MIT2_094S11_lec8

MIT2_094S11_lec8 - 2.094 Finite Element Analysis of Solids...

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2.094 Finite Element Analysis of Solids and Fluids Fall ‘08 Lecture 8 - Convergence of displacement-based FEM Prof. K.J. Bathe MIT OpenCourseWare (A) Find u V such that a ( u , v ) = ( f , v ) v V (Mathematical model) (8.1) a ( v , v ) > 0 v V , v = 0 . (8.2) where (8.2) implies that structures are supported properly. E.g. (B) F.E. Problem Find u h V h such that a ( u h , v h ) = ( f , v h ) v h V h (8.3) a ( v h , v h ) > 0 v h V h , Properties e h = u u h v h = 0 (8.4) (I) a ( e h , v h ) = 0 v h V h (8.5) (II) a ( u h , u h ) a ( u , u ) (8.6) 33
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± ²³ MIT 2.094 8. Convergence of displacement-based FEM (C) Assume Mesh “is contained in” Mesh h 1 h 2 e.g. Mesh not contained in Mesh h 1 h 2 We assume (C), but need another property (independent of (C)) (III) a ( e h , e h ) a ( u v h , u v h ) v h V h (8.7) u h minimizes! (Recall e h = u u h ) Proof: Pick w h V h . a
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MIT2_094S11_lec8 - 2.094 Finite Element Analysis of Solids...

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