Learning constitutive relations from indirect observations using deep neural networks
Not provided.
Computer Science↗
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MULTISTEP AND CONTINUOUS PHYSICS-INFORMED NEURAL NETWORK METHODS FOR LEARNING GOVERNING EQUATIONS AND CONSTITUTIVE RELATIONS
In this work, we investigate the applicability and relative merit of discrete and continuous versions of physics-informed neural network (PINN) methods for learning unknown governing equations or constitutive relations in a nonlinear dynamical system. In the case of unknown dynamics, entire right-hand-side (RHS) equations of the ordinary differential equations are unknown. In the case of unknown constitutive relations, however, the RHS equations are known up to the specification of constitutive relations (that may depend on the state of the system). We use a deep neural network to model unknown governing equations or constitutive relations. The discrete PINN approach combines classical multistep discretization methods for dynamical systems with neural-network-based machine learning methods. On the other hand, the continuous versions utilize deep neural networks to minimize the residual function for the continuous governing equations. We use the case of a fedbatch bioreactor system to study the effectiveness of these approaches and discuss conditions for their applicability. Our results indicate that the accuracy of the trained neural network models is much higher for the cases where we only have to learn a constitutive relation instead of all dynamics. This finding corroborates the well-known fact from scientific computing that building as much structural information as is available into an algorithm can enhance its efficiency and/or accuracy.
97 MATHEMATICS AND COMPUTING↗
Learning Constitutive Relations From Soil Moisture Data via Physically Constrained Neural Networks
Abstract The constitutive relations of the Richardson‐Richards equation encode the macroscopic properties of soil water retention and conductivity. These soil hydraulic functions are commonly represented by models with a handful of parameters. The limited degrees of freedom of such soil hydraulic models constrain our ability to extract soil hydraulic properties from soil moisture data via inverse modeling. We present a new free‐form approach to learning the constitutive relations using physically constrained neural networks. We implemented the inverse modeling framework in a differentiable modeling framework, JAX, to ensure scalability and extensibility. For efficient gradient computations, we implemented implicit differentiation through a nonlinear solver for the Richardson‐Richards equation. We tested the framework against synthetic noisy data and demonstrated its robustness against varying magnitudes of noise and degrees of freedom of the neural networks. We applied the framework to soil moisture data from an upward infiltration experiment and demonstrated that the neural network‐based approach was better fitted to the experimental data than a parametric model and that the framework can learn the constitutive relations.
54 ENVIRONMENTAL SCIENCES↗
Robust deep learning framework for constitutive relations modeling
Modeling the full-range deformation behaviors of materials under complex loading and materials conditions is a significant challenge for constitutive relations (CRs) modeling. Here, we propose a general encoder-decoder deep learning framework that can model high-dimensional stress-strain data and complex loading histories with robustness and universal capability. The framework employs an encoder to project high-dimensional input information (e.g., loading history, loading conditions, and materials information) to a lower-dimensional hidden space and a decoder to map the hidden representation to the stress of interest. We evaluated various encoder architectures, including gated recurrent unit (GRU), GRU with attention, temporal convolutional network (TCN), and the Transformer encoder, on two complex stress-strain datasets that were designed to include a wide range of complex loading histories and loading conditions. All architectures achieved excellent test results with an root-mean-square error (RMSE) below 1 MPa. Additionally, we analyzed the capability of the different architectures to make predictions on out-of-domain applications, with an uncertainty estimation based on deep ensembles. The proposed approach provides a robust alternative to empirical/semi-empirical models for CRs modeling, offering the potential for more accurate and efficient materials design and optimization.
36 MATERIALS SCIENCE↗
Learning constitutive relations using symmetric positive definite neural networks
In this work, we present a new neural-network architecture, called the Cholesky-factored symmetric positive definite neural network (SPD-NN), for modeling constitutive relations in computational mechanics. Instead of directly predicting the stress of the material, the SPD-NN trains a neural network to predict the Cholesky factor of the tangent stiffness matrix, based on which the stress is calculated in incremental form. As a result of this special structure, SPD-NN weakly imposes convexity on the strain energy function, satisfies the second order work criterion (Hill's criterion) and time consistency for path-dependent materials, and therefore improves numerical stability, especially when the SPD-NN is used in finite element simulations. Depending on the types of available data, we propose two training methods, namely direct training for strain and stress pairs and indirect training for loads and displacement pairs. We demonstrate the effectiveness of SPD-NN on hyperelastic, elasto-plastic, and multiscale fiber-reinforced plate problems from solid mechanics. The generality and robustness of SPD-NN make it a promising tool for a wide range of constitutive modeling applications.
