×
Beckett, G.; Mackenzie, J. A.; Robertson, M. L.

A moving mesh finite element method for the solution of two-dimensional Stefan problems. (English) Zbl 1040.65080

J. Comput. Phys. 168, No. 2, 500-518 (2001).
An algorithm to study enthalpy formulation of phase change problems, employing an \(r\)-adaptive moving mesh finite element method, is presented and applied to problems where the phase front is cusp shaped and where the interface changes topology.
Reviewer: S. K. Rangarajan (Yorktown Heights)

Cited in 42 Documents

MSC:

65M60 Finite element, Rayleigh-Ritz and Galerkin methods for initial value and initial-boundary value problems involving PDEs
65M50 Mesh generation, refinement, and adaptive methods for the numerical solution of initial value and initial-boundary value problems involving PDEs
35K05 Heat equation
35R35 Free boundary problems for PDEs
80A22 Stefan problems, phase changes, etc.
80M10 Finite element, Galerkin and related methods applied to problems in thermodynamics and heat transfer

Keywords:

phase change; moving mesh; finite element method; Stefan problems

Software:

NKA
Cite Review PDF
Full Text: DOI

References:

[1] Brackbill, J. U., An adaptive grid with direction control, J. Comput. Phys., 108, 38 (1993) · Zbl 0832.65132
[2] Brackbill, J. U.; Saltzman, J. S., Adaptive zoning for singular problems in two dimensions, J. Comput. Phys., 46, 342 (1982) · Zbl 0489.76007
[3] Budhia, H.; Kreith, F., Heat transfer with melting or freezing in a wedge, Int. J. Heat Mass Transfer, 16, 195 (1973) · Zbl 0261.76057
[4] Cao, W.; Huang, W.; Russell, R. D., An r-adaptive finite element method based upon moving mesh PDEs, J. Comput. Phys., 149, 221 (1999) · Zbl 0923.65062
[5] W. Cao, W. Huang, and, R. D. Russell, A study of monitor functions for two dimensional adaptive mesh generation, SIAM J. Sci. Comput. to appear.; W. Cao, W. Huang, and, R. D. Russell, A study of monitor functions for two dimensional adaptive mesh generation, SIAM J. Sci. Comput. to appear. · Zbl 0937.65104
[6] Carlson, N.; Miller, K., Design and application of a gradient-weighted moving finite element code. Part II. In two dimensions, SIAM J. Sci. Comput., 19, 798 (1998) · Zbl 0911.65088
[7] Chen, S.; Merriman, B.; Osher, S.; Smereka, P., A simple level set method for solving Stefan problems, J. Comput. Phys., 135, 8 (1997) · Zbl 0889.65133
[8] Ciarlet, P. G.; Raviart, P. A., Maximum principle and uniform convergence for the finite element method, Comput. Methods Appl. Mech. Eng., 2, 17 (1973) · Zbl 0251.65069
[9] Crank, J., Free and Moving Boundary Value Problems (1984) · Zbl 0547.35001
[10] Crivelli, L. A.; Idelsohn, S. R., A temperature-based finite element solution for phase-change problems, Int. J. Numer. Methods Eng., 23, 99 (1986) · Zbl 0584.65083
[11] Dvinsky, A. S., Adaptive grid generation from harmonic maps on Riemannian manifolds, J. Comput. Phys., 95, 450 (1991) · Zbl 0733.65074
[12] Egolf, P. W.; Manz, H., Theory and modeling of phase change materials with and without mushy regions, Int. J. Heat Mass Transfer, 37, 2917 (1994) · Zbl 0900.76681
[13] Z.-X. Gong, Y.-F. Zhang, and A. S. Mujumdar, An efficient finite element procedure for the solidification of enthalpy model of phase change heat conduction problems, in Computational Modelling of Free and Moving Boundary Problems. Vol 2. Heat Transfer, edited by L. C. Wrobel and C. A. Brebbia. WIT, Southhampton, 1991, pp. 93-104.; Z.-X. Gong, Y.-F. Zhang, and A. S. Mujumdar, An efficient finite element procedure for the solidification of enthalpy model of phase change heat conduction problems, in Computational Modelling of Free and Moving Boundary Problems. Vol 2. Heat Transfer, edited by L. C. Wrobel and C. A. Brebbia. WIT, Southhampton, 1991, pp. 93-104.
[14] Huang, W.; Russell, R. D., A high-dimensional moving mesh strategy, Appl. Numer. Math., 26, 63 (1997) · Zbl 0890.65101
[15] Huang, W.; Russell, R. D., Moving mesh strategy based on a gradient flow equation for two-dimensional problems, SIAM J. Sci. Comput., 20, 998 (1999) · Zbl 0956.76076
[16] Lang, X., Moving Mesh Control Volume Solution for Two-Dimensional Phase Change Problems in Enthalpy Formulation (1997)
[17] Lynch, D. R.; O’Neill, K., Continuously deforming finite elements for the solution of parabolic problems with and without phase, Int. J. Numer. Meth. Eng., 17, 81 (1981) · Zbl 0473.73076
[18] Mackenzie, J. A.; Robertson, M. L., The numerical solution of one-dimensional phase change problems using an adaptive moving mesh method, J. Comput. Phys., 161, 537 (2000) · Zbl 0965.65105
[19] Miller, K.; Miller, R. N., Moving finite elements. I, SIAM J. Numer. Anal., 18, 1019-1032 (1981) · Zbl 0518.65082
[20] Meyer, G. H., Multidimensional Stefan problems, SIAM J. Numer. Anal., 10, 522 (1973) · Zbl 0256.65054
[21] Nochetto, R. H., Finite elements for free and moving boundary problems, SERC Numerical Analysis Summer School, Lancaster University (1990) · Zbl 0721.65080
[22] Nochetto, R. H.; Paolini, M.; Verdi, C., An adaptive finite element method for two-phase Stefan problems in two space dimensions. Part II: Implementation and numerical experiments, SIAM J. Sci. Comput., 12, 1207 (1991) · Zbl 0733.65088
[23] Nochetto, R. H.; Schmidt, A.; Verdi, C., Error control for phase change problems, IVTAM Symposium on Variations of Domain and Free Boundary Problems, Paris (1997)
[24] Rose, M. E., An enthalpy scheme for Stefan problems in several dimensions, Appl. Numer. Math., 12, 229 (1993) · Zbl 0787.65094
[25] Viswanath, R.; Jaluria, Y., A comparison of different solution methodologies for melting and solidification problems in enclosures, Numer. Heat Transfer, 24, 77 (1993)
[26] Voller, V.; Cross, M., Accurate solutions of moving boundary problems using the enthalpy method, Int. J. Heat Mass Transfer, 24, 545 (1981) · Zbl 0461.76075
[27] Winslow, A., Numerical solution of the quasilinear Poisson equation in a nonuniform mesh, J. Comput. Phys., 2, 149 (1967)
[28] Zerroukat, M.; Chatwin, C. R., A finite-difference algorithm for multiple moving boundary value problems using real and virtual grid networks, J. Comput. Phys., 112, 298 (1994) · Zbl 0806.65131
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.