A Deep Collocation Method for the Bending Analysis of Kirchhoff Plate

Abstract: In this paper, a deep collocation method (DCM) for thin plate bending problems is proposed. This method takes advantage of computational graphs and backpropagation algorithms involved in deep learning. Besides, the proposed DCM is based on a feedforward deep neural network (DNN) and differs from most previous applications of deep learning for mechanical problems. First, batches of randomly distributed collocation points are initially generated inside the domain and along the boundaries. A loss function is built with the aim that the governing partial differential equations (PDEs) of Kirchhoff plate bending problems, and the boundary/initial conditions are minimised at those collocation points. A combination of optimizers is adopted in the backpropagation process to minimize the loss function so as to obtain the optimal hyperparameters. In Kirchhoff plate bending problems, the C1 continuity requirement poses significant difficulties in traditional mesh-based methods. This can be solved by the proposed DCM, which uses a deep neural network to approximate the continuous transversal deflection, and is proved to be suitable to the bending analysis of Kirchhoff plate of various geometries.

• The paper presents a deep collocation method using DNNs that achieves relative errors around 10⁻⁴ for Kirchhoff plate bending analysis.
• It employs randomly distributed collocation points and an optimized loss function to accurately satisfy PDEs and boundary conditions without meshing.
• The method demonstrates robust performance across various geometries and loading conditions, opening new avenues in computational mechanics research.

A Deep Collocation Method for the Bending Analysis of Kirchhoff Plate

The paper "A Deep Collocation Method for the Bending Analysis of Kirchhoff Plate" introduces a novel study on the application of deep learning to solve complex mechanical problems, specifically focusing on Kirchhoff plate bending issues. The authors leverage a deep collocation method (DCM) using feedforward deep neural networks (DNNs) to circumvent the challenges present in traditional mesh-based numerical methods, notably the requirement for $C^1$ continuity in Kirchhoff plate bending problems.

Overview of Methods

The authors propose a technique wherein batches of randomly distributed collocation points are generated within the domain and along the boundaries of the plate structure. A loss function is defined to minimize the residuals of the governing partial differential equations (PDEs) and associated boundary/initial conditions at these points. The DCM employs backpropagation and a combination of optimization methods to fine-tune the neural network, which approximates the continuous transversal deflection of the plate. This framework allows for solving higher-order PDEs without the need for traditional meshing, making the approach truly mesh-free.

Key Numerical Results

The numerical experiments demonstrate the predictive strength of the DCM when applied to plates of various geometries and under different loading conditions. The results are benchmarked against exact solutions where available, and the approach showcases remarkable accuracy and robustness. For instance, in the case of a simply-supported square plate, a comprehensive analysis with varying hidden layers and neurons showed that even a shallow network with a single hidden layer can approximate the solution effectively, achieving relative errors in the order of $10^{-4}$. Such accuracy is consistently replicated across different geometrical configurations, indicating the method's versatility and reliability.

Discussion on Implications and Future Work

The implications of this research extend significantly into the field of computational mechanics. The demonstrated capability of deep neural networks to solve PDEs brings a potential paradigm shift in the way researchers approach complex mechanical simulations. By proving that a neural network can approximate continuous functions necessary for bending analysis, this work paves the way for further explorations of DNNs in solving similar high-dimensional and nonlinear problems.

Future research directions may involve exploring different neural network architectures, activation functions, and optimization strategies that could potentially enhance computational efficiency and accuracy. Additionally, extending this method to more complicated boundary conditions and incorporating other physical effects, like nonlinearities and dynamic loading, could provide a more comprehensive toolset for engineers and scientists dealing with a wide array of mechanical phenomena.

Conclusion

In conclusion, this paper presents a significant stride in employing artificial intelligence for engineering applications, specifically for the bending analysis of Kirchhoff plates. The deep collocation method proposed showcases a promising alternative to conventional numerical techniques, offering a mesh-free and efficient computational strategy. This study underlines the potential of machine learning in transforming the landscape of solving mechanical PDEs and encourages further application and research into deep learning methodologies within the field of structural mechanics.