12a-Bar-Isoparametric-2010s

# 12a-Bar-Isoparametric-2010s - – Linear – Quadratic Now...

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© William Klug MAE M168/CEE M135C Introduction to Finite Element Methods Lecture 12a Isoparametric Elements in 1-D

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© William Klug Problems • Problems with “standard” elements: – Interior nodes evenly spaced. – Shape functions depend on geometry.
© William Klug Solution • Isoparametric mapping – Convenient coordinate transformation between Natural and Physical coordinate systems.

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© William Klug Linear Shape Functions Global (physical) Local (physical) Natural (unitless) Geometry is gone!
© William Klug Redefine Interpolation u ξ ( ) = u i 1 2 1 −ξ ( ) N i ( )    + u j 1 2 1 + ( ) N j ( )   

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© William Klug Coordinate Transformation Natural Geometry Physical Geometry X = X ξ ( ) = X i + X j X i 2 + 1 ( ) = X i 1 2 1 −ξ ( ) N i ( )    + X j 1 2 1 + ( ) N j ( )    Transformation interpolates between nodal positions
© William Klug Isoparametric Elements • Interpolate both displacement and position using the same shape functions

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Unformatted text preview: – Linear – Quadratic Now we can move mid-nodes away from center Nodal displacements Nodal positions © William Klug Compute Derivatives Using the Chain Rule • Need for mechanics: u ξ ( ) = u i 1 2 1 −ξ ( ) N i ( ) + u j 1 2 1 + ( ) N j ( ) X ( ) = X i 1 2 1 −ξ ( ) N i ( ) + X j 1 2 1 + ( ) N j ( ) © William Klug Integration → f X ξ ( ) ( ) − 1 1 ∫ dX d d = f X ( ) ( ) J ( ) d − 1 1 ∫ “Jacobian” of mapping © William Klug Energy and Stiffness For linear shape functions © William Klug Higher Order Elements © William Klug Example (HW) © William Klug Summary and Review • Interpolate both displacement and position using the same shape functions Nodal displacements Nodal positions...
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## This note was uploaded on 08/09/2011 for the course MAE 168 taught by Professor Klug during the Spring '11 term at UCLA.

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12a-Bar-Isoparametric-2010s - – Linear – Quadratic Now...

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