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13a-2D-3D-BVPs

13a-2D-3D-BVPs - MAE M168/CEE M135C Introduction to Finite...

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© William Klug MAE M168/CEE M135C Introduction to Finite Element Methods Lecture 13a 2-D & 3-D BVPs: Steady-state Heat Transfer
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© William Klug Outline: 2-D & 3-D FEA Develop weak form Interpolation in 2-D & 3-D (shape functions) Interpolation + weak form Discrete FE equations
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© William Klug 2-D & 3-D Steady State Heat Transfer Heat flux vector: Heat flow per unit area per unit time across surface
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© William Klug Usually when material is isotropic then, Conductivity Matrix Conductivity matrix
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© William Klug Conservation of Energy: 3-D Q dV V = q n dA V = S = q n dA = S i q dV V Q = i q = −∇ i D T ( ) or i D T ( ) + Q = 0 in V Divergence Theorem
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© William Klug Conservation of Energy: 2-D t i D T ( ) + Qt = 0 Heat supplied per time per area
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© William Klug Weak Form: 3-D v i D T ( ) dV V + vQ dV V = 0 v i D T ( ) + Q dV V = 0 v i D T ( ) = i v D T ( ) − ∇ v i D T ( ) v i D T ( )
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