71 CLASSICAL AND QUANTUM MECHANICS, GENERAL PHYSIC↗
Learning constitutive relations of plasticity using neural networks and full-field data
Not Available
Zhang, Yin (ORCID:0000000291942233)↗
A shear modified enhanced Gurson constitutive relation and implications for localization
Not Available
Khan, I. A.↗
Modified error-in-constitutive-relation (MECR) framework for the characterization of linear viscoelastic solids
Not provided.
Materials Science↗
Modeling the non-Schmid crystallographic slip in MAX phases
We present a crystal plasticity constitutive relation for the description of experimentally observed non-Schmid crystallographic slip in a class of ternary carbides and nitrides commonly referred to as MAX phases. In the constitutive relation, we assume that the evolution of the slip system strength in MAX phases has two components – a classical component that depends on the Taylor cumulative shear strain and a non-Schmid component that depends on the stress normal to the slip plane. The non-Schmid crystal plasticity constitutive relation is then used to carry out finite element simulations of micropillar compression of single crystals of two MAX phases, Ti 2 AlC and Ti 3 AlC 2 . The finite element simulations not only quantitatively predict the stress – strain response of a wide range of crystallographic orientations of the micropillars but also rationalize the non-uniform deformation and the deformed shape of the micropillars observed in the experiments for the two materials. As a result, parametric studies are also carried out to quantify the role of the non-Schmid effect and understand the effects of key experimental parameters on the stress – strain response of the micropillars of the two MAX phases.
36 MATERIALS SCIENCE↗
A combined ensemble-volume average homogenization method for lattice structures with defects under dynamic and static loading
In the study of lattices structures, both experiments and numerical simulations are often conducted with small samples. Using combined ensemble and volume averaging, this work introduces a method to extract a macroscopic constitutive response of a lattice material from numerical simulations performed in periodic domains. The domain size needed to obtain statistically accurate results is investigated. Similar to molecular dynamics, the concept of the virial stress is introduced after homogenized equations are derived using the ensemble averaging method. Under static conditions, the virial stress is shown to agree with the volume averaged solid stress. Using the homogenization method, constitutive relations for this stress can be obtained from systems with uniform strains. Application of such obtained constitutive relations to more general cases results in an error proportional to the square of the ratio between the lattice length scale and the macroscopic length scale. Taking advantage of this property, numerical simulations are performed in systems with a uniform gradient of the average velocity. The volume average method is then used to accelerate convergence when studying lattices with defects. To avoid the artificial numerical time scale from the size of a representative volume element divided by the wave speed, a numerical scheme is developed to enforce a spatially uniform velocity gradient within the computational domain while allowing fluctuations of the velocity or displacement to develop naturally. To account for probability distribution of lattice defects, the stress is calculated as the ensemble-volume averaged value. For dynamic systems, energy dissipation properties are also studied.
36 MATERIALS SCIENCE↗
CTF Improved Drag Model and Flow Regime Transition Criteria
The demand for accurate prediction of two-phase flow behavior in a boiling water reactor (BWR) requires a comprehensive understanding of flow regime, void fraction, heat transfer, and pressure drop. The CTF subchannel code, which is used for the Thermal/Hydraulic (T/H) solution in the Consortium for Advanced Simulation of Light Water Reactors (CASL)-developed Virtual Environment for Reactor Application (VERA) core simulator, is being further developed for BWR applications. In support of this goal, the present work highlights some of the two-phase closure model developments towards improving the CTF void fraction prediction, especially for subcooled boiling. The drift-flux approach has been well-developed for upward dispersed two-phase flows and proven to be accurate in predicting void fraction in bubbly and slug flow regimes. In this work, these kinematic constitutive relations for the drift-flux velocity have been implemented into CTF to describe the interfacial drag of bubbly flow as an alternative to the existing model for better void fraction prediction. The success of these constitutive relations also relies on a good flow regime map that accounts for flow conditions and channel geometry. A more reliable flow regime transition criteria that account for the flow condition has also been implemented in this study for modeling the flow regime transition criteria. The newly implemented models are shown to give improved void fraction predictions in comparison to experimental data.
