Machine learning to accelerate alloy design

Abstract

This invention presents an innovative framework for the application of machine learning for identification of alloys or composites with desired properties of interest. For each output property of interest, we identify the corresponding driving (input) factors. These input factors may include the material composition, heat treatment, process, microstructure, temperature, strain rate, environment or testing mode. Our framework assumes selection of optimization technique suitable for the application at hand and data available, starting with simple linear, or quadratic, regression analysis. We present a physics-based model for predicting the ultimate tensile strength, a model that accounts for physical dependencies, and factors in the underlying physics as a priori information. In case an artificial neural network is deemed suitable, we suggest employing custom kernel functions consistent with the underlying physics, for the purpose of attaining tighter coupling, better prediction, and extracting the most out of theusually limitedinput data available.

Claims

1. An apparatus for predictive analytics, an apparatus that employs a prediction module, for the purpose of efficiently searching composition space of alloys or composites of interest, and hence for accelerating design or manufacturing of alloys or composites with desired material characteristics, an apparatus comprising of a database importing module, for ingesting materials or manufacturing data, an optional data base abstraction module, also referred to as prediction logic, for abstracting the interface between the prediction module and the database, an optional preprocessing module, for unified comparison of materials or manufacturing data across data sets, an optional feature selection module, for extracting features that best describe the material or manufacturing data of interest, a prediction module, also referred to as a prediction engine, for predicting material properties or manufacturing parameters of interest, along with a corresponding composition or composition range, given the materials or manufacturing data ingested, where the prediction engine employs a prediction technique, where the prediction engine combines physics-based models, or analytical models, specific to the alloys or composites of interest, with traditional black-box prediction models, for improved prediction accuracy, and where the prediction technique employed can involve regression analysis, with relatively few unknown model parameters, when limited data is available, but a machine learning predictor, with relatively many unknown model parameters, when sufficiently large data set is available, where the prediction technique is selected such that at least one data point is available for each unknown prediction model parameter, and an optional reporting and verification module, for reporting, evaluating or verifying the materials properties or manufacturing parameters estimated, an apparatus presented either in the form of an integrated or an embedded application.

2. An apparatus according to claim 1, where the data importing module employs an Export, Transform and Load operation for importing data from a source into the destination prediction database logic, when the destination prediction database logic system represents the data differently from the source.

3. An apparatus according to claim 1, where the preprocessing module applies normalization for standardizing the input data ingested, for unified comparison, and where the standardization can involve division of endurance limit with tensile strength, in case of prediction of fatigue behavior of the alloys or composites of interest.

4. An apparatus according to claim 1, where the feature selection module utilizes features related to composition or micro-structure of the alloys or composite of interest, process, heat treatment, temperature, environment, testing mode or loading rate.

5. An apparatus according to claim 1, where the prediction engine is capable of accommodating models capturing physical dependencies, referred to as physics-based models, as a priori information, and correspondingly constructing dependencies within the prediction model, for purpose of expediting training of the prediction model or improving prediction accuracy.

6. An apparatus according to claim 5, where the physics-based models employ thermodynamics, first principle, empirical rules, mesoscale models, or in case of prediction of fatigue life, models for dislocation dynamics or slip band information.

7. An apparatus according to claim 1, where the prediction engine consists of optional forward prediction along with optional inverse prediction, where the forward prediction is responsible for predicting the material properties or manufacturing parameters of interest from source characteristics, where the inverse prediction is responsible for inferring source characteristics, in particular alloy composition, from source characteristics from material properties desired, and where the source characteristics can consist of composition, microstructure, process, heat treatment, temperature, environment, testing mode or loading rate.

8. An apparatus according to claim 1, where the feature selection module employs canonical component analysis or regression, for purpose of selecting features capable of distinguishing between calcium-magnesium-alumino-silicate and calcium sulfate hot corrosion attacks, in air or sea water, in order to develop coatings resistant to calcium-magnesium-alumino-silicate and calcium sulfate hot corrosion.

9. An apparatus according to claim 1, where the prediction engine utilizes a parametrized model, referred to as augmented Statistical Fatigue Life model, a model capable of accounting for multiple sources impacting fatigue life of additively manufactured components, for purpose of accurately predicting the fatigue life.

10. An apparatus according to claim 1, where the prediction engine supports inverse design representations.

11. An apparatus according to claim 1, where the prediction engine can employ custom kernel functions consistent with underlying physics, for the purpose of attaining tighter coupling, better prediction, than with generic, non-custom kernel functions, and for extracting the most out of the materials or manufacturing data ingested.

