Barth, T.J., and Sethian, J.A., Numerical Schemes for the HamiltonJacobi and Level Set Equations on Triangulated Domains, submitted for publication, J. Comp. Phys., Sept. 1997  Home/Search   Document Details and Download   Summary   Related Articles   Check

....of O(N) than O(N log N) 20] Although the algorithm was presented in the context of a planar orthogonal grid, it can easily be extended to the case of triangulated domains in 2D or 3D by modifying the gradient approximation. Suitable upwind approximations are provided by Barth and Sethian [2] and Kimmel and Sethian [8] As shown by Kimmel and Sethian [9] this property can be exploited for the O(N log N) computation of (geodesic) Voronoi diagrams either in the plane or directly on the surface of a curved manifold. 3 Fast Marching farthest point sampling For simplicity, we first ....

....(non uniform) This means gradient approximations such as (8) for the planar case are generally no longer applicable and a suitable monotone and consistent finite di#erence approximation to the Eikonal equation converging to a weak solution needs to be used. Suitable approximations can be found in [2, 8]. The outline of FastFPS for triangulated domains is as follows with speed F ijk , F ijk F (i#x, j#y, k#z) from the sample points outwards using 10 Fast Marching and a finite di#erence approximation for triangulated domains [2, 8] March along the triangles and linearly interpolate the ....

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T. J. Barth and J. A. Sethian. Numerical Schemes for the Hamilton-Jacobi and Level Set Equations on Triangulated Domains. Journal of Computational Physics , 145(1):1--40, 1998.

....are however better discretized starting from their weak form. This was our motivation for using the nite element method for the multiphase ow problem. There are advantages also of using unstructured triangulated meshes for more complex geometries or local mesh re nement. Barth and Sethian [3] have developed numerical schemes, inspired bythetechniques used in nite di erence methods, for the level set equations on triangulated domains. One nite element based level set method for two phase ows is presented in [26] without including surface tension. In that paper, the author compares ....

T.J. Barth and J.A. Sethian. Numerical Schemes for the Hamilton-Jacobi and Level-Set equations on Triangulated Domains. Journal of Computational Physics, 145:1-40, 1998.

....algorithm is sufficient for pre computation of material depths at node points. For more complex objects, we use the fast marching level set method, which quickly computes distance values [35] By utilizing the tetrahedral mesh, we could use the finite element version of the fast marching method [4], which can compute distances even for self intersecting objects. This approach is left for future work. 5.5. Collision detection To accelerate collision detection between tetrahedral and triangular elements, a bounding volume tree is constructed for the tetrahedral mesh. A node of the tree ....

Barth, T.J., and Sethian, J.A., Numerical schemes for the Hamilton-Jacobi and level set equations on triangulated domains. J. Computational Physics 145(1), pp. 1-40, September 1998.

.... many numerical methods have been developed to accurately capture solutions of HamiltonJacobi equations with discontinuous gradients: see [9, 34] for nite di erence schemes of upwind type (see also [28] 1, 26] for nite volume schemes; 36, 37] for ENO schemes; 32, 30] for central schemes; [4, 19] for nite element methods; 20] for relaxation schemes; and [23] for front tracking methods. Using operator splitting, it is also possible to use homogeneous Hamilton Jacobi solvers as building blocks in numerical methods for non homogeneous problems. In the present context, operator splitting ....

T. J. Barth and J. A. Sethian. Numerical schemes for the Hamilton-Jacobi and level set equations on triangulated domains. J. Comput. Phys., 145(1):1-40, 1998.

....Island 02912 (shu cfm.brown.edu) 1 2 CHANGQING HU AND CHI WANG SHU higher order finite volume schemes face the problem of reconstruction on arbitrary triangulation, which is quite complicated. See, e.g. 1] More recently, a continuous finite element method for solving (1. 1) is proposed in [3]. The Runge Kutta discontinuous Galerkin method [6, 7, 8, 9, 10] is a method devised to numerically solve the conservation law (CL) u t f 1 (u) x1 Delta Delta Delta f d (u) xd = 0; u(x; 0) u 0 (x) 1.2) The method has the following attractive properties: ffl It can be designed for ....

T. Barth and J. Sethian, Numerical schemes for the Hamilton-Jacobi and level set equations on triangulated domains, J. Comput. Phys., v145 (1998), pp.1--40. DISCONTINUOUS GALERKIN FOR HAMILTON-JACOBI EQUATIONS 23

....equations. Indeed, many well known shock capturing methods for conservation laws have been extended to Hamilton Jacobi equations, see [8, 35] for nite di erence schemes of upwind type (see also [29] 1, 27] for nite volume schemes; 38, 39, 19] for (W)ENO schemes; 33, 31] for central schemes; [2, 17] for nite element methods; and [20] for relaxation schemes. In contrast to shock capturing schemes just cited, we will in this paper be concerned with extending to Hamilton Jacobi equations (1) a so called front tracking method for conservation laws. The front tracking method was introduced by ....

