J. Durany

ON A DOUBLY NONLINEAR PARABOLIC OBSTACLE PROBLEM MODELLING ICE SHEET DYNAMICS

This paper deals with the weak formulation of a free (moving) boundary problem arising in theoretical glaciology. Considering shallow ice sheet flow, we present the mathematical analysis and the numerical solution of the second order nonlinear degenerate parabolic equation modelling, in the isothermal case, the ice sheet non-Newtonian dynamics. An obstacle problem is then deduced and analyzed. The existence of a free boundary generated by the support of the solution is proved and its location and evolution are qualitatively described by using a comparison principle and an energy method. Then the solutions are numerically computed with a method of characteristics and a duality algorithm to deal with the resulting variational inequalities. The weak framework we introduce and its analysis (both qualitative and numerical) are not restricted to the simple physics of the ice sheet model we consider nor to the model dimension; they can be successfully applied to more realistic and sophisticated models related to other geophysical settings.

Numerical simulation of a lubricated hertzian contact problem under imposed load

Abstract A lot of lubrication problems can be reduced to a contact between an elastic sphere and a rigid plane which leads to a Reynolds–Hertz nonlinear coupled model, where the cavitation phenomenon, the lubricant piezoviscosity and the balance between an imposed load on the device and the hydrodynamic load, must be taken into account. Last aspects add some difficulties: a new piezoviscous nonlinearity and a nonlocal constraint on the main unknown. In this paper, the proposed coupled problem is modelled, normalized and numerically solved. Thus, a numerical scheme which combines fixed point algorithms, the method of characteristics, duality techniques and finite element approximations is developed. Finally, when using a numerical flux conservation test, some computed results for real data are presented.

Numerical computation of free boundary problems in elastohydrodynamic lubrication

Abstract An alternative algorithm has been developed for computing the behavior of thin fluid films in two elastohydrodynamic lubrication problems. The presence of elasticity, lubrication, and cavitation leads to a nonlinear coupled system of partial differential equations. The hydrodynamic part of both problems is governed by the well-known Reynolds equation combined with the cavitation model of Elrod-Adams, which motivates the appearance of a free boundary. Elastic deformations are taken into account by means of the Hertz equation in rolling ball contact problems or the elastic hinged plate biharmonic equation in the case of journal-bearing devices with thin bearing. A numerical method decoupling the hydrodynamic part of the problem and the elastic one is presented. This method also involves an upwind scheme to discretize the lubrication model and finite element approximations.

Numerical approach of temperature distribution in a free boundary model for polythermal ice sheets

Summary. A numerical method for solving the thermal subproblem appearing in the modelization of polythermal ice sheets is described. This thermal problem mainly involves three nonlinearities: a reaction term due to the viscous dissipation, a Signorini boundary condition associated to the geothermic flux and an enthalpy term issued from the two phase Stefan formulation of the polythermal regime. The stationary temperature is obtained as the limit of an evolutive problem which is discretized in time with an upwind characteristics scheme and in space with finite elements. The nonlinearities are solved either by Newton-Raphson method or by duality techniques applied to maximal monotone operators. The application of the algorithms provides the dimensionless temperature distribution approximation and allows to identify the cold and temperate ice regions.

NUMERICAL APPROACH OF THERMOMECHANICAL COUPLED PROBLEMS WITH MOVING BOUNDARIES IN THEORETICAL GLACIOLOGY

In this work we propose a new numerical method for solving a thermomechanical coupled problem which arises as a mathematical model for the evolution of ice sheets profiles and the corresponding temperatures. From the accumulation–ablation ratio, the atmospheric temperature and the geothermic flux, the ice sheet profile is the solution of a moving boundary problem governed by a nonlinear convection–diffusion equation. Moreover, the mathematical model which governs the temperature distribution involves three nonlinearities: a reaction term due to the viscous dissipation, a Signorini boundary condition associated to the geothermic flux and an enthalpy term issued from the two-phase Stefan formulation of the polythermal regime. The problems are discretized in time with upwind characteristics schemes and in space with finite elements. The nonlinearities are solved either by fixed point methods or by duality techniques. Several numerical simulation examples involving real data sets issued from the Antartic ice sheet are shown.

