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Designing alloys for additive manufacturing (AM) presents significant opportunities. Still, the chemical composition and processing conditions required for printability (ie., their suitability for fabrication via AM) are challenging to explore using solely experimental means. In this work, we develop a high-throughput (HTP) computational framework to guide the search for highly printable alloys and appropriate processing parameters. The framework uses material properties from state-of-the-art databases, processing parameters, and simulated melt pool profiles to predict process-induced defects, such as lack-of-fusion, keyholing, and balling. We accelerate the printability assessment using a deep learning surrogate for a thermal model, enabling a 1,000-fold acceleration in assessing the printability of a given alloy at no loss in accuracy when compared with conventional physics-based thermal models. We verify and validate the framework by constructing printability maps for the CoCrFeMnNi Cantor alloy system and comparing our predictions to an exhaustive 'in-house' database. The framework enables the systematic investigation of the printability of a wide range of alloys in the broader Co-Cr-Fe-Mn-Ni HEA system. We identified the most promising alloys that were suitable for high-temperature applications and had the narrowest solidification ranges, and that was the least susceptible to balling, hot-cracking, and the formation of macroscopic printing defects. A new metric for the global printability of an alloy is constructed and is further used for the ranking of candidate alloys. The proposed framework is expected to be integrated into ICME approaches to accelerate the discovery and optimization of novel high-performance, printable alloys., Comment: 46 pages, 20 figures

The inelastic deformation response of NiTi single crystals involves reversible phase transformation and dislocation slip, which is enhanced by the deformation incompatibility among the phases. The phase transformation–plasticity coupling results in decrease in performance, including reduced work output and early fatigue failure. The characterization of the inelastic properties in this material class is crucial for material assessment/ranking and robust performance predictions. Given that direct mesoscale measurements of (coupled) deformation mechanisms are in many cases impractical, top-down characterization of single-crystal properties from limited macroscopic experiments is mostly employed. Here, Bayesian inference and a micromechanics-based continuum single-crystal model are adopted for determining (i) material property values within confidence intervals that allow for a propagation of the quantified uncertainty onto performance predictions, which can be used toward a more efficient design methodology; (ii) ranking of the relative influence of the various material parameters on the deformation response that can further translate to the respective influence of the various deformation mechanisms conditional on the adopted material model; and (iii) a quantitative evaluation of the importance of the deformation incompatibility among the phases in the overall deformation response.

Significant advances in theory, simulation tools, advanced computing infrastructure, and experimental frameworks have enabled the field of materials science to become increasingly reliant on computer simulations. Theory-grounded computational models provide a better understanding of observed materials phenomena. At the same time, computational tools constitute an important ingredient of any framework that seeks to accelerate the materials development cycle. While simulations keep increasing in sophistication, formal frameworks for the quantification, propagation, and management of their uncertainties are required. Uncertainty analysis is fundamental to any effort to validate and verify simulations, which is often overlooked. Likewise, no simulation-driven materials design effort can be done with any level of robustness without properly accounting for the uncertainty in the predictions derived from the computational models. Here, we review some of the most recent works that have focused on the analysis, quantification, propagation, and management of uncertainty in computational materials science and ICME-based simulation-assisted materials design. Modern concepts of efficient uncertainty quantification and propagation, multi-scale/multi-level uncertainty analysis, model selection as well as model fusion are also discussed. While the topic remains relatively unexplored, there have been significant advances that herald an increased sophistication in the approaches followed for model validation and verification and model-based decision support.

Calculation of phase diagrams is one of the fundamental tools in alloy design---more specifically under the framework of Integrated Computational Materials Engineering. Uncertainty quantification of phase diagrams is the first step required to provide confidence for decision making in property- or performance-based design. As a manner of illustration, a thorough probabilistic assessment of the CALPHAD model parameters is performed against the available data for a Hf-Si binary case study using a Markov Chain Monte Carlo sampling approach. The plausible optimum values and uncertainties of the parameters are thus obtained, which can be propagated to the resulting phase diagram. Using the parameter values obtained from deterministic optimization in a computational thermodynamic assessment tool (in this case Thermo-Calc) as the prior information for the parameter values and ranges in the sampling process is often necessary to achieve a reasonable cost for uncertainty quantification. This brings up the problem of finding an appropriate CALPHAD model with high-level of confidence which is a very hard and costly task that requires considerable expert skill. A Bayesian hypothesis testing based on Bayes' factors is proposed to fulfill the need of model selection in this case, which is applied to compare four recommended models for the Hf-Si system. However, it is demonstrated that information fusion approaches, i.e., Bayesian model averaging and an error correlation-based model fusion, can be used to combine the useful information existing in all the given models rather than just using the best selected model, which may lack some information about the system being modelled., Comment: 22 pages, 8 Figures

