Finite element analysis sdfa ggq rsd vqer fas dd sg fa sd qadas casdasc asdac asascasasFEA.pptx

sdfa ggq rsd vqer fas dd sg fa sd qadas casdasc asdac asascasas

1. Piece-wise Con­tin­u­ous Trial Func­tion So­lu­tion of the Weak Form We have seen that the gen­er­al method of weight­ed resid­u­al tech­nique con­sist­ed in as­sum­ing a trial func­tion so­lu­tion& min­imis­ing the resid­u­al in an over­all sense. The Galerkin method gave us a way of choos­ing the ap­pro­pri­ate weight­ing func­tions.

2. We have so far used dif­fer­ent trial func­tions such as poly­no­mi­al and trigono­met­ric se­ries. How­ev­er, in each case the trial func­tion cho­sen was a sin­gle com­pos­ite func­tion, valid over the en­tire so­lu­tion do­main. For ex­am­ple,  c  1  sin (  πx  /  L  ) used in Ex­am­ple 2.4 was valid over the en­tire so­lu­tion do­main 0<  x  <  L  , i.e., the en­tire beam in this case.

3. Sim­i­lar­ly, the trial func­tion  c   11  sin (  π x/a ) sin (  πy  /  b  ) used in Ex­am­ple 2.7 was valid over the en­tire so­lu­tion do­main 0<x<  a  , 0<  y  < b  , i.e., the en­tire plate.

4. Curve fitting When we con­sid­er the essence of the WR method , i.e., as­sum­ing a trial func­tion so­lu­tion and match­ing it as close­ly as pos­si­ble to the exact so­lu­tion of the given dif­fer­en­tial equa­tion and the bound­ary con­di­tions, we re­alise that this is es­sen­tial­ly a pro­cess of"curve fit­ting".

5. It is well known that curve fit­ting is best done"piece-wise"; the more the num­ber of pieces, the bet­ter the fit. Fig­ure 2.​16(a) il­lus­trates this basic idea on a func­tion  f  (  x  ) = sin (  πx  /  L  ) being ap­prox­i­mat­ed using straight line seg­ments.

6. One and two-line segment approximation of a function

7. 1 line segment approximation Since a straight line can be drawn through any two points, we can gen­er­ate one such ap­prox­i­ma­tion by draw­ing a line through the func­tion val­ues at   x  = 0 and  x  =  L  /2. Clear­ly , this is a poor ap­prox­i­ma­tion of the func­tion for  x  >  L /2.

8. 2 line segment approximation If we were to use two straight line seg­ments in­stead of only one, we could draw two line seg­ments— one through the func­tion val­ues at  x  = 0 and  x  =  L  /2; &other through the func­tion val­ues at   x  =  L  /2 and  x  =  L 

9. 2 line segment approximation Each of these is our ap­prox­i­ma­tion to the func­tion with­in that piece of the so­lu­tion do­main—the for­mer seg­ment in the sub-do­main 0 <  x  <  L  /2, and the lat­ter in the sub-do­main   L  /2<  x  <  L.  Clear­ly, this is a bet­ter ap­prox­i­ma­tion to the func­tion

10. 4 line segment approximation It can be fur­ther im­proved by tak­ing four-line seg­ment ap­prox­i­ma­tion as shown in Fig­ure 2.​16(b ).

11. Four-line segment approximation of a function

12. This is the es­sen­tial idea of piece-wise con­tin­u­ous trial func­tion ap­prox­i­ma­tion for the weak form

13. Inherent error in approximate soln. Of course, when we con­sid­er piece-wise trial func­tions, we need to en­sure con­ti­nu­ity of the field vari­able and its deriva­tives at the junc­tions. While the exact so­lu­tion will en­sure con­ti­nu­ity of the field vari­able and ALL the deriva­tives, we ex­pect to be able to sat­is­fy this to deriva­tives of only de­sired de­gree with our ap­prox­i­mate so­lu­tion . This leads to an in­her­ent error in our so­lu­tion.

