# INTRODUCTION. governed by a differential equation Need systematic approaches to generate FE equations

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1 WEIGHTED RESIDUA METHOD INTRODUCTION Drect stffness method s lmted for smple D problems PMPE s lmted to potental problems FEM can be appled to many engneerng problems that are governed by a dfferental equaton Need systematc approaches to generate FE equatons Weghted resal method Energy method Ordnary dfferental equaton (second-order order or fourth-order) order) can be solved usng the weghted resal method, n partcular usng Galerkn method

2 EXACT VS. APPROXIMATE SOUTION Eact soluton Boundary value problem: dfferental equaton + boundary condtons Dsplacements n a unaal bar subject to a dstrbuted force p() d u + p ( ) =, u() = Boundary condtons () = Essental BC: The soluton value at a pont s prescrbed (dsplacement or knematc BC) Natural BC: The dervatve s gven at a pont (stress BC) Eact soluton u(): ( ) twce dfferental functon In general, t s dffcult to fnd the eact soluton when the doman and/or boundary condtons are complcated Sometmes the soluton may not ests even f the problem s well defned EXACT VS. APPROXIMATE SOUTION cont. Appromate soluton It satsfes the essental BC, but not natural BC The appromate soluton may not satsfy the DE eactly Resal: d u p( ) R( ) + = Want to mnmze the resal by multplyng py wth a weght W and ntegrate over the doman RW= ( ) ( ) Weght functon If t satsfes for any W(), then R() wll approaches zero, and the appromate soluton wll approach the eact soluton Dependng on choce of W(): least square error method, collocaton method, Petrov-Galerkn method, and Galerkn method

3 GAERKIN METHOD Appromate soluton s a lnear combnaton of tral functons N u ( ) = c( ) = Tral functon Accuracy depends on the choce of tral functons The appromate soluton must satsfy the essental BC Galerkn method Use N tral functons for weght functons R ( ) ( ) =, =,, N + + p ( ) ( ) =, =,, N d u ( ) ( ) ( ),,, p N = = 5 Galerkn method cont. GAERKIN METHOD cont. Integraton-by-parts: rece the order of dfferentaton n u() d p ( ) ( ),,, N = = Apply natural BC and rearrange d = () () () (),, N p( ) ( ) () () =, + Same order of dfferentaton for both tral functon and appro. soluton Substtute t the appromate soluton N d j d j j = c = p( ) ( ) + () () () (), =,, N 6

4 Galerkn method cont. Wrte n matr form N GAERKIN METHOD cont. Kc,,, = j j = F = N [ K] { c} { F} j= ( N N)( N ) ( N ) K j = d d j F = p ( ) ( ) + () () () () Coeffcent matr s symmetrc; K j = K j N equatons wth N unknown coeffcents 7 EXAMPE Dfferental equaton Tral functons d u + = u() = () =, Boundary condtons ( ) = ( ) = ( ) = ( ) = Appromate soluton (satsfes the essental BC) u ( ) = c ( ) = c+ c = Coeffcent matr and RHS vector K = ( ) = K = K = ( ) = K = ( ) = F = ( ) + () () () = F = ( ) + () () () = 8

5 Matr equaton [ K] = Appromate soluton u ( ) = EXAMPE cont. 9 { } = F { c} = [ K] { F} = 6 8 Appromate soluton s also the eact soluton because the lnear combnaton of the tral functons can represent the eact soluton 9 EXAMPE Dfferental equaton Tral functons d u + = u() = () =, Boundary condtons Coeffcent matr s same, force vector: ( ) = ( ) = ( ) = ( ) = 6 { F} = { c} = [ K] { F } = u ( ) = Eact soluton u ( ) = 6 The tral functons cannot epress the eact soluton; thus, appromate soluton s dfferent from the eact one

6 EXAMPE cont. Appromaton s good for u(), but not good for /.6.. u(), / u-eact u-appro. / (eact) / (appro.) HIGHER-ORDER DIFFERENTIA EQUATIONS w() = dw () = d w ( ), p = dw ( ) = M Beam bendng under pressure load dw ( ) = V N w ( ) = c( ) Fourth-order order dfferental equaton Appromate soluton = Weghted resal equaton (Galerkn method) dw p ( ) ( ) =, =,, N Essental BC Natural BC In order to make the order of dfferentaton same, ntegraton-by-parts must be done twce

