FEA by nitin gokhale

Why 2-d meshing is carried out o n mid surface?+Mathematically element thickness (specifiedby user) is assigned half in Z axis (element top)and half in - Z axis (element bottom). Hence, for appropriate representation of geometry via 2-dmesh its necessary to extract mid surface & generate nodesand elements on the mid surface. Zd element shapes I Tria QuadAlso knownas Also knownasConstant Strain Linear StrainTriangle (CiT). Triangle (LST)* L- Linear element 'P - Paraboiic element*( ) - lndicates number of nodeslelement

7.2 Familyof 2-d Elements1) Plane stress:dofs - 2 / node (Ux,Uy(in-planetranslations))Stress in z direction (thickness)is zero(%= O). \"t 'J.Total dof =8?I': L u*PracticalAa~lications:Thin sheet metalparts, like aircraftskin, narrow beams2) Plain strain:dofs - 2 / node (Ux,Uy(in-planetranslations))Strain in z direction (thickness) is zero (sZ= 0). \"t \"'L\". Totaldof =8PracticalAo~lication:sUndergroundpipes, wide beams.Planestress and plane strain elements are used for 2-d (planner)problems.3) Plate:+dofs - 3 / node (ex,Oy (inplane rotations) Uz(out of plane translation)) Total dof= 12A-

PracticalAr>olication:Bending loadapplication.4) Membrane:dofs - 3 / node (U, U, (inplane translations)+ Oz(out of planerotation)) AITA. A.- A.Totaldof=12PracticalA~olicationB:alloon, Baffles.5) Thin shell:Thinshell element is the most generaltype of element.dofs: 6 dof 1node ( Ux.U, Uz,0,. O, BJ. = Plate = u.,ex,ey = (1T+2R)&&&&&&& :Thin shell elements are most commonly usedelements.6) Axisymmetric Solid:dofs - 2 / node{Ux,Uz(2 in planetranslations, Zaxis is axis of rotation))Why is the word 'solid'in the name of a 2-d element. This is becausethough the elements areplanner but lt actually represent a solid. For generating a cylinder in CAD software we defineaxis of rotation and rectangular cross section. Similarly for an axl-symmetric model we need todefinean axis of rotationandcross section (planer mesh1.The 2-d planer mesh is mathematicallyequivalent to 3-d cylinder.

Practical Au~licationr: Pressure vessels, objects of revolutions subjected to axi-symmetricboundary conditions.7.3 Thin Shen ElementsComparisonof Triangularand Quadrilateral elernenh:We will carryoutplate with circular hole exercise to compare performanceof differentelementswith known analyticalanswer.Analvtical answer:max. stress = 3 N/mmzFor Infiniteplate with very smallcircular hole,Stress Concentration Factor (SCF) =3,SCF = max. stress/ nominal stressNominal stress = FIA= 10,000/(1000*10) =1 Nlmm23 = max. stress I lmax. stress= 3 N/mm2

~ ~ u ~ ç o ~ d t of~nmapS&.modelMedioarammem:Uobalelement length=100, nurnberofetementson hole=â ( m e for al1theRerauon)Since its first wardse, pleasedo not wurv about mmh fiow pattern and quafity check Uuoautomatîcmeshhgoption, apply bwndaryconditions and solvaExact Answer:3 Nlmm' 1 1Stress Displacementfunction , L-I iria 5

U= ,Y (one addiiiond tenn tn cornparisont otria 3, madcer it morearmate) - ~~~ u = a~a,x+a,y+a,x2+a,y2 +a,w (6 nodes- 6 temx in pleammt functian) .. . *., .~ - =-. = : 8 -- L u = a,+a,x+aj+a3xy+a,x2 +a,v +a$y +a+$ (twoadditional t min cornparisai t o tria 6, rnakes it more curat te) Conclusion: Quodelementsore betterthon trlangularelements. Parabdicelementsarebetter than Uneorelements.7.4 Effect of Mesh Densityin the Critical RegionConclusion from first exercise was quad elements are better than trias.The second exercise is t ostudy effect of increasing number of elements in the ctitical areas.In the first exercise number of elements on hole were same but element type was varied, in thecurrent exercise we will keep type of element same Le.Quad 4 and Vary number of elements onhole (cri irea).Hok:4 elemenn 12 ekments 64elementr

2-D Meshing Exact Answer:3 Nlmm2 -dorethe numberofelemei i 1 riticalregionbetter If it i s so then why not to always create very fine mesh with max. possible nodes and eiements? Why the usual guidelinefor meshing is 12/16 elernents around holes in critical area? The reason i s solution time is directly proportional to (dofJ2A. lso large size models are not easy to handle on the computer due to graphicscardmemory limitations.Analyst has to make a fine balance between level of accuracy and elernent size (dof) that could be handled satisfactorily with available hardware configuration. How thumb rulesare made? Usually we get instructions from overseas client about specific number of elements on hole, fillet, specific mesh pattern for bolted weldedjoints etc. How do they decide these things. It is based on simple exercise like above. Results of different rnesh configuration are compared withknownanalyticalanswersandthe one whichgives logicalaccuracywith reasonablesolution tirneis selected. Most of the industriesfollowfollowingthumb rulefor number ofelements on holes: Minimumnumber of elements: critical areas= 12 general areas= 6 7.5 Effpctof Wsing in the Critical Region

