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

Computational Structural Engineering Institute of Korea (한국전산구조공학회)
권호 표지
DOI QR코드

DOI QR Code

On the Improvement of the Accuracy of Higher Order Derivatives in the MLS(Moving Least Square) Difference Method via Mixed Formulation

MLS 차분법의 결정 변수에 따른 정확도 분석 및 혼합변분이론을 통한 미분근사 성능향상

  • Kim, Hyun-Young (Department of Mechanical System Engineering, Kumoh National Institute of Technology) ;
  • Kim, Jun-Sik (Department of Mechanical System Engineering, Kumoh National Institute of Technology)
  • 김현영 (금오공과대학교 기계시스템공학과 대학원) ;
  • 김준식 (금오공과대학교 기계시스템공학과)
  • Received : 2020.04.08
  • Accepted : 2020.08.27
  • Published : 2020.10.31
Download PDF
⟨ Previous Next ⟩

Abstract

In this study, we investigate the accuracy of higher order derivatives in the moving least square (MLS) difference method. An interpolation function is constructed by employing a Taylor series expansion via MLS approximation. The function is then applied to the mixed variational theorem in which the displacement and stress resultants are treated as independent variables. The higher order derivatives are evaluated by solving simply supported beams and cantilevers. The results are compared with the analytical solutions in terms of the order of polynomials, support size of the weighting function, and number of nodes. The accuracy of the higher order derivatives improves with the employment of the mean value theorem, especially for very high-order derivatives (e.g., above fourth-order derivatives), which are important in a classical asymptotic analysis.

본 연구에서는 점근해석 및 논로컬 이론에서 요구하는 4차 이상의 고차 미분근사를 수행하기 위하여 계방정식에 혼합변분이론을 적용하여 MLS 차분법으로부터 구해지는 고차 미분근사의 정확도를 큰 폭으로 향상시킨다. 또한, MLS 차분법에 존재하는 세 가지 조건변수에 따른 고차미분근사의 정확도를 비교·분석한다. 혼합변분이론의 합응력을 후처리하여 변위의 미분을 근사할 경우 기존의 변위장 기반 계방정식의 차분 결과에 비해 미분 차수가 2차 낮아진다. 해석 범위내 절점 수가 과도하게 많거나 기저 차수가 클 경우 MLS 차분법의 영향영역 내에서 과적합(overfitting)이 발생한다. 또한 영향영역이 최적 범위 이상으로 넓어질 경우 근사의 정확도가 떨어진다. 위 내용을 사인 하중을 받는 단순지지보 수치예제로부터 확인하였다.

Keywords

References

  1. Belytschko, T., Lu, Y.Y., Gu, L. (1994) Element-free Galerkin Methods, Int. J. Numer. Methods Eng., 37, pp.229-256. https://doi.org/10.1002/nme.1620370205
  2. Krongauz, Y., Belytschko, T. (1997) A Petrov-Galerkin Diffuse Element Method(PG DEM) and Its Comparison to EFG, Comput. Mech., 19, pp.327-333. https://doi.org/10.1007/s004660050181
  3. Lee, S.H., Yoon, Y.C. (2004) Meshfree Point Collocation Method for Elasticity and Crack Problems, Numer. Methods Eng., 61, pp.22-48. https://doi.org/10.1002/nme.1053
  4. Liu, W.K., Jun, S., Zhang, Y.F. (1995) Reproducing Kernel Particle Methods, Int. J. Numer. Methods Fluid., 20, pp.1081-1106. https://doi.org/10.1002/fld.1650200824
  5. Nayroles, B., Touzot, G., Villon, P. (1992) Generalizing the Finite Element Method: Diffuse Approximation and Diffuse Elements, Comput. Mech., 10, pp.307-318. https://doi.org/10.1007/BF00364252
  6. Onate, E., Idelsohn, S., Zienkiewicz, O.C., Taylor, R.L., Sacco, C. (1996) A Stabilized Finite Point Method for Analysis of Fluid Mechanics Problems, Comp. Methods Appl. Mech. & Eng., 139, pp.315-346. https://doi.org/10.1016/S0045-7825(96)01088-2
  7. Onate, E., Perazzo, F., Miquel, J. (2001) A Finite Point Method for Elasticity Problems, Comput. & Struct., 79, pp.2151-2163. https://doi.org/10.1016/S0045-7949(01)00067-0
  8. Yoon,Y.C., Kim,D.J., Lee, S.H. (2007) A Gridless Finite Difference Method for Elastic Crack Analysis, J. Comput. Struuct. Eng. Inst. Korea, 20(3), pp.321-327.
  9. Yoon. Y.C., Kim, K.H., Lee, S.H. (2012) Dynamic Algorithm for Solid Problems using MLS Difference Method, J. Comput. Struuct. Eng. Inst. Korea, 25(2), pp.139-148. https://doi.org/10.7734/COSEIK.2012.25.2.139
  10. Yoon. Y.C., Seo, C.B., Kim, M.W., Lee, S.H. (2005) Consistent Diffuse Derivative Approximation in Particle Methods for Weak and Strong Formulations (1): Mathematical Foundations and Discretizations, J. Korean Soc. Civil. Eng., 25(5), pp.907-913.