RESUMEN: In this talk I will discuss a geometric constraint, called accessibility constraint, for shape and topology optimization of structures built by additive manufacturing. The motivation comes from the use of sacrificial supports to maintain a structure, submitted to intense thermal residual stresses during its building process. Once the building stage is finished, the supports are of no more use and should be removed. However, such a removal can be very difficult or even impossible if the supports are hidden deep inside the complex geometry of the structure. A rule of thumb for evaluating the ease of support removal is to ask that the contact zone between the structure and its supports can be accessed from the exterior by a straight line which does not cross another part of the structure. It represents the possibility to cut the head of the supports attached to the structure with some cutting tool. We propose a new mathematical approach to evaluate such an accessibility constraint, which is based on distance functions, solutions of eikonal equations. The main advantage is the possibility of computing its shape derivative by the adjoint method, well-knonw in control theory. We numerically demonstrate in 2D and 3D that, in the context of the level-set method for topology optimization, our algorithm can optimize simultaneously the mechanical performance of a structure and the accessibility of its building supports, guaranteeing its additive manufacturing. This is a joint work with M. Bihr, B. Bogosel and M. Godoy.

RESUMEN: In the problem of Bragg-edge elastic strain tomography, measurements are obtained from energy resolved neutron transmission imaging, which provides information about the Longitudinal Ray Transform (LRT) of the elastic strain field. The goal is to recover the elastic strain field by inverting its LRT.

The inversion of the ray transform for tensor fields is a well studied problem [1]. It is known that only the solenoidal part of symmetric tensor fields can be recovered from their LRT, there are inversion formulas available that reconstruct such solenoidal component and there are stability estimates for such reconstruction.

Nonetheless, by taking into account that elastic strain fields additionally satisfy an equilibrium equation, it is possible to also recover the potential part of the elastic strain, for simply connected object [3], or under some extra requirements [2].

In this talk I will present the problem of inverting the LRT for elastic strain tomography that arises from energy resolved neutron transmission imaging,  I will recall the classic results about the inversion of the ray transform for tensor fields, and I will present the contributions of [2,3] dealing with the reconstruction of the potential part of the elastic strain field using the equilibrium equation that is satisfied.

[1] V. A. Sharafutdinov, Integral geometry of tensor fields, Vol. 1, Walter de Gruyter, 2012.

[2] C M Wensrich, S Holman, M Courdurier, W R B Lionheart, A P Polyakova and I E Svetov, Direct inversion of the Longitudinal ray transform for 2D residual elastic strain fields, Inverse Problems, Volume 40, Number 7, 2024.

[3] C M Wensrich, S Holman, W R B Lionheart, M Courdurier, A P Polyakova, I E Svetov, T Doubikin, General reconstruction of elastic strain fields from their Longitudinal Ray Transform, arXiv:2408.10250, 2024.