Monolithic Finite Element Method for the simulation of thixo-viscoplastic flows

This talk is concerned with the application of Finite Element Method (FEM) and Newton-Multigrid solver to simulate thixotropic flows.The thixotropy phenomena are introduced to yield stress material by taking into consideration the internal material micro structure using a structure parameter. Firstly, the viscoplastic stress is modified to include the thixotropic stress dependent on the structure parameter. Secondly, an evolution equation for the structure parameter is introduced to induce the time-dependent processof competition between the destruction (breakdown) and the construction (buildup) inhabited in the material. Substantially, this is done simply by introducing a structure-parameter-dependentviscosity into the rheological model for yield stress material, as for instance the Houska model based on a viscosity approach for the Bingham model [2].The modified viscoplastic stress w.r.t. the structure parameter which is integrated, in quasi-Newtonian manner or lagrangian multiplier manner, into the generalized Navier-Stokes equations and the evolution equation for the structure parameter constitutes the main core of full set of mod-eling equations, which are creditable as the privilege answer to incorporate thixotropy phenomena.The nonlinearity of the problem, related to the dependency of the diffusive stress on the material parameters, is treated with generalized Newton’s method w.r.t. the Jacobian’s singularities having a global convergence property. The linearized systems inside the outer Newton loops are solvedusing the geometrical multigrid methods with a Vanka-like smoother taking into account a stable FEM approximation pair for velocity and pressure with discontinuous pressure and biquadratic velocity spaces.We analyze the accuracy, robustness and efficiency of the Newton-Multigrid FEM solver [1] throughout the solution of thixotropic flow problems of benchmarking character in channel and Couette device [3].