Insttuto Tecnológco de Aeronáutca FIITE ELEMETS I Class notes AE-5
Insttuto Tecnológco de Aeronáutca 5. Isoparametrc Elements AE-5
Insttuto Tecnológco de Aeronáutca ISOPARAMETRIC ELEMETS Introducton What are the condtons to ensure that as the mesh s refned the Galerkn appromate soluton converges to the eact soluton? AE-5
Insttuto Tecnológco de Aeronáutca ISOPARAMETRIC ELEMETS Introducton Shape functons must be Smooth at least C on each element nteror. Contnuous across element boundares Complete AE-5
Insttuto Tecnológco de Aeronáutca ISOPARAMETRIC ELEMETS Introducton Condtons and guarantee fnte jumps across element nterfaces. n n n n AE-5 5
Insttuto Tecnológco de Aeronáutca ISOPARAMETRIC ELEMETS Introducton Shape functons that satsf condtons and are of class C 0 Euler-Bernoull beams requre hgher order of dervatves. Thus shape functons must be of class C C on the element nteror and C across boundares. AE-5 6
Insttuto Tecnológco de Aeronáutca ISOPARAMETRIC ELEMETS Introducton Integrands wth dervatves of order m: C m on the element nteror and C m- across boundares. AE-5 7
Insttuto Tecnológco de Aeronáutca ISOPARAMETRIC ELEMETS Completeness Completeness requres that the element nterpolaton functon s capable of eactl representng an arbtrar lnear polnomal when the nodal degrees of freedom are assgned values n accordance wth t. AE-5 8
Insttuto Tecnológco de Aeronáutca ISOPARAMETRIC ELEMETS Completeness Let u h m d where d u h AE-5 9
Insttuto Tecnológco de Aeronáutca ISOPARAMETRIC ELEMETS Completeness D stuaton: shape functons are complete f mples d c0 c c c u h c0 c c cz z AE-5 0
Insttuto Tecnológco de Aeronáutca ISOPARAMETRIC ELEMETS Completeness Argument: as the mesh s refned the eact soluton and ts dervatves approach constant values over the elements. In elastct completeness means that the element can represent all rgd bod motons and constant stran states. AE-5
Insttuto Tecnológco de Aeronáutca ISOPARAMETRIC ELEMETS Blnear quadrlateral element Straght edge four node quadrlateral mappng AE-5
Insttuto Tecnológco de Aeronáutca AE-5 ISOPARAMETRIC ELEMETS Blnear quadrlateral element Straght edge four node quadrlateral
Insttuto Tecnológco de Aeronáutca AE-5 ISOPARAMETRIC ELEMETS Blnear quadrlateral element Assume blnear epansons β β β β α α α α 0 0
Insttuto Tecnológco de Aeronáutca AE-5 5 ISOPARAMETRIC ELEMETS Blnear quadrlateral element Shape functons
Insttuto Tecnológco de Aeronáutca ISOPARAMETRIC ELEMETS Blnear quadrlateral element Shape functons: propertes Shape functons of class C wthn elements. s alwas smooth n and but t ma be dscontnuous n and. Shape functons of class C 0 boundares. across element AE-5 6
Insttuto Tecnológco de Aeronáutca AE-5 7 ISOPARAMETRIC ELEMETS Blnear quadrlateral element Shape functons: propertes Completeness: c c c c c c c c c c c c d u h 0 0 0 0
