Robust multiscale design of incompressible multi-materials under loading uncertainties

Most of the multiscale topology optimization (TO) approaches, which have been proposed for multi-material designs, are based on the assumption of deterministic loads and the exclusion of uncertain quantities like positions, directions, and magnitudes of loads. This contribution deals with: (1) multiscale design; (2) multi-materials; (3) loading uncertainties; and (4) incompressible materials such as rubbers and biological soft tissues. The proposed approach, in this paper is to exploit polytopal composite finite elements (PCEs) to overcome the inherent volumetric locking phenomenon in incompressible materials and employ adaptive geometric components (AGCs) to model porous multi-materials on an analysis grid of PCEs, which allows simultaneously optimizing incompressible multi-materials at both macro- and micro-scales. Besides the mixed displacement–pressure formulation and traditional concurrent TO, the proposed approach is purely developed from the displacement formulation and the AGCs-based direct-multiscale TO whose only constraint is global volume. The effectiveness of the current technique was demonstrated by solving several examples of incompressible porous multi-material designs under single and multiple random loads.