This note is concerned with the application of Finite Element Methods (FEM) and Newton-Multigrid solvers for the simulation of thixotropic flow problems. The thixotropy phenomena are introduced into viscoplastic material by taking into account the internal material micro structure using a scalar 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 process of competition between the destruction (breakdown) and the construction (buildup) inhabited in the material. Substantially, this is done simply by introducing a structure-parameter-dependent viscosity into the rheological model for yield stress material. The modified thixotropic viscoplastic stress w.r.t. the structure parameter is integrated in quasi-Newtonian manner into the generalized Navier-Stokes equations and the evolution equation for the structure parameter constitutes the main core of full set of modeling equations, which are creditable as the privilege answer to incorporate thixotropy phenomena. A fully coupled monolithic finite element approach has been exercised which manages the material internal micro structure parameter, velocity, and pressure fields simultaneously. The nonlinearity of the corresponding problem, related to the dependency of the diffusive stress on the material parameters and the nonlinear structure parameter models on the other hand, 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 form a typical saddle-point problem which is solved using a geometrical multigrid method with a Vanka-like smoother taking into account a stable FEM approximation pair for velocity and pressure with discontinuous linear pressure and biquadratic velocity spaces. We examine the accuracy, robustness and efficiency of the Newton-Multigrid FEM solver throughout the solution of thixotropic viscoplastic flow problems in Couette device and in 4:1 contraction.
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