한국전산구조공학회논문집 (Journal of the Computational Structural Engineering Institute of Korea)

  • 제26권1호
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  • Pages.39-48
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  • 2013
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  • 1229-3059(pISSN)
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  • 2287-2302(eISSN)
한국전산구조공학회 (Computational Structural Engineering Institute of Korea)
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2차원 융해문제의 해석을 위한 이동최소제곱 차분법

Moving Least Squares Difference Method for the Analysis of 2-D Melting Problem

  • 투고 : 2012.08.28
  • 심사 : 2012.10.31
  • 발행 : 2013.02.28
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⟨ 이전 논문 다음 논문 ⟩

초록

본 논문은 기존의 1차원 Stefan 문제를 해석할 수 있는 이동최소제곱 차분법을 확장하여 복잡한 계면경계 형상을 갖는 2차원 문제에 적용할 수 있는 수치기법을 개발한다. 1차원 경우와 달리 2차원 영역에서 임의로 움직이는 이동경계의 위상변화를 효과적으로 모델링할 수 있는 기법을 제안했으며, 이동경계 모사시 절점만 사용하는 이동최소제곱 차분법의 강점을 그대로 살리면서 이동경계의 불연속 특이성과 kinetics 조건을 정확하게 만족시키는 이동최소제곱 미분근사식을 제시했다. 평형방정식은 implicit(음해)법으로 차분하여 수치 안정성을 확보했으며, 이동경계는 explicit(양해)법으로 update하여 계산효율성의 극대화했다. 몇 가지 수치예제를 통해 개발된 이동최소제곱 차분법이 다양한 계면경계 형상을 갖는 2차원 Stefan 문제를 정확하고 효율적으로 풀 수 있음을 검증했다.

This paper develops a 2-D moving least squares(MLS) difference method for Stefan problem by extending the 1-D version of the conventional method. Unlike to 1-D interfacial modeling, the complex topology change in 2-D domain due to arbitrarily moving boundary is successfully modelled. The MLS derivative approximation that drives the kinetics of moving boundary is derived while the strong merit of MLS Difference Method that utilizes only nodal computation is effectively conserved. The governing equations are differentiated by an implicit scheme for achieving numerical stability and the moving boundary is updated by an explicit scheme for maximizing numerical efficiency. Numerical experiments prove that the MLS Difference Method shows very good accuracy and efficiency in solving complex 2-D Stefan problems.

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참고문헌

  1. Belytschko, T., Lu, Y.Y., Gu, L. (1994) Element-Free Galerkin Methods, International Journal for Numerical Methods in Engineering, 37, pp.229-256. https://doi.org/10.1002/nme.1620370205
  2. Caldwell, J., Kwan, Y.Y. (2004) Numerical Methods for One-dimensional Stefan Problems, Communications in Numerical Methods in Engineering, 20, pp.535-545. https://doi.org/10.1002/cnm.691
  3. Chen, S., Merriman, B. Osher, S., Smereka, P. (1997) A Simple Level Set Method for Solving Stefan Problem, Journal of Computational Physics, 135, pp.8-29. https://doi.org/10.1006/jcph.1997.5721
  4. Chessa, J., Smolinski, P., Belytschko, T. (2002) The Extended Finite Element Method (XFEM) for Solidification Problems, International Journal for Numerical Methods in Engineering, 53, pp.1959-1977. https://doi.org/10.1002/nme.386
  5. Javierre, E., Vuik, C., Vermolen, F.J., van der Zwaag, S. (2006) A Comparison of Numerical Models for One-dimensional Stefan Problems, Journal of Computational and Applied Mathematics, 192, pp.445-459. https://doi.org/10.1016/j.cam.2005.04.062
  6. Juric D., Tryggvason, G. (1996) A Front-tracking Method for Dendritic Solidification, Journal of Computational Physics, 123, pp.127-148. https://doi.org/10.1006/jcph.1996.0011
  7. Kim, D.W., Yoon, Y.-C., Liu, W.K., Belytschko, T. (2007a) Extrinsic Meshfree Approximation Using Asymptotic Expansion for Interfacial Discontinuity of Derivative, Journal of Computational Physics, 221, pp.370-394. https://doi.org/10.1016/j.jcp.2006.06.023
  8. Kim, D.W., Yoon, Y.-C., Liu, W.K., Belytschko, T., Lee, S.-H. (2007b) Meshfree Collocation Method with Intrinsic Enrichment for Interface Problems, Computational Mechanics, 40(6), pp.1037-1052. https://doi.org/10.1007/s00466-007-0162-1
  9. LeVeque, R.J., Li, Z. (1994) The Immersed Interface Method for Elliptic Equations with Discontinuous Coefficients and Singular Sources, SIAM J. Numer. Anal. 31, pp.1019-1044. https://doi.org/10.1137/0731054
  10. Moes, N., Dolbow, J., Belytschko, T. (1999) A Finite Element Method for Crack Growth without Remeshing, International Journal for Numerical Methods in Engineering, 46(1), pp.131-150. https://doi.org/10.1002/(SICI)1097-0207(19990910)46:1<131::AID-NME726>3.0.CO;2-J
  11. Osher, S., Sethian, J.A. (1988) Fronts Propagating with Curvature-dependent Speed: Algorithms Based on Hamilton-Jacobi Formulations, Journal of Computational Physics, 79, pp.12-49 https://doi.org/10.1016/0021-9991(88)90002-2
  12. Tu, C., Peskin, C.S. (1992) Stability and Instability in the Computation of Flows with Moving Immersed Boundaries: A Comparison of Three Methods, SIAM J. SCi. Statist. Comput. 13, pp.1361-1376. https://doi.org/10.1137/0913077
  13. Voller, V.R., Swenson, J.B., Paola, C. (2004) An Analytical Solution for a Stefan Problem with Variable Latent Heat, International Journal of Heat and Mass Transfer, 47, pp.5387-5390. https://doi.org/10.1016/j.ijheatmasstransfer.2004.07.007
  14. Yoon, Y.-C., Kim, D.-J., Lee, S.-H. (2007a) A Gridless Finite Difference Method for Elastic Crack Analysis, J. Comput. Struct. Eng., 20(3), pp.321-327.
  15. Yoon, Y.-C., Kim, D.-W. (2007d) Heat Transfer Analysis of Bi-Material Problem with Interfacial Boundary Using Moving Least Squares Finite Difference Method, J. Comput. Struct. Eng., 20(6), pp.779-787.
  16. Yoon, Y.-C., Kim, D.-W. (2009) Analysis of Moving Boundary Problem Using Extended Moving Least Squares Finite Difference Method, J. Comput. Struct. Eng., 22(4), pp.315-322.
  17. Yoon, Y.-C., Kim, H.-J., Kim, D.-J. Liu, W.K., Belytschko, T., Lee, S.-H. (2007b) Analysis of Stress Concentration Problems Using Moving Least Squares Finite Difference Method (I): Formulation for Solid Mechanics Problem, J. Comput. Struct. Eng., 20(4), pp.493-499.
  18. Yoon, Y.-C., Kim, H.-J., Kim, D.-J. Liu, W.K., Belytschko, T., Lee, S.-H. (2007c) Analysis of Stress Concentration Problems Using Moving Least Squares Finite Difference Method (II): Application to crack and localization band Problems, J. Comput. Struct. Eng., 20(4), pp.501-507.
  19. Yoon, Y.-C., Noh, H.-C. (2011) Extended MLS Difference Method for Potential Problem with Weak and Strong Discontinuities, J. Comput. Struct. Eng., 24(5), pp.577-588.