Newton-Multigrid FEM Solver for the Simulation of Quasi-Newtonian Modeling of Thixotropic Flows

This paper is concerned with the application of Finite Element Methods (FEM) and NewtonMultigrid solvers to simulate thixotropic flows using quasi-Newtonian modeling. The thixotropy phenomena are introduced to yield stress material by taking into consideration the internal material microstructure using a structure parameter. Firstly, the viscoplastic stress is modified to include the thixotropy throughout 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. This is done simply by introducing a structure-parameter-dependent viscosity into the rheological model for yield stress material. The nonlinearity, related to the dependency of the diffusive term 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 solved using the geometrical multigrid with a Vanka-like linear smoother taking into account a stable FEM approximation pair for velocity and pressure with discontinuous pressure and biquadratic velocity spaces. We analyze the application of using the quasi-Newtonian modeling approach for thixotropic flows, and the accuracy, robustness and efficiency of the Newton-Multigrid FEM solver throughout the solution of the thixotropic flows using manufactured solutions in a channel and the prototypical configuration of thixotropic flows in Couette device

Extended One-Step Methods for Solving Delay- Differential Equations

Abstract: We discuss extended one-step methods of order three for the numerical solution of delay-differential equations. A convergence theorem and the numerical studies regarding the convergence factor of these methods are given. Also, we investigate the stability properties of these methods. The results of the theoretical studies are illustrated by numerical examples

Extended One-Step Methods for Solving Delay- Differential Equations

We discuss extended one-step methods of order three for the numerical solution of delay-differential equations. A convergence theorem and the numerical studies regarding the convergence factor of these methods are given. Also, we investigate the stability properties of these methods. The results of the theoretical studies are illustrated by numerical examples