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

Computational Structural Engineering Institute of Korea (한국전산구조공학회)
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Dynamic Algorithm for Solid Problems using MLS Difference Method

MLS 차분법을 이용한 고체역학 문제의 동적해석

  • 윤영철 (명지전문대학 토목과) ;
  • 김경환 (연세대학교 토목환경공학과) ;
  • 이상호 (연세대학교 토목환경공학과)
  • Received : 2011.11.24
  • Accepted : 2012.03.26
  • Published : 2012.04.30
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Abstract

The MLS(Moving Least Squares) Difference Method is a numerical scheme that combines the MLS method of Meshfree method and Taylor expansion involving not numerical quadrature or mesh structure but only nodes. This paper presents an dynamic algorithm of MLS difference method for solving transient solid mechanics problems. The developed algorithm performs time integration by using Newmark method and directly discretizes strong forms. It is very convenient to increase the order of Taylor polynomial because derivative approximations are obtained by the Taylor series expanded by MLS method without real differentiation. The accuracy and efficiency of the dynamic algorithm are verified through numerical experiments. Numerical results converge very well to the closed-form solutions and show less oscillation and periodic error than FEM(Finite Element Method).

MLS(Moving Least Squares) 차분법은 무요소법의 이동최소제곱법과 Taylor 전개를 이용하여 요소망의 제약 및 수치 적분이 없이 절점만을 이용하여 미분방정식을 수치해석할 수 있는 방법이다. 본 연구에서는 고체역학 문제의 동적해석을 위하여 MLS 차분법의 시간이력해석 알고리즘을 제시한다. 개발된 알고리즘은 Newmark 방법으로 시간적분을 하였으며, 강형식을 그대로 이산화하여 해석을 수행했다. 이동최소제곱법을 이용해 Taylor 전개식을 근사하여 실제 미분계산없이 미분근사식을 얻기 때문에 고차까지 Taylor 다항식의 차수를 증가하는 것이 용이하다. 1차원과 2차원 수치예제들을 통하여 동적해석을 위한 MLS 차분법의 정확성과 효율성을 검증하였다. 수치결과들이 정확해에 잘 수렴하였으며, 유한요소법(FEM)의 해석결과와 비교하여 떨림현상(oscillation) 및 주기성(periodicity) 오차에 대해 보다 안정적인 모습을 보였다.

Keywords

References

  1. 윤영철, 김도완 (2009a) 확장된 이동최소제곱 유한차분법을 이용한 이동경계문제의 해석, 한국전산구조공학회 논문집, 22(4), pp.315-322.
  2. 윤영철, 김동조, 이상호 (2007a) 탄성균열해석을 위한 그리드 없는 유한차분법, 한국전산구조공학회 논문집, 20(3), pp.321-327.
  3. 윤영철, 김효진, 김동조, 윙 캠 리우, 테드 벨리치코, 이상호 (2007b) 이동최소제곱 유한차분법을 이용한 응력집중문제 해석(1): 고체문제의 정식화, 한국전산구조공학회 논문집, 20(4), pp.493-499.
  4. 윤영철, 이상호 (2009b) 계면경계를 갖는 포텐셜 문제 해석을 위한 내적확장된 이동최소제곱 유한차분법, 한국전산구조공학회 논문집, 22(5), pp.411-420.
  5. ANSYS (2006) ANSYS 11.0 Theory Reference.
  6. Bathe, K.J., Wilson, E.L. (1973) Stability and Accuracy Analysis of Direct Integration Methods, Earthquake Engineering and Structural Dynamics, 1, pp.283-291.
  7. Belytschko, T., Organ, D., Gerlach, C. (2000) Element-free Galerkin Methods for Dynamic Fracture in Concrete, Computer Methods in Applied Mechanics and Engineering, 187, pp.385-399. https://doi.org/10.1016/S0045-7825(00)80002-X
  8. Belytschko, T., Lu, Y.Y., Gu, L. (1994) Elementfree Galerkin Methods, International Journal for Numerical Methods in Engineering, 37, pp.229-256. https://doi.org/10.1002/nme.1620370205
  9. Belytschko, T., Tabbara M. (1996) Dynamic Fracture using Element-free Galerkin Methods, International Journal for Numerical Methods in Engineering, 39, pp.923-938. https://doi.org/10.1002/(SICI)1097-0207(19960330)39:6<923::AID-NME887>3.0.CO;2-W
  10. Clough, R.W., Penzien J. (1975) Dynamics of Structures, McGraw-Hill.
  11. Dominguez, J. (1993) Boundary Elements in Dynamics, Computational Mechanics Publications.
  12. Gu, Y.T., Liu, G.R. (2005) A Meshfree Weak-Strong (MWS) form Method for Time Dependent Problems, Computational Mechanics, 35(2), pp.134-145. https://doi.org/10.1007/s00466-004-0610-0
  13. Kim, D.W., Kim, Y.S. (2003) Point Collocation Methods using the Fast Moving Least-Square Reproducing Kernel Approximation, International Journal for Numerical Methods in Engineering, 56, pp.1445-1464. https://doi.org/10.1002/nme.618
  14. Lee, S.H., Yoon, Y.C. (2004) Meshfree Point Collocation Method for Elasticity and Crack Problems, International Journal for Numerical Methods in Engineering, 61, pp.22-48. https://doi.org/10.1002/nme.1053
  15. Liu, W.K., Jun, S., Zhang, Y. (1995) Reproducing Kernel Particle Methods, Computer Methods in Applied Mechanics and Engineering, 191, pp.1421-1438.
  16. Newmark, N.M. (1959) A Method of Computation for Structural Dynamics, Journal of the Engineering Mechanics Division, ASCE, 85, pp.67-94.
  17. Perez-Gavilan, J.J, Aliabadi, M.H. (2000) A Galerkin Boundary Element Formulation with Dual Reciprocity for Elastodynamics, International Journal for Numerical Methods in Engineering, 48, pp.1331-1344. https://doi.org/10.1002/1097-0207(20000730)48:9<1331::AID-NME949>3.0.CO;2-E
  18. Sadeghirad, A., Kani, I.M., Rahimian, M., Astaneh, A.V. (2009) A Numerical Approach Based on the Meshless Collocation Method in Elastodynamics. Acta Mechanica Sinica. 25, pp.857-870. https://doi.org/10.1007/s10409-009-0236-8
  19. Samaan, M.F., Rashed, Y.F. (2007) BEM for Transient 2D Elastodynamics using Multiquadric Functions, International Journal of Solids and Structures, 44, pp.8517-8531. https://doi.org/10.1016/j.ijsolstr.2007.06.023
  20. Timoshenko, S.P., Goodier, I.N. (1970) Theory of Elasticity, McGraw-Hil, 3rd Edition.
  21. Wen, P.H., Aliabadi, M.H. (2008) An Improved Meshless Collocation Method for Elastostatic and Elastodynamic Problems. Communications in Numerical Methods in Engineering, 24, pp.635-651.

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