Level-set method

Summary

Level-set methods (LSM) are a conceptual framework for using level sets as a tool for numerical analysis of surfaces and shapes. The advantage of the level-set model is that one can perform numerical computations involving curves and surfaces on a fixed Cartesian grid without having to parameterize these objects (this is called the Eulerian approach). Also, the level-set method makes it very easy to follow shapes that change topology, for example, when a shape splits in two, develops holes, or the reverse of these operations. All these make the level-set method a great tool for modeling time-varying objects, like inflation of an airbag, or a drop of oil floating in water. The figure on the right illustrates several important ideas about the level-set method. In the upper-left corner we see a shape; that is, a bounded region with a well-behaved boundary. Below it, the red surface is the graph of a level set function \varphi determining this shape, and the flat blue r

Official source

https://en.wikipedia.org/wiki/Level-set_method

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In the present work, we investigate mathematical and numerical aspects of interior penalty finite element methods for free surface flows. We consider the incompressible Navier-Stokes equations with variable density and viscosity, combined with a front capturing model using the level set method. We formulate interior penalty finite element methods for both the Navier-Stokes equations and the level set advection equation. For the two-fluid Stokes equations, we propose and analyze an unfitted finite element scheme with interior penalty. Optimal a priori error estimates for the velocity and the pressure are proved in the energy norm. A preconditioning strategy with adaptive reuse of incomplete factorizations as preconditioners for Krylov subspace methods is introduced and applied for solving the linear systems. Different and complementary solutions for reducing the matrix assembly time and the memory consumption are proposed and tested, each of which is applicable in general in the context of either multiphase flow or interior penalty stabilization. As level set reinitialization method, we apply a combination of the interface local projection and a fast marching scheme. We provide for the latter a reformulation of the distance computation algorithm on unstructured simplicial meshes in any spatial dimension, allowing for both an efficient implementation and geometric insight. We present and discuss numerical solutions of reference problems for the one-fluid Navier-Stokes equations and for the level set advection problem. Solutions of benchmark problems in two and three dimensions involving one or two fluids are then approximated, and the results are compared to literature values. Finally, we describe software design techniques and abstractions for the efficient and general implementation of the applied methods.

EPFL2007

Simulation numérique du mouvement des fluides dans une cellule de Hall-Héroult

Sonia Pain

This work aims to investigate, by means of numerical simulation, the evolution of free surface flow in aluminium production cells, and especially the magnetohydrodynamic instabilities. We design a numerical method combining a level-set technique together with a finite element discretization, in order to solve the non-stationnary Navier-Stokes equations and the equation for the motion of the free surface. Our numerical experiments confirms results obtained by other authors and we study the evolution of instabilities calculated by a stationary program, ALUCELL.

EPFL2006

Mass preserving finite element implementations of the level set method

Daniele Antonio Di Pietro, Nicola Parolini

In the last two decades, the level set method has been extensively used for the numerical solution of interface problems in different domains. The basic idea is to embed the interface as the level set of a regular function. In this paper we focus on the numerical solution of interface advection equations appearing in free-surface fluid dynamics problems, where naive finite element implementations are unsatisfactory. As a matter of fact, practitioners in fluid dynamics often complain that the mass of each fluid component is not conserved, a phenomenon which is therefore often referred to as mass loss. In this paper we propose and compare two finite element implementations that cure this ill-behaviour without the need to resort to combined strategies (such as e.g. particle level set). The first relies on a discontinuous Galerkin discretization, which is known to give very good performance when facing hyperbolic problems; the second is a stabilized continuous FEM implementation based on the stabilization method presented in [N. Parolini, Computational fluid dynamics for Naval engineering applications, PhD thesis, Ecole Polytechnique Fédérale de Lausanne, CH-1015 Lausanne, June 2004], which is free from many of the problems that classical methods exhibit when applied to unsteady problems. [All rights reserved Elsevier]

2006