تحليل عددي مسائل الاستيسيته داراي تمركز تنش با استفاده از شبكه‌هاي عصبي فيزيك‌آگاه غني‌سازي شده توسط توابع مجانبي و متغير‌هاي مختلط

Numerical Analysis of Elasticity Problems with Stress Concentration Using Physics-Inform Neural Networks Enriched by Asymptotic Functions an‎d Complex Variables

مهندسي عمران

The analysis an‎d investigation of elasticity problems play a fundamental role in the stability an‎d safety of structures. In elasticity theory, mathematical models are utilized to describe material behavior under loading an‎d to identify stress distribution an‎d critical regions. However, due to geometric complexity, obtaining precise analytical solutions is limited. Consequently, it is essential to develop efficient numerical methods capable of accurately approximating these equations for structural analysis. In this research, the efficiency of the novel Physics-informed Neural Networks (PINNs) method has been studied by examining two problems: the center of dilatation an‎d the central crack, both of which exhibit severe stress variations in the vicinity of geometric discontinuities. To accurately analyze stress concentration at the crack tip an‎d the center of dilatation caused by impurity atoms, this study has employed asymptotic functions to enrich the neural networks. By utilizing these functions, the singularity behavior of stresses near the discontinuity is taken in to account, an‎d the computational load singularities, the neural networks are solely responsible for predicting the intensity factor of the singularity behavior during the training process. The main goal of this research is to eva‎luate the performance of neural networks in predicting an appropriate model of these problems an‎d to precisely calculate the asymptotic correction factor. This process is achieved by integrating asymptotic functions into the loss function an‎d automatically updating the asymptotic function coefficient during training via optimizers. Furthermore, in the central crack problem, the complex potentials of the generalized Westergaard method have been utilized; by adopting this approach, equilibrium an‎d compatibility conditions are automatically satisfied. The performance of the obtained models has been eva‎luated. The extracted results demonstrate the capability of this method in predicting the stress field with an‎d accuracy surpassing that of stan‎dard neural network in solving center of dilation problem. Furthermore, by utilizing these networks, the stress intensity factor for various crack lengths in finite domains have been computed.

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