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## What does Cauchy mean in Double Dogleg solver?

Hello all,

Can anyone tell me what Cauchy stands for in the Double Dogleg solver for a stationary solver? I do not obtain convergence in my model, but am unsure as to what might be the reason because I do not know where to search as long as I do not know what Cauchy stands for (see attached PNG pictures). It seems to almost converge (value of 0.011 before ramping up with tolerance 0.01).

Also, if anyone could pinpoint why my model might not be converging, I would love to know. The model functions well for steps up until the addition of an internal convection (study 1 and study 2 step 1 work, study 2 step 2 doesn't), so I assume my model is properly setup for the other variables. I do not understand why the model is unable to solve for the last step. Does anyone have any experience with internal convection models not working? Might it be due to sharp edges in the geometry? Or do I have to set some extra condition on the boundary between the 'Water' and the 'Air' domain before the program understands that these fluids have seperate internal convection, but heat transfer between the two? I included the model and a schematic of the problem. If anyone could take a look at it, that would be very much appreciated!

Kind regards, Steggink

3 Replies Last Post 23 Oct 2019, 02:54 GMT-4

Posted: 7 months ago

Additionally, I now get the following warning after the first parametric iteration succeeds:

"Jacobian is incomplete. Unable to assemble the complete Jacobian."

The next step does not converge anymore, so I wonder if the above warning might tell anything about the problem. Can anyone say something about this?

Posted: 7 months ago
Updated: 7 months ago

Here is a literal description of what the double dog leg method does. You would have to look into the numerical analysis, particularly the numerical optimization, literature for a rigorous discussion.

After discretization with the finite element method you end up with a system of generally nonlinear algebraic system of equations with the unknowns usually being the solutions at the nodes. You need a nonlinear algebraic solver to solve that.

The Newton method is what is often used and it has a nice quadratic convergence behavior, when it converges. The downside is the iteration has to be close to the solution for this quadratic convergence or even convergence to be achieved. Start too far away from the solution in a strongly nonlinear problem and you can end up even diverging.

On the otherhand residual minimization methods for solving roots are robust in terms of their regions of convergence. But converge slower than the Newton method.

The double dog leg idea is to evaluate those two legs (Newton and Cauchy) and optimally combine the two directions to exploit the advantages of both. The numbers for Newton or Cauchy in the COMSOL solver log indicate how much weight (0 to 1) is given to the method in question.

In a well behaved problem you should be able to take the full Newton step. So usually the Newton weight has to be 1 and Cauchy has to be 0. Sometimes you can have problems where the residual is small but since the double dog leg algorithm could not take the full Newton it keeps iterating even though based on residuals you can say you are good.

This is what is going on as far as the solver is concerned. Try switching the solver to Constant Newton. This is a physics agnostic recommendation. You may have to consult the support team if this doesn't resolve it and you need a closer look.

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We all have opinions. But the COMSOL support team has the truth. I don't know if my opinion is still the truth. Check with that team before you try this at home!