V. Finite Element Method. 5.1 Introduction to Finite Element Method

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1 V. Fnte Element Method 5. Introducton to Fnte Element Method

2 5. Introducton to FEM

3 Rtz method to dfferental equaton Problem defnton k Boundary value problem Prob. Eact : d d, 0 0 0, 0 ( ) ( ) 4 C C * 4 ( ) Varatonal prncple Prob. d Etremze F( ( ) d 0 d subject to (0) 0, () 0 0 0, 0 * F F * 4

4 Rtz method to dfferental equaton Etremzaton of a functon and ts related algebrac equaton y ( )( ) 0 Etremze y ( ) Etremzaton of a functonal and ts related dfferentaton equaton dt, 0, T (0) 0, T () 0 d dt ( ) ( ) 0 Etremze F T T d d T(0) 0, T() 0 ( Prob. A) ( Prob. B) Tral functon: n T ( ) C f ( ), T (0) 0, T () 0

5 Rtz method to dfferental equaton Appromaton ( ) C C Transformaton: Functon space Fnte dmensonal vector space 3 4 F( ) C C 3 C C d 0 d C 3 d C 0 0 Necessary condton for F( C, C ) to be etreme : d C C d C d C F( C, C ) F( ) C C C C C C Lnear equaton: Appromate soluton: Tral functon F C F 0, 0 C 0 5 C 3 C, C C ( ) d Etremze F( ( ) d 0 d subject to (0) 0, () 0 Eact soluton * 4 ( )

6 Weghted resdual approach to dfferental equaton d d 0 0, 0 0, 0 Prob. Prob. b a ( ) ( ) ( ) ( ) ( ) d (0) () 0 Assume (0) () 0 ( ) s arbtrary. b a u v uv uv b a d 0 d 0 0 ( ) d 0 cos tan 0 0, 0 Set of functons e 4 log sn Weak form [ ( ) ( ) ( )] d 0 0 (0) 0, () 0 where ( ) s s arbtrary ecept eptthat that (0) 0 and () 0 cos tan 0 0, 0 * e 4 log sn

7 Galerkn approach Lnear equatons Appro. soluton : ( ) C C ; C and C are unknown Tral functon ( ) W W ; W and W are arbtrary C ( ) C ( 3 ) W ( ) W ( 3 ) d W ( ) W ( ) d 0 3 W ( )( ) d C 0 ( )( 3 ) d C 0 ( ) d d 0 4 W ( 3 )( ) d C 0 ( 3 )( 3 ) d C 0 ( ) d 0 0 W W W are W are arbtrary Appromate weghtng functon 0 5 C 3 C, C C ( ) Basc functon ( ) ( ) ( ) ( ), d, etc , 0 Set of appromate weghtng functons 4 e log sn

8 Error.% * * 0, 7 0,, Error 0% Requrements on basc functon Lnearly ndependent * p H Accuracy of the appromate soluton Comparson between appromate and eact solutons ma , Characterstcs of soluton convergence or C ma p ( ) ( ) ( ) * 4 ( ) 3 Accuracy Accuracy n n ( C ( )) ( C ( )) Eact Appromate n * lm C ( ) ( ) n ma <Comparson of appromate and eact solutons>

9 Basc dea of Fnte Element Method (FEM) C Tral functon : C C for 0 ( ) for Appromate soluton : 7 7 ( ) 9 96 Superconvergence Eact FE soluton <Basc functon = Interpolaton functon> <Comparson of FE soluton wth eact soluton>

10 FE solutons wth dfferent FEA models FEM = Rtz or Galerkn method + FE dscretzaton and nterpolaton (appromaton) FE dscretzaton and nterpolaton functon: N ( ) Technque of makng basc functons N ( ) N ( ) N ( ) N ( ) 3 N ( ) 0 0 Node Element ( ) N ( ) N ( ) n Fnte Element Method 3 3 Fnte element solutons ( ) C ( ) C ( ) n Rtz or Galerkn method Eact FE solutons

11 Rtz method to a beam deflecton Defnton of a beam deflecton problem EIv( ) M ( ), v(0) 0, v( L) 0 b BVP based on beam theory: d d v(0) 0, v() 0, 0 V( ) Varatonal prncple: dv Etremze F( v) ( ) v( ) d 0 d subject to v() 0 Rtz method It should satsfy essental boundary condton, ()=0 [E..] [E..] EI, w, L 0 v( ) = C (-) v( ) = C (- ) C (-) v( ) = C (- ) C (- ) [E. -3] v V M ( ) b Soluton: 0 0 M ( ) b 4 * 5 v ( )

12 Rtz method to a beam deflecton [E..3] : v( ) C ( ) C ( ) 3 F( v) { 0 C ( 3 ) C( )} ( ){ C ( ) C( )} d 5 C 4 C C 6 C C C F( C, C ) C C v Eact: 4 * 5 v ( )

13 5. FEM of Partal Dfferental Equatons

14 Posson s equaton Posson s equaton D d, 0 d k k f (, y) 0 y 0 0, 0 T T on S T D = 3D d Etremze F( ( ) d 0 d subject to (0) 0, () 0 Etremze F( ) (, ) (, ) k k f y y ddy y subject to T T on S (, ) (, ) (, ) T y N y N y N?,? J J J N J J Nodal value

15 Mesh system Mesh Interpolaton functon N (, y) J J η - ξ -

16 Intellgent remeshng of quadrlaterals

17 Intellgent remeshng of tetrahedrals Accurate descrpton wth mnmum elements! As number of elements ncreases, that of remeshngs does so much, whch deterorates soluton accuracy.