12. A method for predictive analytics, one that incorporates a prediction step, for the purpose of efficiently searching composition space of alloys or composites of interest, and hence for accelerating design or manufacturing of alloys or composites with desired material characteristics, a method utilizing a database importing step, for ingesting materials or manufacturing data, an optional preprocessing step, for unified comparison of materials or manufacturing data across data sets, an optional feature selection step, for extracting the features primarily impacting the material properties of interest, a prediction step, for predicting material properties or manufacturing parameters of interest, along with a corresponding composition or composition range, given the materials or manufacturing data ingested, where the prediction step employs a prediction technique, where the prediction step combines physics-based models, or analytical models, specific to alloys or composites of interest, with traditional black-box prediction models, for improved prediction accuracy, and where the prediction technique employed can involve regression analysis, with relatively few unknown model parameters, when limited data is available, but a machine learning predictor, with relatively many unknown model parameters, when a sufficiently large data set is available, where the prediction technique is selected such that at least one data point is available for each unknown prediction model parameter, and a reporting and evaluation step, for reporting or evaluating the materials properties or manufacturing parameters of interest.

13. A method according to claim 12, where the data importing step employs an Export, Transform and Load operation for importing data from a source into the destination predictive analytics system, when the destination system represents the data differently from the source.

14. A method according to claim 12, where the preprocessing step applies a normalization step for standardizing the input data, and where the standardization involves division of endurance limit with tensile strength, in case of prediction of fatigue behavior of alloys or composites of interest.

15. A method according to claim 12, where the feature selection step utilizes features related to composition or microstructure of the alloy or composite of interest, process, heat treatment, temperature, environment, testing mode or loading rate.

16. A method according to claim 12, where the prediction step is capable of accommodating models capturing physical dependencies, referred to as physics-based models, as a priori information, and correspondingly constructing dependencies within the prediction model, for the purpose of expediting training or improving prediction accuracy.

17. A method according to claim 16, where the physics-based models involve thermodynamics, first principle, empirical rules or mesoscale models, or in case of prediction of fatigue life, models for dislocation dynamics or slip band information.

18. A method according to claim 12, where the prediction step consists of an optional forward prediction step along with an optional inverse prediction step, where the forward prediction step is responsible for predicting the material properties or manufacturing parameters of interest from source characteristics, where the inverse prediction step is responsible for inferring source characteristics, in particular alloy composition, from source characteristics from material properties targeted, and where the source characteristics can consist of composition, microstructure, process, heat treatment, temperature, environment, testing mode or loading rate.

19. A method according to claim 12, where the feature selection step employs canonical component analysis or regression analysis, for purpose of selecting features capable of distinguishing between calcium-magnesium-alumino-silicate and calcium sulfate hot corrosion attacks, with or without influence of sea salt, in order to develop coatings resistant to calcium-magnesium-alumino-silicate and calcium sulfate hot corrosion.

20. A method according to claim 16, where the physics-based models involve thermodynamics, first principle, empirical rules or mesoscale models, or in case of prediction of fatigue life, models for dislocation dynamics or slip band information.

21. A method according to claim 12, where the prediction step supports inverse design representations.

Description

DESCRIPTION OF THE DRAWINGS

(1) FIG. 1 presents the primary factors impacting the material properties of alloys, such as high-entropy alloys. The material properties considered include material hardness, yield strength, ultimate strength, plasticity, fatigue, fracture toughness and creep. The process part may include fabrication and machining processes.

(2) FIG. 2 presents a more detailed version of FIG. 1, one specific to fatigue properties, and one that illustrates dependence between the sources.

(3) FIG. 3 presents a figure similar to FIG. 1 of the primary factors impacting the material properties of matrix composites.

(4) FIG. 4 presents an overview over metal AM technologies.

(5) FIG. 5 presents a high-level overview of primary sources contributing to the fatigue life of additively manufactured metallic components.

(6) FIG. 6 presents an overview more complete than FIG. 5, but still not comprehensive, of the sources contributing to the fatigue life of AM metallic components.

(7) FIG. 7 depicts a processing map for metal additive manufacturing.

(8) FIG. 8 illustrates that refractory metals in category HEA-3, yield much superior yield strength at high temperature, compared to alloys conventionally used in turbine blades.