- Practicai Flnite Element AnalysisBiasingis a very usefulfeature provided by commercial softwares. Like in day to day life we usethe word bias (M' y boss is biased, he prefers my colleague even though both of us have equalqualification and efficiency\"),in the samemanner when element length isnot sameon the edgeand biased towardsa point then it is known as biased meshing. Different commercial softwarescalculatebias differently. One of the simple scheme is bias factor = ratio of max. element lengthI minimum element lengthAbove geometry was split along the diagonal and bias defined on the diagonals (at the edgepoint near clrcular hole) BiosedMeshFxercise: DistributionofetemennExact Answer:3 Nlmm2 - Bias 5 - Bias 10 Bias 15Conclusion from previous exercise wos higher number of elements in critlcai area meons higheroccuracy. The exercise on biasing shows even without increasing number of eiernents one couldochieve better resultsjust by oppropriate arrongement of nodes and elements, that to ot no extracomputationalcost(compare number ofnodesin aboveexerciseitssarne throughout).

7.6 Symmetric Boundary ConditionsFull Plate:Half Plate: Verticaledge d rstrsint: wxu, u,exqez 112MSS1=0! Horizontaledge restraint:U, 8,6,(246)= OHow to ao~lvyrnrnetricboundoryconditions:Step 1 :Write plane of symmetryfor half plate Le,x zStep 2: Fix in plane rotations (ex,8J and out of plane translations(US.Quarter Plate: Reshainton vertical edge: u,e,ez(1561=o .\" Horizontaledge: Uy0, €Iz(246]=O

Quarter symmetric model is still free to translate along Z axis. FEA linear static solver can notsolve unless ail the dofs are constraint (exception if inertia relief or kinetic dofs defined in themodel).Rigid body motion of quarter plate could be avoided by constraining Uz(3)=0(at anyedge nodeaway from criticalarea (hole)]Advantage of symmetric boundary conditions is sameaccuracy at lesser computational time &cost.Symmetricboundary conditions should not be used for dynamic analysis (vibrationanalysis). Itcannot calculateantl nodes.7.7 Different ElementType Options for ShellMeshing1) Purequadelernents [7>2) Mixed mode3) Equilateraltria A4) (Right angle) R-tria [1Mixed mode is the commonly preferred due to better mesh pattern(restriction:total tria % <5).Some times for structural analysis or for convergence and better result for non linear analysispure quadrilateralelementmeshingoption is selected.P If quadsare better than trias then why not to mesh usingonly quad elements, why FEA softwares provideoption of tria elements?7) Mesh transition: In structuraland fatigueanalysisrather than uniform mesh what helps is smallelement size in the critical areas and coarse or bigger elements in general areas. This type ofmesh gives good accuracy with manageable dofs. Trias help in smooth mesh transition fromdense to coarse.2 1 Com~lexgeometw: Geometry features like rib ends or sharp cutouts demand for use oftriangular elements. If quads are used instead of trias then it will result in poor qualityelements.3)Bettermeshflow:Forcrashor nonlinearanalysissystematic meshflowlineswithall theelementssatisfyingrequired quality parameters is very important.Mix-mode instead of pure quad helpsto achievebetter flow lines & convergenceof solution.41TetrameshinglconversionfromTria toTetra1:Fortetra meshing, al1theoutersurfaces are meshed

2-D Meshing via 2-d triangular elements & then trias are converted to tetras.This methodology is discussed in detail in next chapter. 5)Mouldflowanalysis: Mould flow analysis requires triangular elements. Comparisonbetween Equilateral tria and Right angle tria meshing. Default tria mesh in commercial softwares produces equilateral triangles while R-tria option generate right angle triangle (generating a rectangular or square mesh and then splitting along diagonal gives 2 trias per elernent). ldeal shape for triangular element is equilateral triangle & is theoretically better than R-tria. But for following specific applications R-trias have advantage over equilateral trias 1 ) Tetra rneshing: For defining contact, similar mesh pattern on the two surfacesis desirable. Equilateral tria option producesziz-zagmesh and also there is no controlover mesh pattern. Similarmesh requirement could be achieved by generating structured quadrilateral mesh (maintaining exactly same number of elements on two contact surfaces) and then splitting it to trias (R-tria). Typical applications are as follows a) Bolt hole and washer area: b) Bearing contact surfaces: Contact surfaces meshed with quad elements (Samemesh pattern and equal no. of elernents)and converted to trias before tetra conversion.

2)Variable thickness of ribsfor mould flow analysisRibs are modelled via quad elements in three layers as shown and then split to A-trias. Averagesection thickness is assigned to different layers.7.8 Geometry Associative MeshIn practice associative mesh is rarely used by CAE groups. But this option is provided by manyCAE softwares for 1\" hand calculations by design / CAD engineers. For getting quick results(rough idea), automatic meshing is carried out by pickingsurfaces or volume of the geometry,simpleboundaryconditions are applied and solutionis obtained. Generatedmesh i s associativewith geometry.Advontaoes:1) If geometry is changed mesh will also change automatically.