18 D mesh systems wth hgher qualty Intal Fnal

19 KSTP Fall 06. 3D mesh systems wth hgher qualty

20 4.5 Fnte element formulaton Equaton of equlbrum q j, g j j j,,t Equaton of heat conducton T

21 Coordnate as Frst and second order Permutaton symbol ε jk j y e u = u or u σ j = σ j = σ σ σ 3 σ σ σ 3 = σ 3 σ 3 σ 33 σ σ y σ z σ y σ yy σ yz σ z σ zy σ zz ε jk 0 f f f = j or j = k or k =, j, k =,,3 or,3, = 3,,, j, k =,3, or,,3 = 3,, e 3 e k 3 z, y, z as,, 3 as Partal dfferentaton φ, = φ, φ,j = φ j, v,j = v, σ j,j = σ j j Eamples W = a b W = a b c = a b c = ε jk a j b k Mechancal quanttes u, u y u, u σ y or τ y σ Unt vector 3 j= Summaton σ j,j + f = 0 σ j,j + f = 0 Free nde: once n a term (cannot be change) Dummy nde: twce n a term grad φ φ, dv v curl v φ v. ε jk v k,j φ,, j, k e, e, e 3 Dvergence theorem Kronecker delta δ j δ j 0 f j f = j Q jk,m..m, dv = Q jk,m..m n ds V S n : Outwardly drected unt normal vector

22 S Fnte element analyss of 3D elastc problems of sotropc materals VV P Fnte element equatons u S Weak form dv t ds f dv 0 V j j S V t u u, 0 on S u j j yy yy zz zz y y yz yz z z, yy, zz, y, yz, z, D yy, j j 3N N U I I I D B U j jj J j j ( D B U )( B W ) j jj J I I W B D B U I ji j jj J T T W B D B U Galerkn appromaton Stffness matr NW I I Shape functon matr B U I I Nodal dsplacement zz 3, 3 y,, yz, 3 3, z 3,, 3 N N N I, I, 3 I, 3 WI BIWI N I, N I, N N K U F K IJ J I B D B dv IJ V I j jj Weghted resdual method N I, 3 3 I, N 3 I, I, 3 Force vector t T j (, j j, ) u, u yy, u zz 3, 3 u y, u, u yz, 3 u 3, u z 3, u, 3 I Stran-dsplacement matr Elastc matr E( ) D ( )( ) /( ), ( ) / ( ) W I BI Dj B jj dv U J t NI ds f NI dv 0 V St V F t N ds f N dv I S I V I N, 0 0 N, 0 0 N N, N, 0 0 N, 0 0 N N, N 0 0 N 0 0 N N, N, 0 N, N, 0 N N, N N, 0 0 N N 0 N N 0 N N N 0 N N 0 N N 0 N, 3, 3 N, 3, 3,, 3, N, 3 N,, 3,, 3, N, 3 N,

23 Rgd-plastc fnte element method Weak form Penalty method Penalty constant dv K dv t ds ds V j j V jj S S t t t c 0 Weak form Lagrange multpler method Lagrange multpler Med formulaton: knowns of velocty and pressure dv p dv f dv v qdv t ds ds j j, t t 0 V V V V S S t c j j ( DB U )( B W ) j jj J I I W B DB U I I j jj J T T W B D B U j j j 3 3 klkl 3 Non-lnear Numercal problem occurs n the elastc regon. Mnmum allowable effectve stran rate s needed. W I BI D jbjj dv U J N V V I, HM dv PM t S NIdS f NIdV t V Q N H dv U M J, M J 0 V Fnte element equatons K j CIQ U J FI CMJ 0 MQ P Q G M Over-constraned problem -Reduced ntegraton -MINI-element K IJ B V I D jbjj dv CIM NI, HM dv V

24 Fnte element equatons of heat conducton Weak form V T ( c kt Q ) dv q h ( T T ) ds h ( T T ) ds S e t 4 4 ( T T ) h ( ) 0 e e T Te ds,, f c c q w S S c q Q C g j j Fnte element equatons T N T N W I I I I T NI TI t TI TI TI TI TI t t t t t T T T t I I I [,, W cn T N kn N T QN dv q h N T T N ds I J J I J I J I f c J J c I V Sc 4 4 h N T T N ds ( N T ) T h N T T N ds 0 Sq q J J w I J J e e J J e I Sc C T ( K K K K K ) T Q Q Q Q IJ J IJ IJ IJ IJ IJ J I I I I 0,,, 3, 4, ( CIJ /( t) KIJ KIJ KIJ KIJ KIJ ) TJ Q Q Q Q C T /( t),, 3, 4, I I I I IJ J C cn N dv IJ V I J K k N N dv K 0 IJ V I, J, IJ S c I J c h N N ds K K ( N N ) N N dv 4 3 IJ S I J I J e K 3 IJ S e I J e IJ S q I J q h N N ds h N N ds Q I V Q N dv I Q ( T h T ) N ds 4 4 I Se e e e I Q ( q h T ) N ds ] I Sc f c c I Q h T N ds 3 I Sq q w I

25 - 5 - Coupled analyss Equaton of equlbrum q j, g j j j,,t Equaton of heat conducton T