(9) FIG. 9 presents qualitative comparison between cost of additive manufacturing and conventional production.

(10) FIG. 10 presents a master architecture (a dependency diagram), one capturing the prediction engine.

(11) FIG. 11 presents the relationship, per the dependency diagram in FIG. 10 between the caller, that initiates a request, and the callee, that responds to a call.

(12) FIG. 12 presents the essential mechanism of Export, Transform and Load.

(13) FIG. 13 outlines the high-level approach to forward prediction of material properties of alloys, such as HEAs. FIG. 13 also illustrates how prediction of material properties of alloys can be extended to matrix composites.

(14) FIG. 14 presents the essentials of averaging in the space comprising the input parameters.

(15) FIG. 15 presents a high-level approach to machine learning with knowledge discovery and data mining.

(16) FIG. 16 outlines a baseline approach for inferring the microstructure from desired material properties.

(17) FIG. 17 summarizes material structure optimization through microstructure-property pairs.

(18) FIG. 18 presents underlying physical dependencies structured in a fashion resembling a neural network. Our intent is to construct models capturing the underlying physics. The model shows that microstructure formed depends on the heat treatment process applied.

(19) FIG. 19 presents variations in tensile strength of Aluminum and Cobalt, based on purity and processing conditions (AsmHandbook 1990).

(20) FIG. 20 presents variations in tensile strength of Chromium and Vanadium, based on processing conditions (AsmHandbook 1990).

(21) FIG. 21: Towards the characterization of temperature dependence of the ultimate tensile strength of HEAs. For a given temperature, the data points listed correspond to the same heat-treatment process.

(22) FIG. 22 presents overall methodology for deriving the output quantity of interest (the endurance limit) from the input sources (compositions, processes and defects).

(23) FIG. 23 presents an example, where a linear prediction model is derived from Data Set A in Table 13 through linear regression analysis.

(24) FIG. 24 presents an example, where a linear prediction model is derived from Data Set B in Table 13 through linear regression analysis.

(25) FIG. 25 presents an example, where a linear prediction model is derived from Data Set C in Table 13 through linear regression analysis.

(26) FIG. 26 captures Step 2 in the prediction methodology outlined in FIG. 22.

(27) FIG. 27 captures analysis of the dependence of the endurance limit of CoCrFeNiMn (the Cantor alloy) on gain size (top) and tensile strength (bottom).

(28) FIG. 28 provides high-level presentation of CALPHAD, in terms of the primary inputs, the primary outputs and association with first-principle calculations.

(29) FIG. 29 illustrates how properties of a ternary system (A-B-C) are associated, at a high level, with the properties of the constituent binary systems (A-B, A-C and B-C).

(30) FIG. 30 presents an illustration of analysis of properties of ternary material systems in CALPHAD.

(31) FIG. 31 illustrates, at a high level, how CALPHAD computes the Gibbs free energy of individual phases from thermochemical measurements and phase equilibria information (from (Thermo-Calc 2014)).

(32) FIG. 32 provides more specific information of the internal structure of CALPHAD (again from (Thermo-Calc 2014)).

(33) FIG. 33 presents corrosion analytics in context with the overall effort to identify high-risk areas for deposit build-up on actual components in gas turbines.

(34) FIG. 34 presents the essential mechanism behind corrosion of the thermal barrier coating instigated by a CMAS hot corrosion attack.

(35) FIG. 35 shows how canonical correlation analysis can be applied to analysis of feature sets corresponding to CMAS and calcium sulfate hot corrosion attacks.

(36) FIG. 36 presents regression analysis applied to the Data Sets from Examples A, B and C.

(37) FIG. 37 presents results from experimental verification of superior tensile strength of the predicted composition (Al0.5Mo0.5Nb1.5Ta0.5Zr1.5) over the reference composition (Al0.5Mo0.5NbTa0.5TiZr). Coupons for both compositions were prepared through arc melting and with no heat treatment applied.

(38) FIG. 38 outlines a classical approach for analyzing fatigue life. The top figure shows the stress applied (from 4-point testing). The middle figure shows the fatigue life obtained from the stress applied. The bottom figure shows the fatigue life remaining (curve fit to data). The classical NASGRO model can yield over- or under-estimation of the fatigue life, due to over- or under-fitting. This results in error magnification during integration. Machine learning may account for all the parameters affecting fatigue life of AM components, not only those modeled by the NASGRO equation, and hence yield more accurate results.

