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Deep Autoencoder based Energy Method for the Bending, Vibration, and Buckling Analysis of Kirchhoff Plates
Xiaoying Zhuang, Hongwei Guo, Naif Alajlan, Timon Rabczuk
TL;DR
Kirchhoff plate analysis requires handling bending, vibration, and buckling through field approximations and energy principles. The paper proposes DAEM, combining deep autoencoder approximation with minimum total potential energy and Rayleigh-based losses. It reports successful discovery of underlying physical patterns, improved stability without compromising performance, and greater simplicity and efficiency than deep collocation, while noting unresolved optimization and scope issues.
Problem
Kirchhoff plate analysis spans bending, vibration, and buckling, motivating an energy-based method that can represent the underlying physical patterns.
Method
DAEM combines a deep autoencoder with the minimum total potential principle and uses Rayleigh-based loss formulations for vibration and buckling analysis.
Results
The method successfully discovers underlying physical patterns and is reported as more simple and efficient than the deep collocation method, with improved stability and alleviated gradient explosion.
Takeaways & Limitations
The deep autoencoder energy framework provides a meshfree approach for Kirchhoff plate analysis that is easily accessible and applicable across the studied cases.
Takeaways & Limitations
The authors describe the studies as preliminary and identify the need for a more suitable global optimization algorithm and applications to more complicated Kirchhoff plate analyses.
Abstract
from arXiv · showhide
In this paper, we present a deep autoencoder based energy method (DAEM) for the bending, vibration and buckling analysis of Kirchhoff plates. The DAEM exploits the higher order continuity of the DAEM and integrates a deep autoencoder and the minimum total potential principle in one framework yielding an unsupervised feature learning method. The DAEM is a specific type of feedforward deep neural network (DNN) and can also serve as function approximator. With robust feature extraction capacity, the DAEM can more efficiently identify patterns behind the whole energy system, such as the field variables, natural frequency and critical buckling load factor studied in this paper. The objective function is to minimize the total potential energy. The DAEM performs unsupervised learning based on random generated points inside the physical domain so that the total potential energy is minimized at all points. For vibration and buckling analysis, the loss function is constructed based on Rayleigh's principle and the fundamental frequency and the critical buckling load is extracted. A scaled hyperbolic tangent activation function for the underlying mechanical model is presented which meets the continuity requirement and alleviates the gradient vanishing/explosive problems under bending analysis. The DAEM can be easily implemented and we employed the Pytorch library and the LBFGS optimizer. A comprehensive study of the DAEM configuration is performed for several numerical examples with various geometries, load conditions, and boundary conditions.
1. Introduction
The paper motivates a deep autoencoder energy method for Kirchhoff plate analysis because shallow neural networks struggle with complex nonlinear patterns. The proposed framework combines deep autoencoder feature learning with energy minimization and exploits higher-order continuity.
- Motivation: Thin plate models are widely used in engineering and mechanics, while their analysis has motivated finite, boundary, meshfree, isogeometric, and deep-learning methods.The introduction lists these approaches as responses to limitations of analytical methods.
- Motivation: Shallow artificial neural networks cannot effectively learn complex nonlinear patterns in partial differential equations.The introduction contrasts shallow networks with deeper architectures for nonlinear approximation.
- Motivation: Deep autoencoders learn latent codes through encoding and decoding layers and are presented as suitable for describing physical patterns from governing equations or potential energy.The paper connects autoencoder feature learning to the underlying physical patterns of the plate problem.
- Proposed method: The proposed deep autoencoder based energy method combines a deep autoencoder with the minimum total potential principle for Kirchhoff plate analysis.The method is proposed for the plate problem and uses the autoencoder as a function approximation framework.
- Proposed method: The method exploits the higher order continuity of the deep autoencoder approximation and introduces a tailored activation function for the mechanical model.The paper presents the activation function as part of the method development before demonstrating benchmark performance.
- Evaluation: The paper evaluates the method on benchmark problems to demonstrate its efficiency and accuracy across the studied plate analyses.The stated demonstrations cover various benchmark problems.
2. Kirchhoff plate model
The Kirchhoff plate model uses transverse mid-plane deflection as the field variable and derives curvatures, moments, boundary conditions, and energy criteria. Rayleigh’s principle supports vibration analysis, while the minimum load parameter determines critical buckling.
- Plate kinematics: The Kirchhoff plate is modeled in Cartesian coordinates, with the transverse deflection of the mid-plane serving as the field variable.The model relates this deflection to the plate’s generalized strains and rotations.
