In solving nonlinear hyperelastic solid mechanics problems, to converge to the correct solution we need to do step-by-step loading which makes the deformation at each step very small (for my particular problem $\mathcal{O}(e^{-11})$). Moreover, due to the material properties, the nonlinear weak form and consequently the linear system that should be solved at each step of the nonlinear solver have very small components. In my experience, this makes the iterative solvers very inefficient especially considering that many components are close to machine epsilon precision and the relative norm is too small and is not a good measure for the convergence. I would appreciate if you can help me to know
1) is what I am observing true or I am making a mistake somewhere?
2) what is the smart way to deal with such matrices that the weak form and consequently the resulted linear system have very small components?
What I am doing now to solve this nonlinear system is to reduce the absolute convergence norm criteria to a very small number $\mathcal{O}(e^{-13})$ when the absolute norm for 0 iteration is $\mathcal{O}(e^{-9})$. And use a relatively small relaxation value.
Thank you, everybody, for your help.
1e-13
, then my question is1e-13
what?1e-13
lightyears? $\endgroup$ – Wolfgang Bangerth Mar 12 '20 at 18:09