# Numerical Study of Viscoplastic Flows Using a Multigrid Initialization Algorithm
## 1 Introduction
The most popular regularization methods in the literature are proposed by [PERSON] and [PERSON] [7], and that by [PERSON] [8]. Many authors have extensively studied viscosity regularization methods for a range of different problems, including the following list [9; 10; 11; 12; 13; 14]. The other solution method in solving viscoplastic flows consists of using variational inequalities whose solutions are equivalent to the solutions of original problems. The underlying problem is solved as the minimization of shear rate or maximization of shear stress, which form the basis of the augmented Lagrangian method. The development of this approach is due to [PERSON] and [PERSON] [15], together with [PERSON] [16]. Numerically solving viscoplastic flows using regularization methods is typically faster and easier to implement than the augmented Lagrangian algorithm, and therefore it has been common to use the first strategy [17]. Understanding the flow behavior of materials with yield remains a challenging problem at both experimental and computational levels. Nevertheless, numerical resolution of viscoplastic flows has significantly improved over the last few decades. Several numerical methods were proposed for the flow analysis of such materials in various bounded domains. The governing equation set of incompressible flows provides, along with momentum equations, a coupled continuity equation that should be satisfied. While every velocity component appears in each momentum equation and in the continuity equation, there is evidently no equation for the pressure, which makes the formulation of an equation for pressure based on the governing flow equations non-trivial. The pressure-velocity coupling gives rise to a constraint in the solution--the applied pressure field in the momentum equations should verify continuity. Both issues associated with the pressure-velocity linkage and the further appearance of non-linearities in the set of equations have lead to a variety of iterative solution algorithms [18; 19]. Historically, iterative solution strategies were widely adopted for incompressible flows in the 1970s, notably through the development of the well-known segregated SIMPLE algorithm (semi-implicit method for pressure linked equations) [20]. Since its formulation, the SIMPLE algorithm has provided the basis for the development of a number of SIMPLE-like algorithms devoted to algorithm improvements, a review of which is reported in [21], whereas detailed descriptions are available in [22]. Although SIMPLE is an old algorithm, it is still very popular and has been used successfully in numerous studies to solve complex flow phenomena, with a recent example [13] involving non-Newtonian flow. However, the reported results of simple iterative techniques to solve flows of fluids with yield show low convergence rates and significant computational costs for the solutions in some cases. SIMPLE is a slowly converging algorithm, but its performance can be greatly enhanced by using it in a multigrid (MG) context [23; 24]. Multigrid techniques are implemented in conjunction with a linear solver using a sequence of coarsening and refining grids. The method is motivated by its capability to offer fast convergence by allowing all wavelengths of the algebraic error to decay uniformly fast [25]. The multigrid method was first adopted in the context of the finite volume methods by [PERSON] and [PERSON] [26] and [PERSON] et al. [27] to solve recirculating Newtonian flows on staggered and co-located grids, respectively. These studies have shown that the efficiency of calculating incompressible flows using implicit methods based on SIMPLE algorithm can be substantially improved using the multigrid method for outer iterations. However, implementing a multigrid algorithm is a non-trivial task that requires considerable thought and care. One of the challenges involves inter-grid transfers, which may be dependent on a dynamic criterion based on the residuals. However, an initial solution can be obtained with relative ease and lower computational cost.