartmentCrack Tip Enrichment FunctionCrack tip enrichment functions1 embed the crack tip singularity into the enriched elementAdditional enrichment functions available for other crack tip conditionsBi-material,branching,cohesive,functionally graded and orthotropic materials2Crack1Fleming et al,1997,Enriched element-free Galerkin methods for crack tip fields,Int.J.Num.Meth.Eng.,40,1483-1504.2Belytschko et al,2009,A review of extended/generalized FEM for material modeling,Int.J.Num.Meth.Eng.,17,043001.9Structural&Multidisciplinary Optimization GroupMechanical and Aerospace Engineering DepartmentHeaviside Enrichment FunctionCrackHeaviside1 function is used in elements which have their support completely cut by the crackPlaces discontinuity directly at crack location within element1Moes et al,1999,A finite element method for crack growth without remeshing,Int.J.Num.Meth.Eng.,46,131-150.10Structural&Multidisciplinary Optimization GroupMechanical and Aerospace Engineering DepartmentCrack Enriched ElementsNodes with Heaviside Enrichment(2 additional DOF)Nodes with Crack Tip Enrichment(8 additional DOF)NnN n裂尖裂尖单元元贯穿穿单元元裂尖裂尖单元元常常规单元元11Structural&Multidisciplinary Optimization GroupMechanical and Aerospace Engineering DepartmentDiscrete EquationscutbntoWeighted ResidualStrain-Displacement RelationshipEquilibrium EquationConstitutive EquationBoundary Conditions12Structural&Multidisciplinary Optimization GroupMechanical and Aerospace Engineering DepartmentWeighted Residual Method13Structural&Multidisciplinary Optimization GroupMechanical and Aerospace Engineering DepartmentXFEM Stiffness Matrixu are traditional DOFa,b are enriched DOFKuu is independent of crack location,traditional FE stiffness matrixKua,Kaa,Kab are components with Heaviside enrichmentKub,Kab,Kbb are components with crack tip enrichmentKua,Kub,Kab are add coupling between traditional,enriched DOF14Structural&Multidisciplinary Optimization GroupMechanical and Aerospace Engineering DepartmentExample:Mixed-Mode Edge CrackHElement Size,hNormalized KINormalized KII1/100.9581.0281/200.9801.0221/400.9871.0181/800.9901.016HWaEvKI,theoKII,theo2 m1 m(0,1)to(0.4,1.4)1 N/m1 MPa0.31.9270.8191Sutradhar,Paulino,Gray.Symmetric Galerkin Boundary Element Method.Springer-Verlag,2008.Structural&Multidisciplinary Optimization GroupMechanical and Aerospace Engineering DepartmentCrack Opening with GrowthDisplacement is magnified to show effect of enrichment.16Structural&Multidisciplinary Optimization GroupMechanical and Aerospace Engineering DepartmentRecent Work on CracksArea of crack tip enrichment to improve convergence rate1Corrected XFEM2 to remove problems with blending elementsIntroduction of harmonic enrichment functions3 to unify enrichment for branching,homogenous,and intersecting cracksIntroduction of optimized enrichment functions4 for homogeneous and cohesive cracksUse of XFEM for interpretation of structural health monitoring(SHM)data through optimization5Use of XFEM for optimization of structure with respect to fatigue life61Laborde et al,2005,High-Order XFEM for Cracked Domains,Int.J.Num.Meth.Eng.,64,354-381.2Fries,2007,A Corrected XFEM Approximation Without Problems in Blending Elements,Int.J.Num.Meth.Eng.,75,503-532.3Mousavi et at,2010,A Unified Treatment of Multiple,Intersecting and Branched Cracks in XFEM,Int.J.Num.Meth.Eng.,In Press.4Abbas et al,2010,A Unified Enrichment Scheme for Fracture Problems,WCCM/APCOM 2010,Sydney,Australia.5Waisman et al,2009,Detection and Quantification of Flaws in Structures by XFEM,10th US Nat.Cong.Comp.Mech.,Columbus,OH.6Edke,Shape Optimization for 2D Mixed-Mode Fracture Using XFEM and LSM,Struct.Mutli.Opt.,In Press.Structural&Multidisciplinary Optimization GroupMechanical and Aerospace Engineering Department18Level Set Representation of Inclusion/VoidIntroduced by Sukumar1 as an extension of the work done by Daux2Define negative values to be interior of inclusion or voidCan be updated if desired using well known methodsNarrow band3,fast marching method4 decrease computational time1Sukumar et al,2001,Modeling Holes and Inclusion by Level Sets in the XFEM,Comp.Meth.App.Mech.Eng,190,6183-6200.2Daux et al,2000,Arbitrary Branched and Intersecting Cracks with XFEM,Int.J.Num.Meth.Eng.,48,1741-1760.3Adalsteinsson et at,1995,A Fast Level Set Method for Propagating Interfaces,J.Comp.Phys.,118,269-277.4Sethian,1996,A Fast Marching Level Set Method for Monotonically Advancing Fronts,Proc.Nat.Acad.Sci.,93,1591-1595.Structural&Multidisciplinary Optimization GroupMechanical and Aerospace Engineering Department19Inclusion Enrichment FunctionInterface1Moes et al,2003,A Comp.Approach to Handle Complex Microstructure Geometries,Comp.Meth.App.Mech.Eng,192,3163-3177.