Welcome!M51MAE Engineering Simulation Analysis Faculty of
Engineering and Computing Summer 20158/13/15 1Lecturers: Dr Iman
DayyaniRoom: EC4 - 35Email: [email protected] Omid
Ramk!a!Room: EC4 - "4Email: ac"[email protected]/13/15 2#odule
leader: Dr C!risto$!e %astienRoom : EC4 - 35email
aa34&[email protected]'8/13/15 35- Matrix Algebra1-
Introduction to FEM2- Discretization Concepts and Accuracy of te
FEM!- FEM Applications in Engineering"- Co##on ele#ent types and
teir for#ulations$- Conclusion4 8/13/151- Introduction to
FEM8/13/15 51- Introduction to FEMCo#putational%olid and
%tructuralMecanics &C%M'8/13/15 61- Introduction to FEM8/13/15
71- Introduction to FEM8/13/15 81- Introduction to FEM8/13/15 91-
Introduction to FEM- Models odies of Comple! S"ape-Can #andle
$eneral %oading&oundary Conditions-Models odies Composed of
Composite and Multip"ase Materials-Model is Easily 'efined for
(mpro)ed Accuracy *y +arying Element Si,e and -ype .Appro!imation
Sc"eme/--ime 0ependent and 0ynamic Effects Can e (ncluded-Can
#andle a +ariety 1onlinear Effects (ncluding Material e"a)ior2
%arge 0eformations2 oundary Conditions2 Etc3 8/13/15 101-
Introduction to FEM8/13/15 111- Introduction to FEM8/13/15 121-
Introduction to FEM8/13/15 131- Introduction to FEM8/13/15 141-
Introduction to FEM8/13/15 151- Introduction to FEM8/13/15 161-
Introduction to FEM8/13/15 171- Introduction to FEM8/13/15 181-
Introduction to FEM8/13/15 192- Discretization Concepts and
Accuracy of the FEM8/13/15 202- Discretization Concepts and
Accuracy of the FEM(ny continuous solution )ield suc! as stress*
dis$lacement* tem$erature* $ressure* etc. can +e a$$ro,imated +y a
discrete model com$osed o) a set o) $iece-ise continuous )unctions
de)ined over a )inite num+er o) su+domains.4ne-0imensional
-emperature 0istri*utionExact Analytical SolutionxTApproximate
Piecewie !inear SolutionxT8/13/15 212- Discretization Concepts and
Accuracy of the FEM()o-Di#ensional Discretization8/13/15 222-
Discretization Concepts and Accuracy of the FEM.0omain
Appro!imation.Element (nterpolation&Appro!imation.1umerical
(ntegration Errors.(ncluding Spatial and -ime (ntegration/.Computer
Errors .'ound-4ff2 Etc32 /8/13/15 232- Discretization Concepts and
Accuracy of the FEMAccuracyError = |(Exact Solution)-(FEM
Solution)|ConvergenceLimit of Error as: um!er of Elements
(h-convergence) orA""roximation #r$er (p-convergence)
%ncreases%$eally& Error ' as um!er of Elements or A""roximation
#r$er 8/13/15 242- Discretization Concepts and Accuracy of the
FEM((iscreti)ation *it+ ,,- Elements)((iscreti)ation *it+ ./,
Elements)(0riangular Element)(o$e)8/13/15 252- Discretization
Concepts and Accuracy of the FEM8/13/15 26- FEM App!ications in
Engineering8/13/15 27- FEM App!ications in Engineering8/13/15 28-
FEM App!ications in Engineering8/13/15 29- FEM App!ications in
Engineering8/13/15 30- FEM App!ications in Engineering8/13/15 31-
FEM App!ications in Engineering8/13/15 32- FEM App!ications in
Engineering"i#$ %i&elity Analyi' %inite Element
(et$o&)*Static Stree+ynamic Stree8/13/15 33- FEM App!ications
in Engineering8/13/15 34"- Co##on e!e#ent types and their
for#u!ations -"e Finite Element E5uation Must (ncorporate t"e
Appropriate 6"ysics of t"e 6ro*lem For 6ro*lems in Structural Solid
Mec"anics2 t"e Appropriate 6"ysics Comes from Eit"er Strengt" of
Materials or -"eory of Elasticity FEM E5uations are Commonly
0e)eloped 7sing Direct2 *ariational-*irtual +or, or +eigted
-esidual Met"ods0irect Met"od ased on p"ysical reasoning and
limited to simple cases2 t"is met"od is 8ort" studying *ecause it
en"ances p"ysical understanding of t"e process+ariational-+irtual
Wor9 Met"odased on t"e concept of )irtual displacements2 leads to
relations *et8een internal and e!ternal )irtual 8or9 and to
minimi,ation of system potential energy for e5uili*riumWeig"ted
'esidual Met"odStarting 8it" t"e go)erning differential e5uation2
special mat"ematical operations de)elop t"e :8ea9 form; t"at can *e
incorporated into a FEM e5uation3-"is met"od is particularly suited
for pro*lems t"at "a)e no )ariational statement3 For stress
analysis pro*lems2 a 'it,-$aler9in W'M 8ill yield a result
identical to t"at found *y )ariational met"ods38/13/15 35"- Co##on
e!e#ent types and their for#u!ations-0omain 0iscreti,ation-Select
Element -ype .S"ape and Appro!imation/-0eri)e Element E5uations
.+ariational and Energy Met"ods/-Assem*le Element E5uations to Form
$lo*al System ?7@ A ?F@ A Stiffness or 6roperty Matri!?7@ A 1odal
0isplacement +ector?F@ A 1odal Force +ector -(ncorporate oundary
and (nitial Conditions -Sol)e Assem*led System of E5uations for
7n9no8n 1odal
0isplacements and Secondary 7n9no8ns of Stress and Strain
+alues8/13/15 36"- Co##on e!e#ent types and their
for#u!ations12$u1u2F1F2Sti1ness Matrixo$al Force 2ector8/13/15 37"-
Co##on e!e#ent types and their for#u!ations} { } ]{ [rm Matrix Fo
in or 2 Node atm Equilibriu 1 Node atm Equilibriu21212 1 22 1 1F u
KFFuuk kk kku ku Fku ku F;';'1]1