7 HIGHER-ORDER DE cont. After ntegraton-by-parts twce dwd dw dwd dw dwd dwd + = p( ) ( ), =,, N = p( ) ( ) +, =,, N Substtute appromate soluton N d j d d w d w d j p( ) ( ) j = c = p( ) ( ) +, =,, N Do not substtute the appro. soluton n the boundary terms Matr form [ K]{ c} = { F} N NN N K j = d d j dw dwd = ( ) ( ) + F p EXMAPE w dw d w, = dw dw () = () = Fourth-order order DE () = () = Two tral functons =, = =, = 6 Coeffcent matr K = ( ) = K = K = ( ) = 6 K = ( ) = 6 [ K ] = 6

8 RHS dw() = + + F V dw() = + + F V EXAMPE cont. dw() () () + M () () = dw() () () + M () () = 6 9 Appromate soluton { c } = [ K] { F } = Eact soluton 7 w ( ) = + w ( ) = 5 EXAMPE cont. w'' w'', w'' (eact) w'' (appro.) w''' (eact) w''' (appro.) 6

9 FINITE EEMENT APPROXIMATION Doman Dscretzaton Weghted resal method s stll dffcult to obtan the tral functons that satsfy the essental BC FEM s to dvde d the entre doman nto a set of smple sub-domans (fnte element) and share nodes wth adjacent elements Wthn a fnte element, the soluton s appromated n a smple polynomal l form u() Appromate soluton Fnte elements Analytcal soluton When more number of fnte t elements are used, the appromated pecewse lnear soluton may converge to the analytcal soluton 7 FINITE EEMENT METHOD cont. Types of fnte elements D D D Varatonal equaton s mposed on each element... = One element 8

10 TRIA SOUTION Soluton wthn an element s appromated usng smple polynomals. n n n n n+ + -th element s composed of two nodes: and +. Snce two unknowns are nvolved, lnear polynomal can be used: u ( ) = a + a, + The unknown coeffcents, a and a, wll be epressed n terms of nodal solutons u( ) and u( + ). 9 Substtute two nodal values TRIA SOUTION cont. u ( ) = u = a + a u ( ) = u = a + a Epress a and a n terms of u and u +. Then, the soluton s appromated by + u ( ) = u () + u () + N ( ) N ( ) Soluton for -th element: + u ( ) = N( u ) + N ( u ), N () and N + (): Shape Functon or Interpolaton Functon

11 Observatons TRIA SOUTION cont. Soluton u() s nterpolated usng ts nodal values u and u +. N () = at node, and = at node +. N () N + () + The soluton s appromated by pecewse lnear polynomal and ts gradent s constant wthn an element. u u u + u + u Stress and stran (dervatve) are often averaged at the node. GAERKIN METHOD Relaton between nterpolaton functons and tral functons D problem wth lnear nterpolaton, ( ) N D N ( ) =, ( ) < u ( ) = u( ) ( ) = () + = N ( ) =, < + (), + < N D Dfference: the nterpolaton functon does not est n the entre doman, but t ests only n elements connected to the node Dervatve, ( ), d ( ) < =, () < +, + < N D ( ) / / ( ) d ( ) ( ) +

12 EXAMPE Solve usng two equal-length length elements u() = d u + =, Boundary condtons () = Three nodes at =,.5,.; dspl at nodes = u, u, u Appromate soluton u ( ) = u( ) + u( ) + u( ),.5,.5 ( ) = ( ) =,.5 <,.5 <,.5 ( ) = +,.5 <.5 _.5 EXAMPE cont. Dervatves of nterpolaton functons d( ),.5 d( ),.5 = =,.5 <,.5 < d ( ),.5 =,.5 < Coeffcent matr RHS d.5 d K = = )() + ()( ) = ( ).5 d.5 d K = = + =.5.5 F = ( ) + () + () () () () =.5 ().5.5 F = + ( ) + () () () () =.5.5

13 Matr equaton u F u =.5 u.5 EXAMPE cont. Consder t as unknown Strkng the st row and strkng the st column (BC) u.5 = u.5 Solve for u =.875, u =.5 Appromate soluton 75.75, 5.5 u ( ) =.5 +.5,.5 Pecewse lnear soluton 5 EXAMPE cont. Soluton comparson 6.6 Appro. soluton has about 8% error Dervatve shows a large dscrepancy Appro. dervatve s constant as the soluton s pecewse lnear u()..8. u-eact u-appro /.5 / (eact) / (appro.)