(39) FIG. 39 presents a high-level overview over optimization of an AM process for HEAs.

(40) FIG. 40 (top) shows an offset strategy, using two beams in a defined offset at the same speed. FIG. 40 (bottom) shows a heating strategy, one that uses a melting beam of scan speed vi and a heating beam of v2 which is much faster and irradiates the layer multiple times while beam 1 melts the tracks.

DETAILED DESCRIPTION OF THE PREFERRED EMBODIMENTS

1. Definitions

(41) Table 5 captures the primary acronyms used in the patent.

(42) TABLE-US-00005 TABLE 5 Summary of the primary definitions and acronyms. Name Definition AM Additive Manufacturing ANN Artificial Neural Network API Application Program Interface CCA Complex, Concentrated Alloy (or Canonical Component Analysis) CCE Carbon Conversion Efficiency CFD Computational Fluid Dynamics CMAS Calcium Magnesium Alumino Silicate CMC Ceramic Matrix Composite DED Direct Energy Deposition DFT Density Functional Theory DMD Direct Metal Deposition DMLS Direct Metal Laser Sintering EBM Electron Beam Melting GE General Electric HEA High-Entropy Alloy HIP Hot Isostatic Pressing ICME Integrated Computational Materials Engineering IDE Integrated Development Environment IM Intermetallic JPL Jet Propulsion Laboratory JSON JavaScript Object Notation k-NN k Nearest Neighbors LENS Laser Engineered Net Shaping LMD Laser Metal Deposition MIG Metal Inert Gas ML Machine Learning MME Mechanical and Materials Engineering MPEA Multi-Principal Element Alloy NIST National Institute of Standards and Technology ODF Oriental Distribution Function PBF Powder-Bed Fusion PMC Polymer Matrix Composite PSU Portland State University RHEA Refractory High-Entropy Alloy SDK Software Development Kit SLM Selective Laser Melting SLS Selective Laser Sintering SMD Shaped Metal Deposition SPS Spark Plasma Sintering SS Solid-Solution TBC Thermal Barrier Coating TIG Tungsten Inert Gas UTS Ultimate Tensile Strength VASP Vienna Ab initio Simulation Package WAAM Wire and Arc Additive Manufacturing

(43) We define artificial intelligence as the use of computers to mimic the cognitive functions of humans. When machines carry out tasks based on algorithms in an intelligent manner, that is Al. Artificial intelligence is a broader concept than machine learning (DataScienceCentral 2018).

(44) We define machine learning as a subset of Al that focuses on the ability of machines to receive a set of data and learn for themselves, and change algorithms as they learn more about the information that they are processing (DataScienceCentral 2018).

(45) We refer to deep learning as a subset of machine learning. We define deep learning in terms of deep neural networks, i.e., neural networks comprising of two or more layers. Deep learning networks need to see large quantities of items in order to be trained (DataScienceCentral 2018).

(46) Supervised learning is a data mining task that involves inference of a function from labeled training data.

(47) Unsupervised learning is a type of machine learning algorithm used to draw inferences from datasets consisting of input data without labeled responses.

(48) Reinforcement learning is an area of machine learning concerned with how software agents ought to take actions in an environment so as to maximize some notion of cumulative reward.

2. Best Mode of the Invention

(49) FIG. 10, FIG. 14, FIG. 15, FIG. 18, FIG. 22, FIG. 26, FIG. 33 and FIG. 35 capture the best mode contemplated by the inventors, according to the concepts of the present invention.

3. System Structure at a High Level

(50) FIG. 10 presents a dependency diagram for the overall system architecture. The architecture has been devised largely based on the requirements for the prediction engine. While the architecture may come across as simplistic, it has been artfully crafted such as not to contain any loops. This offers great value in terms of significantly expediting the process of confining the source of certain behavior (desired or undesired) to given modules.

(51) With the graphical user interface and prediction logic being optional, the architecture supports both embedded (plugin or web service) and integrated applications of the prediction engine.

4. User Interface

(52) 4.1 General Consideration

(53) The GUI is assumed to be based on the traditional Model-View-Controller model (Steingrimsson 2017). The GUI may be integrated into a host application, may be executed through a plugin, or may run through a web interface.

(54) 4.2 Specific Example: Prediction of Fatigue Life of AM Components

(55) Given the large number of parameters that can impact the fatigue life of AM components, it is important to prioritize these parameters, based on importance to the customer, and systematically organize while still keeping the user interface efficient and user friendly.

