lec21 (1)

# lec21 (1) - Introduction to Simulation Lecture 21 Boundary...

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Introduction to Simulation - Lecture 21 Thanks to Deepak Ramaswamy, Michal Rewienski, and Karen Veroy Boundary Value Problems - Solving 3-D Finite-Difference problems Jacob White

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Outline Reminder about FEM and F-D 1-D Example Finite Difference Matrices in 1, 2 and 3D Gaussian Elimination Costs Krylov Method Communication Lower bound Preconditioners based on improving communication
1-D Example Normalized 1-D Equation Normalized Poisson Equation 2 2 ( ) ( ) ( ) s T x u x h f x x x x κ = − ⇒ − = Heat Flow ( ) ( ) xx u x f x =

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SMA-HPC ©2002 MIT FD Matrix properties Finite Differences 1-D Poisson 0 x 1 x 2 x n x 1 n x + 1 2 ˆ ˆ ˆ 2 ( ) j j j j u u u f x x + + = xx u f = 1 1 2 ˆ ( ) 2 1 0 0 1 2 1 1 0 0 1 0 0 1 2 ˆ ( ) n n u f x x u f x = M M L M M O M M M O O O M M M O O O M M L A
SMA-HPC ©2002 MIT Residual Equation Using Basis Functions 2 2 u f x = Partial Differential Equation form (0) 0 (1) 0 u u = = Basis Function Representation Plug Basis Function Representation into the Equation ( ) ( ) ( ) 2 2 1 n i i i d x R x f x dx ϕ ω = = + ( ) ( ) ( ) { 1 Basis Functions n h i i i u x u x x ω ϕ = =

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SMA-HPC ©2002 MIT Basis Weights Using Basis functions Galerkin Scheme Generates n equations in n unknowns ( ) ( ) ( ) 2 1 2 1 0 0 n i l i i d x x f x dx dx ϕ ϕ ω = + = Force the residual to be “orthogonal” to the basis functions { } 1,..., l n ( ) ( ) 1 0 0 l x R x dt ϕ =
SMA-HPC ©2002 MIT Basis Weights Using Basis Functions Galerkin with integration by parts Only first derivatives of basis functions ( ) ( ) ( ) ( ) 1 1 1 0 0 0 n i i i l i x d d x dx x f x dx dx dx ω ϕ ϕ ϕ = = { } 1,..., l n

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Structural Analysis of Automobiles • Equations – Force-displacement relationships for mechanical elements (plates, beams, shells) and sum of forces = 0. – Partial Differential Equations of Continuum Mechanics
Drag Force Analysis of Aircraft • Equations – Navier-Stokes Partial Differential Equations.

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