Weighted Residual Methods

Transcription

1 Weighted Resil Methods

2 Formltion of FEM Model Direct Method Formltion of FEM Model Vritionl Method Weighted Resils Severl pproches cn e sed to trnsform the phsicl formltion of prolem to its finite element discrete nloge. If the phsicl formltion of the prolem is descried s differentil eqtion, then the most poplr soltion method is the Method of Weighted Resils. If the phsicl prolem cn e formlted s the minimiztion of fnctionl, then the Vritionl Formltion is sll sed.

3 Formltion of FEM Model Finite element method is sed to solve phsicl prolems Solid Mechnics Flid Mechnics Het Trnsfer Electrosttics Electromgnetism. Phsicl prolems re governed differentil eqtions which stisf Bondr conditions Initil conditions One vrile: Ordinr differentil eqtion ODE Mltiple independent vriles: Prtil differentil eqtion PDE 3

4 Phsicl prolems Aill loded elstic r = = A = cross section t = od force distrition force per nit length E = Yong s mols = displcement of the r t Differentil eqtion governing the response of the r d AE ; Second order differentil eqtions Reqires ondr conditions for soltion 4

5 Phsicl prolems Aill loded elstic r = = Bondr conditions emples t Dirichlet/ displcement c t t EA F t Nemnn/ force c Differentil eqtion + Bondr conditions = Strong form of the ondr vle prolem 5

6 Phsicl prolems Fleile string S = = S p S = tensile force in string p = lterl force distrition force per nit length w = lterl deflection of the string in the -direction Differentil eqtion governing the response of the r S d p ; Second order differentil eqtions Reqires ondr conditions for soltion 6

7 Phsicl prolems Het conction in fin = = A = cross section t Q = het inpt per nit length per nit time [J/sm] k = therml conctivit [J/ o C ms] T = tempertre of the fin t Q Differentil eqtion governing the response of the fin d Ak dt Q ; Second order differentil eqtions Reqires ondr conditions for soltion 7

8 Phsicl prolems Het conction in fin = = Q Bondr conditions emples T k t dt h t Dirichlet/ displcement c Nemnn/ force c 8

9 Phsicl prolems Flid flow throgh poros medim e.g., flow of wter throgh dm = = Q A = cross section t Q = flid inpt per nit volme per nit time k = permeilit constnt j = flid hed Differentil eqtion d dj k Q ; Second order differentil eqtions Reqires ondr conditions for soltion 9 Bondr conditions emples j t dj k h t Known hed Known velocit

10 Phsicl prolems

11 Phsicl prolems

12 Formltion of FEM Model Oserve:. All the cses we considered led to ver similr differentil eqtions nd ondr conditions.. In D it is es to nlticll solve these eqtions 3. Not so in nd 3D especill when the geometr of the domin is comple: need to solve pproimtel 4. We ll lern how to solve these eqtions in D. The pproimtion techniqes esil trnslte to nd 3D, no mtter how comple the geometr

13 Finite Element Method Integrl Formltion 3

14 Some Mthemticl Concepts Simpl connected domin: If n two points of the domin cn e Joint line ling entirel within the domin Clss of domin: A fnction of severl vriles is sid to e of m Clss C in domin if ll its prtil derivtives p to nd inclding the mth order eist nd re continos in C F is continos i.e. f /, f / eist t m not e continos. Bondr Vle Prolems: A differentil eqtion DE is sid to e BVP if the dependent vrile nd possil its derivtives re reqired to tke specified vles on the ondr. Emple: d f, d, g 4

15 Some Mthemticl Concepts Initil Vle Prolem: An IVP is one in which the dependent vrile nd possil its derivtives re specified initill t t = Emple: d dt Initil nd Bondr Vle Prolem: Emple: f t t,, v dt t, t d 5 t, f, t g for t, t, nd Eigenvle Prolem: the prolem of determining vle l of sch tht Emple: l Eigenvle Eigenfnction d l, t t

16 6 Some Mthemticl Concepts Integrtion--Prt Forml: v w v w dw v dv w dv w v dw wv d Net First w w dw d w Similrl w d v w d v dv w d dv w d v d w d w d v

