1、Lesson content:IntroductionSolid elementsBeam elementsSurface and membrane elementsMesh convergenceWorkshop PreliminariesWorkshop 1a: Balloon-expanded stent geometry and meshingWorkshop 1b: Self-expanding stent geometry and meshingLesson 3: Element Selection and Mesh Convergence1 hoursIntroductionTh

2、e wide range of elements in the Abaqus element library provides flexibility in modeling different geometries and structuresWe will focus on elements commonly used in stent analysesStent element selection and mesh refinement need toYield correct bending propertiesCapture the maximum stress (or strain

3、) that usually occurs on the surfaceslaser-cut stentsresiding vesselswire stentswire stentsexpansion toolscrimping toolsstentsresiding vesselsshell elementsbeam elementsmembrane/surface elementscontinuum (solid elements)Solid Elements (1/10)3D solid elements are commonly used in stent analysesHexahe

4、dral elements are usually used to model stentsBending properties of theseelements are importantIssues to be considered:Shear lockingHourglassing(S)(S)Solid Elements (2/10)Second-order solid elements model bending accuratelyThe axial strain equals the change in length of the initially horizontal line

5、sThe thickness strain is zeroThe shear strain is zeroHowever, these elements are very expansive to useRarely used for stentslines 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 r

6、emains equal to 90o (implies xy= 0).Solid Elements (3/10)First-order fully integrated solid elements (C3D8)Detect shear strains at the integration points NonphysicalOverly stiff Energy going into shearing the element rather than bending it Known as “shear locking”Because the element edges must remai

7、n straight, the angle between the deformed isoparametric lines is not equal to 90 (implies ).integration pointDo not use these elements for stents!Solid Elements (4/10)First-order reduced-integration elements (C3D8R)Eliminate shear locking However, hourglassing is a concernOnly one integration point

8、 at the centroid A single element through the thickness does not detect strain in bendingThe deformation is a zero-energy modeDeforms but no strainKnown as “hourglassing”Change in length is zero (implies no strain is detected at the integration point).Bending behavior for a single first-order reduce

9、d-integration elementSolid Elements (5/10)Hourglassing can propagate easily through a mesh of first-order reduced-integration elements, causing unreliable resultsHourglassing is not a problem if you use multiple elementsat least four through the thicknessEach element captures either compressive or t

10、ensile axial strains but not bothThe axial strains are measured correctlyThe thickness and shearstrains are zeroCheap and effective elementsSolid Elements (6/10)Detecting and controlling hourglassingHourglassing can usually be seen in deformed shape plotsExample: Coarse and medium meshes of a simply

11、 supported beam with a center point loadAbaqus has built-in hourglass controls that limit the problems caused by hourglassing Verify that the artificial energy used to control hourglassing is small (1%) relative to the internal energySame load and displacement magnification (1000)Solid Elements (7/1

12、0)Use the XY plotting capability in Abaqus/Viewer to compare the energies graphicallyinternal energyartificial energyartificial energyinternal energyTwo elements through the thickness: Ratio of artificial to internal energy is 2%Four elements through the thickness: Ratio of artificial to internal en

13、ergy is 0.1%Solid Elements (8/10) patible mode elements (C3D8I)Perhaps the most cost-effective solid continuum elements for bending-dominated problemsCompromise in cost between the first- and second-order reduced-integration elements, with many of the advantages of bothModel shear behavior correctly

14、no shear strains in pure bendingModel bending with only one element through the thicknessNo hourglass modes, and work well in plasticity and contact problemsThe advantages over reduced-integration first-order elements are reduced if the elements are severely distorted; however, all elements perform

15、less accurately if severely distortedSolid Elements (9/10)Example: Cantilever beam with distorted elementsParallel distortionTrapezoidal distortionSolid Elements (10/10)Summary: mended element types for modeling stents patible model element (C3D8I)Most cost-effective element for modeling stentsMust

16、minimize distortion in the initial element shapeFirst-order reduced integration element (C3D8R) with overlaid membrane elements (M3D4R)Lower computational costNeed to control hourglassingBeam Elements (1/2)The main advantages of beam elements are:They are geometrically simpleThey have few degrees of

