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

Hyun-Young Kim; Jun-Sik Kim

doi:10.7734/COSEIK.2020.33.5.279

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2020 Vol.33, Issue 5 Next Page

Research Paper

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

31 October 2020. pp. 279-286

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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.

Keywords

MLS difference method

euler–bernoulli beam

finite differential method

mixed variational theory

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.

키워드

MLS 차분법

혼합변분이론

고전 보 이론

References

Belytschko, T., Lu, Y.Y., Gu, L. (1994) Element-free Galerkin Methods, Int. J. Numer. Methods Eng., 37, pp.229~256. 10.1002/nme.1620370205

Krongauz, Y., Belytschko, T. (1997) A Petrov-Galerkin Diffuse Element Method(PG DEM) and Its Comparison to EFG, Comput. Mech., 19, pp.327~333. 10.1007/s004660050181

Lee, S.H., Yoon, Y.C. (2004) Meshfree Point Collocation Method for Elasticity and Crack Problems, Numer. Methods Eng., 61, pp.22~48. 10.1002/nme.1053

Liu, W.K., Jun, S., Zhang, Y.F. (1995) Reproducing Kernel Particle Methods, Int. J. Numer. Methods Fluid., 20, pp.1081~1106. 10.1002/fld.1650200824

Nayroles, B., Touzot, G., Villon, P. (1992) Generalizing the Finite Element Method: Diffuse Approximation and Diffuse Elements, Comput. Mech., 10, pp.307~318. 10.1007/BF00364252

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. 10.1016/S0045-7825(96)01088-2

Onate, E., Perazzo, F., Miquel, J. (2001) A Finite Point Method for Elasticity Problems, Comput. & Struct., 79, pp.2151~2163. 10.1016/S0045-7949(01)00067-0

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.

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. 10.7734/COSEIK.2012.25.2.139

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.

Information

Publisher :Computational Structural Engineering Institute of Korea
Publisher(Ko) :한국전산구조공학회
Journal Title :Journal of the Computational Structural Engineering Institute of Korea
Journal Title(Ko) :한국전산구조공학회 논문집
Volume : 33
No :5
Pages :279-286
Received Date : 2020-04-08
Revised Date : 2020-08-26
Accepted Date : 2020-08-27
DOI :https://doi.org/10.7734/COSEIK.2020.33.5.279

Journal of the Computational Structural Engineering Institute of Korea ISSN:1229-3059(Print) 2287-2302(Online) 한국전산구조공학회 논문집

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