Details
Title
The modified XFEM for solving problems of a phase change with natural convectionJournal title
Archive of Mechanical EngineeringYearbook
2019Volume
vol. 66Issue
No 3Affiliation
Stąpór, Paweł : Faculty of Management and Computer Modelling, Kielce University of Technology, Kielce, Poland.Authors
Keywords
phase change ; natural convection ; extended finite element method ; level set methodDivisions of PAS
Nauki TechniczneCoverage
273-294Publisher
Polish Academy of Sciences, Committee on Machine BuildingBibliography
[1] A. Faghri and Y. Zhang. Transport Phenomena in Multiphase Systems. Elsevier, 2006.[2] S.C. Gupta. The Classical Stefan Problem: Basic Concepts, Modelling and Analysis. Elsevier, 2003.
[3] V. Alexiades and S.D. Solomon. Mathematical Modeling of Melting and Freezing Processes. Hemisphere Publ. Co, Washington DC, 1993.
[4] O.C. Zienkiewicz, R.L. Taylor, and P. Nithiarasu. The Finite Element Method for Fluid Dynamics, 6th edition. Elsevier Butterworth-Heinemann, Burlington, 2005.
[5] K. Morgan. A numerical analysis of freezing and melting with convection. Computer Methods in Applied Mechanics and Engineering, 28(3):275–284, 1981. doi: 10.1016/0045-7825(81)90002-5.
[6] J. Mackerle. Finite elements and boundary elements applied in phase change, solidification and melting problems. A bibliography (1996–1998). Finite Elements in Analysis and Design, 32(3):203–211, 1999. doi: 10.1016/S0168-874X(99)00007-4.
[7] S. Wang, A. Faghri, and T.L. Bergman. A comprehensive numerical model for melting with natural convection. International Journal of Heat and Mass Transfer, 53(9-10):1986–2000, 2010. doi: 10.1016/j.ijheatmasstransfer.2009.12.057.
[8] G. Vidalain, L. Gosselin, and M. Lacroix. An enhanced thermal conduction model for the prediction of convection dominated solid–liquid phase change. International Journal of Heat and Mass Transfer, 52(7-8):1753–1760, 2009. doi: 10.1016/j.ijheatmasstransfer.2008.09.020.
[9] I. Danaila, R. Moglan, F. Hecht, and S. Le Masson. A Newton method with adaptive finite elements for solving phase-change problems with natural convection. Journal of Computational Physics, 274:826–840, 2014. doi: 10.1016/j.jcp.2014.06.036.
[10] J.M. Melenk and I. Babuska. The partition of unity finite element method: basic theory and application. Computer Methods in Applied Mechanics and Engineering. 139(1-4):289–314, 1996. doi: 10.1016/S0045-7825(96)01087-0.
[11] A. Cosimo, V. Fachinotti, and A. Cardona. An enrichment scheme for solidification problems. Computational Mechanics, 52(1):17–35, 2013. doi: 10.1007/s00466-012-0792-9.
[12] T. Belytschko and T. Black. Elastic crack growth in finite elements with minimal remeshing. International Journal for Numerical Methods in Engineering, 45(5):601–620, 1999. doi: 10.1002/(SICI)1097-0207(19990620)45:5601::AID-NME598>3.0.CO;2-S.
[13] R. Merle and J. Dolbow. Solving thermal and phase change problems with the eXtended finite element method. Computational Mechanics, 28(5):339–350, 2002. doi: 10.1007/s00466-002-0298-y.
[14] J. Chessa, P. Smolinski, and T. Belytschko. The extended finite element method (XFEM) for solidification problems. International Journal for Numerical Methods in Engineering, 53(8):1959–1977, 2002. doi: 10.1002/nme.386.
[15] P. Stapór. The XFEM for nonlinear thermal and phase change problems. International Journal of Numerical Methods for Heat & Fluid Flow, 25(2):400–421, 2015. doi: 10.1108/HFF-02-2014-0052.
[16] N. Zabaras, B. Ganapathysubramanian, and L. Tan. Modelling dendritic solidification with melt convection using the extended finite element method. Journal of Computational Physics, 218(1):200–227, 2006. doi: 10.1016/j.jcp.2006.02.002.
[17] P. Stapór. A two-dimensional simulation of solidification processes in materials with thermodependent properties using XFEM. International Journal of Numerical Methods for Heat & Fluid Flow, 26(6):1661–1683, 2016. doi: 10.1108/HFF-01-2015-0018.
[18] J. Chessa and T. Belytschko. An enriched finite element method and level sets for axisymmetric two-phase flow with surface tension. International Journal for Numerical Methods in Engineering, 58(13):2041–2064, 2003. doi: 10.1002/nme.946.
[19] J. Chessa and T. Belytschko. An extended finite element method for two-phase fluids. Journal of Applied Mechanics, 70(11):10–17, 2003. doi: 10.1115/1.1526599.
[20] M. Li, H. Chaouki, J. Robert, D. Ziegler, D. Martin, and M. Fafard. Numerical simulation of Stefan problem with ensuing melt flow through XFEM/level set method. Finite Elements in Analysis and Design, 148:13–26, 2018. doi: 10.1016/j.finel.2018.05.008.
[21] D. Martin, H. Chaouki, J. Robert, D. Ziegler, and M. Fafard. A XFEM phase change model with convection. Frontiers in Heat and Mass Transfer, 10:1-11, 2018. doi: 10.5098/hmt.10.18.