Insttuto Tecnológco de Aeronáutca ISOPARAMETRIC ELEMETS Isoparametrc element Defnton m u h m d same shape functons The blnear quadrlateral element s soparametrc AE-5 8
Insttuto Tecnológco de Aeronáutca ISOPARAMETRIC ELEMETS Isoparametrc element If the mappng s one-to-one and f ts Jacoban s postve for ever then the convergence condton s met. Isoparametrc elements automatcall satsf the three basc convergence condtons. Isoparametrc elements can assume convenent shapes for practcal analss. AE-5 9
Insttuto Tecnológco de Aeronáutca ISOPARAMETRIC ELEMETS Trangular element Constant stress/stran element. Coalescence of two nodes. AE-5 0
Insttuto Tecnológco de Aeronáutca AE-5 ISOPARAMETRIC ELEMETS Trangular element m
Insttuto Tecnológco de Aeronáutca ISOPARAMETRIC ELEMETS Trangular element Eercses Sketch the shape functons for the trangular element n the doman. Compute the Jacoban determnant at 0 for the element shown. 0 00 0 AE-5
Insttuto Tecnológco de Aeronáutca ISOPARAMETRIC ELEMETS Trlnear heahedral element ζ 6 5 7 8 5 8 6 7 AE-5
Insttuto Tecnológco de Aeronáutca AE-5 ISOPARAMETRIC ELEMETS Trlnear heahedral element ζ α ζ α ζ α α ζ α α α α ζ 7 6 5 0 8 7 6 5 7 6 5 0 α α α α α α α α
Insttuto Tecnológco de Aeronáutca AE-5 5 ISOPARAMETRIC ELEMETS Trlnear heahedral element 8 ζ ζ ζ 8 7 6 5 ζ
Insttuto Tecnológco de Aeronáutca ISOPARAMETRIC ELEMETS Trlnear heahedral element Degenerated wedge 5 6 7 8 5 6 7 8 coalescence AE-5 6
Insttuto Tecnológco de Aeronáutca ISOPARAMETRIC ELEMETS Trlnear heahedral element Degenerated tetrahedral 5 5 6 7 8 6 7 8 coalescence AE-5 7
Insttuto Tecnológco de Aeronáutca ISOPARAMETRIC ELEMETS Hgher order elements Shape functons of order hgher than lnear More accurate representatons Boundares ma be curved Computatonall more epensve AE-5 8
Insttuto Tecnológco de Aeronáutca AE-5 9 ISOPARAMETRIC ELEMETS Hgher order elements Lagrange polnomals m m m j j j m j j j l L L L L th term omtted
Insttuto Tecnológco de Aeronáutca ISOPARAMETRIC ELEMETS Hgher order elements Lagrange polnomals l j δ j m-noded element n one dmenson: l AE-5 0
Insttuto Tecnológco de Aeronáutca ISOPARAMETRIC ELEMETS Hgher order elements Eamples -noded one dmensonal element: -noded one dmensonal element: 0 AE-5
Insttuto Tecnológco de Aeronáutca ISOPARAMETRIC ELEMETS Hgher order elements Lagrange elements: hgher order two- and three- dmensonal elements can be obtaned b takng products of Lagrange polnomals. 7 8 9 6 5 AE-5
Insttuto Tecnológco de Aeronáutca ISOPARAMETRIC ELEMETS Varable number of nodes D quadrlateral element wth 5 to 9 nodes Add ffth node to sde -: 5 AE-5
Insttuto Tecnológco de Aeronáutca ISOPARAMETRIC ELEMETS Varable number of nodes 5 l l ow shape functons and must be modfed to be zero at node 5. 5 5 AE-5
Insttuto Tecnológco de Aeronáutca AE-5 5 ISOPARAMETRIC ELEMETS Varable number of nodes 8 7 7 6 6 5 8 5 8 7 6 5
