An assumed-gradient finite element method for the level set equation

Abstract

SUMMARYThe level set equation is a non-linear advection equation, and standard finite-element and finitedifference strategies typically employ spatial stabilization techniques to suppress spurious oscillations in the numerical solution. We recast the level set equation in a simpler form by assuming that the level set function remains a signed distance to the front/interface being captured. As with the original level set equation, the use of an extensional velocity helps maintain this signed-distance function. For some interface-evolution problems, this approach reduces the original level set equation to an ordinary differential equation that is almost trivial to solve. Further, we find that sufficient accuracy is available through a standard Galerkin formulation without any stabilization or discontinuity-capturing terms. Several numerical experiments are conducted to assess the ability of the proposed assumed-gradient level set method to capture the correct solution, particularly in the presence of discontinuities in the extensional velocity or level-set gradient. We examine the convergence properties of the method and its performance in problems where the simplified level set equation takes the form of a Hamilton-Jacobi equation with convex/non-convex Hamiltonian. Importantly, discretizations based on structured and unstructured finite-element meshes of bilinear quadrilateral and linear triangular elements are shown to perform equally well.

Advances in the numerical treatment of grain-boundary migration: Coupling with mass transport and mechanics

“…Re-initialization should therefore be used judiciously. This issue is examined in detail by Mourad et al [17]. An alternative re-initialization procedure designed to minimize this type of error is described in detail in [35,36].…”

“…An alternative re-initialization procedure designed to minimize this type of error is described in detail in [35,36]. It must be noted however that, in some cases including curvature-driven migration, quadratic convergence in L 2 is achieved with the level-set update formula (41), and importantly, this optimal convergence rate is preserved by the present re-initialization scheme (see [17,37] for details). …”

“…Mourad et al [17] tested the resulting level-set formulation extensively and concluded that it is both accurate and robust despite its remarkable simplicity (for some interface-evolution problems, this approach reduces the original level-set equation, a nonlinear hyperbolic PDE, to an ODE that is almost trivial to solve). They conducted several numerical experiments to assess the ability of the simplified level-set scheme to capture the correct solution, particularly in the presence of discontinuities in the extensional velocity and/or in the gradient of the level-set function.…”

Estudio de las singularidades de frente de grieta y de esquina en grietas tridimensionales mediante el método de los elementos finitos extendido

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