The Finite Element Method - Aalborg Universitet

The Finite Element MethodGeneral Meshing Guidelinesand AccuracyANSYSComputational Mechanics, AAU, Esbjerg

General Considerations in Meshing When choosing elements and creating meshes for FEAproblems users must make sure that– Chosen mesh size and density are optimal for theproblem (to save computational time)– Chosen element types are appropriate for theanalysis type performed (for accuracy)– Element shapes do not result in near singularstiffness matrices– Chosen elements and meshes can represent forcedistributions properlyANSYSComputational Mechanics, AAU, EsbjergMeshing rules2

Symmetry One of the most powerful means of reducing thesize of a FEA problem is the exploitation ofsymmetry Symmetry is said to exist if there is a completesymmetry of geometry, loads and constraintsabout a line or plane of symmetry When exploiting symmetry model needs to bemodified to replace the line or plane of symmetrywithout affecting the resultsANSYSComputational Mechanics, AAU, EsbjergMeshing rules3

Symmetry (cont’d) An simple case of complete symmetryConstraints corresponding to linesof symmetry (LOS) do not allowdisplacements perpendicular to theLOSANSYSComputational Mechanics, AAU, EsbjergMeshing rules4

Symmetry (cont’d) SimilarlyANSYSComputational Mechanics, AAU, EsbjergMeshing rules5

Symmetry (cont’d) There is no symmetry in this caseFANSYSComputational Mechanics, AAU, EsbjergMeshing rules6

Symmetry Meshing Rules Nodes must be placed on lines or planes ofsymmetry In 2D nodes on lines of symmetry (LOS) must beconstrained to have zero displacementsperpendicular to LOS; no rotational constraintson LOS (in-plane) In 3D nodes on the plane of symmetry (POS)must be constrained to have zero displacementsout of the POS; no in-plane rotational constraintson POSANSYSComputational Mechanics, AAU, EsbjergMeshing rules7

Antisymmetry Sometimes the loading or boundaryconditions may be such that antisymmetryexistsANSYSComputational Mechanics, AAU, EsbjergMeshing rules8

Antisymmetry (cont’d) Consider the simple antisymmetry casebelowθz 0θz 0Constraints corresponding to lines ofantisymmetry (LOAS) do not allowdisplacements along the LOAS or anyrotational displacementsANSYSComputational Mechanics, AAU, EsbjergMeshing rules9

Antisymmetry Meshing Rules Nodes must be placed on lines or planes ofantisymmetry In 2D nodes on lines of antisymmetry (LOAS)must be constrained to have zero translationaland rotational displacements along (in-plane)LOAS In 3D nodes on the plane of antisymmetry(POAS) must be constrained to have zero inplane translational and rotational displacementsANSYSComputational Mechanics, AAU, EsbjergMeshing rules10

Symmetry/Antisymmetry in ANSYS ANSYS supports symmetry andantisymmetry constraint setsANSYSComputational Mechanics, AAU, EsbjergMeshing rules11

Discontinuities Nodes must always be placed at locations wheregeometry, loads, or boundary conditions changeabruptly (discontinuities)Concentrated loadAbrupt change in loadAbrupt change ingeometryAbrupt change in supportANSYSComputational Mechanics, AAU, EsbjergMeshing rules12

Correct Choice of Elements Choose element types that are appropriate for theloading and stress conditions of the problem Make sure that the elements chosen capture all possiblesignificant stresses that may result from the givenloading, geometry, and boundary conditionsSlender beam;beam elementsThick beam (shear present);quadrilateral plane stress orplane strain elementsANSYSComputational Mechanics, AAU, EsbjergMeshing rules13

Aspect Ratio For a good mesh all elements must have a lowaspect ratio Specificallybhb 2 4h where b and h are the longest and the shortestsides of an element, respectivelyANSYSComputational Mechanics, AAU, EsbjergMeshing rules14

Element Shape Angles between element sides must notapproach 0 or 180 WorseANSYSComputational Mechanics, AAU, EsbjergBetterMeshing rules15

Mesh Refinement Finer meshing must be used in regions ofexpected high stress gradients (usually occur l Mechanics, AAU, EsbjergMeshing rules16