2) Boundary conditions could be applied on geometry (edges, surfacesinstead of nodesand elements etc.) which is more user friendly. GeomerrybasedmerhGmmetrymodification Auto updateofmeshkot holearthecenter)7.9 QualityChecksWhy quality checks?Result quality = Element qualityldeal shape for quad elements - SquareAldeal shape for triangular elements- Equilateral triangleDifferent quality parameters like skew, aspect ratio, included angles, jacobian, stretch etc. arethe rneasures of howfar a given element deviates from ideal shape. Square means al1angles 90pand equal sides, while equilateral triangle is al1angles 60°and equal sides. Sorne of the qualitychecks are based on angles (like skew, included angles) while others on side ratios & area (likeaspect, stretch).To reduce solution t h e elements are mapped t o local CO-ordinate system (individual for everyelement at the centroid) instead of using a single one coordinate system (global). Effectiveness

of this transformation is checked by jacobian and distortion. ldeally al1 the nodes of quadelement should lie in the same plane but a t curvatures and complicatedgeometry profiles it isnot possible. Measure of out of plane angle is warp angle.Followingare general definitions of various quality checks. Though the namessound same butexact definitionsmay differ from software to software.Warp angle: warp angle is out of plane angle.ldealvalue = OD(Acceptable < IO0).Warp angle is not applicable for triangular elements.Itis defined asanglebetween nomals totwo planesformedby splitting the quad element alongdiagonals. Max. angle out of the two possibilitiesis reported as warp angle.Aspect= max. element edge length / minimum element edge length.Idealvalue = 1 (Acceptable < 5).Skew:ideal value = O (Acce~tablei454

Skewfor quadrilateral element = 90°minus minimum angle between two lines joining oppositemid-sides of the element (a).Skew for Triangular element = 90°minus minimum angle between the lines from each node tothe opposing mid-sideand between the two adjacent mid-sides at each node of the elementJacobian:ldeal value = 1.0 (Acce~table> 0.6)In simple language, jacobian i s a scale factor arising because of transformation of CO-ordinatesystem. Elements are tansformed from global coordinates to local coordinates (defined atcentroid of every element), from faster analysis point of view.Distortion: / Area,,ldeal value =1.O (Acceotable> 0.6)Distortion is defined as - 1 Jacobian 1 * Area,LCS - Local Coordinate systemGCS - Global Coordinate systemStretch:ideal value: 1.0 (Acceptable > 0.2)For quadriiateral elements stretch = Ln,$*\" 11 2 / dmaXStretch forTriangular element = R d 12 / LmaXlncluded angles:Skew i s based on overall shape of element and it does not take in to account individual anglesof quadrilateral or triangular element. lncluded or interior angle check is applied for individualangles.Quad ldeal value = 90\". (Acceptable= 45O < 8 <135')Tria: ldeal value = 60° (Acceotable = 20° < 8 < 120')

- Practical FiniteElement AnalysisTaper:ldealvalue = O (Acceptable c0.5)Minimum element length:Very important check for crash analysis (time step calculations). It is also applied in general tocheck for minimumlength featurecaptured and presence of any zero length element.Chord deviation:Helps in determininghow well curvatures havebeen modelled. Itis defined as distancebetweenmid nodeof element edge to curved surface. It is applicableonly for linear elements.How t o improve quality of poor elements?1) Manualadiustment:By translating the nodesmanually or remeshingin the poor mesh region.This method consumes lot of t h e & was the only technique avallablebefore few years.2) Drag node: User has to drag nodes of failing elernents. It works faster and advantage is, itinstantaneously shows effect of dragging the nodeon al1the attachedelements.3) Autoaualitvimprovementproorams:This is the latestoption for quality irnpmvernent.User hasto submit the rnesh for quality improvement, special software programruns in the backgroundto improve the elementsquality automatically.Thereisawordofcautionforuseofautomesh irnprovementprograms.Forwarpageimprovementof 2-d meshing and Jacobian / distortion improvement of 10 noded tetra meshing, some timessoftware move the nodes out of geometry by considerable amount. This could cause visiblekinks and distortion of geometry.Apart from abovelisted standard checks, the mesh model should also be subjectedto followingadditional checks.7.10 Other Checks for 2-dMeshing1)Element free edges:What is freeedge ?

2-0Meshing Any single quad element has 4 free edges. Two elements In this case middle edge is shared and no more free. For a real life femodel, free edges should match with geometry outer / free edges. Additional free edges is an indication of unconnected nodes. Whitellneindicoterhee/ edge &unconnected nodes. 2) Duplicateelement : Mistakes during operations like reflect, translate etc. results in duplicate elements. These extra duplicateelements donot causeanyerror duringthe analysisbut increasesstiffnessofthe model and results in lesser displacement and stress. For example consider a simpleplate (thickness =2 mm) subjectedto tensile load.Assumethat due to somemeshingoperational1the elements are duplicatedand if analysis is carried out as it is thenit will show half the stress and displacement.102

3) Duplicate node:Operations like copy, translate, orient, reflectetc. results in duplicate nodesat common edge.4) Shell Normal:Consider following example.Alter deformation top surface of cantilever beam is under tension and bottom undercompression.For 2-d meshingmid surface 1sextracted, and analysis is carried out on mid surfacemesh. Nowthe question is, whether result plot as shown below corresponds tomid surface or top surfaceor bottom surface?

