Stress formulation and duality approach in periodic homogenization

Cristian Barbarosie; Anca-Maria Toader

doi:10.3934/dcdsb.2024004

Article Contents

2024, Volume 29, Issue 8: 3276-3296. Doi: 10.3934/dcdsb.2024004

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Stress formulation and duality approach in periodic homogenization

Cristian Barbarosie^{1, 2,} and
Anca-Maria Toader^{1, 2, ,}

1.
Faculdade de Ciências, Universidade de Lisboa, Portugal
2.
Centro de Matemática, Aplicações Fundamentais e Investigação Operacional, Portugal

^*Corresponding author: Anca-Maria Toader
^*Corresponding author: Anca-Maria Toader

Received: November 2023

Revised: December 2023

Early access: January 2024

Published: August 2024

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Abstract

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.

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

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

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

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