Stress-driven design of incompressible multi-materials under frequency constraints

Abstract

This paper presents an efficient and comprehensive numerical approach to address topology optimization challenges within the multi-material framework. The focus encompasses several key aspects: solving various stress-related problems from compressible to incompressible materials, broadening the scope to encompass multi-material systems, adopting a flexible methodology suitable for various elements (triangular, quadrilateral, and polygonal), and expanding to incorporate frequency constraints within the stress-driven system. The core idea to accommodate a broad range of materials, from compressible to nearly incompressible, is to implement a specialized polytopal composite finite element (PCE) technique to mitigate volumetric locking issues often prevalent in nearly incompressible materials. Then, to effectively solve the stress-related multi-material system, a well-known P-norm function integrated with the Moved and Regularized Heaviside function (MRHF) is employed, capitalizing on the correlation between topological phases and material allocation to handle multi-material problems effectively. Additionally, to deal with frequency-related multi-material problems, an extended technique addressing localized mode issues based on the relationship between representative solid and void is also employed. Several numerical examples are tested to validate the method’s efficiency and reliability within a multi-material framework, considering a diverse range of materials from compressible to nearly incompressible.