httpocwmitedu 2094 Finite Element Analysis of Solids and Fluids Spring 2008 For information about citing these materials or our Terms of Use visit httpocwmiteduterms 233 ID: 828864
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MIT OpenCourseWare http://ocw.mit.edu 2.094 Finite Element Analysis of Solids and Fluids Spring 2008 For information about citing these materials or our Terms of Use, visit: http://ocw.mit.edu/terms. | {z } X X MIT 2.0947. Isoparametric elements = Bu^u^ T = u1 u2 v4(7.7) 2 @u 3@x 67 67 6 @v 7 = Bu^(7.8)
2 6@y 7 45 @u + @v @y@x 012
6@y 7 45 @u + @v @y@x 012 301 @ @x @y @ @r @r @r @x @A = 4 5@A(7.9) @ @x @y @ @s @s @s @y J 01 01 @@ @x @r @A = J� 1 @A (7.10) @@ @y @s J must be non-singular which ensures that there is unique correspondence between (x;y) and (r;s). Hence, Z 1 Z 1 K = B T CB t det(J) dr ds(7.11) � 1 � 1 | {z } dV Z
3 1 Z 1 Also, RB = H T f B t det(J) dr
1 Z 1 Also, RB = H T f B t det(J) dr ds(7.12) � 1 � 1 Numerical integration (Gauss formulae) (Ch. 5.5) = t B T det(Jij ) (weight i, j) (7.13) ij K ij CB ij 2x2 Gauss integration, (i = 1; 2) (7.14) (j = 1; 2) (weight i; j = 1 in this case) (7.15) 29 Z MIT 2.094 7. Isoparametric elements Convergence Principle of virtual work: Reading: Z
4 Sec. 5.5.5, 4.3 T C dV =
Sec. 5.5.5, 4.3 T C dV = R ( u ) (7.21) V Find u, solution, in V, vector space (any continuous function that satis\fes boundary conditions), satisfying T C dV = a(u; v) =(f; v) for all v, an element of V. (7.22) | {z } |{z} V bilinear form R (v) Example: Finite Element problem Find uh 2 Vh, where Vh is F.E. vector space such thata(uh; v
5 h)=(f; vh) 8 vh 2 Vh (7.23) Size
h)=(f; vh) 8 vh 2 Vh (7.23) Size of Vh ) # of independent DOFs (here it's 12). Note: a(w; w) � 0 for w 2 V(w = 0) | {z } 6 2x (strain energy when imposing w) Also, a(wh; wh) � 0 for wh 2 Vh (Vh V, wh = 0) 6 31 | {z } MIT 2.094 7. Isoparametric elements Property I De\fne: eh = u � uh. From (7.22) , a(u; vh)=(f; vh) (7.24) From (7
6 .23) , a(uh; vh)=(f; vh) (7.25) Hence,
.23) , a(uh; vh)=(f; vh) (7.25) Hence, a(u � uh; vh) = 0 (7.26) a(eh; vh) = 0 (7.27) (error is orthogonal in that sense to all vh in F.E. space). Property II (7.28) a(uh; uh) a(u; u) Proof: a(u; u)= a(uh + eh; uh + eh) (7.29) = a(u h ; u h )+ : 0 by Prop. I 2a(uh; eh)+ a(eh; eh) (7.30) 0 ) a(u; u) a(uh; uh) (7.31)