Hierarchical model reduction at multiple scales

International Journal for Numerical Methods in Engineering

Volumn 79, Issue 3, 2009, Pages 314-339

  Yuan, Zheng ^a   Fish, Jacob ^a  

^a RENSSELAER POLYTECHNIC INSTITUTE   (United States)

Author keywords

Eigendeformation; Homogenization; Model reduction

Indexed keywords

ANALYTICAL SOLUTIONS; APRIORI; DISCRETIZATIONS; EIGEN-DEFORMATIONS; EIGENDEFORMATION; EQUILIBRIUM EQUATION; HIERARCHICAL MODEL; HOMOGENIZATION; MODEL-REDUCTION; MULTIPLE SCALE; NON-LINEAR; NONLINEAR HOMOGENIZATION; ONE-DIMENSIONAL MODEL; SALIENT FEATURES; SOLUTION STRATEGY; UNIT CELLS;

CELL MEMBRANES; COMPUTATIONAL COMPLEXITY;

HIERARCHICAL SYSTEMS;

EID: 68549125120     PISSN: 00295981     EISSN: 10970207     Source Type: Journal    
DOI: 10.1002/nme.2554     Document Type: Article

References (42)

1. Hill R. A theory of the yielding and plastic flow of anisotropic metals. Proceedings of Royal Society of London 1948; A193:281-297.
2. Babuska I. Homogenization and application. Mathematical and computational problems. In Numerical Solution of Partial Differential Equations-III, SYNSPADE, Hubbard B (ed.). Academic Press: New York, 1975.
3. Benssousan A, Lions JL, Papanicoulau G. Asymptotic Analysis for Periodic Structures. North-Holland: Amsterdam, 1978.
4. Sanchez-Palencia E. Non-homogeneous Media and Vibration Theory. Lecture Notes in Physics, vol. 127. Springer: Berlin, 1980.
5. Chen W, Fish J. A mathematical homogenization perspective of virial stress. International Journal for Numerical Methods in Engineering 2006; 67:189-207.
6. Chen W, Fish J. A generalized space-time mathematical homogenization theory for bridging atomistic and continuum scales. International Journal for Numerical Methods in Engineering 2006; 67:253-271.
7. Fish J, Chen W, Tang Y. Generalized mathematical homogenization of atomistic media at finite temperatures. International Journal for Multiscale Computational Engineering 2005; 4(3):393-413.
8. Fish J, Chen W, Li R. Generalized mathematical homogenization of atomistic media at finite temperatures in three dimensions. Computer Methods in Applied Mechanics and Engineering 2007; 196:908-922.
9. Li A, Li R, Fish J. Generalized mathematical homogenization: from theory to practice. Computer Methods in Applied Mechanics and Engineering 2008; 197:3225-3248.
10. Feynman RP. There's plenty of room at the bottom. Twenty-Ninth Annual Meeting of the American Physical Society, vol. 23, Engineering and Science Magazine of California Institute of Technology, 1960; 22.
11. Terada K, Kikuchi N. Nonlinear homogenization method for practical applications. In Computational Methods in Micromechanics, Ghosh S, Ostoja-Starzewski M (eds), vol. AMD-212/MD-62. ASME: New York, 1995; 1-16.
12. Terada K, Kikuchi N. A class of general algorithms for multi-scale analysis of heterogeneous media. Computer Methods in Applied Mechanics and Engineering 2001; 190:5427-5464.
13. Matsui K, Terada K, Yuge K. Two-scale finite element analysis of heterogeneous solids with periodic microstructures. Computers and Structures 2004; 82(7-8):593-606.
14. Smit RJM, Brekelmans WAM, Meijer HEH. Prediction of the mechanical behavior of nonlinear heterogeneous systems by multilevel finite element modeling. Computer Methods in Applied Mechanics and Engineering 1998; 155:181-192.
15. Miehe C, Koch A. Computational micro-to-macro transition of discretized microstructures undergoing small strain. Archive of Applied Mechanics 2002; 72:300-317.
16. Kouznetsova V, Brekelmans W-A, Baaijens FP-T. An approach to micro-macro modeling of heterogeneous materials. Computational Mechanics 2001; 27:37-48.
17. Feyel F, Chaboche J-L. FE2 multiscale approach for modeling the elastoviscoplastic behavior of long fiber sic/ti composite materials. Computer Methods in Applied Mechanics and Engineering 2000; 183:309-330.
18. Ghosh S, Lee K, Moorthy S. Multiple scale analysis of heterogeneous elastic structures using homogenization theory and Voronoi cell finite element method. International Journal of Solids and Structures 1995; 32:27-62.
