Topological derivatives are defined as the variation of a functional associated with the introduction of an infinitesimal hole in a given domain. In this work, we extend the concept of topological derivatives to extract very efficiently the energy release rate (ERR) field associated with a small edge or surface crack located at any boundary location and orientation [1, 2]. The efficiency of the proposed method is associated with the fact that it is based on a single finite element analysis of the uncracked structure, in contrast with conventional methods, which require a numerical analysis with a crack-tip-refined mesh for each crack length, location and orientation. Computational time savings amounting to multiple orders of magnitude are achieved using the proposed method. 
Building on the efficiency of the ERR computation method and with the aid of a analytical sensitivity approach, we then derive a shape optimization scheme to design fracture-resistant structures. In this gradient-based optimization scheme, the domain is defined using level sets generated by R-functions to reduce the number of design variables and simplify the computation of the design velocity field while allowing complex shapes of the optimized geometry. 
After a presentation of the method, we will summarize examples aimed at illustrating the efficiency of the topological derivative approach used to extract the ERR and its application to shape optimization problems.

[1] Silva, M., Geubelle, P. H., and Tortorelli, D. A., 2011. ‚ÄúEnergy release rate approximation for small surface-breaking cracks using the topological derivative‚Äù. Journal of the Mechanics of Physics of Solids, 59(5), 925‚Äì939.
[2] Alidoost, K., Feng, M., Geubelle, P. H., and Tortorelli, D. A., 2019. ‚ÄúEnergy release rate approximation for small surface cracks in three-dimensional domains using the topological derivative‚Äù. Journal of Applied Mechanics, 12, 1‚Äì16.

Designing fracture-resistant structures: A new approach based on topological derivatives

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