5. Database System

(56) 5.1 General Assumptions

(57) The typical use case assumes a relational database, such as SQL.

(58) In order to be useful, the data needs to be collected into a single repository and have consistent format.

(59) An in-memory implementation of the database stores as inputs vectors, k's, capturing the sources affecting the output quantity of interest, y. The in-memory database also stores the aforementioned output quantity. For further information, refer to the generic system model in Eq. (1). It is preferable that the database supports an in-memory mode. The prediction engine may require millions of comparative operations. Without an in-memory mode, every comparison may require an I/O call. This may introduce significant latency.

(60) 5.2 Extendable Solution for Importing Content from Disparate Databases: SQL and JSON

(61) A key challenge in applying ML algorithms to materials science data is that data can come in many formats. Determining how to featurize and utilize different materials data formats so that prior data can be used as training data for ML algorithms can be difficult. Feature engineering, including extraction, transformation, and selection, is critical for improved ML accuracy. To fully realize data analytics and machine learning tools for materials development, it may be necessary to transform various raw data inputs into information-rich features suitable for modeling. Hence, it may be necessary to unify disparate data sets into a consistent format that can be utilized by the prediction engine.

(62) A SQL server may be an ideal solution for supporting importing of data from disparate databases, for use by the prediction engine. A SQL server, such as a MySQL server, would allow one to import and translate many data types. It would enable one to import data from any database supporting the SQL language, search for data of interest, export into a standardized format, and import into the prediction engine.

(63) JSON data is represented in a logical, organized and easy-to-access manner. JSON can contain multiple levels of objects, arrays and various field data. JSON is supported in order to provide access to open materials data bases, such as the Citerine's JSON-based database (CitrineResearch 2019). Alternatively, the data may originate from the NIST CALPHAD Data Informatics databases (NISTCalPhad 2019), the CHIMAD Polymer Property Predictor Database (CHIMaD 2020), databases associated with the Materials Genome Initiative (NISTmgi 2019) or OPTiMaDe (OPTiMaDe 2020). Both Citrine and OPTiMaDe offer APIs providing convenient access for users.

(64) 5.3 Mechanism for Supporting Multiple Data Formats: ETLExtract, Transform and Load

(65) ETL is a generic principle representing three database functions (extract, transform and load) that are combined into a single tool to pull data out of one database and place into another database (see FIG. 12). Extract is the process of reading from a database. In this stage, the data is collected, often from multiple and different types of sources. Transform is the process of converting the extracted data from its previous form into the form it needs to be in so that it can be placed into another database. Transformation occurs by using rules or lookup tables or by combining the data with other data. Load is the process of writing the data into the target database. Traditional symmetric multi-processing data warehouses use an ETL process for loading the data. Ref. (CodeBurst 2019) shows how to Extract data from mysql, sql-server or firebird, Transform the data, and then Load into a SQL server (a data warehouse) using python 3.6.

(66) 5.4 Preparation of Data Sets for PredictionImplications of Incomplete Data

(67) Ref (AgrawalDeshpande 2014) suggests that one may be able to employ standard, off-the-shelf, free or open-source machine learning libraries, such as TensorFlow (TensorFlow 2020) or scikit-learn (SciKit-Learn 2020), and obtain reasonably accurate prediction results, assuming one has access to data sets that are clean, usable, abundant and trustworthy (and do not contain outliers).

(68) However, in practice material scientists usually find themselves operating in the realm of limited data.

(69) Hence, in practice, it may be of importance to incorporate physics-based models, in order to extract the most out of the (limited) data sets available.

6. Prediction Database Logic

(70) The prediction database logic serves as an interface, or abstraction layer, between the prediction engine and the database. The prediction database logic can provide the ability to store internal data structures in memory and later archive in a data base (e.g., by saving on a hard disk drive or a flash drive). The prediction database logic can represent data in a format convenient to the prediction engine.

7. Prediction Engine

(71) 7.1 General Approach to Prediction of Properties and Feature Sets for Compositions of Interest

(72) FIG. 10-FIG. 11 and FIG. 13-FIG. 17 summarize the general approach for predicting feature sets (or properties) for alloy or composite compositions of interest.

