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A Deep Collocation Method for the Bending Analysis of Kirchhoff Plate
Hongwei Guo, Xiaoying Zhuang, Timon Rabczuk
TL;DR
Kirchhoff plate bending poses a fourth-order continuity challenge for traditional mesh-based methods. The paper proposes a feedforward-neural-network deep collocation method trained on randomly distributed domain and boundary points, reporting accurate solutions across plate geometries and conditions. The authors also identify network, activation, loss, initialization, and optimizer choices as open issues.
Problem
Fourth-order Kirchhoff plate equations make the C1 continuity requirements difficult for traditional mesh-based methods.
Method
The method approximates transverse deflection with a feedforward neural network and minimizes governing-equation and boundary-condition loss at randomly distributed collocation points.
Results
The method performs very well, with accuracy improving as network depth and width increase; circular plates are reported as especially efficient and accurate.
Takeaways & Limitations
The mesh-free deep collocation method is suitable for bending analysis of Kirchhoff plates with various geometries, loads, and boundary conditions.
Takeaways & Limitations
The influence of neural-network types, activation functions, loss forms, weight and bias initialization, and optimizers remains for future study.
Abstract
from arXiv · showhide
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.
1 Introduction
The paper motivates a deep collocation approach for Kirchhoff plate bending, where fourth-order equations and continuity requirements challenge mesh-based methods. It uses feedforward neural networks trained directly from physical-domain collocation points rather than FEM datasets.
- Kirchhoff plate bending is governed by a fourth-order PDE, making globally C1-continuous shape functions difficult to construct in mesh-based methods.
- Feedforward neural networks can approximate continuous functions with arbitrary precision, offering an alternative for Kirchhoff plate bending analysis.
- The proposed method approximates plate deflection with physical-informed feedforward networks using hyperbolic tangent activation functions.
- Training data are randomly distributed collocation points from the plate’s physical domain, and the method does not require background grids.
- The study applies deep learning to engineering mechanics beyond applications confined to datasets generated by finite element methods.
2 Kirchhoff plate bending
Kirchhoff plate bending is formulated through transverse deflection, curvature, constitutive, and equilibrium relations, yielding a fourth-order biharmonic PDE with several boundary-condition types.
- The transverse deflection w(x,y) of the plate’s middle surface is the field variable for the Kirchhoff bending problem.
- Kirchhoff bending geometry relates transverse deflection to bending and twisting curvatures, which act as generalized strains.
- For isotropic thin plates, bending rigidity is D0 = Eh3/[12(1−ν2)], with E, ν, and h denoting Young’s modulus, Poisson ratio, and thickness.
- The governing equation is a fourth-order biharmonic PDE involving the transverse deflection and applied transverse load.
- The considered boundary conditions include prescribed deflection and rotation, simply supported deflection and moment, and edge conditions involving moment, shear force, and load.
3 Deep Collocation Method for solving Kirchhoff plate bending
The deep collocation method represents Kirchhoff plate deflection with a feedforward neural network and trains shared parameters by minimizing PDE and boundary-condition residuals at randomly distributed points.
- 3.1 Feed forward neural network: A fully connected feedforward network contains input, hidden, and output layers, with weights, biases, and activation functions defining an approximation function.
- 3.2 Backpropagation: Backpropagation computes loss gradients through the computational graph, which is essential because the governing Kirchhoff equation requires fourth-order derivatives of the network approximation.
- 3.3 Formulation of deep collocation method: The physical domain and its boundaries are discretized with separate randomly distributed collocation points, while the network approximates the transversal deflection.
- 3.3 Formulation of deep collocation method: The physical-informed network, moments, and shear force share the same parameters and are learned by minimizing the mean-square-error loss.
- 3.3 Formulation of deep collocation method: The loss combines mean-square errors from the governing equation and three boundary-condition sets: L(θ) = MSEG + MSEΓ1 + MSEΓ2 + MSEΓ3.
- 3.3 Formulation of deep collocation method: Training reduces the plate problem to optimization, repeatedly updating parameters with gradient-based methods until convergence.
4 Numerical examples
Numerical examples evaluate the deep collocation method for simply-supported and clamped square plates and clamped circular plates. Across these cases, predicted deflections generally agree closely with analytical or exact solutions, with accuracy improving as network depth or width increases.
- Numerical examples: The examples use a combined L-BFGS and Adam optimizer, with randomly distributed collocation points for plate-bending calculations.L-BFGS is applied first, followed by Adam during linear search when BFGS may fail.
- 4.1 Simply-supported square plate: For the simply-supported square plate, 1000 collocation points and networks varying from 1–4 hidden layers and 20–60 neurons are compared with the exact solution.Maximum central deflection and L2 relative error are evaluated across network configurations.
- 4.1 Simply-supported square plate: The simply-supported square-plate predictions are accurate even for shallow networks, while increasing hidden layers and neurons generally brings results closer to the exact solution.The reported relative-error magnitude is 1 × 10^-4 across the tested hidden-layer types.
- 4.2 Clamped square plate: For the clamped square plate, the method gives results most agreeable with the exact solution, although single-hidden-layer networks are less accurate than deeper networks.The comparison includes Ritz, Galerkin, and Timoshenko-based solutions; increasing neurons improves the shallow-network results.
- 4.3 Clamped circular plate: For the clamped circular plate, predicted maximum deflections are very accurate, relative-error curves converge well, and all tested networks have relative error magnitude 1 × 10^-4.The deflection and error contours also agree well with the exact solution.
- Numerical examples: For the additional plate case, increasing hidden layers and neurons makes maximum deflection more accurate, while contours agree well with the analytical solution.The relative-error curves become flat and close to zero as network capacity increases.
5 Conclusions
The study applies a truly mesh-free deep collocation method to Kirchhoff plate bending across varied shapes, loads, and boundary conditions. The method achieves accurate solutions, with accuracy generally improving as network depth and width increase, while several design choices remain for future study.
- The method analyzes Kirchhoff plates with various shapes, loads, and boundary conditions using a fourth-order biharmonic governing equation.
- The proposed deep collocation method is truly mesh-free and uses neural networks to approximate continuous functions for thin plate bending analysis.
- The method performs well with randomly distributed collocation points, a mean-square-error loss, and combined L-BFGS and Adam optimization.
- Increasing hidden layers and neurons generally improves accuracy toward exact analytical solutions, while circular plates achieve accurate results with only a few layers and neurons.
- The study leaves the effects of neural-network types, activation functions, loss functions, initialization, and optimizers for future research.