Physics-informed DeepONet with stiffness-based loss functions for structural response prediction

Finite element modeling is a well-established tool for structural analysis, yet modeling complex structures often requires extensive pre-processing, significant analysis effort, and considerable time. This study addresses this challenge by introducing an innovative method for real-time prediction of structural static responses using DeepOnet which relies on a novel approach to physics-informed networks driven by structural balance laws. This approach offers the flexibility to accurately predict responses under various load classes and magnitudes. The trained DeepONet can generate solutions for the entire domain, within a fraction of a second. This capability effectively eliminates the need for extensive remodeling and analysis typically required for each new case in FE modeling. We apply the proposed method to two structures: a simple 2D beam structure and a comprehensive 3D model of a real bridge. To predict multiple variables with DeepONet, we utilize two strategies: a split branch/trunk and multiple DeepONets combined into a single DeepONet. In addition to data-driven training, we introduce a novel physics-informed training approaches. This method leverages structural stiffness matrices to enforce fundamental equilibrium and energy conservation principles, resulting in two novel physics-informed loss functions: energy conservation and static equilibrium using the Schur complement. We use various combinations of loss functions to achieve an error rate of less than 5% with significantly reduced training time. This study shows that DeepONet, enhanced with hybrid loss functions, can accurately and efficiently predict displacements and rotations at each mesh point, with reduced training time.

Damage identification for bridges using machine learning: Development and application to KW51 bridge

Over the past few decades, structural health monitoring (SHM) has drawn significant attention to identifying damage in structures. However, there are open challenges related to the efficiency and applicability of the existing damage identification approaches. This paper proposes an effective approach that integrates both modal analysis and dynamic analysis strategies for damage identification and applies it to the KW51 railway bridge in Leuven, Belgium. The ML-based damage identification utilizes four types of features: modal analysis input, frequency characteristics, time-frequency characteristics, and stacked time series feature extraction from forced acceleration response. Signal processing methods such as stacking, Fourier transform, and wavelet transform are employed to extract damage-sensitive features from the acceleration series. The ML methods, including the k-nearest neighbors (kNN) algorithm, stacked gated recurrent unit (stacked GRU) network, and convolutional neural network (CNN), are then combined to assess the existence, extent, and location of damage. The proposed method is applied to the KW51 railway bridge. A finite element model (FEM) of the KW51 bridge is developed, which is validated by modal analysis. Various extents and locations of damage are simulated to generate the "Damaged" data, while the "Intact" data from FEM or measured data serve as a baseline for comparison. The identification results for the KW51 bridge demonstrate the high accuracy and robustness of the proposed approach, confirming its effectiveness in damage identification problems.

A deep learning energy method for hyperelasticity and viscoelasticity

The potential energy formulation and deep learning are merged to solve partial differential equations governing the deformation in hyperelastic and viscoelastic materials. The presented deep energy method (DEM) is self-contained and meshfree. It can accurately capture the three-dimensional (3D) mechanical response without requiring any time-consuming training data generation by classical numerical methods such as the finite element method. Once the model is appropriately trained, the response can be attained almost instantly at any point in the physical domain, given its spatial coordinates. Therefore, the deep energy method is potentially a promising standalone method for solving partial differential equations describing the mechanical deformation of materials or structural systems and other physical phenomena.

Deep learning collocation method for solid mechanics: Linear elasticity, hyperelasticity, and plasticity as examples

Deep learning and the collocation method are merged and used to solve partial differential equations describing structures' deformation. We have considered different types of materials: linear elasticity, hyperelasticity (neo-Hookean) with large deformation, and von Mises plasticity with isotropic and kinematic hardening. The performance of this deep collocation method (DCM) depends on the architecture of the neural network and the corresponding hyperparameters. The presented DCM is meshfree and avoids any spatial discretization, which is usually needed for the finite element method (FEM). We show that the DCM can capture the response qualitatively and quantitatively, without the need for any data generation using other numerical methods such as the FEM. Data generation usually is the main bottleneck in most data-driven models. The deep learning model is trained to learn the model's parameters yielding accurate approximate solutions. Once the model is properly trained, solutions can be obtained almost instantly at any point in the domain, given its spatial coordinates. Therefore, the deep collocation method is potentially a promising standalone technique to solve partial differential equations involved in the deformation of materials and structural systems as well as other physical phenomena.

Machine learning accelerated topology optimization of nonlinear structures

The field of optimal design of linear elastic structures has seen many exciting successes that resulted in new architected materials and designs. With the availability of cloud computing, including high-performance computing, machine learning, and simulation, searching for optimal nonlinear structures is now within reach. In this study, we develop two convolutional neural network models to predict optimized designs for a given set of boundary conditions, loads, and volume constraints. The first convolutional neural network model is for the case of materials with a linear elastic response while the second developed model is for hyperelastic response where material and geometric nonlinearities are involved. For the nonlinear elastic case, the neo-Hookean model is utilized. For this purpose, we generate datasets, composed of the optimized designs paired with the corresponding boundary conditions, loads, and constraints, using topology optimization framework to train and validate both models. The developed models are capable of accurately predicting the optimized designs without requiring an iterative scheme and with negligible computational time. The suggested pipeline can be generalized to other nonlinear mechanics scenarios and design domains.

Prediction and optimization of mechanical properties of composites using convolutional neural networks

In this paper, we develop a convolutional neural network model to predict the mechanical properties of a two-dimensional checkerboard composite quantitatively. The checkerboard composite possesses two phases, one phase is soft and ductile while the other is stiff and brittle. The ground-truth data used in the training process are obtained from finite element analyses under the assumption of plane stress. Monte Carlo simulations and central limit theorem are used to find the size of the dataset needed. Once the training process is completed, the developed model is validated using data unseen during training. The developed neural network model captures the stiffness, strength, and toughness of checkerboard composites with high accuracy. Also, we integrate the developed model with a genetic algorithm (GA) optimizer to identify the optimal microstructural designs. The genetic algorithm optimizer adopted here has several operators, selection, crossover, mutation, and elitism. The optimizer converges to configurations with highly enhanced properties. For the case of the modulus and starting from randomly-initialized generation, the GA optimizer converges to the global maximum which involves no soft elements. Also, the GA optimizers, when used to maximize strength and toughness, tend towards having soft elements in the region next to the crack tip.