CHAPTER 9. Interpolation functions for 2D elements. Numerical Integration. Modeling Considerations

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1 HAPTER 9 Intrpolation functions for D lmnts Numrical Intgration Modling onsidrations

2 Pascal s triangl Dgr of Numbr of Elmnt with th complt trms in th nods polnomial polnomial Triangular Elmnts (Figur not shown) Lagrang and Srndipit Elmnts Pascal s triangl Rctangular arra Lagrang Srndipit of lmnts lmnts lmnts and so on

3 (b) f -D quadratic lmnt f (a) (, ) ψ (, ) ψ Lagrang lmnt (T)

4 ψ ψ Lagrang lmnt (Q9) ψ ψ Srndipit lmnt (Q8) ψ

5 Som Triangular and Rctangular Elmnts (b) (c) Nods with function valus onl (u) Nods with valus of th function (u) and its drivativs u, u, u b a (a) Lagrang triangular lmnts Lagrang rctangular lmnts Hrmit cubic rctangular lmnt

6 Numrical Evaluation of Elmnt officints Govrning Equation a(, ) u b(, ) u + b(, ) u + c(, ) u + d(, )u = f(, ) Finit Elmnt Approimation nx u(, ) = u jψj (, ) Finit Elmnt Modl j= [K ]{u } = {f } + {Q } K ij = f i = q n = Z a(, ) ψ ψ µ i j ψ + b(, ) i ψj Ω + ψ j + c(, ) ψ ψ i j + d(, )ψ i ψj dd Z I f(, )ψi dd, Q i = q n (s)ψi ds Ω Γ µ a u µ + b u n + b u + c u n ψ i

7 Subparamtric, Isoparamtric, and Suprparamtric Formulations Gomtr: Solution: = mx j= ˆψ j j (, ), = m X j= j ˆψ j (, ) u(, ) = nx u jψj (, ) = j= nx u jψj ((, ),(, )) j= = = Ωˆ Ω = (, ), = (, ) d d = J d d = (, ), = (, ) Suprparamtric (m > n): Th polnomial dgr of approimation usd for th gomtr is of highr ordr than that usd for th dpndnt variabl Isoparamtric (m = n): Equal dgr of approimation is usd for both gomtr and dpndnt variabls Subparamtric (m < n): Highr-ordr approimation of th dpndnt variabl is usd

8 Gauss Quadratur Z Z F (, ) dd = Ω Z = Domain of th phsical lmnt = ˆΩ ˆΩ N X F (, )J(, ) dd ˆF (, ) dd N X I= J= Gauss points ˆF ( I, J )W I W J Domain of th mastr lmnt Gauss wights = = = = ( ψ i ψ i ( ψ i ψ i ) = [J] = ) ( ψ i ψ i i ( ψ =[J] ψ i ) ) = = = = = 8 = = = 8 = 8 = 9 = 9 = 8

9 Elmnt alculations Using Numrical Quadratur [J] = = " ˆψ ˆψ ( ψ i ψ i ) ˆψ = ˆψ ( ψ =[J] " P m i= i ˆψ i P m i= i ˆψ i ˆψ m ˆψ M i ψ i ) # P m i= i ˆψ i P m i= i ˆψ i m [J ] m ( ψ i ψ i ) # J = J J, J = J J, J = J J, J = J J J = J J J J Ω = = (,) = (,) Ωˆ Ω = = (, ) = (, ) = (, ), = (, ) Ω Ω Ω d d = J d d = (, ), = (, )

10 Evaluation of Elmnt officint Matri Using th Gauss Quadratur (continud) Z Kij = ˆΩ ( a(, ) Ã J ψi + J ψi!ã J ψj + J ψj "Ã!Ã! + b(, ) J ψi + J ψi J ψ j + ψ j J Ã!Ã!# + J ψi + J ψi J ψ j + J ψ j Ã!Ã! + c(, ) J ψ j + J ψ j J ψ j + J ψ j ) Z + d(, )ψi ψj Jdd ˆF ij (, )dd ˆΩ! Z ˆΩ ˆF ij (, )dd = Z Z N X N X I= J= ˆF ij (, )dd ˆF ij ( I, J )W I W J N =int[ (p +)], N =int[ (q +)]