14 FORMA PROCEDURE Galerkn method s stll not general enough for computer code Apply Galerkn method to one element (e) at a tme Introce a local coordnate = ( ξ) + jξ ξ = = ( e ) j Appromate soluton wthn the element u ( ) = un( ) + un( ) j N( ξ) N( ξ) N ( ξ) = ( ξ) N ( ξ) = ξ j Element e N ( ) = ( e) N ( ) ( e) = dn dnd = ξ = dξ ( e) dn dn d = ξ = + dξ e ( ) ξ () e 7 FORMA PROCEDURE cont. Interpolaton property N( ) =, N( j) = u ( ) = u N =, N = u ( j ) = u ( ) ( j) Dervatve of appro. soluton dn dn = u + + uj dn dn u dn dn u = = e ( ) u d ξ d ξ u Apply Galerkn method n the element level j dn j = p( ) N( ) ( j) N( j) ( ) N( ),, + = j 8

15 FORMA PROCEDURE cont. Change varable from to ξ dn dn dn u d ξ p ( ) N ( ξ) d ξ + ( j) N() ( ) N(), =, ( e) ( e) = d ξ d ξ d ξ u Do not use appromate soluton for boundary terms Element-level matr equaton ( ) ( e) ( e) ( e) [ k ]{ u } = { f } + + ( j ) ( e) ( e) N( ξ) { f } = p( ) dξ N ( ξ) dn dndn e dξ dξ dξ ( ) = d ( e) ( e) ξ = dn dn dn k dξ dξ dξ 9 FORMA PROCEDURE cont. Need to derve the element-level level equaton for all elements Consder Elements and (connected at Node ) () () ( ) k k u f = + k k u f + ( ) () () ( ) k k u f = + k k u f + ( ) Assembly () () () k k u ( ) f () () () () () () k k k k u f f + = + + () () () u k k f ( ) Vanshed unknown term

16 FORMA PROCEDURE cont. Assembly of N E elements (N D = N E + ) () () () k k u f ( ) () () () () () () () () () k k + k k u f + f () () () () () k k k u = + f + f + ( NE ) ( NE ) ( NE ) k k u N f ( ND ) N + ( N ) ( N ) ( N N ) D D [ K ]{ q } = { F } D ( N ) Coeffcent matr [K] s sngular; t wll become non-sngular after applyng boundary condtons D EXAMPE Use three equal-length length elements u u + = = =, (), () All elements have the same coeffcent matr ( ) e k = =, ( e =,,) ( e) Change varable of p() = to p(ξ): RHS p( ξ) = ( ξ) + ξ ( e) ( e) N( ξ) ( e) ξ { f } = p( ) dξ [ ( ξ) jξ] = + dξ N ( ξ) ξ j + ( e) 6 =, ( e =,,), j + 6 j

17 EXAMPE cont. RHS cont. () Assembly 7 = = = () () f f f,, () () () f f f () 5 u + u = + u u () 5 Apply boundary condtons Deletng st and th rows and columns 6 u u = 8 = 6 u 9 5 u = 8 Element Element Element EXAMPE cont. Appromate soluton, 7 u ( ) = +, , 8 7 Eact soluton u ( ) = 6 ( ) Three element solutons are poor Need more elements u() u-appro. u-eact ue...6.8

18 CONVERGENCE Weghted resal of m-th order DE has hghest dervatves of order m Wth eact arthmetc, the followng s suffcent for convergence to true soluton () as mesh s refned: Complete polynomals of at least order m nsde element Contnuty t across element boundares up to dervatves of order m- Element must be capable of representng eactly unform and unform dervatves up to order m-. Beam: -th order DE (m = ) Complete polynomals: v() = a + a + a + a Contnuty on v() and dv()/ across element boundares Unform v() ( ) = a Beam elements wll converge Unform dervatve dv()/ = a upon refnement 5 RIGID BODY MOTION Rgd d body moton for CST can lead to non zero strans! Rgd body moton u a cosθ snθ = v a sn θ cos θ y u The normal stran ε = = cos θ 6

19 CONVERGENCE RATE Quadratc curve u=a+b+c modeled by lnear FE u fe =a+(b+ch) u fe Mamal error at md-pont D e u + u ch A C D = ud ub = ub = = h u 8 Mamal gradent error s mamal at ends e u u h ' C A A = b = hc = h u Error n functon converges faster than n dervatve! '' '' 7 QUADRATIC EEMENT FOR CUBIC SOUTION Eact soluton u = a+ b+ c + Fnte element appromaton u = a + b dh + c + dh Mamal errors e D dh = 6 h = u''' 8 and e ' A dh = h = u''' 8

20 CONVERGENCE RATE Useful to know convergence rate Estmate how much to refne Detect modelng crmes Etrapolate Most studes just do seres of refnements f anythng 9

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