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--- gtspring2009:gibson:localsolns [2009/07/16 11:05]
gibson
+++ gtspring2009:gibson:localsolns [2009/07/20 07:26]
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 ===== 2009-06-12 =====
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 ===== 2009-07-07 =====
-Eigenvalues as function of Reynolds number for the spatially periodic solution, computed for (Lx, Lz) = (2 pi, 4 pi), i.e. four spanwise repeated copies of one spatially periodic state (the Re=150 plot above gives you the picture). The two plots below show the real part of the leading eigenvalues versus Reynolds number. Left: the 10 leading eigenvalues, sorted and colored by magnitude of Re λ. Right: The eigenvalues that cross the real axis, remixed so the color stays continuous as the eigenvalues cross.
+Eigenvalues as function of Reynolds number for the spatially periodic solution, computed for (Lx, Lz) = (4 pi, 8 pi), i.e. four spanwise repeated copies of one spatially periodic state (the Re=150 plot above gives you the picture). The two plots below show the real part of the leading eigenvalues versus Reynolds number. Left: the 10 leading eigenvalues, sorted and colored by magnitude of Re λ. Right: The eigenvalues that cross the real axis, remixed so the color stays continuous as the eigenvalues cross.
 {{:gtspring2009:gibson:local:uglobbifirc2.png?400}}{{:gtspring2009:gibson:local:uglobbifirc.png?400}}
@@ Line 64: / Line 65: @@
 ===== 2009-07-13 =====
-Continuation of localized solution on larger domain. The results above are for a localized solution on a 2pi x 2 x 4pi domain. Note that as the localized lower-branch solution is continued down in Re, it starts to fill the whole box and merges into the globally periodic lower-branch solution around Re=150. Tobias suggested putting the localized solution in a larger box and seeing if it would then continue around a saddle-node bifurcation to give us a localized upper branch. Good news: it does!
+Continuation of localized solution on larger domain. The results above are for a localized solution on a 4pi x 2 x 8pi domain. Note that as the localized lower-branch solution is continued down in Re, it starts to fill the whole box and merges into the globally periodic lower-branch solution around Re=150. Tobias suggested putting the localized solution in a larger box and seeing if it would then continue around a saddle-node bifurcation to give us a localized upper branch. Good news: it does!
 {{:gtspring2009:gibson:local:globlocred3.png?300}}{{:gtspring2009:gibson:local:globlocred4.png?300}}
-These plots show dissipation versus Reynolds numbers for three equilibrium solutions: the "global eqb", a solution that is triply periodic in the (2pi,2,4pi) box (see Re=150 plot above),for which we have both upper and lower branches; the localized solution in (2pi,2,4pi) box (see Re=400 to 150 plots above); and the localized solution fit into a larger box (2pi,2,8pi) (shown below). I'm using a different definition of dissipation here because including the laminar dissipation and normalizing by volume obscures the fact the localized solutions in the two different boxes  are almost identical from Re=400 down to Re=190. The definition of dissipation here is
+These plots show dissipation versus Reynolds numbers for three equilibrium solutions: the "global eqb", a solution that is triply periodic in the (4pi,2,8pi) box (see Re=150 plot above),for which we have both upper and lower branches; the localized solution in (4pi,2,8pi) box (see Re=400 to 150 plots above); and the localized solution fit into a larger box (4pi,2,16pi) (shown below). I'm using a different definition of dissipation here because including the laminar dissipation and normalizing by volume obscures the fact the localized solutions in the two different boxes  are almost identical from Re=400 down to Re=190. The definition of dissipation here is
 <latex>
@@ Line 84: / Line 85: @@
-The computation is pretty slow, because the grid is large 32 x 33 x 384 for (2pi,2,8pi). This discretization was appropriate for the localized lower branch at Re=400 but looking at the spectra 48 x 33 x 256 would be more appropriate for solutions near the saddle-node.
