Efficient FEM solver for quasi-Newtonian flow problems with application to granular materials

Abstract

This thesis is concerned with new numerical and algorithmic tools for flows with pressure and shear dependent viscosity together with the necessary background of the generalized Navier-Stokes equations. In general the viscosity of a material can be constant, e.g. water and this kind of fluid is called as Newtonian fluid. However the flow can be complicated for quasi-Newtonian fluid, where the viscosity can depend on some physical quantity. For example, the viscosity of Bingham fluid is a function of the shear rate. Moreover even further complications can arise when the dependencies of both shear rate and pressure occur for the viscosity as in the case of the granular materials, e.g. Poliquen model. The Navier-Stokes equations in primitive variables (velocity-pressure) are regarded as the privilege answer to incorporate these phenomena. The modification of the viscous stresses leads to generalized Navier-Stokes equations extending the range of their validity to such flow. The resulting equations are mathematically more complex than the Navier-Stokes equations and several problems arise from the numerical point of view. Firstly, the difficulty of approximating incompressible velocity fields and secondly, poor conditioning and possible lack of differentiability of the involved nonlinear functions due to the material laws. The difficulty related to the approximation of incompressible velocity fields is treated by applying the conforming Stokes element Q2/P1 and the lack of differentiability is taken care of by regularization. Then the continuous Newton method as linearization technique is applied and the method consists of working directly on the variational integrals. Next the corresponding continuous Jacobian operators are derived and consequently a convergence rate of the nonlinear iterations independent of the mesh refinement is achieved. This continuous approach is advantageous: Firstly the explicit accessibility of the Jacobian allows a robust method with respect to the starting guess and secondly it avoids the delicate task of choosing the step-length which is required for divided differences approaches. We denote the full Jacobian matrix on the discrete level by A and separate it into two parts: A1 and A2 corresponding to Fixed point and Newton method respectively. A fundamental issue for the continuous Newton method arises when the problem is not ready for it at the initial state due to the poor condition of the 'bad-part' A2 of the Jacobian. Although the Newton method is popular for its local quadratic convergence behavior, however the solver may show unpredictable and undesirable divergent behavior if A2 is poor conditioned. This particular difficulty is handled by our Adaptive Newton method, where we introduce a charateristic function f(Qn), which depends solely on the relative residual change Qn and controls the weighing parameter δn for the 'bad-part' A2 resulting in the swinging back and forth of the solver between Fixed point and Newton state. Finally the new Adaptive Newton method is validated for the Bingham fluid for the benchmark geometry Flow around cylinder and a test case of 2D Couette flow for (modified) Poliquen model having the scope of real world applications is studied to fulfill the objective need of performance.