MIT2_094S11_lec7

MIT2_094S11_lec7 - 2.094 — Finite Element Analysis of...

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Unformatted text preview: 2.094 — Finite Element Analysis of Solids and Fluids Fall ‘08 Lecture 7- Isoparametric elements Prof. K.J. Bathe MIT OpenCourseWare Reading: Sec. 5.1-5.3 We want K = V B T CB dV , R B = V H T f B dV . Unique correspondence ( x,y ) ⇔ ( r,s ) ( r,s ) are natural coordinate system or isoparametric coordinate system. 4 x = h i x i (7.1) i =1 4 y = h i y i (7.2) i =1 where 1 h 1 = (1 + r )(1 + s ) (7.3) 4 1 h 2 = 4 (1 − r )(1 + s ) (7.4) ... 4 u ( r,s ) = h i u i (7.5) i =1 4 v ( r,s ) = h i v i (7.6) i =1 28 MIT 2.094 7. Isoparametric elements = Bu ˆ u ˆ T = u 1 u 2 v 4 (7.7) ··· ⎡ ∂u ⎤ ∂x ⎢ ⎥ ⎢ ⎥ ⎢ ∂v ⎥ = Bu ˆ (7.8) ⎢ ∂y ⎥ ⎣ ⎦ ∂u + ∂v ∂y ∂x ⎛ ⎞ ⎡ ⎤⎛ ⎞ ∂ ∂x ∂y ∂ ∂r ∂r ∂r ∂x ⎝ ⎠ = ⎣ ⎦⎝ ⎠ (7.9) ∂ ∂x ∂y ∂ ∂s ∂s ∂s ∂y J ⎛ ⎞ ⎛ ⎞ ∂ ∂ ∂x ∂r ⎝ ⎠ = J − 1 ⎝ ⎠ (7.10) ∂ ∂ ∂y ∂s J must be non-singular which ensures that there is unique correspondence between ( x,y...
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MIT2_094S11_lec7 - 2.094 — Finite Element Analysis of...

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