Shell element normal helps us in viewing top or bottom side stresses. Every element has elemental (or local) CO-ordinate system. Shell normal is direction of element normal (common practice is t o representnormal via Z axis, assuming element is oriented in xy plane). For viewing stress commercial post processors provide options known as top 1bottom or ZlfZ2 indicating positive and negative direction of shell normal. The top or bottom is not decided by how the +femodel is oriented on the screen but as per Z axis orientation of the elements. Z axis could be displayedon the screen by turning element triad switch or shell normal vector displayoption on. +Top side (or 21)= Z axis (along the direction of arrow as shown in figure) Bottom side (or 22)= - Z axis What would happen if shell normals are not aligned properly 7 From analysis process point of view there is no error. All the calculations will be carried out properly. But at the time of post processing i.e. viewing the results for 2-d elements, software doer not understand tension or compression, what it recognize is shell normal orientation. +It can show either stresses along Z axis or -Z axis. Suppose in the above figure one of the element's shell normal is in opposite direction. While viewing the results on bottom side (22) al1 the elements except the reverse shell normal orientation show Tensile (+) stresses and the odd one compressive (-1 stresses as shown in the following figure.The beginner would interpret from following result that something is wrong with boundary conditions, but an experienced engineer knows this is due t o inconsistent shell normal. How t o correct shell normal's alignment ? FEA software provide special command for consistent shell normal (al1shell normals aligned in one direction).1O4

- Practical Finite ElementAnalysis5) Geometry deviation :After completion of meshing geometry as well as mesh should be viewed together (mesh lineoption off). Mesh should not deviatefrom the geometry.6) Deletefree 1temporary nodes :Free nodes if not deleted results in rigid body motion. When auto singularity option is turnedon,softwarevia spring element of very small stiffness connect free nodeswith parent structure,resultingin warningmessage during the analysis.7) Renumber nodes, elements, properties etc. before export operation:Frequentimport- export operationscouldleadtoverybig numericfigurefor nodesandelementlabels. Some softwaresrefuse to read the file if nodeslelementslabel numbers are greater thanspecific limit. This could be avoidedby renumbering nodes, elements etc.8) Observetype,family & number of elements (element summary for complete model) :Mesh should be checked carefully prior to export operation as well as after importing it in theexternal solver for element type, family, numbers etc. Some times due to translator problem orif properties are not defined properly or for non supportive elements, either elements are notexportedat al1or family is changed (like membrane elements converted to thin shell etc.). Plot,trace lines, element free edges, free faces if any should be deleted.9) Check Mass ( Actual massVsFE model mass) :When prototype / physical model of the component i s available, femodel mass should becompared with actual one. Difference means missing or additional components or impropermaterialor physical properties.10) Free-Free run or dummy linear static analysis:Before delivering the final mesh to client, free-free run should be performed. 6 rigid modesindicate al1 the parts in the assembly are properly connected to each other. In case of singlecomponent meshing job, linear static analysis with dummy boundary conditions should becarried out.11) Request your colleague to checkthe model:Due to continuous working on same project Our mind tends to take some of the things forgranted and there is a possibility of missing some of the points. It is a good practice to get itcross checked via your colleague prior to final delivery.7.11 Haw Not to Mesh1) Back to back triangles should be avoided.Two tria elements should not be connected to eachother directly.

2) On plane surfaces triangular element should be avoided.nof recommended notrecommended recommended5 )NO mesh transitionon constant radius filletsIcurvatures 1 Mesh transition could be carried out on planer surfaces1O6

- Practical Finite Element Analysis4) Avoid tria elementson outer edges, holes5) What is not acceptable at professional level

6)Circular holes should be modelled carefully with washer (1.5 to 2 times d) and minimum two layers around the hole 7) Holeshould be modelled with even number of equally spaced elements: For better representation of hole geometry and smooth mesh flow lines, holes should be modelled with even number of elements (like6,8,12,16 etc. rather than 5,7,9or 13) 7elementsonhole (oddnumber), nof recommended 8) Nodesshould lie properly on the surface, no deviation (&nokinks) Switch off element mesh lines, adjust light options in the software and observe contour (in particular at curvatures), kinks as shown above are not acceptable.108

- Practicai FiniteElementAnaiysis9) Follow thefeature lines (nodes,should lieexactly on theedges).10)lnsteadof zig-zag distribution, srnicturedor smooth mesh is recammended In~desalignedin rtraightlin@Use of \"smooth\" option provided by most of the commercial softwares helps in achievingsystematicmesh.11) Crash analysismesh flow line requirement

2-5Meshing 12) For crash analysis rotating quads are not allowed. Rotating quads recommendedRecommendedforstr~~turala Recemmendedfor crash analphNotrecommendedfor crash onal& Not recommended forstructuralanalysir13) For Crash analysis : Constant mesh size (by using trias) is preferred (due to min. elementlength & time step criteria).Variablemeshsizenot recommendedfor crash but recammendedfor sfrunural anal)