Insttuto Tecnológco de Aeronáutca ISOPARAMETRIC ELEMETS Varable number of nodes ode 9 present Bubble functon: 9-8 do not vansh at node 9. Hence 9 j j 9 j 5 6 7 8 AE-5 6
Insttuto Tecnológco de Aeronáutca ISOPARAMETRIC ELEMETS Element famles Serendpt faml of quadrlateral elements AE-5 7
Insttuto Tecnológco de Aeronáutca ISOPARAMETRIC ELEMETS Element famles Serendpt quadrlaterals m m O O m m AE-5 8
Insttuto Tecnológco de Aeronáutca ISOPARAMETRIC ELEMETS Element famles Lagrange faml of quadrlateral elements AE-5 9
Insttuto Tecnológco de Aeronáutca AE-5 0 ISOPARAMETRIC ELEMETS Element famles Lagrange quadrlaterals m m m m m m m m m m m m O O
Insttuto Tecnológco de Aeronáutca ISOPARAMETRIC ELEMETS Element famles Standard trangular elements AE-5
Insttuto Tecnológco de Aeronáutca ISOPARAMETRIC ELEMETS Element famles Trangles O m m m L O m O m m AE-5
Insttuto Tecnológco de Aeronáutca Integraton of ISOPARAMETRIC ELEMETS umercal ntegraton Ω f dω In one dmenson b a f d AE-5
Insttuto Tecnológco de Aeronáutca ISOPARAMETRIC ELEMETS umercal ntegraton umercal ntegraton methods Trapezodal rule nd order accurate Smpson s rule th order accurate ewton-cotes formulas Gaussan quadrature fewer ponts AE-5
Insttuto Tecnológco de Aeronáutca ISOPARAMETRIC ELEMETS Gaussan quadrature b a d α f α f... α n f f n R n α weghts samplng ponts AE-5 5
Insttuto Tecnológco de Aeronáutca AE-5 6 ISOPARAMETRIC ELEMETS Gaussan quadrature... 0 p f β β β ψ where... n p n n l f l f ψ n n l L L L L
Insttuto Tecnológco de Aeronáutca AE-5 7 ISOPARAMETRIC ELEMETS Gaussan quadrature At the samplng ponts f p f... 0 ψ β β β ψ Integraton of f elds 0 b a n b a b a d p d l f d f β
Insttuto Tecnológco de Aeronáutca ISOPARAMETRIC ELEMETS b Gaussan quadrature Requre that: p d 0 for 0... n a p s of order n. Hence ntegrals of polnomals of order n to n are zero. Samplng weghts: α b l d a AE-5 8
Insttuto Tecnológco de Aeronáutca ISOPARAMETRIC ELEMETS Change of varables Gaussan quadrature b a f d f d Advantage: standardze results AE-5 9
Insttuto Tecnológco de Aeronáutca ISOPARAMETRIC ELEMETS Gaussan quadrature n r α 0.0.0 r smmetrcall postoned ±0.57750698966 ±0.775966698 0.0.0 0.555555555555556 0.888888888888889 α 5 ±0.8665905 ±0.998058856 ±0.90679859866 0.7858575 0.65558656 0.69688505689 ±0.586900568 0.78686709966 0.0 0.568888888888889 6 ±0.969505 0.797970 ±0.6609866665 0.607657089 ±0.869860897 0.67995769 AE-5 50
Insttuto Tecnológco de Aeronáutca ISOPARAMETRIC ELEMETS Gaussan quadrature Two and three dmensons: f r s drds α f r s ds α α j j f r s j AE-5 5
Insttuto Tecnológco de Aeronáutca AE-5 5 ISOPARAMETRIC ELEMETS Dervatves of shape functons Construct stffness matrces Recover strans and stresses Compute and Chan rule
Insttuto Tecnológco de Aeronáutca AE-5 5 ISOPARAMETRIC ELEMETS Dervatves of shape functons Chan rule The Jacoban matr must be computed frst!
