The mathematical homogenization method (MHM) is generally implemented by finite element method, and its calculating accuracy depends completely on the order of perturbation and finite element，the perturbations in uncoupled form are defined as the multiplications of influence functions and the derivatives of homogenized displacements. The order of exact solutions for influence function depend on the order of pseudo load which is the right term of the governing equation whereas the exact solution of homogenized displacements and its different orders derivatives depend on the external loads; The order of elements depends on the exact solution of influence function and homogenized displacements simultaneously while the order of perturbations depend mainly on the accuracy of different order derivatives of homogenized displacements; For the static problems of periodical composite rod, the exact solutions can be obtained using correct order of MHM and finite element for the static problem of periodic composite rod subjected to different order of load. Then two dimensional(2D) periodical composite are explored similarly，and the clamped boundary con-dition of periodical unite cell have great influence for calculating accuracy of MHM. Finally, the potential en-ergy functional is used to evaluate the accuracy of MHM, and numerical comparisons validate the conclu-sions.
CHEN Lei
,
XING Yu-Feng
. Accuracy Analysis of Mathematical Homogenization Method for Periodic Composites[J]. ACTA AERONAUTICAET ASTRONAUTICA SINICA, 0
: 0
-0
.
DOI: 10.7527/S1000-6893.2014.0216
[1]Berthelot JM. Composite materials: mechanical behav-ior and structural analysis[J].New York: Springer, 1999, :-
[2]Kalidindi SR, Abusafieh A.Longitudinal and transverse moduli and strengths of low angle 3D braided compo-sites. [J].J Compos Mater, 1996, 30(8):885-905
[3]Babu?ka I.Solution of interface problems by homoge-nization. Parts I and II. SIAM J Math Anal, 7, 603–645, 1976
[4]A Benssousan, JL Lions.Asymptotic Analysis for Peri-odic Structures. North-Holland: Amsterdam; 1978.
[5]T Strouboulis, IBabu?ka, K Copps.The generalized finite element method: an example of its implementation and illustration of its performance. International Journal for Numerical Methods in Engineering. 47, 1401–1417, 2000
[6]I Babu?ka, J Osborn.Generalized finite element methods: their performance and their relation to mixed methods.SIAM J Numer Anal. 20, 510–536, 1983
[7]T Hou, X Wu.A multiscale finite element method for el-liptic problems in composite materials and porous media.Journal of Computational Physics. 134, 169–189, 1997
[8]T Hou, X Wu, Z Cai.Convergence of a multiscale finite element method for elliptic problems with rapidly os-cillating coefficients.Mathematics of Computation. 68, 913–943, 1999
[9]W E, B Engquis.The heterogeneous multiscale meth-ods.Commun Math Sci. 1, 87–132, 2003
[10]W E, B Engquis.The heterogeneous multiscale meth-ods.Commun Math Sci. 1, 87–132, 2003
[11]W E, B Engquist, XT Li, WQ Ren, E Vanden-Eijnden.Heterogeneous Multiscale Methods: A Review. Com-munications in computational physics. 2(3), 367-450, 2007
[12]YF Xing, Y Yang.An eigenelement method of periodi-cal composite structures.Composite Structures. 93, 502–512, 2011
[13]YF Xing, Y Yang, XM Wang.A MultiscaleEigenele-ment Method and Its Application to Periodical Com-posite Structures.Composite Structures.92, 2265–2275, 2010
[14]O Oleinik, AV Shamaev, GA Yosifian.Mathematical Problems in Elasticity and Homogenization. Amster-dam: North-Holland, 1992
[15]JM Guedes, N Kikuchi.Pre and post processing for materials based on the homogenization method with adaptive finite element methods. Computer Methods in Applied Mechanics and Engineering. 83, 143-98, 1990
[16]B Hassani, E Hinton.A review of homogenization and topology optimization I-homogenization theory for media with periodic structure.Computers and Struc-tures. 69, 707–17, 1998
[17]N Takano, M Zako, M Ishizono.Multi-scale computa-tional method for elastic bodies with global and local heterogeneity.Journal of Computer-Aided Material Design. 7, 111–132, 2000
[18]J Fish, Z Yuan.Multiscale enrichment based on parti-tion of unity. International Journal for Numerical Methods in Engineering. 62, 1341–1359, 2005
[19]JF Bourgat.Numerical experiments of the homogeni-zation method for operators with periodic coefficients. Lecture Notes in Mathematics. 707, 30–356, 1977
[20]PW Chung, KK Tamma, RR Namburu.Asymptotic expansion homogenization for heterogeneous media: computational issues and applications. Composites: Part A. 32, 1291-1301, 2001
[21]CM Chen et al.An enhanced asymptotic homogeniza-tion method of the static and dynamics of elastic composite laminates. Computers and Structures. 82, 373–38
[22]AL Kalamkarov, IV Andrianov, VV Danishevs’kyy.Asymptotic Homogenization of Composite Materials and Structures. Applied Mechanics review. DOI: 10.1115/ 1.3090830, 2009
[23]Y.F. Xing, L. Chen. Accuracy of multiscale asymptotic expansion method. Composite Structures.112, 38-43, 2014
CJA