課件參考文稿iabu-unit99la

1、Lesson content:Elements in AbaqusStructural (Shells and Beams) vs. Continuum ElementsModeling Bending Using Continuum ElementsStress ConcentrationsContact pressible MaterialsMesh GenerationSolid Element Selection SummaryAppendix 2: Element Selection Criteria1.5 hoursElements in Abaqus (1/8)The wide

2、range of elements in the Abaqus element library provides flexibility in modeling different geometries and structures. Each element can be characterized by considering the following:FamilyNumber of nodesDegrees of freedomFormulationIntegrationElements in Abaqus (2/8)Family A family of finite elements

3、 is the broadest category used to classify elements. Elements in the same family share many basic features.There are many variations within a family. special-purpose elements like springs, dashpots, and massescontinuum (solid elements)shell elementsbeam elementsrigid elementsmembrane elementstruss e

4、lementsinfinite elementsElements in Abaqus (3/8)Number of nodes (interpolation)An elements number of nodes determines how the nodal degrees of freedom will be interpolated over the domain of the element.Abaqus includes elements with both first- and second-order interpolation.First-order interpolatio

5、nSecond-order interpolationElements in Abaqus (4/8)Degrees of freedomThe primary variables that exist at the nodes of an element are the degrees of freedom in the finite element analysis. Examples of degrees of freedom are:DisplacementsRotationsTemperatureElectrical potentialSome elements have inter

6、nal degrees of freedom that are not associated with the user-defined nodes.Elements in Abaqus (5/8)FormulationThe mathematical formulation used to describe the behavior of an element is another broad category that is used to classify elements. Examples of different element formulations: IntegrationT

7、he stiffness and mass of an element are calculated numerically at sampling points called “integration points” within the element. The numerical algorithm used to integrate these variables influences how an element behaves.Plane strainPlane stressHybrid elements patible-mode elementsSmall-strain shel

8、lsFinite-strain shellsThick-only shellsThin-only shellsElements in Abaqus (6/8)Abaqus includes elements with both “full” and “reduced” integration.Full integration:The minimum integration order required for exact integration of the strain energy for an undistorted element with linear material proper

9、ties.Reduced integration:The integration rule that is one order less than the full integration rule.Second-orderinterpolationFirst-order interpolationFull integrationReduced integration2 x 23 x 31 x 12 x 2Elements in Abaqus (7/8)Element naming conventions: examplesB21: Beam, 2-D, 1st-order interpola

10、tionCAX8R: Continuum, AXisymmetric, 8-node, Reduced integrationDC3D4: Diffusion (heat transfer), Continuum, 3-D, 4-nodeS8RT: Shell, 8-node, Reduced integration, TemperatureCPE8PH: Continuum, Plane strain, 8-node, Pore pressure, HybridDC1D2E: Diffusion (heat transfer), Continuum, 1-D, 2-node, Electri

11、calElements in Abaqus (8/8)Comparing Abaqus/Standard and Abaqus/Explicit element librariesBoth programs have essentially the same element families: continuum, shell, beam, etc.Abaqus/Standard includes elements for many analysis types besides stress analysis: heat transfer, soils consolidation, acous

12、tics, etc.Acoustic elements are also available in Abaqus/Explicit.Abaqus/Standard includes many more variations within each element family.Abaqus/Explicit includes mostly first-order reduced-integration elements.Exceptions: second-order triangular and tetrahedral elements; second-order beam elements

13、; first-order fully-integrated brick (including patible mode version), shell, and membrane elements.Many of the same general element selection guidelines apply to both programs.Structural (Shells and Beams) vs. Continuum Elements (1/3)Continuum (solid) element models can be large and expensive, part

14、icularly in three-dimensional problems. If appropriate, structural elements (shells and beams) should be used for a more economical solution.A structural element model typically requires far fewer elements than a comparable continuum element model.For structural elements to produce acceptable result

15、s, the shell thickness or the beam cross-section dimensions should be less than 1/10 of a typical global structural dimension, such as:The distance between supports or point loadsThe distance between gross changes in cross sectionThe wavelength of the highest vibration modeStructural (Shells and Bea

