12a-Bar-Isoparametric-2010s

12a-Bar-Isoparametric-2010s - Linear Quadratic Now we can...

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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.
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© 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!
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© 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
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© 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 ( ) = u i 1 2 1 ( ) N i ( ) + u j 1 2 1 + ( ) N j ( ) X ( ) = X i 1 2 1 ( ) N i ( ) + X j 1 2 1 + ( ) N j ( ) William Klug Integration f X ( ) ( ) 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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12a-Bar-Isoparametric-2010s - Linear Quadratic Now we can...

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