- Plate kinematics: Bending and twisting curvatures are treated as generalized strains, from which geometric equations and bending moments are obtained.The bending moments use the plate’s constitutive relations.
- Constitutive model: The bending rigidity is D0 = Eh^3/[12(1−ν^2)], where E is Young’s modulus, ν is Poisson ratio, and h is plate thickness.The model explicitly identifies the material and geometric parameters entering the rigidity.
- Boundary conditions: Boundary conditions are categorized into clamped, simply supported, and free edges using deflection, rotation, moment, and shear-force quantities.The boundary descriptions assign different prescribed quantities to the three edge types.
- Energy formulation: The total potential energy combines plate strain energy with the potential energy of external loads.This energy formulation is used as the basis for the subsequent analyses.
- Vibration analysis: For vibration, Rayleigh’s principle yields the lowest natural frequency through a quotient relating strain and kinetic energy.The formulation assumes harmonic vibration and uses maximum kinetic and strain energies.
- Buckling analysis: For buckling, the critical load factor is the minimum load parameter at which the increment in total potential energy upon buckling equals zero.The in-plane forces are scaled by λ, which becomes the desired buckling load factor under the stated reference-force convention.
3. Basic theory of a deep autoencoder
Deep autoencoders are feedforward networks that encode inputs into lower-dimensional representations and decode them into reconstructions. The paper adapts their architecture and activation function for Kirchhoff plate mechanics.
- Network architecture: Autoencoders combine an encoder and decoder to reconstruct inputs through a lower-dimensional hidden representation.The network maps data through encoding, bottleneck, decoding, and output layers.
- Network architecture: A deep autoencoder uses fully connected neighboring layers, with weights connecting neurons between successive layers.Data flows from the input layer through encoding and hidden layers to the decoder and output.
- Activation function: The paper proposes an improved tanh activation function for Kirchhoff plate analysis because the default tanh can produce unstable DAEM results.The modified activation is designed to address gradient vanishing and exploding problems in deep autoencoder networks.
- Activation function: The modified activation spreads training more uniformly by using steeper mid-range gradients and saturation toward +1 and −1.The paper also relates the activation design to the trigonometric structure of Kirchhoff plate solutions.
- Network configuration: The architecture configuration, including hidden layers and neurons, is selected through numerical study while keeping each layer small.The paper studies configurations to identify an optimum number of hidden layers.
- Function approximation: The autoencoder represents an approximating function parameterized by weights and biases, enabling approximation of unknown field variables.The composition g ◦ f is parameterized by θ = {ω1, b1, ω2, b2}.
4. Deep autoencoder based energy method
The DAEM approximates Kirchhoff plate fields with a deep autoencoder and trains its parameters by minimizing energy-based losses. It uses Monte Carlo integration for bending and Rayleigh-principle losses for vibration and buckling.
- Energy formulation: Unlike deep collocation, the energy method uses a weak formulation requiring less continuity and fewer gradient computations.Natural boundary conditions are automatically satisfied, which is especially useful for fourth-order problems.
- Bending analysis: The DAEM approximates the plate transversal deflection with a deep autoencoder and minimizes total potential energy subject to essential boundary conditions.The induced energy network shares parameters with the deflection approximation.
- Numerical integration: Monte Carlo integration evaluates energy integrals at randomly sampled points inside the physical domain to avoid underfitting at fixed quadrature points.The sampled dataset represents random points in the domain, and the area scales the integral estimate.
- Vibration analysis: For vibration analysis, the loss is based on Rayleigh’s principle and includes normalization so the mode shape is nontrivial.The lowest natural frequency is retrieved by minimizing the Rayleigh quotient while enforcing essential boundary conditions.
- Buckling analysis: Buckling analysis uses a modified loss function based on the deep-autoencoder approximation of transversal deflection.The parameters are optimized so the approximated deflection minimizes the buckling loss.
- Optimization procedure: The DAEM reduces bending, vibration, and buckling solutions to an optimization problem trained with backpropagation and L-BFGS.The algorithm samples domain and boundary points, computes a task-specific loss, and updates weights and biases.
1. Fix the number of neurons on input layer Din, hidden layers H, encoding layers iH,
The DAEM procedure fixes network dimensions, selects an activation and optimizer, iteratively computes activations and task-specific losses, updates parameters, and then performs inference.