Insttuto Tecnológco de Aeronáutca Jacoban matr ISOPARAMETRIC ELEMETS Dervatves of shape functons m m m m m m AE-5 5
Insttuto Tecnológco de Aeronáutca AE-5 55 ISOPARAMETRIC ELEMETS Dervatves of shape functons Chan rule det J
Insttuto Tecnológco de Aeronáutca ISOPARAMETRIC ELEMETS Trangular and tetrahedral elements Elements that model comple geometres Facltate mesh transtons Avod dstorted quadrlaterals/heahedra Area/volume coordnates AE-5 56
Insttuto Tecnológco de Aeronáutca AE-5 57 ISOPARAMETRIC ELEMETS Trangular elements Area coordnates 0 r s s r t s r s s r r s r 0 t constant
Insttuto Tecnológco de Aeronáutca ISOPARAMETRIC ELEMETS Trangular elements Area coordnates s r t AE-5 58
Insttuto Tecnológco de Aeronáutca ISOPARAMETRIC ELEMETS Trangular elements Eample: 6 node trangle 5 6 5 6 rr ss tt rs rt st AE-5 59
Insttuto Tecnológco de Aeronáutca AE-5 60 ISOPARAMETRIC ELEMETS Tetrahedral elements Volume coordnates r s t t s r u t s r t t s r s t s r r t s r
Insttuto Tecnológco de Aeronáutca ISOPARAMETRIC ELEMETS Trangular and tetrahedral elements umercal Integraton n r / s / t / α 0.5 / /6 /6 /6 fdω m Ω α f det J /6 /6 / /5 /5 / /6 / /5 /5 /6 / / /5 /5 /6 /6 7/96 5/96 5/96 /5 /5 /5 5/96 6 0.8687570 0.095765 0.09576 0.0597587 0.09576 0.8687570 0.09576 0.0597587 0.09576 0.09576 0.8687570 0.0597587 0.08008 0.5989 0.5989 0.69798 0.5989 0.08008 0.5989 0.69798 0.5989 0.5989 0.08008 0.69798 AE-5 6
Insttuto Tecnológco de Aeronáutca ISOPARAMETRIC ELEMETS Specal shape functons Elements wth specal propertes Infnte elements Modelng of sngulartes cracks AE-5 6
Insttuto Tecnológco de Aeronáutca ISOPARAMETRIC ELEMETS Specal shape functons One dmensonal eample: 0 / 0 a L u u u u a L AE-5 6
Insttuto Tecnológco de Aeronáutca Obtan stran: ISOPARAMETRIC ELEMETS Specal shape functons 0 / 0 a L a L L a a L If s a purel quadratc functon of then a L/ L AE-5 6
Insttuto Tecnológco de Aeronáutca Obtan stran: ISOPARAMETRIC ELEMETS Specal shape functons 0 / 0 L/ L du du d α α 0 α ε d d / d L L / α 0 L lm 0 ε Sngulart modelng AE-5 65
Insttuto Tecnológco de Aeronáutca Membrane FE code AE-5 66
Insttuto Tecnológco de Aeronáutca Membrane FE code Man routnes nput_data mechancal renum crout_dec crout_sol AE-5 67
Insttuto Tecnológco de Aeronáutca Membrane FE code Routne nput_data Purpose: generate mesh nput materal allocate memor fnd band wdth Features: user defned geometr dependant AE-5 68
Insttuto Tecnológco de Aeronáutca Membrane FE code Routne mechancal Purpose: generate element stffness matr and global stffness matr Features: Gaussan numercal ntegraton bquadratc and 6-node trangular elements AE-5 69
Insttuto Tecnológco de Aeronáutca Membrane FE code Routne renum Purpose: Appl essental boundar condtons Features: Zeroes rows and columns AE-5 70