19. Ghosh S, Lee K, Moorthy S. Two scale analysis of heterogeneous elasticplastic materials with asymptotic homogenization and Voronoi cell finite element model. Computer Methods in Applied Mechanics and Engineering 1996; 132:63-116.
20. Michel JC, Moulinec H, Suquet P. Effective properties of composite materials with periodic microstructure: a computational approach. Computer Methods in Applied Mechanics and Engineering 1999; 172:109-143.
21. Geers MGD, Kouznetsova V, Brekelmans WAM. Gradient-enhanced computational homogenization for the micro-macro scale transition. Journal de Physique IV 2001; 11:145-152.
22. McVeigh C, Vernerey F, Liu WK, Brinson LC. Multiresolution analysis for material design. Computer Methods in Applied Mechanics and Engineering 2006; 195:5053-5076.
23. Miehe C. Strain-driven homogenization of inelastic microstructures and composites based on an incremental variational formulation. International Journal for Numerical Methods in Engineering 2002; 55:1285-1322.
24. Fish J, Shek K, Pandheeradi M, Shephard MS. Computational plasticity for composite structures based on mathematical homogenization: theory and practice. Computer Methods in Applied Mechanics and Engineering 1997; 148:53-73.
25. Fish J, Shek KL. Finite deformation plasticity of composite structures: computational models and adaptive strategies. Computer Methods in Applied Mechanics and Engineering 1999; 172:145-174.
26. Fish J, Yu Q. Multiscale damage modeling for composite materials: theory and computational framework. International Journal for Numerical Methods in Engineering 2001; 52(1-2):161-192.
27. Clayton JD, Chung PW. An atomistic-to-continuum framework for nonlinear crystal mechanics based on asymptotic homogenization. Journal of the Mechanics and Physics of Solids 2006; 54:1604-1639.
28. Fish J, Shek KL. Finite deformation plasticity of composite structures: computational models and adaptive strategies. Computer Methods in Applied Mechanics and Engineering 1999; 172:145-174.
29. Yuan Z, Fish J. Towards realization of computational homogenization in practice. International Journal for Numerical Methods in Engineering 2008; 73:361-380.
30. Zohdi I, Oden T, Rodin J. Hierarchical modeling of heterogeneous bodies. Computer Methods in Applied Mechanics and Engineering 1996; 138:273-298.
31. Feyel F, Chaboche JL. Fe2 multiscale approach for modelling the elastoviscoplastic behaviour of long fibre sic/ti composite materials. Computer Methods in Applied Mechanics and Engineering 2000; 183:309-330.
32. Engquist EWB et al. Heterogeneous multiscale method: a general methodology for multiscale modeling. Physical Review 2003; B67(9):1-4.
33. Hou TY, Wu XH. A multiscale finite element method for elliptic problems in composite materials and porous media. Journal of Computational Physics 1997; 134(1):169-189.
34. Loehnert S, Belytschko T. A multiscale projection method for macro/microcrack simulations. International Journal for Numerical Methods in Engineering 2007; 71(12):1466-1482.
35. Belytschko T, Loehnert S, Song J. Multiscale aggregating discontinuities: a method for circumventing loss of material stability. International Journal for Numerical Methods in Engineering 2008; 73(6):869-894.
36. Hill R. Elastic properties of reinforced solids: some theoretical principles. Journal of the Mechanics and Physics of Solids 1963; 11:357-372.
37. Oskay C, Fish J. Eigendeformation-based reduced order homogenization. Computer Methods in Applied Mechanics and Engineering 2007; 196:1216-1243.
38. Fish J, Yuan Z. N-scale model reduction theory. In Bridging the Scales in Science and Engineering, Fish J (ed.). Oxford University Press: Oxford, 2008.
39. Fish J, Fan R. Mathematical homogenization of nonperiodic heterogeneous media subjected to large deformation transient loading. International Journal for Numerical Methods in Engineering 2007. DOI: 10.1002/nme.2355.
40. Yuan Z. Multiscale design system. Ph.D. Thesis, Rensselaer Polytechnic Institute, Troy, NY, 2008.
41. DeTeresa SJ et al. Experimental results in support of simulating progressive crush in carbon-fiber textile composites. UCRL-ID-143287, Lawrence Livermore National Laboratory, 2001.
42. Beard SJ. Energy absorption of braided composite tubes. Thesis, Stanford University, 2001.