Mesh Refinement (cont’d) Mesh refinement must be gradual withadjacent elements of not too dissimilarsize Mesh refinement must balance accuracywith problem size ANSYS provides various tools for meshrefinement such as refinement at nodes,elements, lines, and volumesANSYSComputational Mechanics, AAU, EsbjergMeshing rules17

Dissimilar Element Types In general different types of elements withdifferent DOF at their nodes should not shareglobal DOF (for example do not use a 3D beamelement in conjunction with plane stresselements) ANSYS allows certain classes of differentelement types to share nodes (e.g. spar andbeam elements) but element and meshingguidelines must always be consulted beforeattempting to combine dissimilar element typesANSYSComputational Mechanics, AAU, EsbjergMeshing rules18

Equilibrium and Compatibility The approximations and discretizations generated bythe FE method enforce some equilibrium andcompatibility conditions but not others– Equilibrium of nodal forces and moments is alwayssatisfied because ofKU F– Compatibility is guaranteed at the nodes because ofthe way K is formed; i.e. the displacements of sharednodes on two elements are the same in the globalframe in which the elements are assembledANSYSComputational Mechanics, AAU, EsbjergMeshing rules19

Equilibrium-Compatibility (cont’d)– Equilibrium is usually not satisfied acrossinterelement boundaries; however discrepanciesdecline with mesh refinementσ x( 1) σ x( 2 )σ y( 1) σ y( 2 )12τ xy( 1) τ xy( 2 )along this boundaryStresses at shared nodesare typically averaged overthe elements sharing thenodeANSYSComputational Mechanics, AAU, EsbjergMeshing rulesANSYS uses stressdisparities at nodes as ameasure of discretizationerror20

Equilibrium-Compatibility (cont’d)– Stresses are most reliable near the centers of elements andleast reliable near their edges– Compatibility may not be satisfied across interelementboundaries (happens with certain types of higher-order elementsand junctures of dissimilar elements); incompatibilities declinewith mesh refinementLower-order elementHigher-order elementgapANSYSComputational Mechanics, AAU, EsbjergMeshing rules21

Equilibrium-Compatibility (cont’d)– Equilibrium is usually not satisfied within elementsbecause KU F does not enforce the relationsproduced by the partial differential equations thatdefine equilibrium at infinitesimal levels; the assumeddisplacement functions that led to KU F only satisfykinematic boundary conditions, not the differentialequations themselves– Compatibility is satisfied within elements (guaranteedby the choice of continuous and single valueddisplacement functions)ANSYSComputational Mechanics, AAU, EsbjergMeshing rules22

Example: Plate with Crack Model the thin aluminum plate shown below usingsymmetry and refine mesh in regions where thediscretization error is large 1 ′′22′′5′′10,000 psi10′′ANSYSComputational Mechanics, AAU, EsbjergMeshing rules23

Example (cont’d) Modeling only the right half of the plate, using PLANE42 elementsand applying symmetry boundary conditions we obtain the followingstress ( σ x ) distribution in ANSYSANSYSComputational Mechanics, AAU, EsbjergMeshing rules24

Example (cont’d) To see the discretization error in ANSYS we plot the variable SDSG(from “Error Estimation” in “Element Solution”)ANSYSComputational Mechanics, AAU, EsbjergMeshing rules25

Example (cont’d) Clearly (and as expected) the worst error occurs aroundthe crack meaning that the elements in that region needto be modified Contour plots of the same stress distribution anddiscretization error estimate are shown in the next pagewith a model that includes (Level 3) refined elementsaround the crack The refined models exhibit smoother distribution ofstress with lower error estimatesANSYSComputational Mechanics, AAU, EsbjergMeshing rules26

Example (cont’d)ANSYSComputational Mechanics, AAU, EsbjergMeshing rules27

Example (cont’d)ANSYSComputational Mechanics, AAU, EsbjergMeshing rules28

Computational Mechanics, AAU, Esbjerg ANSYS Symmetry One of the most powerful means of reducing the size of a FEA problem is the exploitation of symmetry Symmetry is said to exist if there is a complete symmetry of geometry, loads and constraints about a line or plane of symmetry When exploiting symmetry model needs to be