# Optimization of a Coupled Force Intensity by Homogenization Methods

• M. Codegone
Conference paper
Part of the IFIP International Federation for Information Processing book series (IFIPAICT, volume 202)

## Abstract

In this paper, in the framework of a problem related to an elastic non homogeneous medium, we deal with a periodic coupled force $$(f(x)/\varepsilon ^\alpha ) \vec F (x/\varepsilon )$$ with intensity of order 1/ε α. The parameter ε is connected with the period of the non homogeneity of the medium and with the periodicity of the coupled force. The determination of the parameter α is the target of our study to obtain an effect in the microscopic equation. The homogenization technique is used in order to study the equation: $$- (\partial /\partial x_j ) (a_{ijkh} (x/\varepsilon ) e_{kh} (\vec u^{\varepsilon ,\alpha } )) = (f(x)/\varepsilon ^\alpha ) F_i (x/\varepsilon ) + G_i (x,x/\varepsilon )$$, where G i (x,x/ε) is the volume applied force. The limit, when ε → 0, of $$\vec u^{\varepsilon ,\alpha } (x)$$, in the sense of two scale convergence, is $$(\vec u^{0,\alpha } (x), \vec u^{1,\alpha } (x,y))$$ and the microscopic equation becomes: $$- (\partial /\partial y_j ) (a_{ijkh} (y) e_{khx} (\vec u^{0,\alpha } (x))) - (\partial /\partial y_j ) (e_{khy} (\vec u^{1,\alpha } (x,y))) = f(x) F_i (y)$$ if $$\alpha = 1, - (\partial /\partial y_j ) (a_{ijkh} (y) e_{khx} (\vec u^{0,\alpha } (x)) + e_{khy} (\vec u^{1,\alpha } (x,y))) = 0$$ if 0 > α > 1. When α < 1 the solutions are not uniformly bounded respect to ε.

## keywords

Coupled forces Homogenization Elasticity

## References

1. [1]
G. Allaire. Homogenization and two scale convergence. SIAM J. Math. Anal. 23:1482–1518, 1992.
2. [2]
A. Bensoussan, J.L. Lions, G. Papanicolaou. Asymptotic Analysis for Periodic Structures. North-Holland, 1978.Google Scholar
3. [3]
V. Chiadò Piat, M. Codegone. Scattering problem in a perforated domain. Rev. R. Acad. Cien. Serie A Mat. 97:447–454, 2003.
4. [4]
D. Cioranescu, P. Donato. An Introduction to Homogenization. Oxford University Press, Oxford, 1999.
5. [5]
M. Codegone. Problème d’homogènèisation en thèorie de la diffraction. C.R. Acad.Sc. Paris. 288:387–389, 1979.
6. [6]
M. Codegone. G-Convergence and Scattering Problems. Boll. U.M.I. 6.1-A:367–375, 1982.Google Scholar
7. [7]
V.V. Jikov, S.M. Kozlov, O.A. Oleinik. Homogenization of Differential Operators and Integral Functional. Springer-Verlag, Berlin Heidelberg, 1994.Google Scholar
8. [8]
O.A. Oleinik, A.S. Shamaev, G.A. Yosiflan. Mathematical problems in elasticity and homogenization. Studies in mathematics and its applications, Elsevier Science Publishers, Amsterdam, 1992.
9. [9]
G. Nguetseng. A general convergence result for a functional related to the theory of homogenization. SIAM J. Math. Anal. 20:608–623, 1989.
10. [10]
J. Sanchez-Hubert, E. Sanchez-Palencia. Vibration and Coupling of Continuous Systems Asymptotic Methods. Springer-Verlag, Berlin, 1989.
11. [11]
E. Sanchez-Palencia. Comportement local et macroscopique d’un type de milieux physiques hétérogènes. Internat. J. Engrg. Set 12:331–351, 1974.
12. [12]
E. Sanchez-Palencia. Non Homogeneous Media and Vibration Theory. Lecture Notes in Physics, Springer-Verlag, Berlin, 1980.