Numerical computation of ice sheet profiles with free boundary models

Abstract The aim of this work is to develop a numerical method to solve a moving boundary problem governed by a time dependent nonlinear convection–diffusion equation. The mathematical formulation can be framed as a nonlinear parabolic complementarity problem. The model has recently been used to compute ice sheet profiles in theoretical glaciology. After describing the mathematical model of the ice sheet motion and the corresponding dimensionless equations, the proposed numerical method involves an upwind scheme for time semidiscretization, fixed point method for the nonlinear diffusion term, finite elements approximation in space and a duality type algorithm for solving the obstacle like problem at each step. Finally, several numerical simulation examples involving real data sets issued from the Antarctic ice sheet are shown.

Finite elements numerical solution of a coupled profile-velocity-temperature shallow ice sheet approximation model

This work deals with the numerical solution of a complex mathematical model arising in theoretical glaciology. The global moving boundary problem governs thermomechanical processes jointly with ice sheet hydrodynamics. One major novelty is the inclusion of the ice velocity field computation in the framework of the shallow ice model so that it can be coupled with profile and temperature equations. Moreover, the proposed basal velocity and shear stress laws allow the integration of basal sliding effects in the global model. Both features were not taking into account in a previous paper (Math. Model. Methods Appl. Sci. 12 (2) (2002) 229) and provide more realistic convective terms and more complete Signorini boundary conditions for the thermal problem. In the proposed numerical algorithm, one- and two-dimensional piecewise linear Lagrange finite elements in space and a semi-implicit upwinding scheme in time are combined with duality and Newtons methods for nonlinearities. A simulation example involving real data issued from Antarctic shows the temperature, profile and velocity qualitative behaviour as well as the free boundaries and basal effects.

Original article: Some aspects of lubrication in heavy regimes: Thermal effects, stability and turbulence

In this work, a combination of numerical methods applied to thermohydrodynamic lubrication problems with cavitation is presented. It should be emphasized the difficulty of the nonlinear mathematical coupled model involving a free boundary problem, but also the simplicity of the algorithms employed to solve it. So, finite element discretizations for the hydrodynamic and thermal equations combined with upwind techniques for the convection terms and duality methods for nonlinear features are proposed. Additionally, a model describing the movement of the shaft is provided. Considering the shaft as a rigid body this model will consist of an ODE system relating acceleration of the center of gravity and external and pressure loads. The numerical experiments of mechanical stability try to clarify the position of the neutral stability curve. Finally, a rotating machine for ship propulsion involving both axial and radial bearings operating with nonconventional lubricants (seawater to avoid environmental pollution) is analyzed by using laminar and turbulent inertial flows.

GLANUSIT: A software toolbox for the numerical simulation of large ice masses evolution

The here presented GLAciology NUmerical SImulation Toolbox (GLANUSIT) is a software application which provides a user friendly environment for the numerical simulation of large ice masses evolution. The graphical user interface has been developed in MATLAB while the core of GLANUSIT contains the original FORTRAN codes, which develop the specific numerical methods for the solution of the complex shallow ice model. This highly nonlinear model governs the coupled thermodynamical and hydrodynamical processes. The global algorithm mainly consists on a fixed point iteration between the different subproblems. The numerical solution of each subproblem requires specific techniques, which are not common in present software packages, as for example the part of moving boundaries solvers included in the code. Finally, a practical case study with real data is presented.

Numerical methods for a fixed domain formulation of the glacier profile problem with alternative boundary conditions

In this paper we develop a set of numerical techniques for the simulation of the profile evolution of a valley glacier in the framework of isothermal shallow ice approximation models. The different mathematical formulations are given in terms of a highly nonlinear parabolic equation. A first nonlinearity comes from the free boundary problem associated with the unknown basal extension of the glacier region. This feature is treated using a fixed domain complementarity formulation which is solved numerically by a duality method. The nonlinear diffusive term is explicitly treated in the time marching scheme. A convection dominated problem arises, so a characteristic scheme is proposed for the time discretization, while piecewise linear finite elements are used for the spatial discretization. The presence of infinite slopes in polar regimes motivates an alternative formulation based on a prescribed flux boundary condition at the head of the glacier instead a homogeneous Dirichlet one. Finally, several numerical examples illustrate the performance of the proposed methods.