Insttuto Tecnológco de Aeronáutca Membrane FE code Routne crout_dec Purpose: Decompose banded matr K L U upper trangular matr lower trangular matr Features: In-place decomposton smmetrc matr AE-5 7
Insttuto Tecnológco de Aeronáutca Membrane FE code Routne crout_sol Purpose: Solve sstem Kq f L U q f Features: Two step procedure AE-5 7
Insttuto Tecnológco de Aeronáutca Membrane FE code Results: Crcular plate radus.0 m thckness.0 mm E 08 MPa ν 0. AE-5 7
Insttuto Tecnológco de Aeronáutca Membrane FE code Results: Crcular plate.5.5 0.5 0-0.5 - u 6.E-0 5.5E-0.8E-0.50E-0.6E-0.75E-0 8.76E-0 0.00E00-8.76E-0 -.75E-0 -.6E-0 -.50E-0 -.8E-0-5.5E-0-6.E-0 0.5 0-0.5 - v 6.E-0 5.5E-0.8E-0.50E-0.6E-0.75E-0 8.76E-0 0.00E00-8.76E-0 -.75E-0 -.6E-0 -.50E-0 -.8E-0-5.5E-0-6.E-0 -.5 -.5 - - - 0 - - - 0 AE-5 7
Insttuto Tecnológco de Aeronáutca Membrane FE code Results: Crcular plate.5 0.5 0-0.5 - u 6.6E-0 6.0E-0 5.76E-0 5.E-0.87E-0.E-0.99E-0.5E-0.0E-0.66E-0.E-0.77E-0.E-0 8.86E-0.E-0.5 0.5 0-0.5 - v -5.8E-5 -.07E- -.6E- -.6E- -.7E- -.6E- -.8E- -.5E- -.90E- -5.5E- -6.00E- -6.5E- -7.09E- -7.6E- -8.9E- -.5 -.5 - - - 0 - - - 0 AE-5 75
Insttuto Tecnológco de Aeronáutca Membrane FE code Results: Crcular plate Radus.0 m Thckness.0 mm E 08 MPa ν 0. AE-5 76
Insttuto Tecnológco de Aeronáutca Membrane FE code Results: Crcular plate.5 0.5 0-0.5 - u 6.95E00 5.96E00.97E00.97E00.98E00.99E00 9.9E-0 0.00E00-9.9E-0 -.99E00 -.98E00 -.97E00 -.97E00-5.96E00-6.95E00.5 0.5 0-0.5 - v.06e00 9.0E-0 7.58E-0 6.07E-0.55E-0.0E-0.5E-0 0.00E00 -.5E-0 -.0E-0 -.55E-0-6.07E-0-7.58E-0-9.0E-0 -.06E00 -.5 -.5 - - - 0 - - - 0 AE-5 77
Insttuto Tecnológco de Aeronáutca Membrane FE code Results: Crcular plate.5 0.5 0-0.5 - u 8.5E00 7.0E00 6.09E00.87E00.65E00.E00.E00 0.00E00 -.E00 -.E00 -.65E00 -.87E00-6.09E00-7.0E00-8.5E00.5 0.5 0-0.5 - v.06e00 9.0E-0 7.58E-0 6.07E-0.55E-0.0E-0.5E-0 0.00E00 -.5E-0 -.0E-0 -.55E-0-6.07E-0-7.58E-0-9.0E-0 -.06E00 -.5 -.5 - - - 0 - - - 0 AE-5 78
Insttuto Tecnológco de Aeronáutca Membrane FE code Results: Crcular plate.5 0.5 0-0.5 - -.5 u.0e0 8.6E00 7.0E00 5.76E00.E00.88E00.E00 0.00E00 -.E00 -.88E00 -.E00-5.76E00-7.0E00-8.6E00 -.0E0.5 0.5 0-0.5 - -.5 v.8e00.0e00 8.0E-0 6.7E-0 5.0E-0.6E-0.68E-0 0.00E00 -.68E-0 -.6E-0-5.0E-0-6.7E-0-8.0E-0 -.0E00 -.8E00 - - - 0 - - - 0 AE-5 79
Insttuto Tecnológco de Aeronáutca Membrane FE code Results: L shaped plate m P m m thckness.0 mm E 08 MPa ν 0. AE-5 80
Insttuto Tecnológco de Aeronáutca Membrane FE code Results: L shaped plate.75.75.5.5 0.75 0.5 0.5 u 59.. 5.6 8.59-8.9-5.6 -.0-58.9-75.79-9.67-09.5-6. -.9-60.7-77.0.5.5 0.75 0.5 0.5 v 86.5 6.0.5 9.0 96.55 7.07 5.58 9.09 06.60 8. 6.6 9. 6.6-5.85-8. 0 0 0.5.5 0 0 0.5.5 AE-5 8
Insttuto Tecnológco de Aeronáutca Membrane FE code Results: L shaped plate.75.5.5 0.75 0.5 u 6.6. 6.65 9.6-8. -5.8 -.0-60.79-78.7-95.76 -.5-0.7-8. -65.7-8.0.75.5.5 0.75 0.5 v 98.78 75. 5.09 8.7 05.9 8.0 58.69 5.5.00 88.65 65.0.96 8.6 -.7-8.09 0.5 0.5 0 0 0.5.5 0 0 0.5.5 AE-5 8