教程40.流体力学numerical methods for heat,年-peakchina l

1、Lecture 43: Algebraic Multigrid MethodCycling StrategiesLast TimeWeConsidered the algebraic multigrid method Developed coarse-level discrete equations by addion of fine-level equations togetherConsidered an agglomeration strategy based on coefficient sizeThis Time Complete the discussion of coeffici

2、ent-based agglomeration strategyDiscuss cycling strategies for switching between coarse and fine levelsRecall Coefficient-Based Agglomeration At fine level (l) assign each fine-level cell i a coarse-level cell index I. Initialize I=0 for all fine level cellsSet coarse level counter C=1Visit fine-lev

3、el cells one by one Check if already in coarse-level groupIf not, group together the cell and n of its neighbors with the largest coefficients A(l)ijAssign the coarse index C to the group, i.e., set I=CIncrement C=C+1Go to next fine level cellGroup size n=2 can be shown to yield best performanceDoub

4、les necessary coefficient storageUnstructured Mesh Coefficients For unstructured meshes, coefficient agglomeration leads toHere, GI is the set of fine-level cellsbelonging to coarse-level cell IGIGJWhy Does Coefficient-Based Agglomeration Work?Consider conjugate heat transfer in a square domain Air,

5、 k =0.02 W/mkCopper, k =400 W/mkT1T2T3T4Coefficients on Fine-Level MeshUsing harmonic mean interpolationInterior coefficient for near-interface cell:nPNAir/Cu interfaceCoefficients (contd)For cell in copper region at interface, coefficients to air side are very small compared to coefficients to othe

6、r cells in the copper regionHowever, Dirichlet boundary conditions are only available on the air sideCoefficient anisotropy makes it very difficult for information to travel into the copper during normal Gauss-Seidel iterationMulti-grid agglomeration cures this problemCoefficient-Based Agglomeration

7、At coarsest level, only 2 cellsOnly one “neighbor” coeffcient for copper-airNo coefficient anisotropy problemAnisotropic Conductionk 0, k =0Equations in direction agglomeratedThis makes sense because high k in that direction makes cells have similar temperaturebgExampleSet boundary temperatures = 30

8、0KGuess T=500K in the interiorMesh size: 20 x20K ratio =1000T1T2T3T4Residual using Gauss-Seidel IterationTemperature Profiles with Gauss-Seidel Iteration Residual using Multi-Grid SchemeTemperature Profiles Using Multi-GridIter=1Iter=3Multigrid CyclesVarious cycling strategies exist for visiting coa

9、rse levelsThese cycles are most easily written in a language that allows recursionC allows a function to call itself, FORTRAN does notTwo types of cycles Fixed coarse levels visited in a predetermined sequenceFlexible cycle between levels depending on residual reduction ratesFixed Cycles: V CycleTwo

10、 legs a “down leg” and an “up leg”In the down legStart with finest level. Do 1 G-S iterations (sweeps)Restrict residuals to next coarse level. Do 1 G-S sweepsContinue till coarsest level is reachedNow start up legProlongate corrections from coarsest level to next fine level. Do 2 G-S sweepsContinue

11、in this way to finest levelThis completes one MG V-cycle iterationV Cycle (Contd)Value of 1 and 2 do not have to be the sameIn many implementations, 1 =0 but 2 is non-zero Gauss-Seidel SweepsRestriction orProlongationFixed Cycles: CyclesV cycle sometimes not sufficient for very stiff problems such a

12、s the Cu/air example we have just seenCan think of V cycle as consisting of Applying a “down-leg” operation recursively until coarsest level is reachedApplying an “up-leg” operation recursively until finest level is reached cycle Apply V-cycle recursively times at each levelFixed Cycles: W Cycle Can

13、 be thought of as a cycle with =2Spends a lot of effort at coarse levelsGood for very stiff problemsFixed Cycles: F CycleDo the down leg and then apply V cycle once at each successive levelFlexible Cycle: Brandt or Flex CycleTypically used if we do not need as much heft in our multigrid cycleResidua

14、l reduction parameter . At any multigrid level (l)Rk is the residual at iteration k at level (l). R0 is the residual upon entering the levelTermination criterion : if , stop sweeping at current level Flex Cycle (Contd) At finest level, find residual R0Do k iterations. Find residual Rk and therefore If is larger than prescribed value, convergence is not fast enoughHence drop to next coarse level and repeat recursivelyIf is smaller than prescribed, iterations are going well, so continueCheck if R . If finest level, solution is