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Journal Articles Quarterly Journal of Mechanics and Applied Mathematics Year : 2013

The topological derivative in anisotropic elasticity

Abstract

A comprehensive treatment of the topological derivative for anisotropic elasticity is presented, with both the background material and the trial small inhomogeneity assumed to have arbitrary anisotropic elastic properties. A formula for the topological derivative of any cost functional defined in terms of regular volume or surface densities depending on the displacement is established, by combining small-inhomogeneity asymptotics and the adjoint solution approach. The latter feature makes the proposed result simple to implement and computationally efficient. Both three-dimensional and plane-strain settings are treated; they differ mostly on details in the elastic moment tensor (EMT). Moreover, the main properties of the EMT, a critical component of the topological derivative, are studied for the fully anisotropic case. Then, the topological derivative of strain energy-based quadratic cost functionals is derived, which requires a distinct treatment. Finally, numerical experiments on the numerical evaluation of the EMT and the topological derivative of the compliance cost functional are reported.
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Dates and versions

hal-00852248 , version 1 (20-08-2013)
hal-00852248 , version 2 (26-02-2017)

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Marc Bonnet, Gabriel Delgado. The topological derivative in anisotropic elasticity. Quarterly Journal of Mechanics and Applied Mathematics, 2013, 66, pp.557-586. ⟨10.1093/qjmam/hbt018⟩. ⟨hal-00852248v2⟩
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