=Level Set at Node IInclusion function1 is used in elements with multiple materialsNo need to shift enrichment as it is zero at all nodesStructural&Multidisciplinary Optimization GroupMechanical and Aerospace Engineering Department20Void Enrichment FunctionMaterial1Daux et al,2000,Arbitrary branched and intersecting cracks in the XFEM,Int.J.Num.Meth.Eng.,48,1741-1760.Void function1 is used in elements which contain void boundaryNo additional degrees of freedom,modifies displacement directlyOnly perform numerical integration in regions which contain materialVoidStructural&Multidisciplinary Optimization GroupMechanical and Aerospace Engineering Department21Integration of Enriched ElementFor enriched element,subdivide1 quadrilateral into triangles and integrate over each triangle to avoid difficulties with integrating discontinuous domain.1Sukumar et al,2003,Modeling Quasi-Static Crack Growth with XFEM,Part I:Computer Implementation,Int.J.Sol.Str.,40,7513-7537.Structural&Multidisciplinary Optimization GroupMechanical and Aerospace Engineering DepartmentRecent Work on IntegrationMousavi1 presented generalized gauss quadrature rules over arbitrary polygons where optimization is used to identify the location and weights of the gauss pointsMousavi2 presented the Duffy transformation from a triangle(tetrahedron)to a square(cube)for integrationNatarajan3 presented a transformation to a unit disk based on Schwarz-Christoffel mappingPark4 presented a transformation from a triangle(tetrahedron)to a square(cube)where the singularity is placed at the origin of the square(cube)in the transformed space1Mousavi et al,2009,Generalized Gaussian Quadrature Rules on Arbitrary Polygons,Int.J.Num.Meth.Eng.,82,99-113.2Mousavi et al,2010,Generalized Duffy Transformation for Integrating Vertex Singularities,Comp.Meth.,45,127-140.3Natarajan et at,2009,Numerical Integration Over Arbitrary Polygonal Domains Based on Schwarz,Int.J.Num.Meth.Eng.,80,103-134.4Park et al,2009,Integration of Singular Enrichment Functions in the GFEM/XFEM,Int.J.Num.Meth.Eng.,78,1220-1257.22Structural&Multidisciplinary Optimization GroupMechanical and Aerospace Engineering DepartmentXFEM for Fluid Applications1Fries.2009.The intrinsic XFEM for Two-Fliud Flows,Int.J.Num.Meth.Eng.,60,437-471.2Fries et al.2009.On Time Integration in the XFEM.Int.J.Num.Meth.Eng.,79,69-93.3Fries.http:/www.xfem.rwth-aachen.de.Two incompressible fluids Density and viscosity discontinuous across interface Velocity(strong)and pressure(weak)discontinuities Stationary mesh Rising gas bubble in fluid Top bubble more dense than bottomStructural&Multidisciplinary Optimization GroupMechanical and Aerospace Engineering DepartmentXFEM in ABAQUSXFEM implementation of cohesive crack model using phantom node model1Limitations2Only STATIC analysisOnly linear continuum elementsNo parallel processingNo contour integrals(available in 6.9-EF3 and newer)No fatigue crack growthOnly one crack within an elementA crack may not turn more than 90 degrees within an elementA crack may not branchNo implementation in ABAQUS/Explicit21Song et al,2006,Dynamic Crack and Shear Band Propagation with Phantom Nodes,Comp.Meth.App,Mech.Eng,193,3524-3540.22009,Extended Finite Element Method(XFEM),ABAQUS 6.9 Update Seminar,Dassault Systems.32009,ABAQUS 6.9 Extended Functionality(EF)Overview,ABAQUS 6.9-EF Update Webinar,Dassault Systems.Structural&Multidisciplinary Optimization GroupMechanical and Aerospace Engineering Department25XFEM in MATLABCreated 2D plane stress/strain XFEM codeAllows rectangular domain with arbitrary boundary and loading conditionsHomogeneous and bi-material cracks,void and inclusion enrichmentsNarrow band level set for cracks,full level set for inclusions and voidsIntegration of enriched elements by triangular subdivisionContour integrals for calculating SIFsFinite crack growth increment or Paris Law to grow crackPlotting of level sets,mesh,displacement and stressesBenchmarks for enrichments,fatigue crack growth,and optimization implementation1D and 2D MATLAB codes available for Google:Abaqus XFEM or MATLAB XFEMStructural&Multidisciplinary Optimization GroupMechanical and Aerospace Engineering DepartmentReanalysis of XFEMStructural&Multidisciplinary Optimization GroupMechanical and Aerospace Engineering DepartmentFatigue Crack GrowthFatigue cracks form as the result of repeated loading below the yield