T. J. Barth and J. A. Sethian. Numerical schemes for the Hamilton-Jacobi and level set equations on triangulated domains. J. Comput. Phys., 145(1):1-40, 1998.

....order ENO type finite volume scheme is developed in [37] However, higher order finite volume schemes face the problem of reconstruction on arbitrary triangulation, which is quite complicated. See, e.g. 1] More recently, a continuous finite element method for solving (4.1. 1) is proposed in [5]. The Runge Kutta discontinuous Galerkin (RKDG) method [11, 12, 13, 14, 15] is a method devised to numerically solve the conservation law (CL) 8 : u t f 1 (u) x 1 Delta Delta Delta f d (u) x d = 0; u(x; 0) u 0 (x) 4.1.2) The method has the following attractive properties: ....

T. Barth and J. Sethian, Numerical schemes for the Hamilton-Jacobi and level set equations on triangulated domains, Journal of Computational Physics, v145 (1998), pp.1--40.

.... many numerical methods have been developed to accurately capture solutions of HamiltonJacobi equations with discontinuous gradients: see [9, 34] for finite difference schemes of upwind type (see also [28] 1, 26] for finite volume schemes; 36, 37] for ENO schemes; 32, 30] for central schemes; [4, 19] for finite element methods; 20] for relaxation schemes; and [23] for front tracking methods. Using operator splitting, it is also possible to use homogeneous Hamilton Jacobi solvers as building blocks in numerical methods for non homogeneous problems. In the present context, operator splitting ....

T. J. Barth and J. A. Sethian. Numerical schemes for the Hamilton-Jacobi and level set equations on triangulated domains. J. Comput. Phys., 145(1):1--40, 1998.

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Barth, T.J., and Sethian, J.A., Numerical Schemes for the HamiltonJacobi and Level Set Equations on Triangulated Domains, submitted for publication, J. Comp. Phys., Sept. 1997

....The core of our approach is Sethian s Fast Marching Method, 22,19,20] which solves the Eikonal equation on a rectangular orthogonal mesh in O(M log M) steps, where M is the total number of grid points. Contingent upon the triangulated upwind and monotonic update schemes given by Barth and Sethian [1], this technique was extended to triangulated surfaces by Kimmel and Sethian in [11] The triangulated version of the Fast Marching Method has the same computational complexity, and solves the Eikonal equation on triangulated domains in O(M log M) steps, where M is the number of vertices. Using ....

T. Barth and J. A. Sethian. Numerical schemes for the Hamilton-Jacobi and level set equations on triangulated domains. in press, J. Comp. Phys., to appear, 1998.

....scheme from a traditional nite element setting. We also discuss the addition and numerical approximation of a mean curvature term. Finally, we review several forms of mesh adaptation, and present some numerical examples. Complete details about triangulated Hamilton Jacobi solvers may be found in [6]. Citation Information: J.A. Sethian, von Karman Institute Lecture Series, Computational Fluid Mechanics, 1998 1 Introduction Level set methods are numerical techniques for tracking propagating interfaces in many space dimensions. They are work in an arbitrary number of space dimensions with no ....

....with discontinuity capturing operators. We then present some details 3 of the numerical implementation, together with error analysis of the various schemes, aspects of mesh adaptation, and a few computational results. Complete details about triangulated Hamilton Jacobi solvers may be found in [6]. 4 2 Background Fundamentals We begin with a specialized form of the Hamilton Jacobi equation, namely u t H(ru) f(x) x; t) 2 R u(x; 0) u 0 (x) 1) We note that the Eikonal equation can also be modeled by dropping the time derivative term. Let T denote a triangulation set in R d ....

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Barth, T.J. and Sethian, J.A. Numerical Schemes for the HamiltonJacobi and Level Set Equations on Triangulated Domains, submitted for publication, J. Comp. Physics, Sept., 1997

....represented. Second, surfaces are often described by triangulated patches, and hence this discretization is natural. In [13] a Fast Marching Method on triangulated unstructured meshes was introduced, using the unstructured mesh methodology for level set methods developed by Barth and Sethian [2], and applied to the problem of constructing geodesic shortest paths on manifolds. Various examples were given there, and we follow that work closely in this presentation. For further details, see [13] 5.3.1 The update procedure As an introduction, recall the update procedure in the Fast ....

....before, and the heap structure to maintain a list of Trial points, this provides a method for executing the Fast Marching Method on this simple triangulation. 5.3. 3 Fast Marching Methods on triangulated domains Following the construction unstructured mesh upwind approximations to the gradient ([2]) we now to extend this idea to an arbitrary triangulation. Acute triangulations T Figure 20: Acute triangulation around center grid point. We start with an acute triangulation and consider the triangulation around the grid point given in Figure 20. A large number of triangles may 33 share ....

