🤖 AI Summary
This study investigates how the choice of norm in compliance-based objective functions influences structural topology in topology optimization. While the form of the objective function is known to significantly affect optimization outcomes, the mechanistic differences among compliance formulations derived from various norms—specifically ℓ², √ℓ², and spectral ℓ¹ norms induced by the stiffness matrix—remain unclear. Through systematic numerical experiments within a finite element framework, the work reveals that the classical quadratic (ℓ²) compliance promotes uniform load paths, whereas the spectral ℓ¹ norm yields sparse and highly localized structural members. These findings demonstrate that, under identical physical constraints, distinct norm-based formulations can produce markedly different optimization landscapes and topological configurations, offering a principled pathway toward tailoring structural performance to specific design requirements.
📝 Abstract
Compliance minimization is a central objective in structural topology optimization, commonly interpreted as the total strain energy of a system. In this work, we examine the influence of alternative compliance formulations based on different norm representations of structural energy. Specifically, we consider three formulations: the classical quadratic compliance, its square-root form corresponding to an l2 norm, and a spectral l1 -norm based formulation derived from the stiffness weighted displacement field. Although these formulations arise from the same stiffness displacement relationship, they generate markedly different optimization landscapes and result in distinct structural topologies. Numerical results indicate that the classical formulation produces well-distributed load paths, whereas the l1 -based formulation promotes sparse and highly localized structural members. These findings underscore the critical role of objective function selection in topology optimization and offer insights into alternative formulations for achieving tailored structural performance.