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.

已针对多材料设计提出的大多数多尺度拓扑优化（TO）方法都是基于确定性载荷的假设以及对不确定量（例如位置，方向和载荷大小）的排除。该贡献涉及：（1）多尺度设计；（2）多种材料；（3）装载不确定性；（4）不可压缩的材料，例如橡胶和生物软组织。本文提出的方法是利用多拓扑复合有限元（PCE）来克服不可压缩材料中固有的体积锁定现象，并采用自适应几何组件（AGC）在PCE的分析网格上对多孔多材料进行建模，从而实现同时在宏观和微观上优化不可压缩的多种材料。除了混合的位移-压力公式和传统的并发TO之外，所提出的方法纯粹是基于位移公式和基于AGCs的直接多尺度TO的方法而开发的，其唯一的约束是整体体积。通过解决单个和多个随机载荷下不可压缩的多孔多材料设计的几个示例，证明了当前技术的有效性。