(73) 1. Structure of a Generic System Model

(74) We assume a generic system model:
{tilde over (y)}=f({tilde over (x)}).(1)

(75) TABLE-US-00006 TABLE 6 Recommended features for prediction of composition of RHEA and parameters of powder bed AM. Abbrev Category Details Cr Composition % Chromium Hf % Hafnium Mn % Manganese Mo % Molybdenum Nb % Niobium Os % Osmium Re % Rhenium Rh % Rhodium Ru % Ruthenium Ta % Tantalum Tc % Technetium Ti % Titanium V % Vanadium W % Tungsten Zr % Zirconium Al % Aluminum Temp Heat Treatment Temperature HeatTime Treatment time HeatEnvir Heat treatment environment LoadRate Loading Rate 10.sup.5 sec.sup.1- 10.sup.6 sec.sup.1 ColdRoll Environ-ment Cold rolling HotRoll Hot rolling FCC Micro-structure Face centered cubic structure BCC Body centered cubic structure HCP Hexagonal closed packed structure Precip Precipitate Tension Testing Mode Tension Compres Compression Bending Bending Torsion Torsion PowdBT Temperature Powder bed temp PowdFT Powder feeder temp ElevatT Elevated temp LaserPow Process Laser power SpotSize Spot size PulseDur Pulse duration PulseFreq Pulse frequency ScanSpd Scan speed HatchDist Hatch distance ScanPattn Scan pattern PartShape Particle shape PartSize Particle size PartDist Particle distribution

(76) TABLE-US-00007 TABLE 7 Recommended features for prediction of fatigue endurance limit of HEAs. Abbrev. Category Details Ag Composition % Silver Al % Aluminum Au % Gold B % Boron C % Carbon Co % Cobolt Cr % Chromium Cu % Copper Dy % Dysprosium Fe % Iron Gd % Gadolinium Ge % Germanium Hf % Hafnium Li % Lithium Lu % Lutetium Mn % Manganese Mo % Molybdenum Nb % Niobium Nd % Neodymium Ni % Nickel P % Phosphorus Pd % Palladium Rh % Rhodium Ru % Ruthenium S % Sulphur Sc % Scandium Si % Silicon Sn % Tin Ta % Tantalum Tb % Terbium Ti % Titantium Tm % Thulium V % Vanadium W Composition % Tungsten Y % Yttrium Zn % Zinc Zr % Zirconium Temp Heat Treatment Temperature HeatTime Heat treatment time HeatEnvir Heat treatment environment LoadRate Loading Rate 10.sup.5 sec.sup.1- 10.sup.6 sec.sup.1 FCC Micro-structure Face centered cubic structure BCC Body centered cubic structure HCP Hexagonal closed packed structure Precip Precipitate Tension Testing Mode Tension Compres Compression Bending Bending Torsion Torsion CryoT Temperature Cryogenic temp RoomT Room temperature ElevatT Elevated temp DropCast Process Drop casting AddMfg Additive manufacturing Powder Powder metallurgy Sputter Sputtering VolPreci Vol. % of precipitate SizePrec Size of precipitate ShapePre Shape of precipitate ColdRoll Environment Cold rolling HotRoll Hot rolling

(77) The input vector, {tilde over (x)}, can be considered as the definition of a feature set comprising of parameters related to the composition, defect properties, heat treatment, and manufacturing, essentially all the sources that impact the output quantity of interest, {tilde over (y)}. The transformation, (.Math.), can be a non-linear function of the input, x. We present artificial intelligence, regression analysis and supervised learning as options to construct (train) the system model.

(78) 2. Key Characteristics of Forward and Backward Prediction

(79) 2.1 Quantity Predicted

(80) We forward predict the observed properties, such as the properties listed in FIG. 1, FIG. 2, FIG. 3 and FIG. 13. We backward predict the properties comprising the feature set, as shown in FIG. 15. For examples of representative feature sets, refer to Table 6 and Table 7.

(81) 2.2 Preprocessing

(82) For fair comparison, we normalize the input data, as appropriate. In case of the endurance limit, S.sub.e, we normalize with the ultimate tensile strength, UTS:

(83) $\begin{array}{cc}{S}_{e.\mathrm{norm}}=\frac{s_e}{\mathrm{UTS}}.& \left(2\right)\end{array}$
2.3 Feature Selection

(84) We derive the features selected from the driving factors listed in FIG. 1, FIG. 2, FIG. 3 and FIG. 13. Table 6 presents a sample feature selection suitable for identification of RHEAs for high-temperature applications. Note the feature set also captures the AM processing parameters. Hence, the feature set correlates material functionality both with material compositions and manufacturing process conditions. Table 7 presents another representative sample of feature selection suitable for prediction of fatigue endurance limit of HEAs.

