> I calculated the eigenvalues and the condition number. It seems that > it is ok to collapse one vertex and transform the quad into a > triangle. > > Interestingly, applying the constraints with AffineConstraint does not > change the condition number. Even if I fix all the dofs with > add_line() the condition number doesn't change.
Yes. Adding constraints is the same as adding information, which can only make the condition number better. (Not a scientific answer, but intuitively true anyway.) > I collapsed one vertex in a 3D simulation with NedelecSZ and the > solution and the condition number are ok. I have to do more tests with > NelelecSZ. > > In the simple simulation from step-6. I collapse one vertex and then > if I move another vertex and transform the quad into a line the > surface becomes very small and the condition number becomes very > large. Note that I didn't completely squeeze the quad into a line. If > the surface is zero, then as expected the simulation breaks. That's correct too -- if you just squeeze an element but not completely collapse + add constraints, then that's equivalent to a mesh with a poorly shaped cell. Since the condition number of the matrix involves the ratio of the largest to the smallest eigenvalue of the Jacobian of the mapping on each cell, you end up with a large condition number if you have poorly shaped cells. This happens here: > -Collapse vertex a, almost collapse vertex b > * surface_ratio = 1e-8 > * eigenvalues = [ 6.66667000e-01 6.66667000e-01 6.66667000e-01 > 6.66667000e-01 > 1.21307648e+00 1.33333000e+00 1.33333000e+00 1.33333000e+00 > 1.33333000e+00 1.33333000e+00 1.33333000e+00 1.33333000e+00 > 1.33333000e+00 1.33333000e+00 1.35897000e+00 1.66666000e+00 > 1.69231000e+00 8.83186491e+00 1.10689839e+01 1.37682911e+01 > 1.98164659e+01 2.84090528e+01 4.79480606e+01 2.92923117e+05 > 1.70718688e+06] > * condition_number = 3e+06 For this poorly shaped cell, you can compute the eigenvalues of the Jacobian of the mapping, if you want. You'll find that they behave roughly like Delta x and Delta y, respectively, and for your mesh, these are very different. Best W. -- ------------------------------------------------------------------------ Wolfgang Bangerth email: [email protected] www: http://www.math.colostate.edu/~bangerth/ -- The deal.II project is located at http://www.dealii.org/ For mailing list/forum options, see https://groups.google.com/d/forum/dealii?hl=en --- You received this message because you are subscribed to the Google Groups "deal.II User Group" group. To unsubscribe from this group and stop receiving emails from it, send an email to [email protected]. To view this discussion on the web visit https://groups.google.com/d/msgid/dealii/2778fe14-8fac-42c7-ff09-aba23c84c035%40colostate.edu.
