Summary_Ch9 - CHAPTER 9 Interpolation functions for 2D...

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CHAPTER 9 ¾ Interpolation functions for 2D elements ¾ Numerical Integration ¾ Modeling Considerations
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5 4 3 2 2 3 4 5 4 3 2 2 3 4 3 2 2 3 2 2 1 y xy y x y x y x x y xy y x y x x y xy y x x y xy x y x Pascal’s triangle Degree of Number of Element with the complete terms in the nodes polynomial polynomial 0 1 1 3 2 6 3 10 4 15 5 21 (Figure not shown) •• •••• . . . . . . . y xy y x y x y x x y xy y x y x x y xy y x x y xy x y x 5 4 3 2 2 3 4 5 4 3 2 2 3 4 3 2 2 3 2 2 1 Pascal’s triangle Rectangular array Lagrange Serendipity of elements elements elements and so on . . . . . . ... ... ... ... ... ... 4 5 4 4 4 3 4 2 4 4 4 5 4 4 4 3 4 2 4 4 3 5 3 4 3 3 3 2 3 3 2 5 2 4 2 3 2 2 2 2 5 4 3 2 5 4 3 2 1 y x y x y x y x xy y y x y x y x y x xy y y x y x y x y x xy y y x y x y x y x xy y y x y x y x y x xy y x x x x x ●● ●●● Triangular Elements Lagrange and Serendipity Elements
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( b ) 1 2 3 4 5 6 1 2 3 4 5 6 1 2 3 4 6 5 7 8 9 10 11 12 13 14 15 e f ° 1 3 3 4 5 1 1 2 6 2 3 4 5 6 1-D quadratic element e f ° ° 2 ( a ) 1 (, ) xy ψ 1 2 3 4 5 6 4 1 2 3 6 5 4 Lagrange element (T6)
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-1 -0.5 0 0.5 1 -1 -0.5 0 0.5 1 -1 -0.5 0 0.5 1 -1 -0.5 0 0.5 1 -1 -0.5 0 0.5 1 -1 -0.5 0 0.5 1 -1 -0.5 0 0.5 1 -1 -0.5 0 0.5 1 -1 -0.5 0 0.5 1 •• 1 2 3 4 56 78 9 1 ψ 2 5 1 2 3 4 5 6 7 8 -1 -0.5 0 0.5 1 -1 -0.5 0 0.5 1 -1 -0.5 0 0.5 1 -1 -0.5 0 0.5 1 -1 -0.5 0 0.5 1 -1 -0.5 0 0.5 1 1 2 4 1 1 ξ η 1 2 3 4 5 6 7 8 1 2 1 Serendipity element (Q8) Lagrange element (Q9)
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Some Triangular and Rectangular Elements ( b ) 1 3 2 4 η ξ 1 3 2 4 5 6 7 8 9 1 3 2 4 5 6 7 8 ( c ) Nodes with function values only ( u ) Nodes with values of the function ( u ) and its derivatives y x u , y u , x u 2 1 3 2 4 b 2 a 2 1 2 3 ( a ) 1 2 3 4 5 6 Lagrange triangular elements Lagrange rectangular elements Hermite cubic rectangular element
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This note was uploaded on 10/03/2011 for the course MCE 561 taught by Professor Sadd during the Spring '11 term at Rhode Island.

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Summary_Ch9 - CHAPTER 9 Interpolation functions for 2D...

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