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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.

The available tools for damage identification in civil engineering structures are known to be computationally expensive and data-demanding. This paper proposes a comprehensive machine learning based damage identification (CMLDI) method that integrates modal analysis and dynamic analysis strategies. The proposed approach is applied to a real structure - KW51 railway bridge in Leuven. CMLDI diligently combines signal processing, machine learning (ML), and structural analysis techniques to achieve a fast damage identification solver that relies on minimal monitoring data. CMLDI considers modal analysis inputs and extracted features from acceleration responses to inform the damage identification based on the long-term and short-term monitoring data. Results of operational modal analysis, through the analysis of long-term monitoring data, are analyzed using pre-trained k-nearest neighbor (kNN) classifiers to identify damage existence, location, and magnitude. A well-crafted assembly of signal processing and ML methods is used to analyze acceleration time histories. Stacked gated recurrent unit (Stacked GRU) networks are used to identify damage existence, kNN classifiers are used to identify damage magnitude, and convolutions neural networks (CNN) are used to identify damage location. The damage identification results for the KW51 bridge demonstrate this approach's high accuracy, efficiency, and robustness. In this work, the training data is retrieved from the sensor of the KW51 bridge as well as the numerical finite element model (FEM). The proposed approach presents a systematic path to the generation of training data using a validated FEM. The data generation relies on modeling combinations of damage locations and magnitudes along the bridge.

The modern digital engineering design often requires costly repeated simulations for different scenarios. The prediction capability of neural networks (NNs) makes them suitable surrogates for providing design insights. However, only a few NNs can efficiently handle complex engineering scenario predictions. We introduce a new version of the neural operators called DeepOKAN, which utilizes Kolmogorov Arnold networks (KANs) rather than the conventional neural network architectures. Our DeepOKAN uses Gaussian radial basis functions (RBFs) rather than the B-splines. RBFs offer good approximation properties and are typically computationally fast. The KAN architecture, combined with RBFs, allows DeepOKANs to represent better intricate relationships between input parameters and output fields, resulting in more accurate predictions across various mechanics problems. Specifically, we evaluate DeepOKAN's performance on several mechanics problems, including 1D sinusoidal waves, 2D orthotropic elasticity, and transient Poisson's problem, consistently achieving lower training losses and more accurate predictions compared to traditional DeepONets. This approach should pave the way for further improving the performance of neural operators.

Most currently available methods for modeling multiphysics, including thermoelasticity, using machine learning approaches, are focused on solving complete multiphysics problems using data-driven or physics-informed multi-layer perceptron (MLP) networks. Such models rely on incremental step-wise training of the MLPs, and lead to elevated computational expense; they also lack the rigor of existing numerical methods like the finite element method. We propose an integrated finite element neural network (I-FENN) framework to expedite the solution of coupled transient thermoelasticity. A novel physics-informed temporal convolutional network (PI-TCN) is developed and 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; then, the network output is used to compute the other fields of interest using the finite element method. 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 spatiotemporal evolution of the temperature field across the simulation time based on the energy equation and 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 I-FENN's computational efficiency and generalization capability in thermoelasticity through several numerical examples.

Triply periodic minimal surface (TPMS) metamaterials characterized by mathematically-controlled topologies exhibit better mechanical properties compared to uniform structures. The unit cell topology of such metamaterials can be further optimized to improve a desired mechanical property for a specific application. However, such inverse design involves multiple costly 3D finite element analyses in topology optimization and hence has not been attempted. Data-driven models have recently gained popularity as surrogate models in the geometrical design of metamaterials. Gyroid-like unit cells are designed using a novel voxel algorithm, a homogenization-based topology optimization, and a Heaviside filter to attain optimized densities of 0-1 configuration. Few optimization data are used as input-output for supervised learning of the topology optimization process from a 3D CNN model. These models could then be used to instantaneously predict the optimized unit cell geometry for any topology parameters, thus alleviating the need to run any topology optimization for future design. The high accuracy of the model was demonstrated by a low mean square error metric and a high dice coefficient metric. This accelerated design of 3D metamaterials opens the possibility of designing any computationally costly problems involving complex geometry of metamaterials with multi-objective properties or multi-scale applications.

Physics-informed neural networks have gained growing interest. Specifically, they are used to solve partial differential equations governing several physical phenomena. However, physics-informed neural network models suffer from several issues and can fail to provide accurate solutions in many scenarios. We discuss a few of these challenges and the techniques, such as the use of Fourier transform, that can be used to resolve these issues. This paper proposes and develops a physics-informed neural network model that combines the residuals of the strong form and the potential energy, yielding many loss terms contributing to the definition of the loss function to be minimized. Hence, we propose using the coefficient of variation weighting scheme to dynamically and adaptively assign the weight for each loss term in the loss function. The developed PINN model is standalone and meshfree. In other words, it can accurately capture the mechanical response without requiring any labeled data. Although the framework can be used for many solid mechanics problems, we focus on three-dimensional (3D) hyperelasticity, where we consider two hyperelastic models. Once the model is trained, the response can be obtained almost instantly at any point in the physical domain, given its spatial coordinates. We demonstrate the framework's performance by solving different problems with various boundary conditions.

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.

Surrogate neural network-based models have been lately trained and used in a variety of science and engineering applications where the number of evaluations of a target function is limited by execution time. In cell phone camera systems, various errors, such as interferences at the lens-barrel and lens-lens interfaces and axial, radial, and tilt misalignments, accumulate and alter profile of the lenses in a stochastic manner which ultimately changes optical focusing properties. Nonlinear finite element analysis of the stochastic mechanical behavior of lenses due to the interference fits is used on high-performance computing (HPC) to generate sufficient training and testing data for subsequent deep learning. Once properly trained and validated, the surrogate neural network model enabled accurate and almost instant evaluations of millions of function evaluations providing the final lens profiles. This computational model, enhanced by artificial intelligence, enabled us to efficiently perform Monte-Carlo analysis for sensitivity and uncertainty quantification of the final lens profile to various interferences. It can be further coupled with an optical analysis to perform ray tracing and analyze the focal properties of the lens module. Moreover, it can provide a valuable tool for optimizing tolerance design and intelligent components matching for many similar press-fit assembly processes.

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.

The field of optimal design of linear elastic structures has seen many exciting successes that resulted in new architected materials and structural 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 convolutional neural network models to predict optimized designs for a given set of boundary conditions, loads, and optimization constraints. We have considered the case of materials with a linear elastic response with and without stress constraint. Also, we have considered the case of materials with a 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 a topology optimization framework to train and validate the neural network models. The developed models are capable of accurately predicting the optimized designs without requiring an iterative scheme and with negligible inference computational time. The suggested pipeline can be generalized to other nonlinear mechanics scenarios and design domains.

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.