17 7 Some Mthemticl Concepts Grdient Theorem Fds n F d F d grd ˆ Bt j n i n n j F i F F ˆ, Ths Fds j n i n d j F i F or ds Fn d F ds Fn d F

18 Some Mthemticl Concepts Divergence Theorem divg d. G d nˆ. Gds G G d ng ng Using grdient nd divergence theorem, the following reltions cn Be derived! Eercise G wd w Gd nˆ wgds * nd G wd w. G d ds G wds n 8

19 Some Mthemticl Concepts The components of eqtion * re: G G wd wd w Gd w Gd n n wgds wgds 9

20 Some Mthemticl Concepts Fnctionls An integrl in the form of I F,, ',, ' where integrnd F,, is given fnction of rgments,, is clled fnctionl fnction of fnction. A fnctionl is sid to e liner if nd onl if: I v I I v, re sclrs A fnctionl B,v is sid to e iliner if it is liner in ech of its rgments B, v B, v B, v inerit in the first rgment B, v v B, v B, v inerit in the second rgment

21 Some Mthemticl Concepts Fnctionls A iliner form B,v is smmetric in its rgments if B, v B v, Emple of liner fnctionl is I v vf Emple of iliner fnctionl is B v, w dv dv M dw

22 Some Mthemticl Concepts

23 Some Mthemticl Concepts 3

24 Some Mthemticl Concepts 4

25 Some Mthemticl Concepts 5

26 Some Mthemticl Concepts 6

27 Some Mthemticl Concepts 7

28 Some Mthemticl Concepts 8

29 Some Mthemticl Concepts The Vritionl Smol Consider the fnction F F,, ' 9 for fied vle of, F onl depends on The chnge v in, where is constnt nd v is fnction, is clled vrition of nd denoted : Vritionl Smol F F v In nlog with the totl differentil of fnction Note tht df F F ' ' F F ' ','

30 3 Some Mthemticl Concepts F F n F F F F F F F F F F F F F F F F F F n n Also Frthermore v dv v d d ' ' The Vritionl Smol

31 Wek Formltion of BVP Weighted integrl nd wek formltion Consider the following DE d, q Q Trnsverse deflection of cle Ail deformtion of r Het trnsfer Flow throgh pipes Flow throgh poros medi Electrosttics 3

32 Wek Formltion of BVP There re 3 steps in the development of wek form, if eists, of n DE. STEP : Move ll epression in DE to one side, mltipl w weight fnction nd integrl over the domin. w d q + Weighted-integrl or weighted-resil N U N c j j j N linerl independent eqtion for w nd otin N eqtion for c,,c N 3

33 Wek Formltion of BVP STEP -The integrl + llows to otin N independent eqtions - The pproimtion fnction,, shold e differentile s mn times s clled for the originl DE. 3- The pproimtion fnction shold stisf the BCs. 4- If the differentition is distrited etween w nd then the reslting integrl form hs weker continit conditions. Sch weighted-integrl sttement is clled wek form. The wek form formltion hs two min chrcteristics: -reqires weker continit on the dependent vrile nd often reslts in smmetric set of lgeric eqtions. - The ntrl BCs re inclded in the wek form, nd therefore the pproimtion fnction is reqired to stisf onl the essentil BCs. 33

34 Wek Formltion of BVP Retrning to or emple: w d wq dw wq w Secondr Vrile SV: Coefficient of weight fnction nd its derivtives Q n Ntrl Bondr Conditions NBC Primr Vrile PV: The dependent vrile of the prolem Essentil Bondr Conditions EBC 34

35 Wek Formltion of BVP dw wq w dw wq w n w n dw wq wq wq Note tht n n 35

36 Wek Formltion of BVP STEP 3: The lst step is to impose the ctl BCs of the prolem w hs to stisf the homogeneos form of specified EBC. In wek formltion w hs the mening of virtl chnge in PV. If PV is specified t point, its vrition is zero. w Ths dw dw n wq w wq w Q n Q w NBC n 36