- Configuration: The procedure fixes the hidden-layer and encoding-layer counts before training begins.The algorithm also specifies the input, hidden, encoding, and output dimensions.
- Training loop: Each training iteration computes hidden-layer activations and reconstructs the output from them.These computations use the activation and reconstruction mappings defined for the autoencoder.
- Task-specific losses: The algorithm selects a bending, vibration, or buckling loss according to the analysis task.The three loss choices correspond to Equations 4.7, 4.12, and 4.13.
- Parameter updates: Backpropagation updates the network weights and biases after the loss is computed.The procedure repeats this optimization over the specified training iterations.
- Inference: After training, the deep autoencoder is used to make predictions and infer the field variables.The algorithm outputs the predicted field variables.
5. Numerical Experiments
Numerical experiments evaluate DAEM for plate bending across activation functions, network configurations, quadrature sizes, and geometries, then extend it to vibration analysis. The results generally support stable, accurate predictions, with deeper networks often preferred over wider shallow ones.
- Bending analysis: DAEM experiments cover plate bending, including benchmark plates, an annular plate, and a plate on a Winkler foundation.The studies use varied geometries, loading conditions, and boundary conditions, with analytical solutions available for comparison in several cases.
- Activation functions: The modified hyperbolic tangent activation function always yields stable results, whereas the original Tanh sometimes produces NaN values from exploding gradients.The modified function is also less dependent on the number of hidden layers than the original activation function.
- Computational cost: More encoding layers increase computational cost, while the two tested activation schemes have comparable computational cost.The reported computational cost includes training cost.
- Network configuration: Increasing encoding layers generally improves agreement with analytical solutions, but increasing neuron counts does not necessarily improve accuracy.The experiments report that results are already quite accurate with one encoding layer and favor deeper, narrower structures over shallow, wider ones.
- Additional bending examples: For annular plates and Winkler-foundation plates, DAEM-predicted deflections agree well with analytical solutions across tested configurations.The foundation model adds the foundation potential energy to the total potential energy, with reaction p(x, y) = kw.
- Vibration analysis: DAEM-predicted nondimensional fundamental frequencies and mode shapes agree well with reference methods for square plates with cutouts and other boundary conditions.The comparisons include HBM, Modified Ritz, FEM, and Discrete Ritz results.
CSCS 35,4996
DAEM is applied to skew plates under uniaxial compressive loading with varying skew angles, aspect ratios, and boundary conditions. Predicted buckling parameters and mode shapes agree well with analytical and reference solutions.
- Buckling analysis: The buckling study examines skew plates under uniaxial in-plane compression with simply-supported and clamped boundary conditions.The effects of skew angle and aspect ratio on the critical buckling load factor are compared with reference solutions.
- Simply-supported plates: For simply-supported skew plates, DAEM results are compared with Rayleigh-Ritz, FEM, CQUAD4, and CQUAD8 methods.The adopted nonlinear encoding configuration is [60, 20], selected as suitable for varying geometries.
- Clamped plates: Predicted critical buckling loads agree well with analytical results for clamped skew plates at fixed aspect ratio ξ = 1.The corresponding results are reported in Table 6.
- Buckling mode shapes: DAEM-predicted buckling mode shapes agree well with reference results for simply-supported and clamped skew plates.The mode-shape studies vary aspect ratio for simply-supported plates and skew angle for clamped plates.
- Practical implication: Once neural-network hyperparameters are obtained, DAEM can predict similar problems quickly without target solution values during training.This follows from the method's unsupervised-learning setup.
6. Conclusions
The paper concludes that DAEM can analyze Kirchhoff-plate bending, vibration, and buckling while learning underlying physical patterns through an energy-based, meshfree framework. Numerical studies support its accuracy, efficiency, activation-function stability, and practical applicability, although broader optimization and engineering applications remain open.
- DAEM combines unsupervised feature extraction with the minimum total potential energy principle for Kirchhoff-plate mechanical analysis.
- The tailored activation function is more stable, alleviates gradient explosion, and preserves computational efficiency.
- The fitted Monte Carlo integration makes DAEM meshfree and readily accessible, while trained models can predict physical features directly.
- The method extracts fundamental frequencies, critical buckling loads, and corresponding mode shapes using Rayleigh’s principle.
- Numerical examples cover plates with varied geometries, cutouts, boundary and load conditions, and Winkler foundations, validating the proposed approach.
- The preliminary results indicate favorable performance and simplicity relative to deep collocation, but improved global optimization and more complicated engineering applications remain future work.