Optimization of a Coupled Force Intensity by Homogenization Methods

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


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.


Coupled forces Homogenization Elasticity 


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© International Federation for Information Processing 2006

Authors and Affiliations

  • M. Codegone
    • 1
  1. 1.Dipartimento di MatematicaPolitecnico di TorinoTorinoItaly

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