The accurate prediction of the thermal histories and melt pool characteristics during additive manufacturing (AM) is necessary to understand the factors responsible for the quality and integrity of the manufactured part. More importantly, the determination of optimal process windows and even feed-forward and in-line feedback control of the manufacturing process require computationally cheap, fast-acting, quantitative models connecting (local) processing parameters to melt and solidification conditions. Initially developed in the context of welding, the Eagar-Tsai (E-T) model stands out among the most widely used computationally cheap models to predict melt pool characteristics during AM. Despite its widespread use, its statistical validity in the context of AM has yet to be thoroughly verified. In this work, we study the E-T model in a systematic manner, from an uncertainty quantification/propagation (UQ/UP) perspective. E-T model parameters are calibrated against high quality single-track experimental data on the melt pool geometries of several materials through Markov Chain Monte Carlo (MCMC) sampling. Posterior distributions of the model parameter values are then propagated. We find that there are considerable discrepancies between predicted and measured melt pool depths when process conditions correspond to keyholing. We then apply a physics-based correction and find that it is possible to achieve much better agreement with experiments without increasing significantly the complexity of the E-T model. Although there might be some uncertainties due to the missing physics and assumptions in the model, the model accuracy and trend are satisfactory for the purpose of accelerated product design under uncertainty.

Often times, the development of physical models of materials behavior is hindered not only by the incompleteness of the theoretical approach, but also by uncertainties that arise from limitations in the experimental observations used to validate and calibrate these models. In this work, we present a Bayesian framework for both the calibration of physical models as well as the quantification of likely uncertainty in experimental observations and apply it to a model for the plastic response of multi-phase Transformation Induced Plasticity (TRIP) steels. The model is based on a formulation of irreversible thermodynamics of plastic deformation and accounts for the presence of multiple phases through homogenization theories based on the iso-work approximation. Bayesian calibration through Metropolis-Hastings Markov Chain Monte Carlo has been used to calibrate a subset of the model parameters against experimental data sequentially and simultaneously. The calibrated parameters obtained from sequential training were in turn used to assess the uncertainty in a subset of experimental data—i.e. phase volume fractions—used as input to the models themselves. The viability of the calibration approach has also been examined using synthetic data generated from simultaneous calibrated model.

In this work, thermodynamic and kinetic diffusivities of uranium–niobium (U–Nb) are re-assessed by means of the CALPHAD (CALculation of PHAse Diagram) methodology. In order to improve the consistency and reliability of the assessments, first-principles calculations are coupled with CALPHAD. In particular, heats of formation of γ-U–Nb are estimated and verified using various density-functional theory (DFT) approaches. These thermochemistry data are then used as constraints to guide the thermodynamic optimization process in such a way that the mutual-consistency between first-principles calculations and CALPHAD assessment is satisfactory. In addition, long-term aging experiments are conducted in order to generate new phase equilibria data at the γ 2 / α + γ 2 boundary. These data are meant to verify the thermodynamic model. Assessment results are generally in good agreement with experiments and previous calculations, without showing the artifacts that were observed in previous modeling. The mutual-consistent thermodynamic description is then used to evaluate atomic mobility and diffusivity of γ-U–Nb. Finally, Bayesian analysis is conducted to evaluate the uncertainty of the thermodynamic model and its impact on the system's phase stability.

ICME approaches provide decision support for materials design by establishing quantitative process-structure-property relations. Confidence in the decision support, however, must be achieved by establishing uncertainty bounds in ICME model chains. The quantification and propagation of uncertainty in computational materials science, however, remains a rather unexplored aspect of computational materials science approaches. Moreover, traditional uncertainty propagation frameworks tend to be limited in cases with computationally expensive simulations. A rather common and important model chain is that of CALPHAD-based thermodynamic models of phase stability coupled to phase field models for microstructure evolution. Propagation of uncertainty in these cases is challenging not only due to the sheer computational cost of the simulations but also because of the high dimensionality of the input space. In this work, we present a framework for the quantification and propagation of uncertainty in a CALPHAD-based elasto-chemical phase field model. We motivate our work by investigating the microstructure evolution in Mg$_2$(Si$_x$Sn$_{1-x}$) thermoelectric materials. We first carry out a Markov Chain Monte Carlo-based inference of the CALPHAD model parameters for this pseudobinary system and then use advanced sampling schemes to propagate uncertainties across a high-dimensional simulation input space. Through high-throughput phase field simulations we generate 200,000 time series of synthetic microstructures and use machine learning approaches to understand the effects of propagated uncertainties on the microstructure landscape of the system under study. The microstructure dataset has been curated in the Open Phase-field Microstructure Database (OPMD), available at \href{http://microstructures.net}{http://microstructures.net}.

