In hydrology, projected climate change impact assessment studies typically rely on ensembles of downscaled climate model outputs. Due to large modeling uncertainties, the ensembles are often averaged to provide a basis for studying the effects of climate change. A key issue when analyzing averages of a climate model ensemble is whether to weight all models in the ensemble equally, often referred to as the equal‐weights or unweighted approach, or to use a weighted approach, where, in general, each model would have a different weight. Many studies have advocated for the latter, based on the assumption that models that are better at simulating the past, that is, the models with higher hindcast accuracy, will give more accurate forecasts for the future and thus should receive higher weights. To examine this issue, observed and modeled daily precipitation frequency (PF) estimates for three urban areas in the United States, namely Boston, Massachusetts; Houston, Texas; and Chicago, Illinois, were analyzed. The comparison used the raw output of 24 Coupled Model Intercomparison Project Phase 5 (CMIP5) models. The PFs from these models were compared with the observed PFs for a specific historical training period to determine model weights for each area. The unweighted and weighted averaged model PFs from a more recent testing period were then compared with their corresponding observed PFs to determine if weights improved the estimates. These comparisons indeed showed that the weighted averages were closer to the observed values than the unweighted averages in nearly all cases. The study also demonstrated how weights can help reduce model spread in future climate projections by comparing the unweighted and weighted ensemble standard deviations in these projections. In all studied scenarios, the weights actually reduced the standard deviations compared to the equal‐weights approach. Finally, an analysis of the results' sensitivity to the areal reduction factor used to allow comparisons between point station measurements and grid‐box averages is provided.

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.

A graph convolutional network (GCN) is employed in the deep energy method (DEM) model to solve the momentum balance equation in three‐dimensional space 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. The method's 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.

This paper explores the possibilities of applying physics-informed neural networks (PINNs) in topology optimization (TO) by introducing a fully self-supervised TO framework based on PINNs. This framework solves the forward elasticity problem by the deep energy method (DEM). Instead of training a separate neural network to update the density distribution, we leverage the fact that the compliance minimization problem is self-adjoint to express the element sensitivity directly in terms of the displacement field from the DEM model. Thus, no additional neural network is needed for the inverse problem. The method of moving asymptotes is used as the optimizer for updating density distribution. The implementation of Neumann, Dirichlet, and periodic boundary conditions is described in the context of the DEM model. Three numerical examples are presented to demonstrate framework capabilities: (i) compliance minimization in 2D under different geometries and loading, (ii) compliance minimization in 3D, and (iii) maximization of homogenized shear modulus to design 2D metamaterial unit cells. The results show that the optimized designs from the DEM-based framework are very comparable to those generated by the finite element method and shed light on a new way of integrating PINN-based simulation methods into classical computational mechanics problems.

Using recent advancements in high-performance computing data assimilation to combine satellite InSAR data with numerical models, the prolonged unrest of the Sierra Negra volcano in the Galápagos was tracked to provide a fortuitous, but successful, forecast 5 months in advance of the 26 June 2018 eruption. Subsequent numerical simulations reveal that the evolution of the stress state in the host rock surrounding the Sierra Negra magma system likely controlled eruption timing. While changes in magma reservoir pressure remained modest (<15 MPa), modeled widespread Mohr-Coulomb failure is coincident with the timing of the 26 June 2018 moment magnitude 5.4 earthquake and subsequent eruption. Coulomb stress transfer models suggest that the faulting event triggered the 2018 eruption by encouraging tensile failure along the northern portion of the caldera. These findings provide a critical framework for understanding Sierra Negra’s eruption cycles and evaluating the potential and timing of future eruptions.

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.

Iterative methods are widely used for solving sparse linear systems of equations and eigenvalue problems. Their performances are relevant to the conditioning of the linear systems. This work explores factors which affects the conditioning of the discretized system, including material heterogeneity, different constitutive characteristics and element sizes, and reveals the dependencies among solvers performance and the conditioning of linear systems. Results show that multiple materials can alter the eigenvalue distributions significantly, while lowering Young’s modulus results in higher condition numbers but has less effects on the spectral scope, additionally, there is a approximately reciprocal square linear relation between element size and condition numbers. These entangled effects along with the chosen pre-conditioners render that there is no simple monotonic increasing dependency among condition numbers and solving time, except with specific conditions. It is hoped that this work will provide more understanding of the iterative sparse linear solver behavior used in similar structural problems.

The solidifying steel follows highly nonlinear thermo-mechanical behavior depending on the loading history, temperature, and metallurgical phase fraction calculations (liquid, ferrite, and austenite). Numerical modeling with a computationally challenging multiphysics approach is used on high-performance computing to generate sufficient training and testing data for subsequent deep learning. We have demonstrated how the innovative sequence deep learning methods can learn from multiphysics modeling data of a solidifying slice traveling in a continuous caster and correctly and instantly capture the complex history and temperature-dependent phenomenon in test data samples never seen by the deep learning networks.

Abstract Predicting history-dependent materials’ responses is crucial, as path-dependent behavior appears while characterizing or geometrically designing many materials (e.g., metallic and polymeric cellular materials), and it takes place in manufacturing and processing of many materials (e.g., metal solidification). Such phenomena can be computationally intensive and challenging when numerical schemes such as the finite element method are used. Here, we have applied a variety of sequence learning models to almost instantly predict the history-dependent responses (stresses and energy) of a class of cellular materials as well as the multiphysics problem of steel solidification with multiple thermo-viscoplasticity constitutive models accounting for substantial temperature, time, and path dependencies, and phase transformation. We have shown the gated recurrent unit (GRU) as well as the temporal convolutional network (TCN), can both accurately learn and almost instantly predict these irreversible, and history- and time-dependent phenomena, while TCN is more computationally efficient during the training process. This work may open the door for the broader adoption of data-driven models in similar computationally challenging constitutive models in plasticity and inelasticity.

This article introduces a computational design framework for obtaining three‐dimensional (3D) periodic elastoplastic architected materials with enhanced performance, subject to uniaxial or shear strain. A nonlinear finite element model accounting for plastic deformation is developed, where a Lagrange multiplier approach is utilized to impose periodicity constraints. The analysis assumes that the material obeys a von Mises plasticity model with linear isotropic hardening. The finite element model is combined with a corresponding path‐dependent adjoint sensitivity formulation, which is derived analytically. The optimization problem is parametrized using the solid isotropic material penalization method. Designs are optimized for either end compliance or toughness for a given prescribed displacement. Such a framework results in producing materials with enhanced performance through much better utilization of an elastoplastic material. Several 3D examples are used to demonstrate the effectiveness of the mathematical framework.