Hizoum, Belgacem↗
Modeling the viscoplastic behavior of a semicrystalline polymer
In this study, a complex constitutive relation is identified using inverse modeling with the nominal mechanical response as sole experimental input. The methodology is illustrated for a semicrystalline thermoplastic in the presence of strain localization at finite deformations. The experimental database includes cylindrical tensile bars, compression pins and round notched bars loaded at strain rates spanning up to five decades and temperatures below and above T g . The data is organized into a calibration set and a validation set. The response of tensile specimens is determined using finite element analyses and a two-phase constitutive relation for semicrystalline polymers that accounts for temperature- and rate-sensitive plastic flow, pressure-sensitivity, small-strain softening and large-strain orientational hardening of the amorphous phase, along with the evolution of crystallinity. The large number of constitutive parameters is identified using an optimization tool coupled with the finite element solver and the calibration set from experiments. The methodology is shown to be successful in predicting the response of round notched bars and replicating the effects of temperature and strain rate on the severity of necking in tensile bars. The proposed model identification strategy is both simple and effective in comparison with other elaborate methods that attempt to access intrinsic behavior directly from high-fidelity experimental measurements.
36 MATERIALS SCIENCE↗
Linear and Nonlinear Solvers for Simulating Multiphase Flow within Large-Scale Engineered Subsurface Systems
Simulation of multiphase flow in the subsurface is well-known to be computationally challenging. While there have been many studies that have explored approaches to overcoming these challenges, they often utilize relatively simple case studies. In this paper, we focus on the unique numerical challenges posed by modeling large-scale engineered subsurface systems, characterized by discrete features embedded in a heterogeneous natural subsurface setting. The man-made features such as shafts, tunnels, and barriers often cause multiple challenges in modeling the domain for multiphase porous media flow. This flow scenario can have a wide range of applications such as nuclear waste repositories, enhanced recovery of a petroleum reservoir, geothermal engineering, and carbon sequestration. An example of these severe numerical challenges is the case of performance assessment (PA) for Waste Isolation Pilot Plant (WIPP), the only operating deep geological repository in the US, which simulates extreme material properties of bedded salt rock formation and extreme contrast due to open excavation next to the formation. The models have extremes not only of permeability and porosity but also of the constitutive models needed for multiphase flow; additionally, they have process models like salt creep closure reducing porosity over time, fracturing in clay and anhydrite interbeds of the bedded salt, gas generation from the waste materials, and unintentional human borehole intrusions in some scenarios. Numerical simulations require the solution of coupled systems of nonlinear PDEs; in our work, we use the open-source simulator PFLOTRAN which is based on Finite Volume discretization. The solution of the nonlinear equations requires use of the Newton-Raphson iteration at each time step, which entails the solution of the linearized Jacobian system at each iteration. The effects of all the processes (i.e., large number of unknowns, highly nonlinear constitutive relations, large contrasts in material properties in short distances) lead to an ill-conditioned Jacobian matrix that severely challenges traditional linear solver, i.e., stabilized biconjugate gradient with block Jacobi incomplete LU preconditioner (BCGS-ILU) leading to non-convergence for traditional Newton-Raphson nonlinear solver causing unacceptably long computation time for each model. This paper presents linear solvers such as constrained pressure residual (CPR) two-stage preconditioner with alternate-block-factorization (ABF) and quasi- implicit pressure and explicit saturation (QIMPES) decouplers and flexible generalized residual solver (FGMRES). The new general-purpose nonlinear solver, Newton trust-region dogleg Cauchy (NTRDC), is also introduced to resolve extreme nonlinearities in the models. We demonstrate the effectiveness of each method relative to the default BCGS-Newton solver. The two best cases had nearly 50 times speed-up and achieved completion of a simulation in 14 hours that never completed due to non-convergence with the default solver. We also investigate the strong scalability of each method and discuss some of the deficiencies found for Block Jacobi preconditioner using parallel domain decomposition, and node packing effects of modern processor architecture.
Preconditioner, Nonlinear, Porous media, Multiphas↗
Learning macroscopic internal variables and history dependence from microscopic models
This paper concerns the study of history dependent phenomena in heterogeneous materials in a two-scale setting where the material is specified at a fine microscopic scale of heterogeneities that is much smaller than the coarse macroscopic scale of application. Here, we specifically study a polycrystalline medium where each grain is governed by crystal plasticity while the solid is subjected to macroscopic dynamic loads. The theory of homogenization allows us to solve the macroscale problem directly with a constitutive relation that is defined implicitly by the solution of the microscale problem. However, the homogenization leads to a highly complex history dependence at the macroscale, one that can be quite different from that at the microscale. In this paper, we examine the use of machine-learning, and especially deep neural networks, to harness data generated by repeatedly solving the finer scale model to: (i) gain insights into the history dependence and the macroscopic internal variables that govern the overall response; and (ii) to create a computationally efficient surrogate of its solution operator, that can directly be used at the coarser scale with no further modeling. We do so by introducing a recurrent neural operator (RNO), and show that: (i) the architecture and the learned internal variables can provide insight into the physics of the macroscopic problem; and (ii) that the RNO can provide multiscale, specifically FE 2 , accuracy at a cost comparable to a conventional empirical constitutive relation.
