Topology optimization approach using a training-dataset-free neural network reparameterization framework

This paper proposes a novel topology optimization method based on the reparameterization framework of the fractional Kolmogorov-Arnold Network (fKAN). This method is different from the previous topology optimization that required training with datasets; instead, it directly employs fKAN for topology optimization. It adopts Jacobi polynomials as the basis functions for the activation functions of fKAN, utilizing the learnable parameters within the activation functions to reparameterize the design variables of topology optimization. An external finite element solver is used to compute structural responses, and a loss function is constructed by weighting the objective function and volume constraints of the topology optimization model. The backpropagation of the neural network replaces the manual calculation of sensitivities in topology optimization, and AdamW is used to update the learnable parameters of the fKAN activation functions. Learnable parameters within the activation functions are updated through a machine learning process that minimizes the loss function, thereby updating the topology optimization design variables, thus achieving the optimal topology. Through topology optimization examples of continuum structures and compliant mechanisms, as well as the comparative validation of different Jacobi polynomial degrees and different algorithms, the effectiveness of the proposed method is demonstrated, and applied to the structural design of engineering cases.