+The computation is pretty slow, because the grid is large 32 x 33 x 384 for (4pi,2, 16pi). This discretization was appropriate for the localized lower branch at Re=400 but looking at the spectra 48 x 33 x 256 would be more appropriate for solutions near the saddle-node.
@@ Line 129: / Line 130: @@
 {{:gtspring2009:gibson:local:globlocred7.png?400}}{{:gtspring2009:gibson:local:globlocred8.png?400}}
-These fields are points on the green curve. All are for 2pi,2,8pi.
+These fields are points on the green curve. All are for 4pi,2,16pi.
 {{:gtspring2009:gibson:local:re_179.98_d_1.28.png?400}}
@@ Line 142: / Line 143: @@
 **Resolution** These computations are expensive enough that it's worthwhile finding the minimum acceptable resolution.
-I recomputed the Re=174.36, D=1.484 solution localized solution at variety of resolutions. Spectra and properties of solutions are shown below.
+I recomputed the Re=174.36, D=1.484 4pi x 2 x 16pi localized solution at variety of resolutions. Spectra and properties of solutions are shown below.
-{{:gtspring2009:gibson:local:spectra0_32_33_192.png?300}} {{:gtspring2009:gibson:local:spectra0_32_33_256.png?300}}
+{{:gtspring2009:gibson:local:spectra0_32_33_192.png?300}} {{:gtspring2009:gibson:local:spectra0_32_33_256.png?300}} {{:gtspring2009:gibson:local:spectra0_32_33_384.png?300}} {{:gtspring2009:gibson:local:spectra0_32_49_256.png?300}}
-{{:gtspring2009:gibson:local:spectra0_32_33_384.png?300}}
+{{:gtspring2009:gibson:local:spectra0_48_33_256.png?300}}  {{:gtspring2009:gibson:local:spectra0_48_33_384.png?300}}
 The quantity shown in the Fourier spectrum is
@@ Line 170: / Line 171: @@
 | 48 33 384 | 1.48390 | 8e-06 | 2e-14 | 099 | 231 |
-x 33 x 256 for 2pi x 2 x 8pi seems to be the sweet spot. To get a few rungs on the snaking ladder, I am starting continuations of the localized solution on boxes with Lz = 12pi and 16pi, with Nz = 384 and 512 respectively.
+x 33 x 256 for 4pi x 2 x 16pi seems to be the sweet spot. To get a few rungs on the snaking ladder, I am starting continuations of the localized solution on boxes with Lz = 24pi and 32pi, with Nz = 384 and 512 respectively.
 ===== 2009-07-16 =====
-A few more rungs on the ladder in the 4pi x 24pi box (green) and a few points on 4pi x 24 pi (magenta)
+A few more rungs on the ladder in the 4pi x 16pi box (green) and a few points on 4pi x 24 pi (magenta)
-and 4pi x 32 pi (cyan). I've marked a few spots on the 4pi x 24pi curve that show corresponding points
+and 4pi x 32pi (cyan). I've marked a few spots on the 4pi x 16pi curve that show corresponding points
 on successive rungs of the ladder. Midplane velocity fields for those solutions are shown below. Tobias,
 John, please add comments to remind me what we are looking for between successive rungs.
@@ Line 186: / Line 187: @@
 {{:gtspring2009:gibson:local:loc2pi8pid1.358.png?400}}
-{{:gtspring2009:gibson:local:loc2pi8pid1.306.png?400}}
+{{:gtspring2009:gibson:local:loc2pi8pid1.306.png?400}} **Warning: all these should be labeled 4pi x 16 pi!**
 BTW, I've confirmed that the globally periodic solution is the same as the Nagata lower branch EQB, by continuing down one copy from 4pi x 2pi to 2pi/1.14 x 2pi/2.5, for which I have the Nagata EQB.