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Barth, T.J., and Sethian, J.A., Numerical Schemes for the HamiltonJacobi and Level Set Equations on Triangulated Domains, J. Comp. Phys., 145, 1, pp. 1-40, 1998.

....O ce of Energy Research under contract DE AC03 76SF00098, and the National Science Foundation and DARPA under grant DMS 8919074. approximation is consistent in that it produces the correct shortest path on an orthogonal grid. For details about Fast Marching Methods, see [8, 9] Barth and Sethian [2] have recently constructed operators for viscosity solutions to both the Eikonal equation and Hamilton Jacobi equations on arbitrary triangulated domains. These operators exploit the upwind nature to construct the correct entropy satisfying weak solutions. In this paper, we extend these ideas and ....

....such a simple triangulation. 4 Fast Marching Methods on Triangulated Domains Our goal now is to extend this idea to an arbitrary triangulation. Here, we are essentially following the construction of upwind approximations to the gradient on triangulated meshes developed by Barth and Sethian in [2]. 4.1 A Construction for Acute Triangulations We start with an acute triangulation, and consider the triangulation around the grid point given in Fig. 4. T Figure 4: Acute triangulation around center grid point. A large number of triangles may share the center vertex. Our procedure, motivated by ....

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Barth T., and Sethian, J.A., Numerical Schemes for the Hamilton-Jacobi and Level Set Equations on Triangulated Domains, submitted for publication, J. Comp. Phys., 1997.

.... to the gradient is due to Godunov, and was used, for example, by Rouy and Tourin [12] to solve the Eikonal equation, namely max(D x ij u; D x ij u; 0) 2 max(D y ij u; D y ij u; 0) 2 1=2 = F ij ; 7) Additional schemes for solving Hamilton Jacobi equations may be found in [11, 5]. The central idea behind the Fast Marching Method is to systematically advance the front in an upwind fashion to produce the solution u. The key idea is the observation that the upwind di erence structure of Equation (7) means that information propagates one way , that is, from smaller values of ....

Barth, T.J., and Sethian, J.A., Numerical Schemes for the Hamilton-Jacobi and Level Set Equations on Triangulated Domains, to appear, J. Comp. Physics, August, 1998.

....the next point in the update sweep. As such, the technique is a reminiscent of Dijkstra s method [4] however, the resulting approximation is consistent in that it produces the correct shortest path on an orthogonal grid. For details about Fast Marching Methods, see [9, 10] Barth and Sethian [2] have recently constructed operators for viscosity solutions to both the Eikonal equation and Hamilton Jacobi equations on arbitrary This work was supported in part by the Applied Mathematical Science subprogram of the Office of Energy Research, U.S. Department of Energy, under Contract Number ....

....such a simple triangulation. 4 Fast Marching Methods on Triangulated Domains Our goal now is to extend this idea to an arbitrary triangulation. Here, we are essentially following the construction of upwind approximations to the gradient on triangulated meshes developed by Barth and Sethian in [2]. 4.1 A Construction for Acute Triangulations We start with an acute triangulation, and consider the triangulation around the grid point given in Fig. 4. T Figure 4: Acute triangulation around center grid point. A large number of triangles may share the center vertex. Our procedure, motivated by ....

[Article contains additional citation context not shown here]

Barth T., and Sethian, J.A., Numerical Schemes for the Hamilton-Jacobi and Level Set Equations on Triangulated Domains, submitted for publication, J. Comp. Phys., 1997.

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Barth T.J., Sethian J.A. "Numerical Schemes for the Hamilton-Jacobi and Level Set Equations on Triangulated Domains." Journal of Computational Physics, vol. 145, 1--40, 1998

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Barth, T., and J. A. Sethian, Numerical schemes for the Hamilton-Jacobi and level set equations on triangulated domains, J. Comp. Phys. 145(1) (1998), 1--40.

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Timothy J. Barth and James A. Sethian. "Numerical schemes for the Hamilton-Jacobi and level set equations on triangulated domains." J. Comput. Phys., 145(1):1--40, 1998.

No context found.

T. Barth and J. Sethian, Numerical schemes for the Hamilton-Jacobi and level set equations on triangulated domains, Journal of Computational Physics, v145 (1998), pp.1--40.

No context found.

Timothy J. Barth and James A. Sethian. Numerical schemes for the Hamilton-Jacobi and level set equations on triangulated domains. J. Comput. Phys., 145(1):1--40, 1998.

No context found.

T. J. Barth and J. A. Sethian. Numerical Schemes for the Hamilton-Jacobi and Level Set Equations on Triangulated Domains. Journal of Computational Physics , 145(1):1--40, 1998.

No context found.

T. J. BARTH AND J. A. SETHIAN, Numerical schemes for the Hamilton-Jacobi and level set equations on triangulated domains, J. Comp. Phys., 145 (1998), pp. i 40.

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