Ni-Ti alloys are the most popular shape memory alloys in different industrial applications due to their especial properties provided, for at least some variants, by the precipitation of secondary strengthening phases. Therefore, precipitation modeling and associated model calibration coupled with uncertainty estimations of their performance seem to be crucial for these alloys, particularly when the focus is on the design of precipitation schemes leading to optimal performance. In this work, the calibration and uncertainty quantification of influential model parameters (i.e. the parameters with strong effects on the response of the model) have been performed against individual and all available experimental data using a Bayesian approach based on the Markov chain Monte Carlo-Metropolis Hastings algorithm. Then, the resulting parameters’ uncertainties have been propagated to the uncertainty in model results through a forward model analysis. An empirical relationship for matrix/precipitate interfacial energy in terms of aging temperature and nominal composition has been introduced according to the values of interfacial energy obtained from the model calibration with each given experimental data individually. Using this equation for interfacial energy in the model, all experimental data has been used together to calibrate the other influential parameters in the model. After the probabilistic calibration of the model, the discrepancies between the model outcomes and the experimental data were the main reasons to apply co-Kriging surrogate modeling that takes advantage of an error correlation-based fusion of the model results and the experimental data for more precise prediction of precipitation behavior in this system.

The plastic deformation of multiphase steels is described employing an irreversible thermodynamics formulation. Transformation induced plasticity and dual phase grades are described within a single theoretical framework. The approach describes the plastic deformation of each individual phase in terms of the evolution of dislocation density, subject to dissipative mechanisms associated to dislocation generation, glide and annihilation. The collective behaviour of the ensemble of phases into a single microstructure is ensured through a self-consistent approach based on the iso-work approximation. The parameterised model shows very good agreement with several alloys studied experimentally and available in the literature.

Design is about making decisions bounded by a quantifiable degree of certainty. In the context of alloy design, Integrated Computational Materials Engineering (ICME) provides the framework whereby performance requirements are ultimately transformed into alloy/processing specifications through the combination of (complex) computational models connecting process-structure-property-performance relationships and experiments. Most ICME approaches consider the models used as deterministic and thus do not provide the means to make alloy design decisions with proper confidence measures. At the root of ICME lie CALPHAD models that describe the thermodynamics and phase stability of phases under specific thermodynamic boundary conditions. To date, the vast majority of efforts within the CALPHAD community have been deterministic in that thermodynamic models and the resulting thermodynamic properties and phase diagram features do not explicitly account for the uncertainties inherent in the model formulation or in the experimental/computational data used. In this contribution, we provide an overview of the state of the field. We review major efforts thus far and we then provide a (brief) tutorial on basic concepts of uncertainty quantification and propagation (UQ/UP) in CALPHAD. We discuss the major features of frequentist and Bayesian interpretations of uncertainty and proceed with a discussion of recent case studies in which UQ has been used to parameterize models for the thermodynamic properties of phases. We follow our discussion by presenting frameworks and demonstrating the propagation of uncertainty in thermodynamic properties and phase diagram predictions and briefly discuss how we can use Bayesian frameworks for rigorous model selection as well as for model fusion. We close our contribution by providing context for what has been done and what remains to be accomplished in order to fully embrace the management of uncertainty in CALPHAD modeling, a foundational element of ICME.

Rivera’s physical model [Modeling Simul. Mater. Sci. Eng. 22 (2014) 015009 (22pp)] based on irreversible thermodynamics of dislocation evolution has been used in this work to predict the stress-strain behavior of TRIP steels during plastic deformation. This model has been applied to account for plastic deformation of individual phases, and the iso-work approximation has been used as a homogenization framework to simulate the behavior of the composite system. Contribution to plastic behavior due to strain-induced martensitic transformation in austenite has also been taken into account. Calibration and analysis of parameter uncertainty is performed using a Bayesian method based on Metropolis-Hastings Markov Chain Monte Carlo algorithms. Using this approach, the model has been trained with different experimental data to estimate parameters and their uncertainties. The parameter posterior probability distribution obtained from is considered as the prior probability distribution for subsequent training. The stress-strain curves obtained from the model with new estimated parameters show good agreement with the experimental data in literature.