8.1 When to Use 3-d ElementsAll dimensionsare comparablex - y-zElement s h a ~ e-Tetra, penta, hex, pyramidAdditional data from user - NothingElement t v ~ -eSolidpracticala~~licatio:nsGear box, engine block, crankshaftetc. CrankShah TetmMeshing (Image Source:AltairCafendar2006. Courtesy:BharatForgeLtd.J

3 D Mesliing 3-dElement types: 3-d Elementsma Pemaor Wedae ex O pamid I I ILinear Linear Penta6 Linear Hex 8Tara 4ParabolicTetralO8.2 DOFsFor Solid Elements2-d thin shell and 1-d beam element supports 6 dofs, but al1 solid elements have only 3translationaldofs (no rotational dof) Le. a10 noded tetra element has total 10 x 3 = 30 dofsWhy solidhave only 3 translations & no rotational dofs (Physicalinterpretation)?Consider a piece of paper (2-d geometry) or long steel scale (1-d geometry), it could be easilybended and twisted (rotationaldof) but now considera solid object like duster or paper weightit could not be i.e. very high bending and torsion stiffness. Hence solid elements have beenformulated with 3 translationaldofs andno rotationaldofs.

=PractkalFiniteElement Analysls8.3 Tetra MeshingTechniquesThere are two methods of terrameshing1) Automatic mesh:This approachis limited to simplegeometriesand pre-requislteis ermr freeCAD model. User has to just selectthevolumeand softwareautomaticallycarriesout meshing asper specified elementlength, quality criteria etc.Advantaae: Very quick, no meshing effortsDisadvantaae:Results in very high number of nodesand elements.There is no control on meshflow lines and specific mesh pattern requirement (likebolted, welded joints or contact surfacesimulation)2 ) 2d (Tria)t o 3d (Tetra):Most commonlyused method. Quad or tria meshing is carried out onal1the outer surfaces of the geometry, quads split to trias and then converted to tetras.Stepsfor 2-d (Tria) to3-d(Tetra)mesh aeneration:Steps 1) Study the geometryStep 2) Separate (isolate)surfaces, split the job among engineers(if there is tlme constraint)a. CAE engineer1

3 DMeshing b. CAE engineer 2 Step 3) Combine the mesh Step 4) Quality checks for triangular element (Min. tria angle > 159 Max. tria angle < 1204 jacobian > 0.6). zero free edges, noT-connection Step 5) ConvertTria mesh toTetra Step 6) Quality checks for tetra elements (tet collapse > 0.1, Jacobian & distortion > 0.5, stretch > 0.2 etc.). improvequality of mesh if required. Step 7) Free-Free run or otherwise linear static analysis with dummy BC's. Common algorithmsforTria toTetraConversion: 1) Advancing Front :Very powerful and most commonly used algorithm 2) Delaunay Algorithm 3)Tria-Quadmesh All Algorithms providefollowing two options forTria toTetra conversion- 1) Floating trias:Not recommended, original tria mesh generated by user on outer surfaces might not match with software produced tetra. Selection of this algorithm gives freedom to software to change thetriangularmeshpattern (incase of anyproblemintetra mesh generation).

This option could be used for meshing of general components or areas (not high stress, used forrepresenting stiffness or massetc.).2) Fixed trias: Recommended.Original tria mesh and generated tetra mesh pattern matches..Wgiginal tria Y 7 mesh Tr;a & Tetromesh togathe, -Tria &Tetra toaather (floating triamethod) Tria&tetramesh same Tria&Tetramesh notsame (fixedtria method) fatcrosspatternofmerhllncase if you are getting the job done from a service provider, always request for 2d shellmesh along with 3d tetra.8.4 Quality Checks for Tetra Meshingldeal shape for tetrahedron element is equilateral tetrahedron (al1 equilateral triangle faces).Various quality parameters check how far a given element deviates from the ideal shape.Tetra Collapse:idealValue= 1.O (Acce~table> 0.1)Tetra collapse= h * 1.24lA(Defined as distance of node from opposite facedivided by area of the face multiplied by 1.24)Volumetric Skew:Create a sphere passing through corner nodes oftetra, fit an ideal (equilateral) tetra in it. Find volumeof idealand actual tetra elements.ldeal value =0 (Acce~table< 0.7)

3 DMeshing Stretch: ldeal value = 1.O (Acce~table> 0.2) Stretch = RXd241Lm R = Radius of largest possible sphere insidegiven tetra element. Distortion: ldeal value= 1.O (Acce~table> 0.5) Distortion = 1J 1 XVolmKs/Voim,s LCS - Local CoordinateSystem GCS -GlobalCoordinateSystem Jacobian: ldeal value = 1.O (Acce~table> 0.5) In simple language, jacobian is a scale factor arising because of transformation of co-ordinate system. Elements are transformed from global co-ordinates to local co-ordinates for reducing solution time 8.5 Other Checks forletra Meshing 1) Quality checks for 2-d tria elements Beforeconvertingtrias to tetrasall thequality checks as discussedin chapter 7 should be applied. 2) Free edges: Conversion from tria to tetra is possible only when there is no free edge. NO free edge indicates mesh is forming enclosed volume. 3) T-connections: Mesh model should not contain any T-connection. T-connection is internai (volume) loop formation in the outer enclosedvolume. T-connecfion,notecceptable Green elementrshouldnot be thereln themodel