16、ms) vs. Continuum Elements (2/3)Shell elementsShell elements approximate a three-dimensional continuum with a surface model.Model bending and in-plane deformations efficiently.If a detailed analysis of a region is needed, a local three-dimensional continuum model can be included using multi-point co

17、nstraints or submodeling.3-D continuumsurface modelshell model of a hemispherical dome subjected to a projectile impactStructural (Shells and Beams) vs. Continuum Elements (3/3)Beam elementsBeam elements approximate a three-dimensional continuum with a line model.Model bending, torsion, and axial fo

18、rces efficiently.Many different cross-section shapes are available.Cross-section properties can also be specified by providing engineering constants.line modelframed structure modeled using beam elements3-D continuumModeling Bending Using Continuum Elements (1/10)Physical characteristics of pure ben

19、dingPlane cross-sections remain plane throughout the deformation.The axial strain xx varies linearly through the thickness.The strain in the thickness direction yy is zero if = 0.No membrane shear strain.Implies that lines parallel to the beam axis lie on a circular arc.xxModeling Bending Using Cont

20、inuum Elements (2/10)Modeling bending using second-order solid elements (CPE8, C3D20R, )Second-order full- and reduced-integration solid elements model bending accurately:The axial strain equals the change in length of the initially horizontal lines. The thickness strain is zero.The shear strain is

21、zero.lines that are initially vertical do not change length (implies yy= 0).Because the element edges can assume a curved shape, the angle between the deformed isoparametric lines remains equal to 90o (implies xy= 0).Modeling Bending Using Continuum Elements (3/10)Modeling bending using first-order

22、fully-integrated solid elements (CPS4, CPE4, C3D8)These elements detect shear strains at the integration points. Nonphysical; present solely because of the element formulation used.Overly stiff behavior results from energy going into shearing the element rather than bending it (called “shear locking

23、”).Because the element edges must remain straight, the angle between the deformed isoparametric lines is not equal to 90 (implies ).integration pointDo not use these elements in regions dominated by bending!Modeling Bending Using Continuum Elements (4/10)Modeling bending using first-order reduced-in

24、tegration elements (CPE4R, )These elements eliminate shear locking. However, hourglassing is a concern when using these elements.Only one integration point at the centroid. A single element through the thickness does not detect strain in bending.Deformation is a zero-energy mode (deformation but no

25、strain; called “hourglassing”).Change in length is zero (implies no strain is detected at the integration point).Bending behavior for a single first-order reduced-integration elementModeling Bending Using Continuum Elements (5/10)Hourglassing can propagate easily through a mesh of first-order reduce

26、d-integration elements, causing unreliable results.Hourglassing is not a problem if you use multiple elementsat least four through the thickness.Each element captures either compressive or tensile axial strains but not both.The axial strains are measured correctly.The thickness and shear strains are

27、 zero.Cheap and effective elements.Modeling Bending Using Continuum Elements (6/10)Detecting and controlling hourglassingHourglassing can usually be seen in deformed shape plots.Example: Coarse and medium meshes of a simply supported beam with a center point load.Abaqus has built-in hourglass contro

28、ls that limit the problems caused by hourglassing. Verify that the artificial energy used to control hourglassing is small ( 0.475). Examples include:RubberMetals at large plastic strains Conventional finite element meshes often exhibit overly stiff behavior due to volumetric locking, which is most

29、severe when these materials are highly confined.overly stiff behavior of an elastic-plastic material with volumetric lockingcorrect behavior of an elastic-plastic materialExample of the effect of volumetric locking pressible Materials (2/3)For an pressible material each integration points volume mus

30、t remain almost constant. This overconstrains the kinematically admissible displacement field and causes volumetric lockingFor example, in the limit of mesh refinement, for a three-dimensional mesh of 8-node hexahedra, there ison average1 node with 3 degrees of freedom per element. The volume at eac

31、h integration point must remain fixed. Fully integrated hexahedra use 8 integration points per element; thus, in this example, we have as many as 8 constraints per element, but only 3 degrees of freedom are available to satisfy these constraints.The mesh is overconstrainedit “l(fā)ocks.”Volumetric locki

