CE 60: Numercl Methods Lecture 36 Fnte Element Methods Course Coordntor: Dr. Suresh A. Krth, Assocte Professor, Deprtment of Cvl Engneerng, IIT Guwht.
In the lst clss, we dscussed on the ppromte methods to solve oundry-vlue prolem (sy) d y Qy F; Approprte BCs They were Rylegh-Rtz method sed on clculus of vrtons. Collocton method sed on resduls. Glerkn weghted resdul method. In Glerkn weghted resdul method, the methodology nvolves: Identfy the dfferentl equton to e solved (e.g. here d y Qy F) Appromte the ctul soluton y ( ) wth pprmte soln. y( ), where y( ) C y y I re trl functons usully lnerly ndependent polynomls tht stsfy the gven oundry condtons.
Defne the resdul functons W ( );,,3, glerkn suggested tht results wll e etter f we choose the trl functons y ( ) s the weghng functons. Perform the ntegrton usng ech weghng functon.e. W ( ) R( ) 0;,,3, Solve the system of weghted resdul ntegrls for the coeffcents C; (,,3, ) Usully collocton method s not tht much used. If the vrtonl functonl s known pror nd f we wnt to solve n the domn (usully sold mechncs prolem), Rylegh-Rtz method preferred. If the dfferentl equton s lredy specfed n the domn, the Glerkn weghted resdul method s preferred (flud mechncs).
The Fnte-Element Method for B.V. prolems The Rylegh-Rtz method nd Glerkn weghted resdul method ppromte the soluton y ( ) for the entre domn. As lnerly ndependent trl functons re ppled for the whole domn D ( ), the ccurcy flters for lrger domns. Or else you mght hve to use hgher degree polynomls s trl functons. It s nturl pproch to ncrese the degree of the polynoml trl functons to ncrese the ccurcy. However on ncrese of polynoml degree, completes my rrve s the ppromton my not work wthn two desred ponts. Recll n the clss POLYNOMIAL APPROXIMATIONS, we hve seen tht t s etter to use pece-wse lower degree polynomls n smller domns rther tht gong for hgher degree polynoml for the entre domn (splnes).
We wll use the sme phlosophy for the ppromte solutons of oundry-vlue prolem whle usng Rylegh-Rtz (RR) or Glerkn Weghted Resdul method (GWRM). How? Tht s the entre soluton domn D ( ) s dscretsed nto smll smll peces - clled elements - nd the RR or GWRM re ppled n ech of these elements. Ths s the Fnte-Element Method. Note: If the vrtonl functonl of prolem s lredy known n dvnce (sold mechncs) t s etter to use Rylegh-Rtz FEM. If the governng dfferentl equton s known n dvnce (flud mechncs) t s etter to use Glerkn Weghted Resdul FEM.
Glerkn FEM We now descre Glerkn FEM to solve the smple lner BV-ODE d y Qy F (wth pproprte BCs) The concept nvolves: The overll outlne nvolves: The ctul GWRM for the entre domn. M y C y ( ) C y ( ) C y ( ) C y ( ) M M W ( ) R( ) 0 I( C ) R( ) s evluted y usng y( ).
In FEM: The domn s dscretsed nto dscrete nodes (), (), (3),...,( I) Agn the domn conssts of I - elements (elements cn e lner, qudrtc etc.) () (element length) Also totl ntegrton, I I I I () () ( I ) Summton of ntegrls for ll the elements,,..., I. ().e. I W ( ) R( ), ntegrton over n element. On pplyng ths equton for ll the weghts W ( ) wthn the element, we wll get set of equtons reltng the nodl vlues wthn ech element.
Domn dscretston D( ) nto I nodes, I elements () The ect soluton y( ) y( ) y y I () ( ) Sum of seres of locl nterpoltng polynomls ( );,,3,.., These re vld wthn ech element. I () ( ) ( ) y y Locl nterpoltng polynoml y ( ) y N ( ) y N ( ) ( ) ( ) ( ) ( ) ( ) where y Nodl vlues of y t node & y Nodl vlues of y t node N ( ) & N ( ) Lner nterpoltng polynomls wthn element. N ( ).0 t () N ( ) 0.0 t () N ( ) 0.0 t () N ( ).0 t ()
N y ( ) y y () ( ) ( ) re clled shpe functons. To solve the BV-ODE d y d y d y Defne R( ) Qy F I( y( )) W( ) Qy F 0 W ( )( " ) y Qy F W y" Qy F Agn, I( y( )) () () ( I ) I I I y ' WI ( ) y ' WI ( ) 0 y ' W ' y ' W y ' W ' y ' W ( ) y ' W ( ) If the Neumnn.c.'s re gven: y ' 0, y ' 0 You hve I( y( )) ( y ' W ') ( QyW FW ) 0
() dw I y ' QyW FW () Aslo, y ( ) y ( ) y ( ) As n Glerkn method, the weghng functons re sme s shpe functons N ( ) nd N ( ). N () ( ) ( ) ( ) 0.0 for & N ( ) 0.0 for & () ( ) d ( ) ( ) ( ) I y ' ( N ( )) QyN FN 0 ( A) ( ) d ( ) ( ) ( ) Also, I y ' ( N ( )) QyN FN 0 ( B) (A) nd (B) re element equtons.