Journal of the Korean Society for Industrial and Applied Mathematics

  • 제25권4호
  • /
  • Pages.196-220
  • /
  • 2021
  • /
  • 1226-9433(pISSN)
  • /
  • 1229-0645(eISSN)
한국산업응용수학회 (Korean Society for Industrial and Applied Mathematics)
권호 표지
DOI QR코드

DOI QR Code

NOVEL GEOMETRIC PARAMETERIZATION SCHEME FOR THE CERTIFIED REDUCED BASIS ANALYSIS OF A SQUARE UNIT CELL

  • LE, SON HAI (DEPARTMENT OF AEROSPACE ENGINEERING, PUSAN NATIONAL UNIVERSITY) ;
  • KANG, SHINSEONG (DEPARTMENT OF AEROSPACE ENGINEERING, PUSAN NATIONAL UNIVERSITY) ;
  • PHAM, TRIET MINH (DEPARTMENT OF AEROSPACE ENGINEERING, PUSAN NATIONAL UNIVERSITY) ;
  • LEE, KYUNGHOON (DEPARTMENT OF AEROSPACE ENGINEERING, PUSAN NATIONAL UNIVERSITY)
  • 투고 : 2021.10.17
  • 심사 : 2021.12.21
  • 발행 : 2021.12.25
PDF 다운로드
⟨ 이전 논문 다음 논문 ⟩

초록

This study formulates a new geometric parameterization scheme to effectively address numerical analysis subject to the variation of the fiber radius of a square unit cell. In particular, the proposed mesh-morphing approach may lead to a parameterized weak form whose bilinear and linear forms are affine in the geometric parameter of interest, i.e. the fiber radius. As a result, we may certify the reduced basis analysis of a square unit cell model for any parameters in a predetermined parameter domain with a rigorous a posteriori error bound. To demonstrate the utility of the proposed geometric parameterization, we consider a two-dimensional, steady-state heat conduction analysis dependent on two parameters: a fiber radius and a thermal conductivity. For rapid yet rigorous a posteriori error evaluation, we estimate a lower bound of a coercivity constant via the min-θ method as well as the successive constraint method. Compared to the corresponding finite element analysis, the constructed reduced basis analysis may yield nearly the same solution at a computational speed about 29 times faster on average. In conclusion, the proposed geometric parameterization scheme is conducive for accurate yet efficient reduced basis analysis.

키워드

과제정보

The authors would like to express our gratitude to Dr. Francesco Ballarin at Catholic University of the Sacred Heart for his advice on utilizing RBniCS for certified reduced basis analysis. This work was supported by the 2-year Research Grant from Pusan National University.

참고문헌

  1. C.T. Sun and R.S. Vaidya. Prediction of composite properties from a representative volume element. Composites Science and Technology, 56(2):171-179, January 1996. https://doi.org/10.1016/0266-3538(95)00141-7
  2. Siva Bhaskara Rao Devireddy and Sandhyarani Biswas. Effect of fiber geometry and representative volume element on elastic and thermal properties of unidirectional fiber-reinforced composites. Journal of Composites, 2014:1-12, November 2014.
  3. Hui Wang and Qing-Hua Qin. A new special coating/fiber element for analyzing effect of interface on thermal conductivity of composites. Applied Mathematics and Computation, 268:311-321, October 2015. https://doi.org/10.1016/j.amc.2015.06.077
  4. Jan S Hesthaven, Gianluigi Rozza, and Benjamin Stamm. Certified Reduced Basis Methods for Parametrized Partial Differential Equations. Springer International Publishing, 2016.
  5. G. Rozza, D. B. P. Huynh, and A. T. Patera. Reduced basis approximation and a posteriori error estimation for affinely parametrized elliptic coercive partial differential equations. Archives of Computational Methods in Engineering, 15(3):229-275, May 2008. https://doi.org/10.1007/s11831-008-9019-9
  6. Minh Triet Pham. Geometrical parameterization of unit cell configurations for reduced basis approximation. Master thesis, The Graduate School of Pusan National University, Busan, Korea, February 2021.
  7. Maxime Barrault, Yvon Maday, Ngoc Cuong Nguyen, and Anthony T. Patera. An 'empirical interpolation' method: application to efficient reduced-basis discretization of partial differential equations. Comptes Rendus Mathematique, 339(9):667-672, November 2004. https://doi.org/10.1016/j.crma.2004.08.006
  8. Saifon Chaturantabut and Danny C. Sorensen. Discrete empirical interpolation for nonlinear model reduction. In Proceedings of the 48h IEEE Conference on Decision and Control (CDC) held jointly with 2009 28th Chinese Control Conference. IEEE, December 2009.
  9. Md. R. Islam and A. Pramila. Thermal conductivity of fiber reinforced composites by the FEM. Journal of Composite Materials, 33(18):1699-1715, September 1999. https://doi.org/10.1177/002199839903301803
  10. Annalisa Buffa, Yvon Maday, Anthony T. Patera, Christophe Prud'homme, and Gabriel Turinici. A prioriconvergence of the greedy algorithm for the parametrized reduced basis method. ESAIM: Mathematical Modelling and Numerical Analysis, 46(3):595-603, January 2012. https://doi.org/10.1051/m2an/2011056
  11. D.B.P. Huynh, G. Rozza, S. Sen, and A.T. Patera. A successive constraint linear optimization method for lower bounds of parametric coercivity and inf-sup stability constants. Comptes Rendus Mathematique, 345(8):473-478, October 2007. https://doi.org/10.1016/j.crma.2007.09.019
  12. Ronald Gibson. Principles of composite material mechanics. CRC Press, Boca Raton, FL, 2007.
  13. William Callister. Materials science and engineering : an introduction. John Wiley & Sons, Inc, Hoboken, NJ, 2018.
  14. Ronald Joven and Bob Minaie. Thermal properties of autoclave and out-of-autoclave carbon fiber-epoxy composites with different fiber weave configurations. Journal of Composite Materials, 52(29):4075-4085, May 2018. https://doi.org/10.1177/0021998318774608
  15. Anders Logg and Garth N. Wells. DOLFIN. ACM Transactions on Mathematical Software, 37(2):1-28, April 2010.
  16. Martin Alnaes, Jan Blechta, Johan Hake, August Johansson, Benjamin Kehlet, Anders Logg, Chris Richardson, Johannes Ring, Marie E Rognes, and Garth N Wells. The fenics project version 1.5. Archive of Numerical Software, 3, 2015.