Laplace s Equation FEM Methods. Jacob White. Thanks to Deepak Ramaswamy, Michal Rewienski, and Karen Veroy, Jaime Peraire and Tony Patera

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1 Introduction to Simulation - Lecture 19 Laplace s Equation FEM Methods Jacob White Thanks to Deepak Ramaswamy, Michal Rewienski, and Karen Veroy, Jaime Peraire and Tony Patera

2 Outline for Poisson Equation Section Why Study Poisson s equation Heat Flow, Potential Flow, Electrostatics Raises many issues common to solving PDEs. Basic Numerical Techniques basis functions (FEM) and finite-differences Integral equation methods Fast Methods for 3-D Preconditioners for FEM and Finite-differences Fast multipole techniques for integral equations SMA-HPC 2003 MIT

3 Outline for Today Why Poisson Equation Reminder about heat conducting bar Finite-Difference And Basis function methods Key question of convergence Convergence of Finite-Element methods Key idea: solve Poisson by minimization Demonstrate optimality in a carefully chosen norm SMA-HPC 2003 MIT

4 Drag Force Analysis of Aircraft Potential Flow Equations Poisson Partial Differential Equations. SMA-HPC 2003 MIT

5 Engine Thermal Analysis Thermal Conduction Equations The Poisson Partial Differential Equation. SMA-HPC 2003 MIT

6 Capacitance on a microprocessor Signal Line Electrostatic Analysis The Laplace Partial Differential Equation. SMA-HPC 2003 MIT

7 HeatFlow 1-D Example Incoming Heat T ( 0) Near End Temperature Unit Length Rod T () 1 Far End Temperature T ( 0) Question: What is the temperature distribution along the bar T T x () 1

8 HeatFlow 1-D Example Discrete Representation 1) Cut the bar into short sections 2) Assign each cut a temperature T (0) T (1) T1 T2 TN 1 TN

9 HeatFlow 1-D Example Constitutive Relation T i Heat Flow through one section x Ti 1 Ti T i + 1 hi 1, i heat flow κ + + = = x h i + 1, i Limit as the sections become vanishingly small T( x) lim x 0 hx ( ) κ = x

10 HeatFlow control volume 1-D Example Two Adjacent Sections Incoming Heat ( h s ) Conservation Law Ti 1 hii, 1 Ti h i+ 1, i T i + 1 x Heat Flows into Control Volume Sums to zero h h = h x i+ 1, i i, i 1 s

11 HeatFlow h 1-D Example Conservation Law Heat Flows into Control Volume Sums to zero Incoming Heat ( h s ) h = h x i+ 1, i i, i 1 s Ti 1 hii, 1 Ti h i+ 1, i T i + 1 x Heat in from left Heat out from right Incoming heat per unit length Limit as the sections become vanishingly small lim h ( x) x 0 s hx ( ) T( x) = = κ x x x

12 HeatFlow 1-D Example CircuitAnalogy Temperature analogous to Voltage Heat Flow analogous to Current T1 1 R κ = x TN + - vs = T(0) is = h x s + - vs = T(1) SMA-HPC 2003 MIT

13 HeatFlow 1-D Example Normalized 1-D Equation Normalized Poisson Equation T( x) u( x) κ = h = s x x x 2 2 f ( x) u ( x) = f ( x) xx

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16 Using Basis Functions Residual Equation Partial Differential Equation form 2 u = 2 x Basis Function Representation SMA-HPC 2003 MIT f u(0) = 0 u(1) = 0 u( x) u ( ) ( ) h x ωi i x { ϕ Plug Basis Function Representation into the Equation d ϕ ( x) R x f x ( ) n = i= 1 n 2 i = ωi + 2 i= 1 dx Basis Functions ( )

17 Using Basis Functions Example Basis functions Introduce basis representation u h ( x) is a weighted sum of basis functions The basis functions define a space n Xh= v Xh v= βϕ i i for some βi's i= 1 Hat basis functions ϕ 2 ϕ ϕ4 6 Example u( x) u ( ) ( ) h x ωi ϕi x { i= 1 Basis Functions Piecewise linear Space n = ϕ 1 ϕ3 ϕ5 SMA-HPC 2003 MIT

18 Using Basis functions Basis Weights Galerkin Scheme Force the residual to be orthogonal to the basis functions 1 ϕ ( ) ( ) l x R x dt = 0 Generates n equations in n unknowns ϕ l ( ) d ( ) n 2 ϕi x ω ( ) i f x dx 2 i= 1 dx x + = 0 l { 1,..., n} SMA-HPC 2003 MIT

19 Using Basis Functions Basis Weights Galerkin with integration by parts Only first derivatives of basis functions ( x) n d ωϕ ( x) i i i= 1 dx ϕ ( x) f ( x) dx = i dx 1 dϕ 1 l dx l { 1,..., n} SMA-HPC 2003 MIT

20 Convergence Analysis u The question is u h How does u u decrease with refinement? 123h SMA-HPC 2003 MIT error This time Finite-element methods Next time Finite-difference methods

21 Heat Equation Convergence Analysis Overview of FEM Partial Differential Equation form 2 u = 2 x Nearly Equivalent weak form SMA-HPC 2003 MIT f u(0) = 0 u(1) = 0 u v dx = f v dx for all v x x Ω Ω auv (, ) lv ( ) Introduced an abstract notation for the equation u must satisfy auv (, ) = lv ( ) for all v

22 Heat Equation Introduce basis representation u h Convergence Analysis Overview of FEM ( x) is a weighted sum of basis functions The basis functions define a space n Xh= v Xh v= βϕ i i for some βi's i= 1 Hat basis functions ϕ 2 ϕ ϕ4 6 Example u( x) u ( ) ( ) h x ωi ϕi x { i= 1 Basis Functions Piecewise linear Space n = ϕ 1 ϕ3 ϕ5 SMA-HPC 2003 MIT

23 Heat Equation Convergence Analysis Overview of FEM Key Idea auu (, ) defines a norm auu (, ) u U is restricted to be 0 at 0 and1!! Using the norm properties, it is possible to show If au (, ϕ ) = l( ϕ ) h i i for all ϕ ϕ, ϕ,..., ϕ i { } 1 2 n Then u uh = min wh X u w h h Solution Projection Error Error SMA-HPC 2003 MIT

24 Heat Equation Convergence Analysis Overview of FEM The question is only u u h How well can you fit u with a member of X h But you must measure the error in the norm For piecewise linear: SMA-HPC 2003 MIT 1 u uh = O n error

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63 Summary Why Poisson Equation Reminder about heat conducting bar Finite-Difference And Basis function methods Key question of convergence Convergence of Finite-Element methods Key idea: solve Poisson by minimization Demonstrate optimality in a carefully chosen norm SMA-HPC 2003 MIT