[22] S. Osher and J.A. Sethian. Fronts propagating with curvature dependent speed: Algorithms based on Hamilton–Jacobi formulations. Journal of Computational Physics, 79(1):12–49, 1988. doi: 10.1016/0021-9991(88)90002-2.
[23] M. Stolarska, D.L. Chopp, N. Möes, and T. Belytschko. Modelling crack growth by level sets in the extended finite element method. International Journal for Numerical Methods in Engineering, 51(8):943–960, 2001. doi: 10.1002/nme.201.
[24] M. Sussman, P. Smereka, and S. Osher. A level set approach for computing solutions to incompressible two-phase flow. Journal of Computational Physics, 114(1):146–159, 1994. doi: 10.1006/jcph.1994.1155.
[25] M.Y. Wang, X. Wang, and D. Guo. A level set method for structural topology optimization. Computer Methods in Applied Mechanics and Engineering, 192(1-2):227–246, 2003. doi: 10.1016/S0045-7825(02)00559-5.
[26] N. Peters. Turbulent Combustion. Cambridge University Press, Cambridge, 2000.
[27] Y.H. Tsai and S. Osher. Total variation and level set methods in image science. Acta Numerica, 14:509–573, 2005. doi: 10.1017/S0962492904000273.
[28] V. Alexiades and J.B. Drake. A weak formulation for phase-change problems with bulk movement due to unequal densities. In J.M. Chadam and H. Rasmussen editors, Free Boundary Problems Involving Solids, pages 82–87, CRC Press, 1993.
[29] S. Chen, B. Merriman, S. Osher, and P. Smereka. A simple level set method for solving Stefan problems. Journal of Computational Physics, 135(1):8–29, 1997. doi: 10.1006/jcph.1997.5721.
[30] H. Sauerland. An XFEM Based Sharp Interface Approach for Two-Phase and free-Surface Flows. Ph.D. Thesis, RWTH Aachen University, Aachen, Germany, 2013.
[31] J.E. Tarancòn, A.Vercher, E. Giner, and F.J. Fuenmayor. Enhanced blending elements forXFEM applied to linear elastic fracture mechanics. I nternational Journal for Numerical Methods in Engineering, 77(1):126–148, 2009. doi: 10.1002/nme.2402.
[32] T.P. Fries. A corrected XFEM approximation without problems in blending elements. International Journal for Numerical Methods in Engineering, 75(5):503–532, 2008. doi: 10.1002/nme.2259.
[33] N. Moës, M. Cloirec, P. Cartraud, and J.F. Remacle. A computational approach to handle complex microstructure geometries. Computer Methods in Applied Mechanics and Engineering, 192(28-30):3163–3177, 2003. doi: 10.1016/S0045-7825(03)00346-3.
[34] G. Zi and T. Belytschko. New crack-tip elements for XFEM and applications to cohesive cracks. International Journal for Numerical Methods in Engineering, 57(15):2221–2240, 2003. doi: 10.1002/nme.849.
[35] G. Ventura, E. Budyn, and T. Belytschko. Vector level sets for description of propagating cracks in finite elements. International Journal for Numerical Methods in Engineering, 58(10):1571–1592, 2003. doi: 10.1002/nme.829.
[36] P. Stąpór. An improved XFEM for the Poisson equation with discontinuous coefficients. Archive of Mechanical Engineering, 64(1):123–144, 2017. doi: 10.1515/meceng-2017-0008.
[37] H.G. Choi, H. Choi, and J.Y. Yoo. A fractional four-step finite element formulation of the unsteady incompressible Navier-Stokes equations using SUPG and linear equal-order element methods. Computer Methods in Applied Mechanics and Engineering, 143(3-4):333–348, 1997. doi: 10.1016/S0045-7825(96)01156-5.
[38] R. Codina. Pressure stability in fractional step finite element methods for incompressible flows. Journal of Computational Physics, 170(1):112–140, 2001. doi: 10.1006/jcph.2001.6725.
[39] Z. Chen. Finite Element Methods and Their Applications. Springer, 2005.
[40] T. Belytschko,W.K. Liu, and B. Moran. Nonlinear Finite Elements for Continua and Structures. Wiley, 2000.
[41] T.A. Kowalewski and M. Rebow. Freezing of water in differentially heated cubic cavity. International Journal of Computational Fluid Dynamics, 11(3-4):193–210, 1999. doi: 10.1080/10618569908940874.
[42] T. Michałek and T.A. Kowalewski. Simulations of the water freezing process – numerical benchmarks. Task Quarterly, 7(3):389–408, 2003.
[43] M. Giangi, T.A.Kowalewski, F. Stella, and E. Leonardi.Natural convection during ice formation: numerical simulation vs. experimental results. Computer Assisted Mechanics and Engineering Sciences, 7(3):321–342, 2000.
[44] P. Stąpór. An enhanced XFEM for the discontinuous Poisson problem. Archive of Mechanical Engineering, 66(1):25–37, 2019. doi: 10.24425/ame.2019.126369.
[45] Thermal-FluidCentral. Thermophysical Properties: Phase Change Materials, 2010 (last accessed January 14, 2016). https://thermalfluidscentral.org.
[46] M. Okada. Analysis of heat transfer during melting from a vertical wall. I nternational Journal of Heat and Mass Transfer, 27(11):2057–2066, 1984. doi: 10.1016/0017-9310(84)90192-3.
[47] Z. Ma and Y. Zhang. Solid velocity correction schemes for a temperature transforming model for convection phase change. International Journal of Numerical Methods for Heat & Fluid Flow, 16(2):204–225, 2006. doi: 10.1108/09615530610644271.