# Resources for viscous behavior in simple FEM

I am working on a simple explicit-integration lumped-mass elastic FEM code which implements CST+DKT triangles (plate+shell) and constant-strain tetrahedra (http://woodem.eu/doc/theory/membrane-element.html, http://woodem.eu/doc/theory/tet4-element.html). The code focuses on contact dynamics, so FEM is there only to model flexible boundaries. I would like to add some kind of viscous damping to the model, and I am looking for some resource which is not overly complicated.

I independently thought I could use the elastic stiffness matrix $\mathbf{K}$ (as in $f=\mathbf{K}u$), scaled by some viscosity factor $\eta'$ (that would be computed from material's $E$ and $\eta$), to compute viscous resisting force as $f_v=-\eta'\mathbf{K}\dot u$. It this formulation something known in literature? Or is it plain wrong?

Thanks for pointers.

• Did you search the literature for viscoelastic models? What did you find? – Wolfgang Bangerth Feb 23 '15 at 14:25
• @WolfgangBangerth: yes I did; the problem is not about viscoelastic models, but about how to plug those into explicit FEM. I searched scholar.google.com, but found references which did not really treat what I need. Perhaps I am just missing the right keywords. – eudoxos Feb 23 '15 at 14:41
• By "explicit", I assume you mean explicit as opposed to implicit time-stepping? When you incorporate viscosity, explicit time-stepping tends to fare quite poorly, no matter what space discretization you're using (FEM, FDM, ...) because the computational costs necessary to guarantee stability of the numerical scheme are very steep. – Daniel Shapero Feb 23 '15 at 17:52
• What you're suggesting seems to be a particular case of Rayleigh damping, with only a stiffness matrix-proportional component. It wouldn't be explicit in any case. You can try a Rayleigh damping matrix with only a mass-matrix-proportional component, which you would lump similarly to the mass matrix. However, as mentioned above, the stability may deteriorate. – DanielRch Feb 23 '15 at 18:07
• @DanielShapero: yes, explicit time-stepping. Critical timestep is low, but the focus is on contact problems (like youtube.com/watch?v=cOLMNqtCy1c) which have their own constraints on timestep. – eudoxos Feb 23 '15 at 19:03