(85) 3. Approach to Building a Generic System Model Through Forward Prediction

(86) We present a scalable solution for deriving the system model, one that accounts for the application at hand and the input data available. In the case of a small set of input data, we present regression as a suitable tool for deriving (constructing) the system model. But for a large set of input data, say, hundreds, thousands, or millions of (x, y) duplets, we present feed-forward neural networks as a suitable tool for constructing the system model.

(87) Our approach is founded in part on observations of Agrawal et. al. (AgrawalDeshpande 2014). Table 2 and FIG. 5 of (AgrawalDeshpande 2014) illustrate that there is at most a few percentage difference between the techniques applied to predict the fatigue strength of stainless steel. Table 2 of (AgrawalDeshpande 2014) shows that both the simple linear regression and pace regression yield the coefficient of determination, R.sup.2, of 0.963, while the artificial neural network, a traditional ML approach, results in R.sup.2 of 0.972.

(88) 4. More on Forward Prediction: Review of Predictive Modeling Techniques Considered

(89) The predictive modeling techniques considered include, but are not limited to, linear regression, pace regression, regression post non-linear transformation of select input variables, robust fit regression, multivariate polynomial regression (including quadratic regression), decision tables, support vector machines, artificial neural networks, reduced error pruning trees and M5 model trees.

(90) 4.1. Statistical Regression

(91) For background information on statistical regression, refer to (SteingrimssonJonesKisialiou 2018).

(92) 4.2. k-Nearest Neighbor Averaging

(93) FIG. 13 present a simple framework for predicting the observed properties for alloys or composites of interest. For each observed property of interest, we identify the corresponding driving factors. These input factors may include the material composition, heat treatment, process, microstructure, temperature, strain rate, environment, and testing mode. We then carry out the prediction through a customized averaging process in the space comprising the input parameters.

(94) FIG. 14 explains, at a high level, how we are going to populate the input parameter space and carry out the prediction. In case of continuous-valued parameter space, one can apply k nearest neighbor (k-NN) averaging for determining the parameters corresponding to point A in FIG. 14, p.sub.A:

(95) $\begin{array}{cc}{p}_A=\frac{1}{k}{.Math.}_{i=1}^{k-\mathrm{NN}}{p}_i& \left(3\right)\end{array}$

(96) Here, p.sub.i represents the parameter vector corresponding to the i-th nearest neighbor of point A in FIG. 14.

(97) 4.3. Feed-Forward Neural Network

(98) For further information on the neural networks, refer to (SteingrimssonJonesKisialiou 2018).

(99) 4.4 Alternative Methods Considered, but not Chosen

(100) 1. Multi-Class ML or Multi-Class Neural Networks: More appropriate for classification problems. 2. Evolutionary Methods: May exhibit problems with curse of dimensionality in high dimensional search spaces. 3. Combinatorial search methods: Do not provide insight desired into the physics and numerical aspects.
5. Example: Forward Prediction of Fatigue Endurance Limit

(101) As a simple example, one can look to forward predict the fatigue endurance limit as follows:
endurance limit=f(UTS,process,defect(process),grain(process),microstructure(process),T . . . )(4)
wherein the composition is specified in terms of the % strength of constituent elements, and UTS represents the ultimate tensile strength. In general, we may be looking at a complex, combinatorial optimization problem.
6. General Approach to Inferring the Feature Set (e.g., Composition): Backward Prediction

(102) Backward prediction (identification of candidate compositions) is accomplished through an inverse design framework, one that suggests candidate compositions to test next, based on a set of property specifications and design goals. Through a sequential learning workflow, the inverse design tool is used to suggest test candidates, and the data from those tests are then used to retrain the model, presumably leading to iterative refinements and convergence. For efficiency, we allow for polynomial fit, during the iterative refinements, when the underlying physics are not expected to result in major nonlinearity.

(103) 7. Specifics on the Approach to the Backward Prediction

(104) To develop the backward prediction, we will be considering a few approaches.

(105) 1. Starting Point: Microstructure of the Nearest Neighbor

(106) The simplest approach consists of identifying the neighbor, B, to the desired HEA, A, in FIG. 14, and simply using the microstructure of B as a starting point. Other options exist as well.