32、ng is most pronounced in fully integrated elements. Reduced-integration elements have fewer volumetric constraints.Reduced integration effectively eliminates volumetric locking in many problems with nearly pressible material.The “hybrid” formulation (elements whose names end with the letter“H”) may

33、also be used to alleviate volumetric locking.In this formulation the pressure stress is treated as an independently interpolated basic solution variable, coupled to the displacement solution through the constitutive theory. pressible Materials (3/3)In general, the use of hybrid elements often greatl

34、y improves convergence behavior when a material is nearly pressible. To encourage the proper use of hybrid elements, Abaqus/Standard issues:a warning message if the effective initial Poissons ratio exceeds 0.48an error message if the effective initial Poissons ratio exceeds 0.495In particular, note

35、that all meshes with strictly pressible materials, such as rubber, require the use of hybrid elementsHybrid elements are also strongly mended for:Second-order meshes with approximately pressible materials.Refined meshes of reduced-integration elements that still show volumetric locking problems. Suc

36、h problems are possible with elastic-plastic materials strained far into the plastic range.A mesh consisting of first-order hybrid triangles or tetrahedra will be overconstrained because the number of constraints imposed in the hybrid formulation is too high to prevent locking. Hence, for pressible

37、materials these elements are mended only as “fillers” in quadrilateral or brick-type meshes.Mesh Generation (1/5)MeshesTypical element shapes are shown at right.Most elements in Abaqus are topologically equivalent to these shapes.For example, CPE4 (stress), DC2D4 (heat transfer), and AC2D4 (acoustic

38、s) are topologically equivalent to a linear quadrilateral.Mesh Generation (2/5)Quad/hex vs. tri/tet elementsOf particular importance when generating a mesh is the decision regarding whether to use quad/hex or tri/tet elements.Quad/hex elements should be used wherever possible.They give the best resu

39、lts for the minimum cost. When modeling complex geometries, however, the analyst often has little choice but to mesh with triangular and tetrahedral elements.Turbine blade with platform modeled with tetrahedral elementsMesh Generation (3/5)First-order tri/tet elements (CPE3, CPS3, CAX3, C3D4, C3D6)

40、are poor elements; they have the following problems:Poor convergence rate. They typically require very fine meshes to produce good results. Volumetric locking with pressible or nearly pressible materials, even using the “hybrid” formulation.These elements should be used only as fillers in regions fa

41、r from any areas where accurate results are needed.Second-order tri/tet elements (C3D10, C3D10I, etc.)Suitable for general usageLess sensitive to initial element shape that quads/hex but convergence rate is slowerGuidelines for contact analysisSurface-to-surface contact discretizationNo restriction

42、on element type (use C3D10, C3D10I, C3D10M, etc.)Node-to-surface contact discretizationRestrict usage to modified second-order elements (e.g., C3D10M)Mesh Generation (4/5)Mesh refinement and convergenceUse a sufficiently refined mesh to ensure that the results from your Abaqus simulation are adequat

43、e.Coarse meshes tend to yield inaccurate results.The computer resources required to run your job increase with the level of mesh refinement.It is rarely necessary to use a uniformly refined mesh throughout the structure being analyzed.Use a fine mesh only in areas of high gradients and a coarser mes

44、h in areas of low gradients.Can often predict regions of high gradients before generating the mesh.Use hand calculations, experience, etc.Alternatively, you can use coarse mesh results to identify high gradient regions.Some mendations:Minimize mesh distortion as much as possible.A minimum of four qu

45、adratic elements per 90o should be used around a circular hole.A minimum of four elements should be used through the thickness of a structure if first-order, reduced integration solid elements are used to model bending. Other guidelines can be developed based on experience with a given class of prob

46、lem. Mesh Generation (5/5)It is good practice to perform a mesh convergence study.Simulate the problem using progressively finer meshes, and compare the results.The mesh density can be changed very easily using Abaqus/CAE since the definition of the analysis model is based on the geometry of the structure.When two meshes yield nearly identical results, th