A multiscale eigenelement method and its application to periodical composite structures

Composite Structures

Volumn 92, Issue 9, 2010, Pages 2265-2275

  Xing, Y F ^a   Yang, Y ^a   Wang, X M ^a  

^a BEIHANG UNIVERSITY   (China)

Author keywords

Eigenelement method; Frequency; Multiscale method; Periodical composite structure; Stress

Indexed keywords

COMPARISON STUDY; EIGENELEMENT METHOD; FREQUENCY; GENERALIZED FINITE ELEMENT METHODS; HETEROGENEOUS MULTISCALE METHOD; MATHEMATICAL HOMOGENIZATION; MULTI SCALE ANALYSIS; MULTISCALE FINITE ELEMENT METHOD; MULTISCALE METHOD; MULTISCALES; NUMERICAL EXPERIMENTS;

HOMOGENIZATION METHOD; STRUCTURE (COMPOSITION);

FINITE ELEMENT METHOD;

COMPOSITE; DEFORMATION; FINITE ELEMENT ANALYSIS; STRESS; STRUCTURAL ANALYSIS;

EID: 77950866655     PISSN: 02638223     EISSN: None     Source Type: Journal    
DOI: 10.1016/j.compstruct.2009.08.006     Document Type: Article

References (50)

1. Strouboulis T., Babuška I., Copps K. The generalized finite element method: an example of its implementation and illustration of its performance. Int J Numer Methods Eng 2000, 47:1401-1417.
2. Babuška I. Solution of interface problems by homogenization, parts I, II. SIAM J Math Anal 1976, 7:603-645.
3. Benssousan A., Lions J.L. Asymptotic analysis for periodic structures 1978, Amsterdam, North-Holland.
4. Oleinik O., A Shamaev A.V., Yosifian G.A. Mathematical problems in elasticity and homogenization 1992, North-Holland, Amsterdam.
5. Guedes J.M., Kikuchi N. Pre and post processing for materials based on the homogenization method with adaptive finite element methods. Comput Methods Appl Mech Eng 1990, 83:143-198.
6. Hassani B., Hinton E. A review of homogenization and topology optimization I-homogenization theory for media with periodic structure. Comput Struct 1998, 69:707-717.
7. Takano N., Zako M., Ishizono M. Multi-scale computational method for elastic bodies with global and local heterogeneity. J Computer-Aided Mater Des 2000, 7:111-132.
8. Fish J., Yuan Z. Multiscale enrichment based on partition of unity. Int J Numer Methods Eng 2005, 62:1341-1359.
9. Bourgat J.F. Numerical experiments of the homogenization method for operators with periodic coefficients. Lect Notes Math 1977, 707:330-356.
10. Kalamkarov A.L., Andrianov I.V., Danishevs'kyy V.V. Asymptotic homogenization of composite materials and structures. Appl Mech Rev 2009, 10. 10.1115/1.3090830.
11. E W., Engquist B. The heterogeneous multiscale methods. Commun Math Sci 2003, 1:87-132.
12. E W., Ming P.B., Zhang P.W. Analysis of the heterogeneous multiscale method for elliptic homogenization problems. J Am Math Soc 2005, 18:121-156.
13. Ming P.B., Zhang P.W. Analysis of the heterogeneous multiscale method for parabolic homogenization problems. Math Comput 2007, 76:153-177.
14. Ming P.B., Yue X.Y. Numerical methods for multiscale elliptic problems. J Comput Phys 2006, 214:421-445.
15. E W., Engquist B. Multiscale modeling and computation. Notice Am Math Soc 2003, 50:1062-1070.
16. Abdulle A. Heterogeneous multiscale methods with quadrilateral finite elements. Numer Math Adv Appl 2007, 20:743-751.
17. E W., Engquist B., Li X.T., Ren W.Q., Vanden-Eijnden E. Heterogeneous multiscale methods: a review. Commun Comput Phys 2007, 2(3):367-450.
18. E W., Engquist B. The heterogeneous multi-scale method for homogenization problems. Lect Notes Comput Sci Eng: Multiscale Methods Sci Eng 2006, 44:89-110.
19. Chen Z.X. On the heterogeneous multiscale method with various macroscopic solvers. Nonlinear Anal 2009, 10.1016/j.na.2009.01.22.
20. Hou T., Wu X. A multiscale finite element method for elliptic problems in composite materials and porous media. J Comput Phys 1997, 134:169-189.
21. Efendiev Y., Hou T. Multiscale finite element methods for porous media flows and their applications. Appl Numer Math 2007, 57:577-596.
22. Zhang H.W., Fu Z.D., Wu J.K. Coupling multiscale finite element method for consolidation analysis of heterogeneous saturated porous media. Adv Water Resour 2009, 32:268-279.
23. Efendiev Y.R., Hou T., Wu X. Convergence of a nonconforming multiscale finite element method. SIAM J Numer Anal 2000, 37:888-910.
24. Hou T., Wu X., Cai Z. Convergence of a multiscale finite element method for elliptic problems with rapidly oscillating coefficients. Math Comput 1999, 68:913-943.
25. Babuška I., Osborn J. Generalized finite element methods: their performance and their relation to mixed methods. SIAM J Numer Anal 1983, 20:510-536.