11 Fortran Statmnts to omput Elmnt officint Matri and Sourc Vctor SUBROUTINE ELEMKF(NPE,NN,INTF) IMPLIIT REAL*8(A-H,O-Z) OMMON/STF/ELF(7),ELK(7,7),ELXY(9,) OMMON/PST/A,AX,AY,B,BX,BY,,X,Y,F,FX,FY OMMON/SHP/SF(9),GDSF(,9) DIMENSION GAUSPT(,),GAUSWT(,) DATA GAUSPT/*D, -777D, 777D, *D, -7797D, D, 7797D, *D, -8D, -998D, 998D, 8D, D, -9798D, -89D,D,89D,9798D/ DATA GAUSWT/D, *D, *D, *D, D, D, D, *D, 788D, *D, 788D, D, 988D, 7887D, D, 7887D, 988D/ NDF = NN/NPE Initializ th arras DO I =,NN ELF(I) = DO J =,NN ELK(I,J)= Do-loops on numrical (Gauss) intgration bgin hr Subroutin SHPRT (SHaP functions for RTangular lmnts) is calld hr DO NI =,INTF DO NJ =,INTF XI = GAUSPT(NI,INTF) ETA = GAUSPT(NJ,INTF) ALL SHPRT (NPE,XI,ETA,DET,ELXY) NST = DET*GAUSWT(NI,INTF)*GAUSWT(NJ,INTF)

12 Fortran Statmnts to omput Elmnt officint Matri and Sourc Vctor (continud) X= Y= DO I=,NPE X=X+ELXY(I,)*SF(I) Y=Y+ELXY(I,)*SF(I) SOURE=F+FX*X+FY*Y AA=A+AX*X+AY*Y BB=B+BX*X+BY*Y =+X*X+Y*Y II= DO 8 I=,NPE JJ= DO J=,NPE S=SF(I)*SF(J)*NST S=GDSF(,I)*GDSF(,J)*NST S=GDSF(,I)*GDSF(,J)*NST S=GDSF(,I)*GDSF(,J)*NST S=GDSF(,I)*GDSF(,J)*NST ELK(I,J) = ELK(I,J) + AA*S + BB*(S + S) + *S JJ = NDF*J+ ELF(I) = ELF(I)+NST*SF(I)*SOURE 8 II = NDF*I+ ONTINUE RETURN END

13 MODELING ONSIDERATIONS Elmnts with unaccptabl vrt angls Msh orintation and msh rfinmnts h and p rfinmnts, and accptabl and unaccptabl msh rfinmnts Rang of accptabl location of th `midsid nods Accptabl (compatibl) connctions btwn lmnts of diffrnt kind Various tps of incompatibl connctions Msh rfinmnts with compatibl and incompatibl connctions Handling of mathmatical singularitis in th application of boundar conditions

14 Unaccptabl Vrt Angls Too larg Too larg Too small Too larg (a) Too small Too larg (b) Too small Too larg (c)

15 h h 8 Placmnt of Midsid Nods h 7 h Rang of nods and 8 Rang of nods and 7 7 h h h h Rang of nod h h 8 h h ompatibl onnctions Btwn Elmnts (a) (b)

16 Msh Rfinmnts (a) A msh rfinmnt must contain th prvious msh as a subst (b) h rfinmnt p rfinmnt (a) (b) (c) (d)

17 (a) Linar lmnt Transition lmnt Quadratic lmnt (b) Linar lmnt Us constraint conditions Quadratic lmnt (a) 7 8 s u u = u(s) u () u ( s) () ( s) onstraint condition: u u = s ( u ) 7 = u u + (b) (c) 7 8 s 7 s 8 7 s s 8 u u = u u u u = () u(s) u(s) ( s) u () 7 = u7 u u = () u ( s) () u ( s) 7 u () ( s) u7 u u = ( s) s u u = s

18 Handling of Mathmatical Singularitis in Appling Boundar onditions