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Stress formulation and duality approach in periodic homogenization

  • *Corresponding author: Anca-Maria Toader

    *Corresponding author: Anca-Maria Toader
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  • This paper describes several different variational formulations of the so-called "cellular problem'' which is a system of partial differential equations arising in the theory of homogenization, subject to periodicity boundary conditions. These variational formulations of the cellular problem, all of them equivalent, have as main unknown the displacement, the stress or the strain, respectively. For each of these three cases, an equivalent minimization problem is introduced. The variational formulation in stress proves to have a distinguished role and it gives rise to two dual formulations, one in displacement-stress and another one in strain-stress. The corresponding Lagrangians may be used in numerical optimization for developing algorithms based on alternated directions, of Uzawa type.

    Mathematics Subject Classification: Primary: 35B27; Secondary: 74Q05, 74Q15, 74Q99.


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  • Figure 1.  A periodic function with $ u'' = -2 $ in $ {]}0,1{[} $

    Figure 2.  Relationships between different variational formulations of the cellular problem

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