4) ConsistentShellnormals: Beforeconvertingtriasto tetras,shell normals shouldbe corrected.Some softwares do not allow shell to solid conversion unless normals of al1the elements areproperly aligned.5) Geometry deviation: After completion of meshing, geometry as well as mesh should beviewed together (mesh line option off). Mesh should not deviate from the geometry. In theprocessofqualityimprovement(in particularfor dlstortion/jacobianon curvedsurfaces orfillets)sometimes nodesget translated too far away from the geometryand not acceptable.6 )2-d tria elements should be deleted before final submission: It's a common mistake toexport 2-d shell elements along with tetra mesh as final delivery.> How t o improve quality of tetra mesh:Softwares provideauto algorithms / localizedremesh facility for improvingmesh quality.Theseprogramsimprove most oftheelementsbut some mightremainas it is. For suchelementsqualityimprovement is carried out via manual shiftingl translation of nodes. Unlike 2-d shell elements,'drag node'facility is not available for 3-d elements.P Linear Vs. ParabolicTetraElements:LinearTetraelements not recommended for structural analysis.They are very stiff &inaccurateincomparisonto parabolic elements.If lineartetrasare not recommendedforstructural analysisthenwhythisoptionisprovidedi n commercial FEA softwarest1) For structural analysis of very big assemblies, to reduce overall dofs, components away from critical areas are meshedwith linear tetras.2) For thermal analysis linear tetra is convenient since temperatureis the only variable (1 dof unlike 3 for structural)use of tetra 10 would unnecessarilyincrease dofs.3) CFD calculations are mostly basedon linear elements.4) Mouldflow analysis

3D Meshing 8.6 BrickMeshing Brick meshing (also known as Hex meshing) is al1about planning, hard work & patience. Brick meshing support only manual and semi automatic command options. Automatic meshing or optionlike quad to brick (tria to tetra) isnot supportedby commercialsoftwares. Rearaxleassembly wilh brakedrumand wheelhub, Brickmesh (1mageBorce:Altair CaIendarZW6, Courtesy:AshokLeylond) Procedurefor brick meshing of complicatedparts is to shellmesh the surfaces and then convert tohex by usingcommandsextrude, spin,sweep, linear solidetc.Freeface checkisvery important after completionof hex meshing.. -Extrude I drag

- Practical Finite Element Anaiysis Rotate/ spinSweep 1line dragLinear solid

Tlps for brick meshing:Even experienced engineers fear and do not willingly accept job of brick meshing. Don't youbelieve, ok let us have some fun, try it on 1\" of April, tell your experienced Company colleaguethatjustnow in themeetinghehasbeenassignedjobofengineblockbrickmeshingandobserveexpressionson his face. it will tell entire story about brick meshing.Inphilosophytheysay \"knowtheone (God)&youwillknoweverythingofthisworld: Philosophyfor meshing would be\"Know the brickmeshing& you will know al1about meshing\"No bookor consultantor universitycan teach you brick meshing.Therealteacher is determined....approach & hours of sitting in front of the computer, making mistakes and learning from themistakes. Are you readyHere are some tips for brick meshing1)Proper planning before starting the job : Sufficient time should be spent in studying thegeometry & meshing should not be started unlessand until meshing steps and how to proceedisvisualizedin the mind. Symmetry, sub symmetryor repetitivefeatures if any could save time.2) 2-d quad mesh should be systematic (ruledor mapped), avoid 2-d auto mesh:Flow linesshould be maintained with minimum trias, diamond or rotating quads should be avoided. Useof auto mesh on surfaces results in zig-zag or random mesh which might lead to unexpectedproblems later.3)Do not hurry to convert the sheil mesh t o brick: One should not convert 2-d mesh to 3-dimmediately instead,proceeding further with quads and checkingforanypossibleproblem withthe current pattern is recommended.4) Start from the complicatedfeature and not the simple one or corner of the part: Duringthe exams for effective time management basic thumb rule told to students is\"Attack simpleproblems first and then the complicated one.\"Thumb rulefor brick meshingis exactlyreverse i.e.\"Attack complicatedfeatures first and then simple ones': Beginners makea common mistake tomesh the simple and outer corners of the part first.5) Use llnear solid command: Linear solid and morphare very powerfui commands for brickmeshingand should be utilized.> Brick andTetrameshingcomparison No.ofelements&nodesgeneratedbybrickmeshareoftheorderof 112to1/50 incomparison to tetra mesh. Brick meshreducessolutiontime andresults in ease of handling the model on workstation(pre & post display).

