Integrated computational materials engineering (ICME) models have been a crucial building block for modern materials development, relieving heavy reliance on experiments and significantly accelerating the materials design process. However, ICME models are also computationally expensive, particularly with respect to time integration for dynamics, which hinders the ability to study statistical ensembles and thermodynamic properties of large systems for long time scales. To alleviate the computational bottleneck, we propose to model the evolution of statistical microstructure descriptors as a continuous-time stochastic process using a non-linear Langevin equation, where the probability density function (PDF) of the statistical microstructure descriptors, which are also the quantities of interests (QoIs), is modeled by the Fokker-Planck equation. We discuss how to calibrate the drift and diffusion terms of the Fokker-Planck equation from the theoretical and computational perspectives. The calibrated Fokker-Planck equation can be used as a stochastic reduced-order model to simulate the microstructure evolution of statistical microstructure descriptors PDF. Considering statistical microstructure descriptors in the microstructure evolution as QoIs, we demonstrate our proposed methodology in three integrated computational materials engineering (ICME) models: kinetic Monte Carlo, phase field, and molecular dynamics simulations.
Uncertainty quantification (UQ) plays a major role in verification and validation for computational engineering models and simulations, and establishes trust in the predictive capability of computational models. In the materials science and engineering context, where the process-structure-property-performance linkage is well known to be the only road mapping from manufacturing to engineering performance, numerous integrated computational materials engineering (ICME) models have been developed across a wide spectrum of length-scales and time-scales to relieve the burden of resource-intensive experiments. Within the structure-property linkage, crystal plasticity finite element method (CPFEM) models have been widely used since they are one of a few ICME toolboxes that allows numerical predictions, providing the bridge from microstructure to materials properties and performances. Several constitutive models have been proposed in the last few decades to capture the mechanics and plasticity behavior of materials. While some UQ studies have been performed, the robustness and uncertainty of these constitutive models have not been rigorously established. In this work, we apply a stochastic collocation (SC) method, which is mathematically rigorous and has been widely used in the field of UQ, to quantify the uncertainty of three most commonly used constitutive models in CPFEM, namely phenomenological models (with and without twinning), and dislocation-density-based constitutive models, for three different types of crystal structures, namely face-centered cubic (fcc) copper (Cu), body-centered cubic (bcc) tungsten (W), and hexagonal close packing (hcp) magnesium (Mg). Our numerical results not only quantify the uncertainty of these constitutive models in stress-strain curve, but also analyze the global sensitivity of the underlying constitutive parameters with respect to the initial yield behavior, which may be helpful for robust constitutive model calibration works in the future.
This report documents the results of an FY22 ASC V&V level 2 milestone demonstrating new algorithms for multifidelity uncertainty quantification. Part I of the report describes the algorithms, studies their performance on a simple model problem, and then deploys the methods to a thermal battery example from the open literature. Part II (restricted distribution) applies the multifidelity UQ methods to specific thermal batteries of interest to the NNSA/ASC program.
Determining process–structure–property linkages is one of the key objectives in material science, and uncertainty quantification plays a critical role in understanding both process–structure and structure–property linkages. In this work, we seek to learn a distribution of microstructure parameters that are consistent in the sense that the forward propagation of this distribution through a crystal plasticity finite element model matches a target distribution on materials properties. This stochastic inversion formulation infers a distribution of acceptable/consistent microstructures, as opposed to a deterministic solution, which expands the range of feasible designs in a probabilistic manner. To solve this stochastic inverse problem, we employ a recently developed uncertainty quantification framework based on push-forward probability measures, which combines techniques from measure theory and Bayes’ rule to define a unique and numerically stable solution. This approach requires making an initial prediction using an initial guess for the distribution on model inputs and solving a stochastic forward problem. To reduce the computational burden in solving both stochastic forward and stochastic inverse problems, we combine this approach with a machine learning Bayesian regression model based on Gaussian processes and demonstrate the proposed methodology on two representative case studies in structure–property linkages.
