ECE257 Numerical Methods andECE257 Numerical Methods andScientific ComputingScientific ComputingPartial Differential EquationsPartial Differential EquationsECE 257 Numerical Methods and Scientific ComputingFall 2004Lecture 22John A. ChandyDept. of Electrical and Computer EngineeringUniversity of ConnecticutTodayToday’’s class:s class:••Finite Element MethodFinite Element Method––One DimensionOne Dimension––Two DimensionTwo DimensionECE 257 Numerical Methods and Scientific ComputingFall 2004Lecture 22John A. ChandyDept. of Electrical and Computer EngineeringUniversity of ConnecticutFinite Element MethodFinite Element Method••Finite Difference MethodFinite Difference Method––Solution domain is divided into a grid ofSolution domain is divided into a grid ofdiscrete pointsdiscrete points––At each point, the PDE is evaluated byAt each point, the PDE is evaluated byapproximating derivatives with finite-dividedapproximating derivatives with finite-divideddifferencesdifferences––DoesnDoesn’’t work well with irregular geometriest work well with irregular geometriesECE 257 Numerical Methods and Scientific ComputingFall 2004Lecture 22John A. ChandyDept. of Electrical and Computer EngineeringUniversity of ConnecticutFrom Numerical Methods for Engineers, Chapra and Canale, Copyright © The McGraw-Hill Companies, Inc.ECE 257 Numerical Methods and Scientific ComputingFall 2004Lecture 22John A. ChandyDept. of Electrical and Computer EngineeringUniversity of ConnecticutFinite Element MethodFinite Element Method••Divide solution domain into simply shaped regionsDivide solution domain into simply shaped regionsinstead of rectangular gridsinstead of rectangular grids••Each region is a Each region is a ““finite elementfinite element”” connected to each connected to eachother at nodal points and boundary surfacesother at nodal points and boundary surfaces••Solve each region independentlySolve each region independently••Assemble the region solutions and make sure theAssemble the region solutions and make sure theboundary continuity requirements are satisfiedboundary continuity requirements are satisfiedECE 257 Numerical Methods and Scientific ComputingFall 2004Lecture 22John A. ChandyDept. of Electrical and Computer EngineeringUniversity of ConnecticutFrom Numerical Methods for Engineers, Chapra and Canale, Copyright © The McGraw-Hill Companies, Inc.ECE 257 Numerical Methods and Scientific ComputingFall 2004Lecture 22John A. ChandyDept. of Electrical and Computer EngineeringUniversity of ConnecticutSix Steps in the Finite Element MethodSix Steps in the Finite Element Method••Step 1 - Step 1 - DiscretizationDiscretization: The problem domain is : The problem domain is discretizeddiscretizedinto a collection of simple shapes, or elements.into a collection of simple shapes, or elements.••Step 2 - Develop Element Equations: Developed using theStep 2 - Develop Element Equations: Developed using thephysics of the problemphysics of the problem••Step 3 - Assembly: The element equations for each elementStep 3 - Assembly: The element equations for each elementin the FEM mesh are assembled into a set of global equationsin the FEM mesh are assembled into a set of global equationsthat model the properties of the entire system.that model the properties of the entire system.••Step 4 - Application of Boundary Conditions: Solution cannotStep 4 - Application of Boundary Conditions: Solution cannotbe obtained unless boundary conditions are applied. Theybe obtained unless boundary conditions are applied. Theyreflect the known values for certain primary unknowns.reflect the known values for certain primary unknowns.Imposing the boundary conditions modifies the globalImposing the boundary conditions modifies the globalequations.equations.••Step 5 - Solve for Primary Unknowns: The modified globalStep 5 - Solve for Primary Unknowns: The modified globalequations are solved for the primary unknowns at the nodes.equations are solved for the primary unknowns at the nodes.••Step 6 - Calculate Derived Variables: Calculated using theStep 6 - Calculate Derived Variables: Calculated using thenodal values of the primary variables.nodal values of the primary variables.ECE 257 Numerical Methods and Scientific ComputingFall 2004Lecture 22John A. ChandyDept. of Electrical and Computer EngineeringUniversity of ConnecticutElement EquationsElement Equations••How do you determine the solution acrossHow do you determine the solution acrossthe finite element?the finite element?••Consider a one-dimensional ODEConsider a one-dimensional ODE ! "2T"x2+ f x( )= 0 T 0( )= T1,T L( )= T2DiscretizationECE 257 Numerical Methods and Scientific ComputingFall 2004Lecture 22John A. ChandyDept. of Electrical and Computer EngineeringUniversity of ConnecticutElement EquationsElement Equations••Individual ElementIndividual Element••Approximate solutionApproximate solutionwith a linear equationwith a linear equation! ˜ T = N1T1+ N2T2 ! N1=x2" xx2" x1N2=x " x1x2" x1 interpolationfunctionsECE 257 Numerical Methods and Scientific ComputingFall 2004Lecture 22John A. ChandyDept. of Electrical and Computer EngineeringUniversity of ConnecticutElement EquationsElement Equations••Direct methodDirect method••Look at heat flux at each nodeLook at heat flux at each node ! At node 1 q1= k'T1" T2x2" x1= "k'dT x1( )dxAt node 2 q2= k'T2" T1x2" x1= k'dT x2( )dx! 1x2" x11 "1"1 1# $ % & ' ( T1T2) * + , - . ="dT x1( )dxdT x2( )dx) * / + / , - / . /ECE 257 Numerical Methods and Scientific ComputingFall 2004Lecture 22John A. ChandyDept. of Electrical and Computer EngineeringUniversity of ConnecticutMethod of Weighted ResidualsMethod of Weighted Residuals••Start with an approximate solutionStart with an approximate solution••When you plug it into the differential equation youWhen you plug it into the differential equation youget an error or get an error or residualresidual••The method of weighted residuals tries to minimizeThe method of weighted residuals tries to minimizethis residualthis residual! ˜ T = N1T1+ N2T2 ! "2˜ T "x2+ f x( )= RECE 257 Numerical Methods and Scientific ComputingFall 2004Lecture 22John A. ChandyDept. of Electrical and Computer EngineeringUniversity of ConnecticutMethod of Weighted ResidualsMethod of Weighted Residuals••The idea is to minimize The idea is to minimize RR by minimizing the by minimizing thefollowing integralfollowing