14. Aim H ow­ev­er , we aim to ob­tain a rea­son­ably ac­cu­rate so­lu­tion using sev­er­al con­ve­nient trial func­tions (e.g. sim­ple poly­no­mi­als) de­fined in a piece-wise man­ner over the en­tire so­lu­tion do­main .

15. Func­tion val­ues at the ends of the sub-do­main Let us de­fine these trial func­tions , each valid in its own sub-do­main, in terms of the func­tion val­ues at the ends of the sub-do­main. For ex­am­ple, for Fig­ure 2.​16, we de­fine the trial func­tions in each case as fol­lows:

16. For the one-line seg­ment (Fig­ure 2.​16(a)) ap­prox­i­ma­tion ,

17. Local Coordinate System For two-line and four-line seg­men­ta­tion, we can, in fact, more con­ve­nient­ly rewrite these trial func­tions if we de­fine a local co­or­di­nate  x  with the ori­gin fixed at the left end of each sub-do­main as fol­lows ( ref. Fig­ure 2.​16(c)):

18. Domain with Local coordinate frame

19. Two-line segment approximation

20. Four-Line segment approximation

21. char­ac­ter­is­tic func­tions These char­ac­ter­is­tic func­tions dic­tate the con­tri­bu­tion of a given  f  k  to the value of the func­tion at any point  P  with­in the do­main 0<  X  <  L  . Fig­ure 2.​17 shows these in­ter­po­la­tion func­tions for the two-line seg­ment case dis­cussed ear­li­er.

22. Fig­ure 2.​17(a) shows the in­ter­po­la­tion func­tions sep­a­rate­ly for the two sub-do­mains while the Fig­ure 2.​17(b) shows them more com­pact­ly for the whole do­main 0<  X  <  L  . Fig­ure 2.​18 shows these shape func­tions for the four-line seg­ment case.

23. We ob­serve that we have as many shape func­tions  N  k  as there are func­tion val­ues  f  k  used in the in­ter­po­la­tion. Since we have cho­sen lin­ear shape func­tions, each func­tion   N  k  ramps up and down with­in each sub-do­main. Each  N  k  takes on the value of unity at the point  x  =  x k   and goes to zero at all other end-points x =  x  i  (i  ≠  k  ). This is to be ex­pect­ed since, at  x  =  x k   , the full con­tri­bu­tion to the value of the func­tion comes from  f  k  alone and none else.

24. Equal­ly im­por­tant­ly, the na­ture of the in­ter­po­la­tion we have used is such that each  f  k  con­tributes to the value of the func­tion only with­in the sub-do­mains on its ei­ther side or, in other words, only in those sub-do­mains to which it is"con­nect­ed ".

25. Once the func­tion val­ues  f  k  at the ends of the sub-do­mains have been de­ter­mined in this man­ner, the value of the func­tion at any in­te­ri­or point can be ob­tained using the ap­pro­pri­ate in­ter­po­la­tion func­tions. This is the essence of the fi­nite el­e­ment method.

26. Each of the sub-do­mains is called a  fi­nite el­e­ment  —to be dis­tin­guished from the dif­fer­en­tial el­e­ment used in con­tin­u­um me­chan­ics. The ends of the sub-do­main are re­ferred to as the  nodes  of the el­e­ment. We will later on dis­cuss el­e­ments with nodes not nec­es­sar­i­ly lo­cat­ed at only the ends, e.g. an el­e­ment can have mid-side nodes, in­ter­nal nodes, etc.

27. The un­known func­tion val­ues  f  k  at the ends of the sub-do­mains are known as the  nodal de­grees of free­dom  (  d.o.f .  )  .  A gen­er­al fi­nite el­e­ment can admit the func­tion val­ues as well as its deriva­tives as nodal d.o.f .