26. Babuška I., Caloz G., Osborn J. Special finite element methods for a class of second order elliptic problems with rough coefficients. SIAM J Numer Anal 1994, 31:945-981.
27. Strouboulis T., Babuška I., Copps K. The generalized finite element method. Comput Methods Appl Mech Eng 2001, 190:4081-4193.
28. Babuška I., Nistor V., Tarfulea N. Generalized finite element method for second-order elliptic operators with Dirichlet boundary conditions. J Comput Appl Math 2008, 218:175-183.
29. Strouboulis T., Babuška I., Copps K. The design and analysis of the generalized finite element method. Comput Methods Appl Mech Eng 2000, 181:43-69.
30. Strouboulis T., Zhang L., Babuška I. Generalized finite element method using mesh-based handbooks: application to problems in domains with many voids. Comput Methods Appl Mech Eng 2003, 192:3109-3161.
31. Strouboulis T., Zhang L., Babuška I. p-version of the generalized FEM using mesh-based handbooks with applications to multiscale problems. Int J Numer Methods Eng 2004, 60:1639-1672.
32. Duarte C.A., Kim D.J. Analysis and applications of a generalized finite element method with global-local enrichment functions. Comput Methods Appl Mech Eng 2008, 197:487-504.
33. Strouboulis T., Babuška I., Hidajat R. The generalized finite element method for Helmholtz equation: theory, computation, and open problems. Comput Methods Appl Mech Eng 2006, 195:4711-4731.
34. Strouboulis T., Hidajat R., Babuška I. The generalized finite element method for Helmholtz equation part II: effect of choice of handbook functions, error due to absorbing boundary conditions and its assessment. Comput Methods Appl Mech Eng 2008, 197:364-380.
35. O'Hara P., Duarte C.A., Eason T. Generalized finite element analysis of three-dimensional heat transfer problems exhibiting sharp thermal gradients. Comput Methods Appl Mech Eng 2009, 198:1857-1871.
36. Srinivasan K.R., Matouš K., Geubelle P.H. Generalized finite element method for modeling nearly incompressible bimaterial hyperelastic solids. Comput Methods Appl Mech Eng 2008, 197:4882-4893.
37. Berthelot J.M. Composite materials: mechanical behavior and structural analysis 1999, Springer, New York.
38. Fish J., Belsky V. Multigrid method for a periodic heterogeneous medium part I: convergence studies for one-dimensional case. Comput Methods Appl Mech Eng 1995, 126:1-16.
39. Fish J., Belsky V. Multigrid method for a periodic heterogeneous medium part 2: multiscale modeling and quality control in multidimensional case. Comput Methods Appl Mech Eng 1995, 126:17-38.
40. Oden J.T., Vemaganti K.S. Estimation of local modeling error and global-oriented adaptive modeling of heterogeneous materials I: error estimates and adaptive algorithms. J Comput Phys 2000, 164:22-47.
41. Giancarlo Sangalli Capturing small scales in elliptic problems using a residual-free bubbles finite element method. Multiscale Model Simul 2003, 1(3):485-503.
42. Babuška I., Melenk J.M. The partition of unity method. Int J Numer Methods Eng 1997, 40:727-758.
43. Babuška I., Melenk J.M. The partition of unity finite element method: basic theory and applications. Comput Methods Appl Mech Eng 1996, 139:289-314.
44. Oh H.S., Kim J.G., Hong W.T. The piecewise polynomial partition of unity functions for the generalized finite element methods. Comput Methods Appl Mech Eng 2008, 197:3702-3711.
45. Eigel M., et al. The partition of unity meshfree method for solving transport-reaction equations on complex domains: implementation and applications in the life sciences. Lect Notes Comput Sci Eng, Meshfree Methods Partial Differ Equat 2008, 65(IV):69-93.
46. Riker C., Stefan M.H. The mixed-cell-complex partition-of-unity method. Comput Methods Appl Mech Eng 2009, 198:1235-1248.
47. Ladeveze P., Loiseau O., Dureisseix D. A micro-macro and parallel computational strategy for highly heterogeneous structures. Int J Numer Methods Eng 2001, 52:121-138.
48. Xing Y.F., Tian J.M., Zhu D.C., Xie W.J. The homogenization method based on eigenvector expansions for woven fabric composites. Int J Multiscale Comput Eng 2006, 4(1):197-206.
49. Xing YF, Yang Y. A new rod eigenelement and its application to structural static and dynamic analysis. In: Proceedings of international symposium on computational mechanics, Tsinghua University press and Springer-Verlag, Beijing, China; 2007.
50. Xing Y.F., Wang X.M. An eigenelement method and two homogenization conditions. Acta Mech Sinica 2009, 25(3):345-351.