A parallel FreeFEM framework for topology optimization of structures into three spatial dimensions

This article presents a simple and compact parallel FreeFEM implementation for topology optimization of structures into three spatial dimensions. The topology optimization algorithm relies on the topological derivative concept together with a level-set domain representation method, in which the topological derivative is used as a steepest descent direction evolving the level-set function within the optimization procedure. In addition, adaptive mesh refinement is performed in the optimization scheme for enhancing the boundary representation of the resulting structure and reducing the computational cost of structured mesh refinements. In addition, a rich and thorough discussion of the main aspects of the parallel FreeFEM implementation is given in full detail. In particular, a version of the resulting code is provided for the reader convenience, including two benchmark examples for structural stiffness maximization and for the design of compliant mechanisms, showing the simplicity and effectiveness of the proposed implementation. Besides, we also show how the resulting code can be easily converted into a SIMP-based topology design algorithm. Since all the background knowledge on the topological derivative method is presented and the complete FreeFEM implementation is provided as supplementary material, this paper may also be used as educational/pedagogical material for graduate students and/or newcomers to the field of topology optimization.