17、 freedomThis is achieved by assuming that the beams deformation can be estimated entirely from variables that are functions of position along the beam axis onlyRequires cross-sectional dimensionssmaller than 1/10 of typical axialdimensionsFor wire stents, the distance between two wire crosses repres

18、ents a typical axial dimensionLlRequirement for using beam elements: l 1/10 LBeam Elements (2/2)Types of beam elementsEuler-Bernoulli (slender) beams (B33, and B33H)No transverse shear deformationBeam must be slenderTypical dimensions in the cross-section should be less than about 1/15 of typical ax

19、ialOnly available in Abaqus/StandardTimoshenko (shear flexible) beams (B31 and B32)Model transverse shear deformationCan be used for thick (“stout”), as well as for slender, beamsSurface and Membrane ElementsSurface and membrane elements are used to model expansion and crimping tool surfacesThey are

20、 used to define contact surfacesThey are interchangeable for this purposeMembrane elements (M3D4R)Surface elements that transmit in-plane forces only (no moments)Have no bending stiffnessSurface elements (SFM3D4R)Same as membrane elements exceptHave no inherent stiffnessMay have mass per unit areaHa

21、ve zero thickness Mesh Convergence (1/7)Results extrapolation and integration point locationsMaximum stress/strain of a stent usually lies on stent surfaceBending dominated loading conditionCalculations are carried out at integration pointsErrors are introduced if integration points are not close to

22、 the surfaceDue to extrapolation of analysis resultsWith the same element dimensions, results of reduced integration elements are extrapolated more than fully integrated elementsIntegration point locationsMesh Convergence (2/7)Overlaying membrane elements on reduced integration elementsAdd integrati

23、on points on the surfaceSimilar to adding “strain gages” on stent surfacesTake advantage of the lower computational cost of reduced integration elementsReduce results extrapolation and capture surface stress/strain more accuratelyMembrane elements properties Same material as underlying solid element

24、sYield correct stress results Minimum thicknessReduce impact on the solutionMesh Convergence (3/7)Mesh convergence studyAre important to achieve a balance between accurate and efficient solutionExampleObjectivesDetermine mesh refinement required to capture the maximum principal stressModel highlight

25、sSingle-ring stentMade of stainless steelExpanded by a rigid cylinderIntegration point results evaluatedMesh Convergence (4/7)Element type and mesh refinementCompared C3D8R and C3D8I elementsWith and without overlaid membrane elementsFour levels of refinement studiedDistinguished by three numbersNum

26、ber of elements through strut thickness4 x 4 x 6Number of elements across strut widthNumber of elements along a 90 degree arc4 x 4 x 68 x 8 x 1216 x 16 x 2432 x 32 x 48Mesh Convergence (5/7)ResultsCritical maximum principal stress normalized by that of the model with 32x32x48 refinement of C3D8I wit

27、h membrane overlayMesh Convergence (6/7)Cost comparisonTotal number of variable normalized by the that of the model with 4x4x6 refinement of C3D8R elementsNote: adding membrane elements doesnt increase number of variablesMesh Convergence (7/7)ConclusionsC3D8RUnderestimates surface stress without mem

28、brane overlayThe discrepancy decreases as the mesh is refinedBut the cost increases significantlyConservative estimate is achieved with membrane overlayC3D8ICaptures the surface stress accurately even with low mesh refinement and no membrane overlayA cost-effective element type mended element type f

29、or modeling stentsHowever, element shape need to be as close to a perfect cube as possible to maximize the advantage of this elementObjectivesWhen you complete this exercise you will be able to extract all the files necessary to complete the demonstrations and workshops associated with this courseWo

30、rkshop file setup (option 1: installation via plug-in)From the main menu bar, select Plug-insTools Install Courses.In the Install Courses dialog box:Specify the directory to which the files will be written.Chooses the course(s) for which the files will be extracted.Click OK.Workshop Preliminaries (1

31、/2)5 minutesWorkshop file setup (option 2: manual installation)Find out where the Abaqus release is installed by typingabqxxx whereamiwhere abqxxx is the name of the Abaqus execution procedure on your system. It can be defined to have a different name. For example, the command for the 6.131 release might be aliased to abq6131.This command will give the ful