FOUR-FIELD FINITE ELEMENT SOLVER AND SENSITIVITIES FOR QUASI-NEWTONIAN FLOWS

摘要

A computationally efficient finite element algorithm for power-law fluid is elaborated in view of extensive direct and inverse simulations. We adopt a splitting technique to simplify the nonlinear structure of the fluids equations and derive a four-field saddle-point formulation for which we prove the existence of a solution. The resolution of the corresponding variational inequalities is based on an augmented Lagrangian method and a mixed finite element discretization. The resulting iterative solver proves to be fast and robust with low memory consumption. The time-saving provided by the algorithm compared to the standard algorithms of fixed point and Newton increases with the number of degrees of freedom and the nonlinearity of the problem. It is therefore well suited for the solution of large problems with a great number of elements and for corresponding adjoint-based computations. Bidimensional numerical experiments are performed on two realistic situations of gravity flows: an experimental viscoplastic steady wave and a continental glacier. In the present study, results emphasize that for both cases, the modeling at bottom plays a strongly dominant role. Using surface velocity observations, the sensitivity analysis with respect to a spatially varying power-law exponent highlights the importance of an accurate knowledge of the rheology at high shear rate. The one on the basal sliding allows one to detect the presence of a short wavelength (two times the thickness) free-slip area undetectable from surface velocities.