On the use of graph neural networks and shape-function-based gradient computation in the deep energy method

A graph neural network (GCN) is employed in the deep energy method (DEM) model to solve the momentum balance equation in 3D for the deformation of linear elastic and hyperelastic materials due to its ability to handle irregular domains over the traditional DEM method based on a multilayer perceptron (MLP) network. Its accuracy and solution time are compared to the DEM model based on a MLP network. We demonstrate that the GCN-based model delivers similar accuracy while having a shorter run time through numerical examples. Two different spatial gradient computation techniques, one based on automatic differentiation (AD) and the other based on shape function (SF) gradients, are also accessed. We provide a simple example to demonstrate the strain localization instability associated with the AD-based gradient computation and show that the instability exists in more general cases by four numerical examples. The SF-based gradient computation is shown to be more robust and delivers an accurate solution even at severe deformations. Therefore, the combination of the GCN-based DEM model and SF-based gradient computation is potentially a promising candidate for solving problems involving severe material and geometric nonlinearities

I-FENN for thermoelasticity based on physics-informed temporal convolutional network (PI-TCN)

We propose an integrated finite element neural network (I-FENN) framework to expedite the solution of coupled multiphysics problems. A physics-informed temporal convolutional network (PI-TCN) is embedded within the finite element framework to leverage the fast inference of neural networks (NNs). The PI-TCN model captures some of the fields in the multiphysics problem, and their derivatives are calculated via automatic differentiation available in most machine learning platforms. The other fields of interest are computed using the finite element method. We introduce I-FENN for the solution of transient thermoelasticity, where the thermo-mechanical fields are fully coupled. We establish a framework that computationally decouples the energy equation from the linear momentum equation. We first develop a PI-TCN model to predict the temperature field based on the energy equation and available strain data. The PI-TCN model is integrated into the finite element framework, where the PI-TCN output (temperature) is used to introduce the temperature effect to the linear momentum equation. The finite element problem is solved using the implicit Euler time discretization scheme, resulting in a computational cost comparable to that of a weakly-coupled thermoelasticity problem but with the ability to solve fully-coupled problems. Finally, we demonstrate the computational efficiency and generalization capability of I-FENN in thermoelasticity through several numerical examples

A deep learning energy-based method for classical elastoplasticity

The deep energy method (DEM) has been used to solve the elastic deformation of structures with linear elasticity, hyperelasticity, and strain-gradient elasticity material models based on the principle of minimum potential energy. In this work, we extend DEM to elastoplasticity problems involving path dependence and irreversibility. A loss function inspired by the discrete variational formulation of plasticity is proposed. The radial return algorithm is coupled with DEM to update the plastic internal state variables without violating the Kuhn-Tucker consistency conditions. Finite element shape functions and their gradients are used to approximate the spatial gradients of the DEM-predicted displacements, and Gauss quadrature is used to integrate the loss function. Four numerical examples are presented to demonstrate the use of the framework, such as generating stress-strain curves in cyclic loading, material heterogeneity, performance comparison with other physics-informed methods, and simulation/inference on unstructured meshes. In all cases, the DEM solution shows decent accuracy compared to the reference solution obtained from the finite element method. The current DEM model marks the first time that energy-based physics-informed neural networks are extended to plasticity, and offers promising potential to effectively solve elastoplasticity problems from scratch using deep neural networks

Novel DeepONet architecture to predict stresses in elastoplastic structures with variable complex geometries and loads

A novel deep operator network (DeepONet) with a residual U-Net (ResUNet) as the trunk network is devised to predict full-field highly nonlinear elastic-plastic stress response for complex geometries obtained from topology optimization under variable loads. The proposed DeepONet uses a ResUNet in the trunk to encode complex input geometries, and a fully-connected branch network encodes the parametric loads. Additional information fusion is introduced via an element-wise multiplication of the encoded latent space to improve prediction accuracy further. The performance of the proposed DeepONet was compared to two baseline models, a standalone ResUNet and a DeepONet with fully connected networks as the branch and trunk. The results show that ResUNet and the proposed DeepONet share comparable accuracy; both can predict the stress field and accurately identify stress concentration points. However, the novel DeepONet is more memory efficient and allows greater flexibility with framework architecture modifications. The DeepONet with fully connected networks suffers from high prediction error due to its inability to effectively encode the complex, varying geometry. Once trained, all three networks can predict the full stress distribution orders of magnitude faster than finite element simulations. The proposed network can quickly guide preliminary optimization, designs, sensitivity analysis, uncertainty quantification, and many other nonlinear analyses that require extensive forward evaluations with variable geometries, loads, and other parameters. This work marks the first time a ResUNet is used as the trunk network in the DeepONet architecture and the first time that DeepONet solves problems with complex, varying input geometries under parametric loads and elasto-plastic material behavior

Sequential Deep Operator Networks (S-DeepONet) for Predicting Full-field Solutions Under Time-dependent Loads

Deep Operator Network (DeepONet), a recently introduced deep learning operator network, approximates linear and nonlinear solution operators by taking parametric functions (infinite-dimensional objects) as inputs and mapping them to solution functions in contrast to classical neural networks that need re-training for every new set of parametric inputs. In this work, we have extended the classical formulation of DeepONets by introducing sequential learning models like the gated recurrent unit (GRU) and long short-term memory (LSTM) in the branch network to allow for accurate predictions of the solution contour plots under parametric and time-dependent loading histories. Two example problems, one on transient heat transfer and the other on path-dependent plastic loading, were shown to demonstrate the capabilities of the new architectures compared to the benchmark DeepONet model with a feed-forward neural network (FNN) in the branch. Despite being more computationally expensive, the GRU- and LSTM-DeepONets lowered the prediction error by half (0.06\% vs. 0.12\%) compared to FNN-DeepONet in the heat transfer problem, and by 2.5 times (0.85\% vs. 3\%) in the plasticity problem. In all cases, the proposed DeepONets achieved a prediction $R^2$ value of above 0.995, indicating superior accuracy. Results show that once trained, the proposed DeepONets can accurately predict the final full-field solution over the entire domain and are at least two orders of magnitude faster than direct finite element simulations, rendering it an accurate and robust surrogate model for rapid preliminary evaluations