+**Yikes! I've been off by a factor of two on my box sizes through most of this discussion! I went back and changed all the values in the text but some figures have box sizes in labels. I won't change those; it's too much trouble. Box sizes in figures above this point should be doubled, below this line they are correct.**
+Also, the only symmetry satisfied by this solution is s_xyz, where s_xyz : [u,v,w](x,y,z) -> [-u,-v,-w](-x,-y,-z), with a suitably chosen origin. I've checked in detail. The globally periodic solutions have more symmetries, due to the possibility of combining phase shifts with s_z : [u,v,w](x,y,z) -> [u,v,-w](x,y,-z) and s_xy : [u,v,w](x,y,z) -> [-u,-v,w](-x,-y,z).
+===== 2009-07-18 =====
+Further progress on continuing solutions in 4pi,16pi and 4pi,24pi.
+{{:gtspring2009:gibson:local:globlocred10.png?300}} {{:gtspring2009:gibson:local:globlocred11.png?300}}
+Some comments on computation. After determining the symmetries of the solutions I optimized the phase shifts in x,z so that the center of symmetry is exactly (0,0) or equivalently (Lx/2, Lz/2) and also made sure the solutions were in the same orientation about the center. We had a brief power outage last night that caused my machine to reboot; this gave a good opportunity to restart the computations with the optimized phase shift, with sxyz symmetry enforced during time integration and continuation, and with the continuation program set to give up and try a smaller continuation step if the Newton-search steps are poor enough that the trust-region hookstep mechanism kicks in. Observations suggest that continuation is more efficient in total computation time if we never leave the convergence region of pure Newton iteration. Also, I reduced the discretization of the 4pi,16pi continuation to 32 x 33 x 256.
+I started Arnoldi iteration on the phase-shift optimized versions the previously calculated solutions for 4pi,16pi. I didn't want to start these without symmetry optimization, since I need to examine symmetry to sort the eigenvalues, and it's a lot easier to detect the symmetry of the eigenfunction if you know the center of symmetry of the solution. That's running now. Looks like about an hour per solution at 100 Arnoldi iterations per solution. With two CPUs going and about fifty solutions (every other dot on the green curve), it'll take 24 hrs or so. It's clear from the first calculation that 100 Arnoldi iterations is not overkill. The first calc is at the 66th iteration, and the second zero eigenvalue is not yet resolved well enough to distinguish it clearly from the small non-zero eigenvalues. Arnoldi iteration generally requires more iterations to get accurate eigenvalues than GMRES does to get an sufficiently accurate solution. No great accuracy is required in GMRES solutions to Newton steps, since the Newton procedure is iterative itself.
+===== 2009-07-20 =====
+Further progress on continuing the localized solutions 4pi,16pi and 4pi,24pi. 4pi,16pi appears to be headed to the upper branch of the Nagata solution.
+The Arnoldi iterations are still cooking. I had to restart them with a longer integration time (T=20) for the finite-time map u-f^T(u) in order to get better separation of the eigenvalues near zero. I've set up a bash script to do Arnoldi iteration on every other data point in the 4pi,16pi continuation; those should finish today, and then maybe I'll start it on the intervening points. Following that I'll have to solve the bookkeeping problem to track individual eigenvalues between continuation steps. In the meantime I'm going to work on some Poincare sections that have been stalled for some time...
+BTW, the limiting factor on these computations is going to be memory in the Arnoldi iteration. The algorithm requires an LU decomp of an M x N matrix, where M is the number of independent spectral coefficients in the discretization of the velocity field, and N is the number of Arnoldi iterations. For the 4pi,16pi box with 32 x 33 x 256 gridpoints, that's a 200,000 x 100 matrix, and the process size gets to 1.2 GB during the LU decomp. My machine has 4 GB DDR3 1600 RAM and 4 cores, so I can do a couple Arnoldis at a time without trouble. But I expect to hit my upper limit pretty soon (a factor of 2 or 4) in domain size or Reynolds, or even sooner with less smooth solutions, which will need finer resolution. The Nagata LB is unusually smooth. I could put in two more 2 GB sticks to move up to 8 GB RAM, but then I'd have to downclock the RAM to 1033. I think the PACE machines are all 4 GB.
+{{:gtspring2009:gibson:local:globlocred12.png?300}}

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