A multi-scale spectral stochastic method for homogenization of multi-phase periodic composites with random material properties

International Journal for Numerical Methods in Engineering

Volumn 83, Issue 1, 2010, Pages 59-90

  Tootkaboni, M ^a   Graham Brady, L ^a  

^a JOHNS HOPKINS UNIVERSITY   (United States)

Author keywords

Hermite chaos expansion; Homogenization theory; Karhunen loeve transform; Multi phase composites; Solids; Stochastic galerkin

Indexed keywords

GALERKIN; HERMITE-CHAOS EXPANSION; HOMOGENIZATION THEORY; KARHUNEN-LOEVE TRANSFORM; MULTIPHASE COMPOSITES;

COMPUTATIONAL GEOMETRY; COMPUTER SIMULATION; GALERKIN METHODS; HOMOGENIZATION METHOD; MATERIALS PROPERTIES; MONTE CARLO METHODS; NANOELECTRONICS; STOCHASTIC SYSTEMS; WAVELET TRANSFORMS;

CHAOS THEORY;

EID: 77954271074     PISSN: 00295981     EISSN: 10970207     Source Type: Journal    
DOI: 10.1002/nme.2829     Document Type: Article

References (31)

1. 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; 157:109-143.
2. Gibson L, Ashby M. Cellular Solids: Structure and Properties. Pergamon Press: Oxford, 1988.
3. Nemat-Nasser S, Iwakuma T, Hejazi M. On composites with periodic structure. Mechanics of Materials 1982; 1:239-267.
4. Sanchez-Palencia E. Non-Homogeneous Media and Vibration Theory. Lecture Notes in Physics, vol. 127. Springer: Berlin, 1980.
5. Hollister SJ, Kikuchi N. A comparison of homogenization and standard mechanics analyses for periodic porous composites. Computational Mechanics 1992; 10(2):73-95.
6. Oskay C, Fish J. Eigendeformation-based reduced order homogenization for failure analysis of heterogeneous materials. Computer Methods in Applied Mechanics and Engineering 2007; 196:1216-1243.
7. Yuan Z, Fish J. Multiple scale eigendeformation-based reduced order homogenization. Computer Methods in Applied Mechanics and Engineering 2009; 198:2016-2038.
8. Li A, Li R, Fish J. Generalized mathematical homogenization: from theory to practice. Computer Methods in Applied Mechanics and Engineering 2008; 197:3225-3248.
9. Kaminski M, Kleiber M. Perturbation based stochastic finite element method for homogenization of two-phase elastic composites. Computers and Structures 2000; 78:811-826.
10. Kaminski M. Stochastic finite element method homogenization of heat conduction problem in fiber composites. Structural Engineering and Mechanics 2001; 11(4):373-392.
11. Kaminski M. Homogenization of 1D elastostatics by the stochastic second order approach. Mechanics Research Communications 2000; 27(3):273-280.
12. Sakata S, Ashida F, Kojima T, Zako M. Three-dimensional stochastic analysis using a perturbation-based homogenization method for elastic properties of composite material considering microscopic uncertainty. International Journal of Solids and Structures 2007; 45:894-907.
13. Sakata S, Ashida F, Kojima T, Zako M. Influence of uncertainty in microscopic material property on homogenized elastic property of unidirectional fiber reinforced composites. Theoretical Applied Mechanics 2007; 56:67-76.
14. Cameron JR, Martin W. The orthogonal development of nonlinear functionals in series of Fourier-Hermite functionals. Annals of Mathematics 1947; 48:385-392.
15. Wiener N. The homogeneous chaos. American Journal of Mathematics 1938; 60(4):897-936.
16. Ghanem R, Spanos PD. Stochastic Finite Element: A Spectral Approach. Dover: New York, 2002.
17. Bensoussan A, Lions J, Papanicolao G. Asymptotic Analysis for Periodic Structures. North-Holland: Amsterdam, 1978.
18. Suquet PM. Elements of homogenization for inelastic solid mechanics. In Homogenization Techniques for Composite Media, Sanchez-Palencia E, Zaoui A (eds). Lecture Notes in Physics, vol. 272. Springer: Berlin, 1987.
19. Kakutani S. Spectral analysis of stationary Gaussian processes. Proceedings of the 4th Berkeley Symposium on Mathematical Statistics and Probability, vol. 2, University of California, 1961; 239-247.
20. Puig B, Poirion F, Soize C. Non-Gaussian simulation using Hermite polynomial expansion: convergences and algorithms. Probabilistic Engineering Mechanics 2002; 17(3):253-264.
21. Shohat JA, Tamarkin JD. The problem of moments. Bulletin of the American Mathematical Society 1945; 51(11):860-863.
22. Devroy L. Non-Uniform Random Variate Generation. Springer: New York, 1986.
23. Abramowitz M, Stegun IA. Handbook of Mathematical Functions with Formulas, Graphs, and Mathematical Table. Dover: New York, 1964.
24. Xiu D, Karniadakis GE. The Wiener-Askey polynomial chaos for stochastic differential equations. SIAM Journal on Scientific Computing 2002; 24(2):619-644.
25. Sakamoto S, Ghanem R. Simulation of multi-dimensional non-Gaussian non-stationary random fields. Probabilistic Engineering Mechanics 2002; 17:167-176.
26. Sakamoto S, Ghanem R. Polynomial Chaos decomposition for the simulation of non-Gaussian non-stationary stochastic processes. Journal of Engineering Mechanics (ASCE) 2002; 128:190-201.
27. Declercq D. Apport des polynomes d'Hermite à la modélisation non-Gaussienne et tests statistiques associés. Ph.D. Thesis (in French), 1998.
28. Loeve M. Probability Theory. Springer: New York, 1977.
29. Fukunaga S. Introduction to Statistical Pattern Recognition. Academic Press Inc.: San Diego, CA, 1990.
30. Ghanem R. Ingredients for a general purpose stochastic finite element implementation. Computer Methods in Applied Mechanics and Engineering 1999; 168:19-34.
31. Doostan A, Ghanem R, Red-Horse J. Stochastic model reduction for chaos representations. Computer Methods in Applied Mechanics and Engineering 2007; 196:3951-3966.