Analysis types like crash or nonlinear give preference to brick mesh due to no. of nodes & mesh flow lines. Previously al1 the organizations used to brick mesh solid parts. So in case if objective is to compare existing model with the previous one then it has to be modelled using same type of element and length for a logical conclusion. Time consumed in brick meshing is more & requires experience, hard work & lot of patience too! Over the years algorithmfortetra hasiriiproved & accuracy wise there i s not much difference in tetra 10&brick 8 elements.ldeal shape for brick element is cube. Various quality criteria check how far a given elementdeviate from ideal shape.Warp angle:ldeal value = 0 (Acceptable<309Warp angle is calcuiated on faces (quadrilateral)of hex element. It i s angle between the planesform by splitting quad element.Jacobian:ldeal value = 1.0 (Acceotable > 0.5)In simple language, jacobian is a scale factor arising because of transformation of CO-ordinatesystem. Elements are transformed from global to local coordinates for reducing solution tirne.Distortion:ldeal value = 1.O (Acceotable > 0.5)Distortion = IJ 1 *Volm,, /Volm,,,LCS - Local Coordinate SystemGCS - Global Coordinate SystemStretch:ldeal value = 1.0 (Acceutable> 0.20)Stretch= min. edge length * \/ 3 / max.diagonal lengthAspect ratio:ldeal value =1.0 (Acceptable < 5)Aspect ratio = max. edge iength / minimum edge lengthSkew:ldeal value = O0(Acceutable < 45O)Skew is checked on ali the faces (quadrilateral).For skew definition please refer chapter 7.Quad face included angles: 4S0< 9 < 13S0Tria face (wedge 1penta elements) included angles: 20° < 9 < 120°% o f Pentas: Acceptable < 5 %

3 DMeshing 8.8 Other Checks for Brick Meshing Free faces: Free face check is the most important check for brick meshing. A singlebrick element has 6 free faces, Free faces of the mesh should match with outer surfaces [skin) of solid part. Any extra (inside) faces indicates either nodes are not connected properly or otherwise mismatching elements. - Free focer Outerloyer elementsdlsplayedlnltnemode tocheck intemal faces Convertingfree faces to tetras: For complicated geometries checking internal free faces could consume lot of t h e , a quick shortcuti s toconvertfreefaces (Splitquadsto trias)to tetramesh. Successfulconversionindicates brick meshing is ok and there are no internalfaces.

- Practical FiniteElementAnalysisHidden llne or dynamicviewlngoption:By displaying the brick mesh mode1in hidden line or fast dynamic viewing mode m e canimrnedîatefyrecognizenode/ element conneaivity problemif any.Apart from above lisredchecks othef checks Ilikeduplicate element~,duplicate noder, detetetemp. / free nodesetc.) as discussedin chrrpter 7, should also be applied.8.9 How Not am Math1i Mid nodesshould lie exaitly on the geomeby:Nor acceptable RecommendedFor parabolic tetra meshing task, many CAE engineers prefer to start with linear tria (insteadof

parabolic) meshing and then covert it to parabolic. In the conversion process mid nodes might not get projected automatically on the curved surfaces and fillets.If so, it should be projectedon corresponding surfaces before conversion to tetras. 2)Whenthe job i s split among severalengineers,element length and overall meshpattern should be consistent. \ Above job was split arnong 3 engineers due very short duration provided by the client. Same mesh size and pattern was not followed by engineers working independently on sub parts of geometry. 3) Minimum 2 elements on the fillets for tetra meshing: Elementsat fillets and curved surfaces usually fail in jacobianldistortion. Manualadjustment for improving the quality results in mesh deviation from geometry and visible kinks.This could be avoided by modeling the fillets with 2 or more elements. 4) For brick meshing minimum 2 elements across the thickness: Single element leadto poor interpolation & thus affect accuracyof results. Minimum 2 elements124

across any thickness is recommended. Exception is NVH applications where not the stress butrepresentationof mass and stiffness(with least dofs) is main criteria...5)Use of tetra 1pyramid . .- w.>hile brick meshing: elements .' . . . -. ,-Some clients allow fewtetra elements for brickmeshing. Also some software5and analysis typessupport pyramid elements. Use of tetra and pyramid can make life of brick mesher tolerable ifnot easyl It's a good practice to clarify instructionsfor use of these elements from the client.6) Modelinga sheet metalpart with 3-d elements:For sheet metal or very small thickness parts 2-d shell elements are better suited andrecommended.I t s not like we can not use 3-d mesh atall, but it will result in very high numberof nodes & elements.Consider following sheet metal part 200 x 200 x 2 mm. We will mesh the same part with 3 4paraboiic tetra elements and 2-d quad-4 (linear) elements using same element length andcompare number of nodesand elements.

3 D Meshing 3-d Terramesh Elements =lOO Nodes= 14% Elements= 6897 )Limitation of 1-d element and advantageof 3-d meshing:Fillets, cutouts and complicated geometry features can not be represented accurately by 1-delement.3-d elements becauseof 3 dimensions cancapture al1the minute detailsaccurately. For exampleconsider following shaft. lt is very difficult to capture key way slot and variable fillet via 1-delements, instead 3-d meshing is recommended for such applicat.lon>. . .

