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

  • 제21권3호
  • /
  • Pages.233-238
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  • 2008
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  • 1229-3059(pISSN)
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  • 2287-2302(eISSN)
한국전산구조공학회 (Computational Structural Engineering Institute of Korea)
권호 표지

등기하 해석법을 이용한 형상 최적설계

Shape Design Optimization Using Isogeometric Analysis

  • 하승현 (서울대학교 조선해양공학과) ;
  • 조선호 (서울대학교 조선해양공학과 및 RIMSE)
  • 발행 : 2008.06.30
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초록

본 논문에서는 등기하 해석법을 이용하여 선형 탄성문제에 대한 형상 최적설계 기법을 개발하였다. 실용적인 공학문제에 대한 많은 최적설계 문제에서는 초기의 데이터가 CAD 모델로부터 주어지는 경우가 많다. 그러나 대부분의 설계 최적화 도구들은 유한요소법에 기초하고 있기 때문에 설계자는 이에 앞서 CAD 데이터를 유한요소 데이터로 변환해야 한다. 이 변환과정에서 기하 모델의 근사화에 따른 수치적 오류가 발생하게 되고, 이는 응답 해석뿐만 아니라 설계민감도 해석에 있어서도 정확도 문제를 발생시킨다. 이러한 점에서 등기하 해석법은 형상 최적설계에 있어서 유망한 방법론 중 하나가 될 수 있다. 등기하 해석법의 핵심은 해석에 사용되는 기저 함수와 기하 모델을 구성하는 함수가 정확히 일치한다는 것이다. 이러한 기하학적으로 정확한 모델은 설계민감도 해석 및 형상 최적설계에 있어서도 사용된다. 이로 인해 높은 정확도의 설계민감도를 얻을 수 있으며, 이는 설계구배 기반의 최적화에 있어서 매우 중요하게 작용한다. 수치 예제를 통하여 본 논문에서 제시된 등기하 해석 기반의 형상 최적설계 방법론이 타당함을 확인하였다.

In this paper, a shape design optimization method for linearly elastic problems is developed using isogeometric approach. In many design optimization problems for practical engineering models, initial raw data usually come from a CAD modeler. Then, designers should convert the CAD data into finite element mesh data since most of conventional design optimization tools are based on finite element analysis. During this conversion, there are some numerical errors due to geometric approximation, which causes accuracy problems in response as well as design sensitivity analyses. As a remedy for this phenomenon, the isogeometric analysis method can be one of the promising approaches for the shape design optimization. The main idea of isogeometric approach is that the basis functions used in analysis is exactly the same as the ones representing the geometry. This geometrically exact model can be used in the shape sensitivity analysis and design optimization as well. Therefore the shape design sensitivity with high accuracy can be obtained, which is very essential for a gradient-based optimization. Through numerical examples, it is verified that the shape design optimization based on an isogeometic approach works well.

키워드

참고문헌

  1. 조선호, 정현승, 양영순 (2002) 기하학적 비선형 구조물의 설계민감도 해석 및 위상 최적설계, 한국전산구조공학회 02 봄 학술발표회 논문집, pp.335-342
  2. 하승현, 조선호 (2007) 등기하 해석법을 이용한 설계민감도 해석, 한국전산구조공학회논문집, 20(3), pp.339-345
  3. Arora, J.S., Lee, T.H., Cardoso, J.B. (1992) Structural shape design sensitivity analysis: Relationship between material derivative and control volume approaches, AIAA Journal, 30(6), pp.1638-1648 https://doi.org/10.2514/3.11112
  4. Azegami, H., Kaizu, S., Shimoda, M., Katamine, E. (1997) Irregularity of shape optimization problems and an improvement technique, Computer Aided Optimization Design of Structures, V, pp.309-326
  5. Braibant, V., Fluery, C. (1984) Shape optimal design using B-splines, Computer Methods in Applied Mechanics and Engineering, 44, pp.247-267 https://doi.org/10.1016/0045-7825(84)90132-4
  6. Cho, M., Roh, H.Y. (2003) Development of geometrically exact new shell elements based on general curvilinear coordinates, International Journal for Numerical Methods in Engineering, 56(1), pp.81-115 https://doi.org/10.1002/nme.546
  7. Cho, S., Ha, S.H. (2007) Shape design optimization of geometrically nonlinear structures using isogeometric analysis, 9th United States National Congress on Computational Mechanics, San Francisco, California, U.S.A., July 22-26
  8. Choi, K.K., Chang, K.H. (1994) A study of design velocity field computation for shape optimal design, Finite Elements in Analysis and Design, 15, pp. 317-341 https://doi.org/10.1016/0168-874X(94)90025-6
  9. Choi, K.K., Duan, W. (2000) Design sensitivity analysis and shape optimization of structural components with hyperelastic material, Computer Methods in Applied Mechanics and Engineering, 187, pp.219-243 https://doi.org/10.1016/S0045-7825(99)00121-8
  10. Choi, K.K., Kim, N.H. (2004) Structural Sensitivity Analysis and Optimization: Volume 1, Linear Systems & Volume 2, Nonlinear Systems and Applications, Springer, New York, NY
  11. Cottrell, J.A., Hughes, T.J.R., Reali, A. (2007) Studies of refinement and continuity in isogeometric structural analysis. Computer Methods in Applied Mechanics and Engineering, 196, pp.4160-4183 https://doi.org/10.1016/j.cma.2007.04.007
  12. Cottrell, J.A., Reali, A., Bazilevs, Y., Hughes, T.J.R. (2006) Isogeometric analysis of structural vibrations, Computer Methods in Applied Mechanics and Engineering, 195, pp.5257-5296 https://doi.org/10.1016/j.cma.2005.09.027
  13. Farin, G. (2002), Curves and Surfaces for CAGD: A Practical Guide, Academic Press
  14. Ha, S.H., Cho, S. (2007) Shape design optimization of structural problems based on isogeometric approach, 7th World Congress on Structural and Multidisciplinary Optimization, COEX Seoul, Korea, May 21-25
  15. Hughes, T.J.R., Cottrell, J.A., Bazilevs, Y. (2005) Isogeometric analysis: CAD, finite elements, NURBS, exact geometry and mesh refinement, Computer Methods in Applied Mechanics and Engineering, 194, pp.4135-4195 https://doi.org/10.1016/j.cma.2004.10.008
  16. Lindby, T., Santos, J.L.T. (1997) 2-D and 3-D shape optimization using mesh velocities to integrate analytical sensitivities with associative CAD, Structural Optimization, 13, pp.213-222 https://doi.org/10.1007/BF01197449
  17. Piegl, L., Tiller, W. (1997) The NURBS Book (Monographs in Visual Communication), second ed., Springer-Verlag, New York
  18. Rogers, D.F. (2001) An Introduction to NURBS With Historical Perspective. Academic Press, San Diego, CA
  19. Roh, H.Y., Cho, M. (2004) The application of geometrically exact shell elements to B-spline surfaces, Computer Methods in Applied Mechanics and Engineering, 193, pp.2261-2299 https://doi.org/10.1016/j.cma.2004.01.019
  20. Roh, H.Y., Cho, M. (2005) Integration of geometric design and mechanical analysis using B-spline functions on surface, International Journal for Numerical Methods in Engineering, 62(14), pp. 1927-1949 https://doi.org/10.1002/nme.1254