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Level-set-based deformation methods for adaptive grids. (English) Zbl 0958.65100
A new method for generating adaptive moving grids is formulated based on physical quantities. Level set functions are used to construct the adaptive grids, which are solutions of the standard level set evolution equation with the Cartesian coordinates as initial values. The intersection points of the level sets of the evolving functions form a new grid at each time. The velocity vector in the evolution equation is chosen according to a monitor function and is equal to the node velocity.
A uniform grid is then deformed to a moving grid with desired cell volume distribution at each time. The method achieves precise control over the Jacobian determinant of the grid mapping as the traditional deformation method does. The new method is consistent with the level set approach to dynamical moving interface problems.
Reviewer: L.G.Vulkov (Russe)

MSC:
65M50 Mesh generation, refinement, and adaptive methods for the numerical solution of initial value and initial-boundary value problems involving PDEs
35K15 Initial value problems for second-order parabolic equations
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[1] Brackbill, J.U.; Saltzman, J.S., Adaptive zoning for singular problems in two dimensions, J. comput. phys., 46, 342, (1982) · Zbl 0489.76007
[2] Thompson, J.F.; Warsi, Z.U.A.; Mastin, C.W., Numerical grid generation, (1985) · Zbl 0598.65086
[3] Zegeling, Paul A., Moving grid methods, (1992)
[4] Knupp, P.; Steinberg, S., The fundamentals of grid generation, (1994) · Zbl 0855.65123
[5] Carey, G., Computational grid generation, (1997)
[6] Liseikin, V., Grid generation methods, (1999) · Zbl 0949.65098
[7] Castillo, J.; Steinberg, S.; Roache, P.J., Mathematical aspects of variational grid generation, J. comput. appl. math., 20, 127, (1987) · Zbl 0631.65118
[8] Anderson, D.A., Grid cell volume control with an adaptive grid generator, Appl. math. comput., 35, 35, (1990) · Zbl 0721.65068
[9] K. Miller, Recent results on finite element methods with moving nodes, in, Accuracy Estimates and Adaptive Methods in Finite Element Computations, edited by, Babuska, Zienkiewicz, Gago, and Oliveira, Wiley, New York, 1986, p, 325.
[10] Arney, D.; Flaherty, J., ACM trans. math. software, 16, 48, (1990)
[11] Hawken, D.F.; Gottlieb, J.J.; Hansen, J.S., Review of some adaptive node-movement techniques in finite-element and finite-difference solutions of partial difference equations, J. comput. phys., 95, 254, (1991) · Zbl 0733.65057
[12] Huang, W.; Ren, Y.; Russell, R., Moving mesh methods based on moving mesh partial differential equations, J. comput. phys., 113, 279, (1994) · Zbl 0807.65101
[13] Moser, J., Volume elements of a Riemann manifold, Trans. AMS, 120, 286, (1965) · Zbl 0141.19407
[14] Dacorogna, B.; Moser, J., On a PDE involving the Jacobian determinant, Ann. inst. H. Poincaré, 7, (1990)
[15] Liao, G.; Anderson, D., A new approach to grid generation, Applicable anal., 44, 285, (1992) · Zbl 0794.65085
[16] Liao, G.; Su, J., A moving grid method for (1+1) dimension, Appl. math. lett., 8, 47, (1995) · Zbl 0826.65086
[17] Semper, B.; Liao, G., A moving grid finite element method using grid deformation, Numer. methods pdes, 11, 603, (1995) · Zbl 0838.65093
[18] Bochev, P.; Liao, G.; dela Pena, G., Analysis and computation of adaptive moving grids by deformation, Numer. methods pdes, 12, 489, (1996) · Zbl 0856.65109
[19] Liu, F.; Ji, S.; Liao, G., An adaptive grid method with cell-volume control and its applications to Euler flow calculations, SIAM J. sci. comput., 20, 811, (1998)
[20] Liu, F.; Jameson, A., Multi-grid navier – stokes calculation for three-dimensional cascades, Aiaa j., 31, 1785, (1993) · Zbl 0800.76395
[21] Liu, F.; Zheng, A strongly coupled time-marching method for solving the navier – stokes and turbulance model equations with multigrid, J. comput. phys., 128, 289, (1996) · Zbl 0862.76064
[22] Liao, G., Proceedings of the 15th IAMCS world congress, 2, 155, (August 1997)
[23] G. Liao, and, G. dela Pena, A moving grid finite difference algorithm for PDEs, preprint.
[24] Sethian, J.A., Curvature flow and entropy conditions to grid generation, J. comput. phys., 115, 440, (1994) · Zbl 0837.65134
[25] D. Peng, B. Merriman, S. Osher, H. Zhao, and, M. Kang, A PDE based fast level set method, UCLA CAM Report 98-25 (1998). · Zbl 0964.76069
[26] Chen, S.; Merriman, B.; Osher, S.; Smereka, P., A simple level set method for solving Stefan problems, J. comput. phys., 134, 236, (1997) · Zbl 0889.65133
[27] Osher, S.; Sethian, J., Fronts propagating with curvature dependent speed: algorithms based on hamilton – jacobi formulations, J. comput. phys., 79, 12, (1988) · Zbl 0659.65132
[28] Chorin; Marsden, A mathematical introduction to fluid dynamics, (1993) · Zbl 0774.76001
[29] Meyer, R., Introduction to mathematical fluid dynamics, (1971) · Zbl 0225.76001
[30] Osher, S.; Shu, C.W., High order essentially non-oscillatory schemes for hamilton – jacobi equations, SIAM J. numer. anal., 28, 907, (1991) · Zbl 0736.65066
[31] Sussman, M.; Smereka, P.; Osher, S., A level set method for computing solutions to incompressible two-phase flow, J. comput. phys., 114, 146, (1994) · Zbl 0808.76077
[32] B. Merriman, R. Caflisch, and, S. Osher, Level set methods with an application to modelling the growth of thin films, Proc of 1997 Congress on Free Boundary Problems, Heraklion, Crete, Greece (1997), to appear. · Zbl 0929.35192
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