Augmented Lagrangian for shallow viscoplastic flow with topography

Abstract

In this paper we have developed a robust numerical algorithm for the visco-plastic Saint-Venant model with topography. For the time discretization an implicit (backward) Euler scheme was used. To solve the resulting nonlinear equations, a four steps iterative algorithm was proposed. To handle the non-differentiability of the plastic terms an iterative decomposition-coordination formulation coupled with the augmented Lagrangian method was adopted. The proposed algorithm is consistent, i.e. if the convergence is achieved then the iterative solution satisfies the nonlinear system at each time iteration. The equations for the velocity field are discretized using the finite element method, while a discontinuous Galerkin method, with an upwind choice of the flux, is adopted for solving the hyperbolic equations that describe the evolution of the thickness. The algorithm permits to solve alternatively, at each iteration, the equations for the velocity field and for the thickness.

The iterative decomposition coordination formulation coupled with the augmented Lagrangian method works very well and no instabilities are present. The proposed algorithm has a very good convergence rate, with the exception of large Reynolds numbers ($\text{Re}\gg 1000$), not involved in the applications concerned by the shallow viscoplastic model. The discontinuous Galerkin technique assure the mass conservation of the shallow system. The model has the exact C-property for a plane bottom and an asymptotic C-property for a general topography.

Some boundary value problems were selected to analyze the robustness of the numerical algorithm and the predictive capabilities of the mechanical model. The comparison with an exact rigid flow solution illustrates the accuracy of the numerical scheme in handling the non-differentiability of the plastic terms. The influence of the mesh and of the time step are investigated for the flow of a Bingham fluid in a talweg. The role of the material cohesion in stopping a viscoplastic avalanche on a talweg with barrier was analyzed. Finally, the capacities of the model to describe the flow of a Bingham fluid on a valley from the broken wall of a reservoir situated upstream were investigated.

Introduction

Modeling avalanche formation of soils, snow, granular materials or other geomaterials, is a difficult task. Since the flow involves flowing zones and zones at rest (see [1], [29]), the material is best represented by a viscoplastic model. This material model, which uses an yield (flow/no flow) criterion to switch between a fluid behavior to a solid one, is associated to a non smooth mathematical problem. Moreover, since the problems of interest are fully three dimensional, the numerical integration of the associated boundary value problems is very complex and poses many challenges. That is why, reduced 2-D models, called also Saint-Venant models, are generally considered to model shallow flows. Such models are able to capture the principal features of the flow: onset, dynamic propagation and arrest.

One class of reduced (or Saint-Venant) models were deduced from a lubrication-style asymptotic approximation. To model the mud flow, lubrication models were introduced by Liu and Mei [22], [23] for two-dimensional (sheet) flow, by Balmforth et al. [3] for axisymmetric flow, and extended to three dimensions in [2], [4], [5], [13]. A second class of reduced (or Saint-Venant) models concern fluids which exhibit an effective slip at the bottom (free liquid threads and films [27], ice shelves and streams [24] or snow avalanches). The shear stresses, which are dominant in the lubrication models become small, while the extensional and in-plane shear stress becomes important [5], and the flow is plug-like across the film thickness. A depth integrated theory, obeying a Mohr–Coulomb type yield criterion, was introduced by Savage and Hutter [32], [33] and developed thereafter by many authors (see for instance [19], [25], [26], [29]). These models take into account the frictional dissipation through an anisotropy factor which depends on the friction angles. The importance of this anisotropy factor is still an open question and an accurate derivation of the equations is still lacking (see e.g. [28]). Moreover, these models cannot be used for fluids which exhibit cohesional properties, hence they exclude the classical Bingham model (Von-Mises plasticity).

Very recently, another depth integrated theory was obtained in [17] for fluids with a Drucker–Prager type yield criterion (which includes the Bingham model) flowing down inclined planes. The resulting shallow equations of this last theory have the same structure as the three dimensional ones. The 2-D (tangent) momentum balance law and the thickness evolution equation are completed with a “shallow constitutive equation” which links the in-plane averaged stresses to the in-plane rate of deformations.

Natural avalanches and debris flows are often associated with complicated mountain topologies, which makes the prediction very difficult. That is why a lot of studies attempt to include the bottom curvature effect into the classical Saint-Venant or Savage–Hutter [32] equations. One of the first study in this direction is perhaps due to Dressler [11]. Other generalizations concern channelized flows along talwegs [16], [36], [30], [35] (see also the review [31]) and flows on more general basal geometries [21], [7], [8]. Using a local base given by the bottom geometry and the associated differential operators, the extension of the Saint-Venant, obtained in [17] for plane slopes, was extended in [18] to the case of a general basal topography.

The main goal of this paper is to develop a robust numerical algorithm for the visco-plastic Saint-Venant model proposed in [18]. We will use an implicit (backward) Euler scheme for time discretization, which gives a coupled system of nonlinear equations for the velocities and thickness fields. At each time iteration, an iterative algorithm is developed to solve these nonlinear equations. Specifically, a mixed finite-element and discontinuous Galerkin strategy is proposed. The equations for the velocity field are discretized using the finite element method, while a discontinuous Galerkin method, with an upwind choice of the flux, is adopted to solve the hyperbolic equations that describe the evolution of the thickness.

To handle the non-differentiability of the plastic terms an iterative decomposition-coordination formulation coupled with the augmented Lagrangian method (see [15], [14], or more recently [10]) is adapted. Since for the “shallow constitutive equation” there is no co-axiality between the stress and the rate of deformation tensors, the original method, developed for the Bingham model [15], [14], cannot be used directly. This problem is specific to the general case (two dimensions) and it is not present in the one dimensional case, where the stress and rate of deformation are scalars (see the well balanced method proposed in [9]). To overcome this difficulty we have written here the shallow visco-plastic model as a constitutive law in three dimensions and then we have used the decomposition-coordination formulation. It is worth noting that this type of algorithm permits also to solve alternatively, at each iteration, the thickness equations.

This paper is organized as follows: in Section 2 the boundary-value problem for the viscoplastic Saint-Venant model with topography is stated. To do that, we have used a system of coordinates adapted to the geometry of the flow through a general parametric representation of the bottom surface. A system of coordinates, defined as the loci of points at constant distances along the normals of the bottom surface, is considered in Section 2.1 to describe the three dimensional domain occupied by the viscoplastic fluid. The boundary conditions and the geometric decomposition of the velocity vector and of the stress tensor are discussed in Section 2.2. The scaling assumptions of the shallow model are presented in Section 2.3, where a complete set of governing equations and boundary conditions is given. The expressions of the shallow momentum balance law and of the evolution equations for the thickness are given in terms of tangential differential operators given in Appendix. In Section 3 we describe the numerical strategy. After the time discretization (Section 3.1) we present in Section 3.2, the four steps of the proposed iterative algorithm. We prove that the iterative algorithm is consistent (i.e. it provides a solution if the convergence is achieved). In Section 4 we analyze some conservation properties of the numerical scheme. We prove that the algorithm has the exact mass conservation property and the asymptotic C-property. The robustness of the method is analyzed in Section 5. The comparison with an exact rigid flow solution illustrates the accuracy of the numerical scheme in handling the non-differentiability of the plastic terms. A boundary value problem (the flow of a Bingham fluid on a talweg) was selected to investigate the influence of the mesh and of the time step. In Section 6 we use the numerical method proposed here to investigate two shallow visco-plastic flows. Firstly, we analyze the role of barriers in stopping a viscoplastic avalanche on a talweg and then we investigate the flow of a Bingham fluid from a reservoir.