37 iner nd Biliner Forms dw wq w Q B w, l w B w, lw B w, Biliner nd smmetric in w nd lw iner Therefore, prolem ssocited with the DE cn e stted s one of finding the soltion sch tht B w, l w holds for n w stisfies the homogeneos form of the EBC nd continit condition implied the wek form 37

38 iner nd Biliner Forms Assme Vritionl soltion Stisf EBC * w Actl soltion Stisf EBC+NBC Stisf the homogeneos Form of EBC ooking t the definition of the vritionl smol, w is the vrition of the soltion, i.e. w Then B w, l w B, l # B, l q Q q Q 38 [ l ] B,

39 39 iner nd Biliner Forms,,, l B I I l B l B Sstitting in #, we hve: In generl, the reltion,, B B holds onl if, w B is iliner nd smmetric nd w l is liner If Bw, is not liner t smmetric the fnctionl I cn e derived t not from ##. see Oden & Redd, 976, Redd 986 ##

40 iner nd Biliner Forms Eqtion I represents the necessr condition for the fnctionl I to hve n etremm vle. For solid mechnics, I represents the totl potentil energ fnctionl nd the sttement of the totl potentil energ principle. Of ll dmissile fnction, tht which mkes the totl potentil energ I minimm lso stisfies the differentil eqtion nd ntrl ondr condition in +. 4

41 Some Emples Emple Consider the following DE which rise in the std of the deflection of cle or het trnsfer in fin when c =. d c for, Step w d c Step dw cw w w w EBC NBC 4

42 Some Emples Emple Step 3 or dw B w, l w cw w dw w cw B is iliner nd smmetric nd l is liner! prove Ths we cn compte the qdrtic fnctionl form w w B w, l w I c 4

43 Some Emples Emple Consider the following forth-order DE elstic ending of em Step Step d w d w dw d v dv d f, d w 43 d w f for M d w vf v, d d d w d w

44 Some Emples Emple d v d w vf v d d w dv d w d d w d w M V Sher force Bending moment B.C dw w dv v d d w d w M 44

45 Some Emples Emple Step 3 d v d w dv vf M 45 B v, w or B v, w l v where l v Smmetric&Biliner iner The fnctionl I w cn e written s: d v d w dv vf d w dw I w wf M M

46 Some Emples Emple 3 Consider D het trnsfer prolem T k T T T n T k qˆ Sted het conction in two-dimensionl domin q A B Y in T T 46 ˆ q q : niform het genertion k : conctivit of the isotropic mteril T : tempertre Inslted C D T k T T Convection

47 Some Emples Emple 3 Step Step on on on on 3 4 T w k T 47 q d w T w T T T k wq d wk n n ds T k AB n BC CD DA n n n n, n, n, n, n T n qˆ T T k n T k T T n T n q n T=Primr vrile =Secondr vrile het fl q n *

48 Some Emples Emple 3 Step 3 w shold stisf the EBC T wk wq n ds w, n T n T k n qˆ d ds ds Sstitting in * we hve 3 w, T wk n w T T T, ds ds T 4 d w ds w k T w T wq d w, qˆ d w, T, T d B w, T l w 48

49 49 Emple 3 Some Emples d T w d T w T w k w T B,,, d T w d q w d wq w l, ˆ, The qdrtic fnctionl is given : d T T T d q T d Tq d T T k T I,, ˆ,

50 Conclsions iner nd Biliner Forms - The wek form of DE is the sme s the sttement of the totl potentil energ. - Otside solid mechnics I m not hve mening of energ t it is still se mthemticl tools. 3- Ever DE dmits weighted-integrl sttement, or wek form eists for ever DE of order two or higher. 4- Not ever DE dmits fnctionl formltion. For DE to hve fnctionl formltion, its iliner form shold e smmetric in its rgment. 5- Vritionl or FE methods do not reqire fnctionl, wek form of the eqtion is sfficient. 6- If DE hs fnctionl, the wek form is otined tking its first vrition. 5

51 References - An Introction to the Finite Element Method, : J. N. Redd, 3rd ed., McGrw-Hill Ection 5. chpter - Energ Principles nd Vritionl Methods in Applied Mechnics, : J. N. Redd, nd ed., John Wile. chpter 7 5