Ch_27 - ELEMENT MATRICES Higher order elements have curved...

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Chapter 27 Page 1 G. Subbarayan ELEMENT MATRICES x Higher order elements have curved boundaries Integration is a challenge x Integrate in natural coordinate system! x Need numerical integration!
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Chapter 27 Page 2 G. Subbarayan Changing the Variables of Integration x One-dimensional integrals x [J]: Jacobian matrix for the transformation [ ] 1 1 ( ) ( ) ( ) ( ) m i x x dx f x dx g d d dx J d ξ ξ ξ ξ ξ ξ = =
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Chapter 27 Page 3 G. Subbarayan x Consider the linear element with two nodes: 1 1 2 2 1 1 1 2 2 2 1 2 2 1 1 1 ( ) ( ) ( ) (1 ) (1 ) ( ) 2 2 2 2 ( ) ( ) 2 m i x x x N X N X X X X X X X dx L d L f x dx g d ξ ξ ξ ξ ξ ξ ξ ξ ξ = + = + + = − + = = =
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Chapter 27 Page 4 G. Subbarayan x Two-dimensional integrals [ ] [ ] 1 1 1 1 1 1 1 1 1 2 1 2 0 0 2 2 ( , ) ( , )det[ ] ( , ) ( , )det[ ] where or, in area coordinates where A A x y f x y dxdy g J d d J x y x y L L f x y dxdy g L L J dL dL J x y L L ξ ξ ξ η ξ η η η = = = = ∫ ∫ ∫∫
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Chapter 27 Page 5 G. Subbarayan 1 2 1 1 2 2 3 3 1 1 2 2 1 2 3 1 2 1 1 2 2 1 2 3 1 3 1 3 1 1 2 3 2 3 2 2 ( , ) (1 ) ( , ) (1 ) x L L L X L X L X L X L X L L X y L L L Y L Y L L Y x y X X Y Y L L x y X X Y Y L L = + + = + + = + + = = = =
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Chapter 27 Page 6 G. Subbarayan 1 3 1 3 2 3 2 3 [ ] det[ ] 2 X X Y Y J X X Y Y J A = =
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Chapter 27 Page 7 G. Subbarayan Example
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Chapter 27 Page 8 G. Subbarayan 1 1 1 1 1 1 4 4 4 2 2 1 1 1 2 4 4 4 3 3 1 1 1 3 4 4 4 4 1 1 4 4 4 (1 )(1 ) (1 ) (1 ) (1 )(1 ) (1 ) (1 ) (1 )(1 ) (1 ) (1 ) (1 )(1 ) (1 N N N N N N N N N N N ξ η η ξ ξ η ξ η η ξ ξ η ξ η η
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