Special Elements and Special TechniquesB.T connecfion of SolidElements with Beamsand ShellsNIsolid elemenk have only 3 translationaldofswhile thin shellandbeam elements have 6 dofs@translations+ 3 rotations). -.. , -., 8eam:6 dof 13Tt 3RjSalid:3 dof (3TfWhen a shell or beamis connectedto solid, at thejunction nodesrotationaldofs remain freeandhence moments would not be transmitted.The joint would work as a bal1joint (only forces aretransmitted and no moments).Exercise to study connection of solidand shell:Problem definition:

SpecialElementsandSpecia1Techniques Case 1: Pure brick mesh - ..-. .'Case 2r Shell Solid[brick) çonnection (direct conmlion at the nides)Direct connection would result in very high displacement and stress [solution possible if autosingularity option is switchedon) and work as a bail or hinge joint.Case 3: One layer of shelicoating on adjoiningbrick faces:The most commoniy used and recommended technique

Could you notice, this is a special trick to impose 3 rotational dofs atcommon nodes. Additionallayer of quad elements of same thickness as parent shell elements i s created on the faces ofadjoining brick elements (on both sides of junction). Thus the common nodes are now sharedby quad as well brick (6 dofs) and will transfer the moments appropriately.Case 4: One layer o fcoat insldethe element (acrossthe thickness o f brick element):3 Rotationaldofs at the common nodescould also be imposedby inserting quad element insidethe brickelement (acrossthe thickness).Though one element insidecould solvethe purposebut2 elements are recommended for better results.Case 5: Connection of Tetra 10 t o quad 8Tria 6 elements coating should be provided on parabolic tetra 10 element faces, connectingquad elements should also be parabolic i.e. quad 8 (parabolicthin shell meshing of plate).

SpeciolElernent~ondSpeciol Techniques 9.2 Linear to Parabolic and Brick toTetra Connection Why do we need to connect dissimilar mesh? For a big assembly, effective strategy to keep control on dofs (withoutcompromising accuracy) is to mesh criticalcomponents fine or with parabolic elements and remaining ones with coarse or linear elements. Please note that use of dissimilar element techniques isnot recommended when it is possible to -manaae with samet.w. e of element.Thouq-h dissimilar mesh does not haveany significant effect on stress results but it could affect displacement results. Commercial softwares orovide several outions for dissimilar mesh connection such as MPCs, Rigid, Specialelements (e.g. rsscon)etc. After the analysis displacement and stresscontour plots should be observedcarefully in the regionof dissimilar mesh connectivity. As a thumb ruletransition should be carried out in areasaway from the critical. Quad 8 to Quad 4: parabolic toLin@urdirectconnection Many theoretical books do not recommend direct connection between linear and parabolic element as shown above but we found very little effect in particular on stress magnitude, provided the connectionis carried out away from critical areas.

Exercise to study effect of joining dissimilar meshBrick 8 to Tetra 4:For visible clarity purpose there is large gap between two dissimilar meshes. For real lifeapplications the gap should be as srnall as possible.Boundary conditions:Back face clamped, Force =10,000 NTetra 10 toTetra 4:

Specinl E h e n r s ondSpeefnlTechniques Pure bricka: Pure Tetra 10:

- Practical FiniteElement AnalysisSummary table: ..S.tress, Displacement Nodes Description9.3 Hybrid Meshing(Hex-Pyram-Tetra) :Hybrid meshing is a very special option and not supported by al1 the softwares. Basic conceptis to use hex (linear) elements in critical areas and tetra meshing (parabolic) in general areas. Ifmeshing is carried out using pure hex elements then it will take lot of time, hybrid mesh savesmeshing time substantially without compromising quality of results. There arethree options forconnecting hex 8 and tetra 10 elements :1) Hex8 - Special hexelement (8nodesonone faceand 4on rest)- Pyram parabolic (connectedto 8 node face side of hex element) -Tetra 10 elements.2)Hex 8 - Special Pyram element (4 nodesat quadrilateral base 6 nodeson al1tria faces) - Tetra10 elements. Mixed Formulation PyramidElement

SpeciolElementsandSpecio1Techniques 3) ) Hex 8 - Linear Pyrarn element -Special tetra element (3 nodeson sorne edgesand 6 on some) - Tetra 10 elements. Mixed FarmulationTetrahedralElement 9.4 GAP Element .4 Gap is nothing but literally nothing. Nothing means ernpty air between components Consider following two cantilever plates with physical gap in between. -J - iwoploteswithoutgap elemenfs/contoct In real life when force is applied on upper plate, it will deflect till it cornes in contact with the bottorn plate and then both the plates will defiect together.When this problem i s solved in FEA, without defining gap elements or contact, result will be as shown below.Interference ofp/ates .--1 LPlates will physically intersect. Bonom plate not taking any load (stress = 0).FEA is based on rnathematicalequations. Itdoesnot understand / recognize any other physicallyseparated component/element unless and until rnathematical relation is defined between

Gap element defines mathematical equation as follows:Assume gap between two parts is 4 mm, Stiffnessof top plate is Ktw and bottom plate is K:-,Till4mm: F = Yw*6After4mm: F= ( K , + L ) * 6 Gap elemenfs dehned1( Description hl!...-* 1 1Animation Resultslnoutforaaoelement:Gap distance, Contact direction, Contact stiffness, Co-ordinatesystem andFriction (rubbing of two surfaces when in contact).Aoolications:1) Bearingand Shaft2) Press/shrinkfitproblems (negativegap)3) In generalto simulatecontact betweenany two surfaces4) Specialapplications liketo simulate ropebehaviour by using beamIrodelements.