The Semi-Penalized Updated Properties Model and Its Algorithm to Impose the Volume Fraction

Intricate structures with minimal weight and maximum stiffness are demanded in many practical engineering applications. Topology optimization is a method for designing these structures, and the rise of additive manufacturing technologies has opened the door to their production. In a recently published paper, a novel topology optimization algorithm, named the Updated Properties Model (UPM), was developed with the homogenization of strain level as an objective function and an updating Young modulus as the design variable. The UPM method optimizes mechanical structures without applying any constraints. However, including constraints such as volume, mass, and/or stress in topology optimization is prevalent. This paper uses the density-dependent Young modulus concept to incorporate the volume fraction in the UPM method. We address the critical problem of constraint-aware design without the complexity of constraint-handling formulations. We show the proposed methodology's success and functionality by plotting the algorithm's results in two- and three-dimensional benchmark structures. Key results present that adjusting algorithmic parameters can yield both binary (single-material) and graded-material solutions, offering flexibility for different applications. These findings suggest that the UPM can effectively replicate constraint-driven outcomes without explicitly enforcing constraints. The main novelty of this work lies in extending the constraint-free UPM framework to allow for controlled material distribution using a physically meaningful update rule. This extends the applicability of the UPM beyond previous efforts in the literature. We have also created a Julia package for our proposal.