28. The sub-do­main level con­tri­bu­tions to the weak form are typ­i­cal­ly re­ferred to as  el­e­ment level equa­tions.  The pro­cess of build­ing up the en­tire co­ef­fi­cient ma­tri­ces [  A  ] and {  b  } is known as the pro­cess of as­sem­bly   , i.e ., as­sem­bling or ap­pro­pri­ate­ly plac­ing the in­di­vid­u­al el­e­ment equa­tions to gen­er­ate the sys­tem level equa­tions.

29. We will now il­lus­trate the fi­nite el­e­ment for­mu­la­tion based on the weak form through a sim­ple ex­am­ple. Through this ex­am­ple we will be for­mu­lat­ing the one-di­men­sion­al bar fi­nite el­e­ment which is a line el­e­ment with two nodes and the axial dis­place­ment '  u  ' at each node as the nodal d.o.f .

30. This is a very use­ful el­e­ment to solve prob­lems of com­plex net­work of truss­es. When com­bined with a one-di­men­sion­al beam fi­nite el­e­ment, it can be used for solv­ing any com­pli­cat­ed prob­lem in­volv­ing frame struc­tures, au­to­mo­bile chas­sis, etc.

31. One-di­men­sion­al Bar Fi­nite El­e­ment Let us re­con­sid­er Ex­am­ple 2.9 where we solved the prob­lem using the weak form, but this time we will use piece-wise de­fined in­ter­po­la­tion func­tions. From Eq. (2.​85), the weak form of the dif­fer­en­tial equa­tion can be writ­ten as  

32. One dimensional bar element

33. Generalizing Formulation Method Our method of formulation.

34. where we have used the fact that the term  AE  (  d û /dx  ) stands for the axial force  P  in the bar at that sec­tion. Thus ,  P  0  and  P  l  stand for the forces at ei­ther end of the el­e­ment.

35. It is ob­served that these des­ig­nate the al­ge­bra­ic sum of the ex­ter­nal­ly ap­plied forces  F  at these nodes and the in­ter­nal re­ac­tion forces  R  (i.e., those forces that are ex­ert­ed on this el­e­ment by the ad­join­ing el­e­ments), that get ex­posed when we con­sid­er this el­e­ment alone as a free body.

36. Com­bin­ing Eqs . (2.​130)–(2.​134), we can write the con­tri­bu­tions of the  k  th el­e­ment to the over­all sys­tem equa­tions given in Eq. (2.​125) as

37. These two sets of terms con­sti­tute the " char­ac­ter­is­tic matrices "or " char­ac­ter­is­tic equations "for the bar el­e­ment.

38. We ob­serve that (  AE/l  ) rep­re­sents the axial stiff­ness of a rod; the first force vec­tor rep­re­sents the nodal forces equiv­a­lent to the dis­tribut­ed force  q  0  , and the el­e­ment equa­tions are sim­ply the nodal force equi­lib­ri­um equa­tions.

39. In view of the fact that these equa­tions rep­re­sent the force-de­flec­tion re­la­tions for the bar el­e­ment, the LHS ma­trix is termed as the  el­e­ment stiff­ness ma­trix  and the RHS vec­tors as the  el­e­ment nodal force vec­tors.

40. Now we need to sum up all the el­e­ment level con­tri­bu­tions to gen­er­ate the sys­tem level equa­tions. As al­ready men­tioned, this pro­cess would sim­ply in­volve repet­i­tive use of the gener­ic el­e­ment ma­tri­ces given above.

41. To il­lus­trate how the el­e­ment ma­tri­ces get sim­ply"placed ap­pro­pri­ate­ly"in sys­tem level equa­tions, we show the de­tailed com­pu­ta­tions for a two-el­e­ment mesh. Based on this il­lus­tra­tion, we pro­vide a gen­er­al method of"as­sem­bling el­e­ment equa­tions"for later use.

42. Example 2.11 Ex­am­ple 2.​11.  Il­lus­tra­tion of as­sem­bly.  Con­sid­er a rod of total length  L  = 2  l  , dis­cre­tised into two el­e­ments, as shown in Fig­ure 2.​21. The weak form of the dif­fer­en­tial equa­tion is given by