Applications of Scientific Machine Learning for the Analysis of Functionally Graded Porous Beams

TL;DR

The paper addresses bending analysis of functionally graded porous beams by introducing DeepNetBeam (DNB), a SciML framework that unifies three paradigms: Physics-Informed Neural Networks (PINN), Deep Energy Method (DEM), and Neural Operators (e.g., Fourier Neural Operator). It leverages a displacement-based beam formulation with $u_x$ and $u_z$ driven by neural outputs $ N_x$ and $ N_z$, enforcing physics via PDE residuals, energy functionals, or operator mappings, while incorporating functionally graded material properties through Lamé parameters derived from $E$, $ u$, and porosity. The work provides detailed numerical demonstrations including cantilever validation against analytical solutions, energy-based analysis of FG porous beams under various porosity patterns and boundary conditions, and a neural-operator framework (FNO) capable of rapid predictions for arbitrary traction and porosity distributions, with substantial speedups over isogeometric analysis. Overall, DNB shows that SciML can deliver accurate, flexible, and fast analyses of FG porous beams, potentially reducing reliance on classical beam theories in design and optimization contexts. The public code base enables replication and extension to other FG structures and loading scenarios.

Abstract

This study investigates different Scientific Machine Learning (SciML) approaches for the analysis of functionally graded (FG) porous beams and compares them under a new framework. The beam material properties are assumed to vary as an arbitrary continuous function. The methods consider the output of a neural network/operator as an approximation to the displacement fields and derive the equations governing beam behavior based on the continuum formulation. The methods are implemented in the framework and formulated by three approaches: (a) the vector approach leads to a Physics-Informed Neural Network (PINN), (b) the energy approach brings about the Deep Energy Method (DEM), and (c) the data-driven approach, which results in a class of Neural Operator methods. Finally, a neural operator has been trained to predict the response of the porous beam with functionally graded material under any porosity distribution pattern and any arbitrary traction condition. The results are validated with analytical and numerical reference solutions. The data and code accompanying this manuscript will be publicly available at https://github.com/eshaghi-ms/DeepNetBeam.

Figures (17)

• Figure 1: Cantilever Beam Example
• Figure 2: Examples of 400 points generated in $[0, 2]\times [0, 8]$ using different sampling methods
• Figure 3: Convergence of the loss function in different sampling methods
• Figure 4: Computed solution for the Cantilever beam $(cm)$.
• Figure 5: Two porosity distribution patterns.