stress,eventually leading to failureHigh cycle fatigue,Nfail 106+cyclesModels predict growth with an ordinary differential equation,da/dN=f(K)Many models provide f(K)in varying levels of complexityNKStructural&Multidisciplinary Optimization GroupMechanical and Aerospace Engineering DepartmentIncidents from Fatigue Failure1954 South African Airways Flight 201,21 deaths1954 BOAC Flight 781,35 deaths1957 Cebe Douglas crash,25 deaths1968 Helicopter crash in Compton,21 deaths1980 Alexander L.Keilland oil platform,123 deaths1985 Japan Flight 123,521 deaths1988 Aloha Airlines Flight 243,1 death1989 United Flight 232,112 deaths1992 El Al Flight 1862,43 deaths1998 ICE train crash,101 deaths2002 China Airlines Flight 611,225 deaths2005 Chalks Ocean Airways Flight 101,20 deaths2007 Missouri Air National Guard crashed,pilot ejected2009 Southwest Airlines Flight 2294,football sized holeMetal fatigue has been and still is a challenge for engineering applications and can lead to the loss of life.28Structural&Multidisciplinary Optimization GroupMechanical and Aerospace Engineering DepartmentMotivationForward Euler method is conditionally stable,becomes unstable for some a or NSeries of linear approximations to exponential functionCrack path becomes function of h,a or N29Structural&Multidisciplinary Optimization GroupMechanical and Aerospace Engineering Department30Observation of K for Quasi-Static GrowthFor same boundary conditions:Kuu are equal for A and BKaa,Kua grow,old portion constantKbb,Kub,Kab change based on tip locationABStructural&Multidisciplinary Optimization GroupMechanical and Aerospace Engineering DepartmentReanalysis TechniquesExisting reanalysis techniques may be applied to XFEM for modeling quasi-static crack growthDeveloped in design and optimization fieldsConsider small changes to system of equations by modifying KSavings in assembly and factorization of the system of equationsApproximate methodsIterative solver1 for adding DOFs to system of equationsExact methodsIncremental Cholesky2 factorizationSherman-Morrison-Woodbury3 formula1Wu et al,2006,Static reanalysis of structures with added degrees of freedom,Comm.Num.Meth.Eng.,22,269-281.2Chapra et al,2002,Numerical methods for engineers,4th edition,McGraw Hill,New York,NY.3Woodbury,1950,Inverting modified matrices,Mem.Rpt.42,Stat.Res.Gr.,Princeton University,4pp MR38136.31Structural&Multidisciplinary Optimization GroupMechanical and Aerospace Engineering DepartmentUpdating Stiffness MatrixConstant32Structural&Multidisciplinary Optimization GroupMechanical and Aerospace Engineering DepartmentUpdating Factorized Stiffness MatrixConstantFactorAppend new Cholesky factorization of Heaviside enriched elements to end of constant Cholesky factorization.33Structural&Multidisciplinary Optimization GroupMechanical and Aerospace Engineering Department34Example of Assembly Savings40.5Consider 30 increments of a=0.1,structured mesh of square quadrilaterals with length=1/80,200,000 traditional DOFInitial iteration 150 enriched DOF,final iteration 1000 enriched DOFAssembly time with traditional algorithm:588 secondsAssembly time with reanalysis algorithm:39 seconds Structural&Multidisciplinary Optimization GroupMechanical and Aerospace Engineering Department35Sensitivity of Assembly Reanalysis to MeshThe reanalysis of the XFEM stiffness matrix is largely independent of the mesh densityGeneral trend is linear compared to quadratic trend for traditional assembly techniquesAllows smaller time step in integration of fatigue law for comparable computing timeStructural&Multidisciplinary Optimization GroupMechanical and Aerospace Engineering DepartmentConclusionsThe extended finite element method(XFEM)models discontinuities independent of the finite element mesh by incorporating discontinuous functions into the displacement approximationThe level set method is coupled with the XFEM to track the discontinuities as they are not explicitly defined by the meshChallenges with the method include blending elements,convergence rate,and integration of enriched elementsWork was presented to increase the accuracy of fatigue crack growth simulations through reanalysis of the XFEMFuture work will focus on the realization of the factorization and solution savings in addition to the assembly savingsIn addition to modeling fatigue,the use of XFEM in optimization may be better enabled though the proposed reanalysis algorithm36Structural&Multidisciplinary Optimization GroupMechanical and Aerospace Engineering DepartmentThank you for your time!Questions?