A material interpolation technique using the simplex polytope

Summary

The Discrete Material Optimization (DMO) and the Shape Function with Penalization (SFP) constitute the state-of-the-art material interpolation techniques for identifying from a list of pre-defined candidate materials the most suitable one(s) for the structural domain. The candidate materials are represented on this list through their mechanical properties, and are interpolated within the domain of interest (DOI), whether that is the finite element (FE) domain or groups of FEs, so-called patches. Depending on the technique preferred to interpolate the mechanical properties within the DOI, a different type of weights is selected. Goal of the discrete material optimization problem (MOP) is to solve for these weights and determine for each FE/patch a unique material from the list. The current work extends the concept of the SFP technique by employing as weights the shape functions of the hyper-tetrahedral FE, the dimension of which is dynamically adapted depending on the number of candidate materials considered for the structural domain. This generalized hyper-tetrahedral FE constitutes what is defined as a simplex, and similar to the SFP technique each of its nodes is tied to a specific candidate material. In the context of discrete optimization and utilizing the shape functions of an abstract high-dimensional FE as weights for the candidate materials, the proposed interpolation technique secures the continuity between the number of candidate materials that can be considered for the structure, a feature lacking in the SFP technique. Additionally, given that the number of nodes forming the simplex FE is always one unit greater than the dimension of the space it is defined within, the dimension of the resulting MOP drops by one per DOI. The developed material interpolation technique is combined with the topology optimization problem (TOP) to formulate the concurrent material and topology optimization problem for compliance minimization of the structure. Finally, the latter is examined on the academic case study of the 3D Messerchmitt-B¨olkow-Blohm (MBB) beam for the case of the concurrent topology and discrete fiber orientation optimization problem.

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