Learning solution of nonlinear constitutive material models using physics-informed neural networks: COMM-PINN

We applied physics-informed neural networks to solve the constitutive relations for nonlinear, path-dependent material behavior. As a result, the trained network not only satisfies all thermodynamic constraints but also instantly provides information about the current material state (i.e., free energy, stress, and the evolution of internal variables) under any given loading scenario without requiring initial data. One advantage of this work is that it bypasses the repetitive Newton iterations needed to solve nonlinear equations in complex material models. Additionally, strategies are provided to reduce the required order of derivation for obtaining the tangent operator. The trained model can be directly used in any finite element package (or other numerical methods) as a user-defined material model. However, challenges remain in the proper definition of collocation points and in integrating several non-equality constraints that become active or non-active simultaneously. We tested this methodology on rate-independent processes such as the classical von Mises plasticity model with a nonlinear hardening law, as well as local damage models for interface cracking behavior with a nonlinear softening law. Finally, we discuss the potential and remaining challenges for future developments of this new approach

Hardness of nitric acid treated polyethylene followed by recrystallization

The hardness variation of mek crystallized polyethylene as a consequence of controlled fuming nitric exposure has been investigated using the microindentation technique. This study complements previous results obtained using other reagents (H2SO4, C1HSO3). After HNO3 exposure the microhardness of polyethylene decreases very rapidly, instead of increasing after the first hours of treatment. The hardness decrease is correlated to the volume fraction of interlamellar microvoids arising through selective acid digestion. For longer treatment times (t > 40 h) the fragility of the material increases and the sample collapses under the indenter. The hardening of the degraded material after recrystallization from the melt is followed as a function of treatment time. The results are discussed in the light of the molecular mechanisms involved. Comparison of the experimental data with hardness calculations for ideal PE lamellar structures and chain extended dicarboxylic crystals implies that the major contribution to hardening is due to electron dense gfroups attachment at the surface of a mixed lamellar structure.Peer reviewe

Mixed formulation of physics-informed neural networks for thermo-mechanically coupled systems and heterogeneous domains

Physics-informed neural networks (PINNs) are a new tool for solving boundary value problems by defining loss functions of neural networks based on governing equations, boundary conditions, and initial conditions. Recent investigations have shown that when designing loss functions for many engineering problems, using first-order derivatives and combining equations from both strong and weak forms can lead to much better accuracy, especially when there are heterogeneity and variable jumps in the domain. This new approach is called the mixed formulation for PINNs, which takes ideas from the mixed finite element method. In this method, the PDE is reformulated as a system of equations where the primary unknowns are the fluxes or gradients of the solution, and the secondary unknowns are the solution itself. In this work, we propose applying the mixed formulation to solve multi-physical problems, specifically a stationary thermo-mechanically coupled system of equations. Additionally, we discuss both sequential and fully coupled unsupervised training and compare their accuracy and computational cost. To improve the accuracy of the network, we incorporate hard boundary constraints to ensure valid predictions. We then investigate how different optimizers and architectures affect accuracy and efficiency. Finally, we introduce a simple approach for parametric learning that is similar to transfer learning. This approach combines data and physics to address the limitations of PINNs regarding computational cost and improves the network's ability to predict the response of the system for unseen cases. The outcomes of this work will be useful for many other engineering applications where deep learning is employed on multiple coupled systems of equations for fast and reliable computations

A mixed formulation for physics-informed neural networks as a potential solver for engineering problems in heterogeneous domains: comparison with finite element method

Physics-informed neural networks (PINNs) are capable of finding the solution for a given boundary value problem. We employ several ideas from the finite element method (FEM) to enhance the performance of existing PINNs in engineering problems. The main contribution of the current work is to promote using the spatial gradient of the primary variable as an output from separated neural networks. Later on, the strong form which has a higher order of derivatives is applied to the spatial gradients of the primary variable as the physical constraint. In addition, the so-called energy form of the problem is applied to the primary variable as an additional constraint for training. The proposed approach only required up to first-order derivatives to construct the physical loss functions. We discuss why this point is beneficial through various comparisons between different models. The mixed formulation-based PINNs and FE methods share some similarities. While the former minimizes the PDE and its energy form at given collocation points utilizing a complex nonlinear interpolation through a neural network, the latter does the same at element nodes with the help of shape functions. We focus on heterogeneous solids to show the capability of deep learning for predicting the solution in a complex environment under different boundary conditions. The performance of the proposed PINN model is checked against the solution from FEM on two prototype problems: elasticity and the Poisson equation (steady-state diffusion problem). We concluded that by properly designing the network architecture in PINN, the deep learning model has the potential to solve the unknowns in a heterogeneous domain without any available initial data from other sources. Finally, discussions are provided on the combination of PINN and FEM for a fast and accurate design of composite materials in future developments

A thermo-mechanical phase-field fracture model: application to hot cracking simulations in additive manufacturing

Thermal fracture is prevalent in many engineering problems and is one of the most devastating defects in metal additive manufacturing. Due to the interactive underlying physics involved, the computational simulation of such a process is challenging. In this work, we propose a thermo-mechanical phase-field fracture model, which is based on a thermodynamically consistent derivation. The influence of different coupling terms such as damage-informed thermomechanics and heat conduction and temperature-dependent fracture properties, as well as different phase-field fracture formulations, are discussed. Finally, the model is implemented in the finite element method and applied to simulate the hot cracking in additive manufacturing. Thereby not only the thermal strain but also the solidification shrinkage are considered. As for thermal history, various predicted thermal profiles, including analytical solution and numerical thermal temperature profile around the melting pool, are regarded, whereas the latter includes the influence of different process parameters. The studies reveal that the solidification shrinkage strain takes a dominant role in the formation of the circumferential crack, while the temperature gradient is mostly responsible for the central crack. Process parameter study demonstrates further that a higher laser power and slower scanning speed are favorable for keyhole mode hot cracking while a lower laser power and quicker scanning speed tend to form the conduction mode cracking. The numerical predictions of the hot cracking patterns are in good agreement with similar experimental observations, showing the capability of the model for further studies