36 MATERIALS SCIENCE↗
A unified finite strain gradient-enhanced micropolar continuum approach for modeling quasi-brittle failure of cohesive-frictional materials
Here, in this work, a novel framework for modeling quasi-brittle crack propagation and shear band dominated failure of cohesive-frictional materials like concrete, mortar, rock, tough ceramics, energetic materials, but also granular materials like sands or powders in terms of a unified continuum approach is proposed. It is based on a combination of the gradient-enhanced continuum with gradients of internal variables for representing quasi-brittle cracking, and the micropolar continuum, accounting for the deformation of the microstructure. For developing the gradient-enhanced micropolar framework, the set of balance equations and the kinematic relations are derived, and the constitutive relations are established in a general manner. The framework is formulated in a geometrically exact setting, based on the thermodynamically sound theory of hyperelasto-plasticity, and the numerical implementation by means of the finite element method is discussed. For assessing the approach, realizations of this new approach in terms of constitutive models for particular materials are developed. They are applied to numerical benchmark examples, investigating various loading conditions, and the obtained results are validated by means of a comparison with experiments from the literature.
36 MATERIALS SCIENCE↗
Space-time dependent thermal conductivity in nonlocal thermal transport
Nonlocal thermal transport is generally described by the Peierls-Boltzmann transport equation (PBE). However, solving the PBE for a general space-time dependent problem remains a challenging task due to the high dimensionality of the integro-differential equation. In this work, we present a direct solution to the space-time dependent PBE with a linearized collision matrix using an eigendecomposition method. We show that there exists a generalized Fourier-type relation that links heat flux to the local temperature, and this constitutive relation defines a thermal conductivity that depends on both time and space. Combining this approach with ab initio calculations of phonon properties, we demonstrate that the space-time dependent thermal conductivity gives rise to an oscillatory response in temperature in a transient grating geometry in high thermal conductivity materials. The present solution method allows us to extend the reach of our computational capability for heat conduction to space-time dependent nondiffusive transport regimes. Here, this capability will not only enable a more accurate interpretation of thermal measurements that observe nonlocal thermal transport, but also enhance our physical understanding of nonlocal thermal transport in high thermal conductivity materials that are promising candidates for nanoscale thermal management applications.
75 CONDENSED MATTER PHYSICS, SUPERCONDUCTIVITY AND↗
Closures and Simulation for Thermal Radiation Transport in Stochastic Media with Nonlinear Temperature Dependence
Because of the practical challenge of rendering very complex realistic spatial structures for numerical work, it is common practice to resort to characterizing such media as stochastic mixtures of materials, ideally parametrized with low order statistics such as the mean, variance, and correlation functions of the now random material properties. This enables realizations of the medium to be repeatedly generated and radiation transport computations to, in principle, be performed for a large ensemble of these realizations to obtain a statistically well-characterized radiation field. Statistical post-processing yields desired quantities such as conditional and unconditional mean radiation flux and probability distributions of transmitted radiation. However, such computations prove expensive for all but the simplest stochastic geometries and are most suited for benchmarking approximate models. The most common approximations lead to homogenized media so that transport computations are required only on a single medium realization but by construct provide only limited statistical information on the radiation field. Almost all approximate approaches to this problem attempt to develop equations for low order moments of the radiation intensity (mean, second moment, correlation function) but inevitably encounter a closure problem: the equation for any statistical moment will contain terms depending on unknown higher-order moments. Thus, the challenge shifts to one of developing closure relations that relate the unknown moments to the lower order moments. Under very special conditions, an exact closure can be derived but in general closures are heuristically stated constitutive relations. Also, closure approaches depend on whether the mixing statistics are spatially and/or temporally continuous as in fluctuating turbulent fields, or discontinuous as in randomly mixed solid chunks of material. Thus, unconditional averaging is generally applied in the former case but conditional averaging is more appropriate when the mixing is discontinuous. In this work, the emphasis is on binary statistical mixtures of immiscible fluids as as such the quantities of interest are averages (flux, temperature) conditioned on the material type.
73 NUCLEAR PHYSICS AND RADIATION PHYSICS↗