Numerical Schemes for the Hamilton–Jacobi and Level Set Equations on Triangulated Domains

Barth, Timothy J.; Sethian, James A.

doi:10.1006/jcph.1998.6007

Cited by 236 publications

(198 citation statements)

References 26 publications

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“…Another approach to obtaining a "time" dependent H-J equation from the static H-J equation is using the so called paraxial formulation in which a preferred spatial direction is assumed in the characteristic propagation [21,17,29,36,37]. High order numerical schemes are well developed for the time dependent H-J equation on structured and unstructured meshes [34,25,51,24,33,7,26,31,35,1,3,4,6,8]; see a recent review on high order numerical methods for time dependent H-J equations by Shu [46]. Due to the finite speed of propagation and the CFL condition for the discrete time step size, the number of time steps has to be of the same order as that for one of the spatial dimensions so that the solution converges in the entire domain.…”

Section: Introductionmentioning

confidence: 99%

A second order discontinuous Galerkin fast sweeping method for Eikonal equations

Shu

Zhang

et al. 2008

Journal of Computational Physics

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Section: Introductionmentioning

confidence: 99%

A second order discontinuous Galerkin fast sweeping method for Eikonal equations

Shu

Zhang

et al. 2008

Journal of Computational Physics

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“…There is also a volume preserving version of the mean curvature flow, for which one subtracts the average of the mean curvature from the normal velocity. A plethora of analytical results and theories of weak solutions exists, for example, see [20,22,30,35,36,37,43,44,52,53,57,67], and for numerical solutions [1,9,13,14,15,60,69], to name but a few.The algorithm proposed below is a front-tracking boundary-integral method. It has the advantage that one does not have to differentiate across the front, as compared to a level-set approach.…”

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confidence: 99%

A numerical scheme for moving boundary problems that are gradient flows for the area functional

Mayer

2000

Eur. J. Appl. Math

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Many moving boundary problems that are driven in some way by the curvature of the free boundary are gradient flows for the area of the moving interface. Examples are the Mullins-Sekerka flow, the Hele-Shaw flow, flow by mean curvature, and flow by averaged mean curvature. The gradient flow structure suggests an implicit finite differences approach to compute numerical solutions. The proposed numerical scheme will allow to treat such free boundary problems in both IR 2 and IR 3 . The advantage of such an approach is the re-usability of much of the setup for all of the different problems.As an example of the method we will compute solutions to the averaged mean curvature flow that exhibit the formation of a singularity. IntroductionIn this paper we will study geometric evolution problems for surfaces driven by curvature. These moving boundary problems arise from models in physics and the material sciences, and they describe phase changes, or more generally, phenomena in fluid flow.The Hele-Shaw model (named after H.S. ) describes the pressure of two immiscible viscous fluids trapped between two parallel glass plates and has attracted considerable attention in the literature, both on the analytical side [23,27,28,29,33,38,51,54,63] and on the numerical side [2,6,7,8,12,25,26,49,50,65]. This list of references also encompasses work related to the one-sided Hele-Shaw model, which arises as a limit when the viscosity of one of the fluids approaches zero. Furthermore, the Hele-Shaw model has classically been considered on a bounded domain, but many of the references above also address the problem on an unbounded domain. We shall only consider the problem on all of IR n ; for the precise formulation see (4.4).The Mullins-Sekerka model was first proposed by (and much later named after) Mullins and Sekerka to study solidification of materials of negligible specific heat [58]. For a while this model was also called the Hele-Shaw model, leading to a somewhat confused literature. Pego [61], and then Alikakos, Bates, and Chen [3], and also Stoth [68], established this model as a singular limit of the Cahn-Hilliard equation [17], a fourth order partial differential equation modelling nucleation and coarsening phenomena in a melted binary alloy. The Mullins-Sekerka model is considered to be a good model to describe the stage of Ostwald ripening in phase transitions, which is the stage after the initial nucleation has 2 U. F. Mayer essentially been completed and where some particles grow at the cost of others in an effort to decrease interfacial energy (surface area). The literature focuses mainly on the modelling and analytical aspects of the problem [11,16,18,19,21,32,34,46,56,66,71,72] though there are numerical simulations for the two-dimensional case [10,73]. We shall consider the two-phase problem on all of IR n , see (4.3).The mean curvature flow is probably the most studied geometric moving boundary problem, it is in some way also the simplest: the normal velocity is equal to the mean curvature of the interface. This sha...

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“…The iso-/ surfaces are planes parallel to the plane (OxOy) and the iso-w surfaces are coaxial cylinders whose axes go through O and are supported by the line (Oz). The signed distance functions are updated and reorthogonalized by level set methods (see [24][25][26][27][28][29][30] for details on finite difference based level set methods).…”

Section: Discretized Equilibrium Equationsmentioning

confidence: 99%

Enriched finite elements and level sets for damage tolerance assessment of complex structures

Bordas

Moran

2006

Engineering Fracture Mechanics

153

116

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The extended finite element method (X-FEM) has recently emerged as an alternative to meshing/remeshing crack surfaces in computational fracture mechanics thanks to the concept of discontinuous and asymptotic partition of unity enrichment (PUM) of the standard finite element approximation spaces. Level set methods have been recently coupled with X-FEM to help track the crack geometry as it grows. However, little attention has been devoted to employing the X-FEM in real-world cases. This paper describes how X-FEM coupled with level set methods can be used to solve complex threedimensional industrial fracture mechanics problems through combination of an object-oriented (C++) research code and a commercial solid modeling/finite element package (EDS-PLM/I-DEAS Ò ). The paper briefly describes how object-oriented programming shows its advantages to efficiently implement the proposed methodology. Due to enrichment, the latter method allows for multiple crack growth scenarios to be analyzed with a minimal amount of remeshing. Additionally, the whole component contributes to the stiffness during the whole crack growth simulation. The use of level set methods permits the seamless merging of cracks with boundaries. To show the flexibility of the method, the latter is applied to damage tolerance analysis of a complex aircraft component.

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Numerical Schemes for the Hamilton–Jacobi and Level Set Equations on Triangulated Domains

Cited by 236 publications

References 26 publications

A second order discontinuous Galerkin fast sweeping method for Eikonal equations

A second order discontinuous Galerkin fast sweeping method for Eikonal equations

A numerical scheme for moving boundary problems that are gradient flows for the area functional

Enriched finite elements and level sets for damage tolerance assessment of complex structures

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