Process-structure linkage is one of the most important topics in materials science due to the fact that virtually all information related to the materials, including manufacturing processes, lies in the microstructure itself. Therefore, to learn more about the process, one must start by thoroughly examining the microstructure. This gives rise to inverse problems in the context of process-structure linkages, which attempt to identify the processes that were used to manufacturing the given microstructure. In this work, we present an inverse problem for structure-process linkages which we solve using asynchronous parallel Bayesian optimization which exploits parallel computing resources. We demonstrate the effectiveness of the method using kinetic Monte Carlo model for grain growth simulation.
We present a scale-bridging approach based on a multi-fidelity (MF) machine-learning (ML) framework leveraging Gaussian processes (GP) to fuse atomistic computational model predictions across multiple levels of fidelity. Through the posterior variance of the MFGP, our framework naturally enables uncertainty quantification, providing estimates of confidence in the predictions. We used density functional theory as high-fidelity prediction, while a ML interatomic potential is used as low-fidelity prediction. Practical materials' design efficiency is demonstrated by reproducing the ternary composition dependence of a quantity of interest (bulk modulus) across the full aluminum-niobium-titanium ternary random alloy composition space. The MFGP is then coupled to a Bayesian optimization procedure, and the computational efficiency of this approach is demonstrated by performing an on-the-fly search for the global optimum of bulk modulus in the ternary composition space. The framework presented in this manuscript is the first application of MFGP to atomistic materials simulations fusing predictions between density functional theory and classical interatomic potential calculations.
Determining a process–structure–property relationship is the holy grail of materials science, where both computational prediction in the forward direction and materials design in the inverse direction are essential. Problems in materials design are often considered in the context of process–property linkage by bypassing the materials structure, or in the context of structure–property linkage as in microstructure-sensitive design problems. However, there is a lack of research effort in studying materials design problems in the context of process–structure linkage, which has a great implication in reverse engineering. In this work, given a target microstructure, we propose an active learning high-throughput microstructure calibration framework to derive a set of processing parameters, which can produce an optimal microstructure that is statistically equivalent to the target microstructure. The proposed framework is formulated as a noisy multi-objective optimization problem, where each objective function measures a deterministic or statistical difference of the same microstructure descriptor between a candidate microstructure and a target microstructure. Furthermore, to significantly reduce the physical waiting wall-time, we enable the high-throughput feature of the microstructure calibration framework by adopting an asynchronously parallel Bayesian optimization by exploiting high-performance computing resources. Case studies in additive manufacturing and grain growth are used to demonstrate the applicability of the proposed framework, where kinetic Monte Carlo (kMC) simulation is used as a forward predictive model, such that for a given target microstructure, the target processing parameters that produced this microstructure are successfully recovered.
Bayesian optimization (BO) is an effective surrogate-based method that has been widely used to optimize simulation-based applications. While the traditional Bayesian optimization approach only applies to single-fidelity models, many realistic applications provide multiple levels of fidelity with various levels of computational complexity and predictive capability. In this work, we propose a multi-fidelity Bayesian optimization method for design applications with both known and unknown constraints. The proposed framework, called sMF-BO-2CoGP, is built on a multi-level CoKriging method to predict the objective function. An external binary classifier, which we approximate using a separate CoKriging model, is used to distinguish between feasible and infeasible regions. Finally, the sMF-BO-2CoGP method is demonstrated using a series of analytical examples and a flip-chip application for design optimization to minimize the deformation due to warping under thermal loading conditions.
We present a formulation to simultaneously invert for a heterogeneous shear modulus field and traction boundary conditions in an incompressible linear elastic plane stress model. Our approach utilizes scalable deterministic methods, including adjoint-based sensitivities and quasi-Newton optimization, to reduce the computational requirements for large-scale inversion with partial differential equation (PDE) constraints. Here, we address the use of regularization for such formulations and explore the use of different types of regularization for the shear modulus and boundary traction. We apply this PDE-constrained optimization algorithm to a synthetic dataset to verify the accuracy in the reconstructed parameters, and to experimental data from a tissue-mimicking ultrasound phantom. In all of these examples, we compare inversion results from full-field and sparse data measurements.