Comparative analysis of phase-field and intrinsic cohesive zone models for fracture simulations in multiphase materials with interfaces: Investigation of the influence of the microstructure on the fracture properties

This study evaluates four widely used fracture simulation methods, comparing their computational expenses and implementation complexities within the Finite Element (FE) framework when employed on heterogeneous solids. Fracture methods considered encompass the intrinsic Cohesive Zone Model (CZM) using zero-thickness cohesive interface elements (CIEs), the Standard Phase-Field Fracture (SPFM) approach, the Cohesive Phase-Field fracture (CPFM) approach, and an innovative hybrid model. The hybrid approach combines the CPFM fracture method with the CZM, specifically applying the CZM within the interface zone. A significant finding from this investigation is that the CPFM method is in agreement with the hybrid model when the interface zone thickness is not excessively small. This implies that the CPFM fracture methodology may serve as a unified fracture approach for multiphase materials, provided the interface zone's thickness is comparable to that of the other phases. In addition, this research provides valuable insights that can advance efforts to fine-tune material microstructures. An investigation of the influence of the interface material properties, morphological features and spatial arrangement of inclusions showes a pronounced effect of these parameters on the fracture toughness of the material

Introducing a microstructure-embedded autoencoder approach for reconstructing high-resolution solution field data from a reduced parametric space

In this study, we develop a novel multi-fidelity deep learning approach that transforms low-fidelity solution maps into high-fidelity ones by incorporating parametric space information into a standard autoencoder architecture. This method's integration of parametric space information significantly reduces the need for training data to effectively predict high-fidelity solutions from low-fidelity ones. In this study, we examine a two-dimensional steady-state heat transfer analysis within a highly heterogeneous materials microstructure. The heat conductivity coefficients for two different materials are condensed from a 101 x 101 grid to smaller grids. We then solve the boundary value problem on the coarsest grid using a pre-trained physics-informed neural operator network known as Finite Operator Learning (FOL). The resulting low-fidelity solution is subsequently upscaled back to a 101 x 101 grid using a newly designed enhanced autoencoder. The novelty of the developed enhanced autoencoder lies in the concatenation of heat conductivity maps of different resolutions to the decoder segment in distinct steps. Hence the developed algorithm is named microstructure-embedded autoencoder (MEA). We compare the MEA outcomes with those from finite element methods, the standard U-Net, and various other upscaling techniques, including interpolation functions and feedforward neural networks (FFNN). Our analysis shows that MEA outperforms these methods in terms of computational efficiency and error on test cases. As a result, the MEA serves as a potential supplement to neural operator networks, effectively upscaling low-fidelity solutions to high fidelity while preserving critical details often lost in traditional upscaling methods, particularly at sharp interfaces like those seen with interpolation

Hybrid Modeling of Lithium-Ion Battery: Physics-Informed Neural Network for Battery State Estimation

Accurate forecasting of the lifetime and degradation mechanisms of lithium-ion batteries is crucial for their optimization, management, and safety while preventing latent failures. However, the typical state estimations are challenging due to complex and dynamic cell parameters and wide variations in usage conditions. Physics-based models need a tradeoff between accuracy and complexity due to vast parameter requirements, while machine-learning models require large training datasets and may fail when generalized to unseen scenarios. To address this issue, this paper aims to integrate the physics-based battery model and the machine learning model to leverage their respective strengths. This is achieved by applying the deep learning framework called physics-informed neural networks (PINN) to electrochemical battery modeling. The state of charge and state of health of lithium-ion cells are predicted by integrating the partial differential equation of Fick’s law of diffusion from a single particle model into the neural network training process. The results indicate that PINN can estimate the state of charge with a root mean square error in the range of 0.014% to 0.2%, while the state of health has a range of 1.1% to 2.3%, even with limited training data. Compared to conventional approaches, PINN is less complex while still incorporating the laws of physics into the training process, resulting in adequate predictions, even for unseen situations

Hybrid modeling of lithium-ion battery : physics-informed neural network for battery state estimation

Accurate forecasting of the lifetime and degradation mechanisms of lithium-ion batteries is crucial for their optimization, management, and safety while preventing latent failures. However, the typical state estimations are challenging due to complex and dynamic cell parameters and wide variations in usage conditions. Physics-based models need a tradeoff between accuracy and complexity due to vast parameter requirements, while machine-learning models require large training datasets and may fail when generalized to unseen scenarios. To address this issue, this paper aims to integrate the physics-based battery model and the machine learning model to leverage their respective strengths. This is achieved by applying the deep learning framework called physics-informed neural networks (PINN) to electrochemical battery modeling. The state of charge and state of health of lithium-ion cells are predicted by integrating the partial differential equation of Fick’s law of diffusion from a single particle model into the neural network training process. The results indicate that PINN can estimate the state of charge with a root mean square error in the range of 0.014% to 0.2%, while the state of health has a range of 1.1% to 2.3%, even with limited training data. Compared to conventional approaches, PINN is less complex while still incorporating the laws of physics into the training process, resulting in adequate predictions, even for unseen situations.German Federal Ministry for Economic Affairs and Climate Action (BMWK