(107) 2. Baseline Approach: Generalization of (LiuKumarChen 2015)

(108) Our baseline approach for inferring the microstructures from the properties desired, shown in FIG. 16 and FIG. 17, is based on (LiuKumarChen 2015). The microstructure design of polycrystals can be performed by tailoring the distribution of various crystal orientations [the oriental distribution function] in the microstructure (LiuKumarChen 2015). Structural optimization is carried out along different crystallographic directions to attain favorable properties. The multiple crystallographic directions embedded in the multi-dimensional ODF are used as control variables and the theoretical functions for properties are the objectives (LiuKumarChen 2015). ML allows us to explore multiple design solutions and diminish search time in the high dimensional space of microstructure design problems, where the number of distinct design candidates can be infinite.

(109) Two crucial ML steps, namely, search path refinement and search space reduction, are designed to develop heuristics that tour the search force to a much smaller preferable space (LiuKumarChen 2015). A ML-based preprocessing is designed to locate critical region of a search space with a small overhead, so that the search force can be consciously concentrated.

(110) 8. Towards SimplificationSuitability of ML vs. Polynomial Fit in Absence of Discrete Jumps in Data

(111) One can study if the alloy candidates of interest exhibit discrete jumps in the material properties observed. With increasing x, the AlxCoCrFeNi alloys change from the single face-centered-cubic (FCC) phase to single body-centered-cubic (BCC) phase with a transition duplex FCC/BCC region. If the data does involve continuous change (continuous 1.sup.st and 2.sup.nd derivatives), and there are not discrete jumps, similar to the on-set of superconductivity (where the conductivity suddenly exhibits a discrete jump to infinity), one can employ polynomial fit (of much lower complexity), in conjunction with ML, or at least as a part of a hybrid solution.

(112) 9. More on a Custom, Hybrid Solution for Backward Prediction: Polynomial Fit Employed for Complexity Reduction

(113) One can employ ML to predict if you are close to a close to a state transition. If not, then polynomial fit may suffice. If you are indeed close to a state transition, then a ML approach may be necessitated.

(114) Similar to (LiuKumarChen 2015) and (LingAntonoBajaj 2018), one can employ ensample prediction, i.e., build many models (often hundreds) in order to predict a given quantity of interest. Each model will make a prediction for a given new test point, and the final ensemble prediction will be given by the average value of all the individual model predictions. A polynomial function with numerical representation can be inverted. If the polynomial is monotonically increasing or decreasing, the inverse is unique. Otherwise, more than one inverse may exist, and the designer is at liberty of picking the best one.

(115) 10. Prediction of Distributions (Mean and Variance)Stochastic Prediction

(116) It is important to keep in mind that we are looking to develop a stochastic, not deterministic, predictors. The outputs of the predictors will involve mean and variance, not a single scalar quantity. Our intent is to determine what % of data falls within a single std. dev (.Math.), what within 26's, etc.

(117) 7.2 Specific Approach to Prediction Incorporating Physics-Based ModelingCapturing of Physical Dependencies

(118) 1. Generic Approach to Incorporating a Physics-Based Model

(119) We start with developing qualitative understanding of the physics and dependencies underlying the data available. We then present a generic mathematical model describing the data. In case of limited data, we start with a simple, linear model, but if supported by sufficient data for training, we employ a more sophisticated model. Next, we introduce non-linearity into the model, based on the underlying physics. The kernel functions of the non-linear models may utilize tanh(.Math.), log(.Math.), or exp(.Math.) functions, based on applications. In the case of reliability analysis, we may choose exponential functions. By carefully formulating the structure of the models, such as to capture underlying dependencies, together with a priori knowledge derived from the physics at play, one can expect to infer more from theusually limiteddata available than when directly using the same data to train generic, out-of-the-box models (with no apriori knowledge incorporated).

(120) 2. Overview Over the Primary Physics-Based Models Incorporated

(121) One of the primary, unique aspects of the innovation involves incorporation of physics-based, metallurgical prediction models. There can be great benefits derived from combining ML with physics-based modeling approaches for design of alloys or composites (in particular HEAs), for improved prediction accuracy. These modeling approaches may include 1. Thermo-dynamics (CalPHAD) (MiracleSenkov 2017), (FengGaoZhangGuo 2018), (ZhangZhangDiaoGao 2016), (ShiCollinsFengZhang 2018). 2. First principle calculations of bond energies and phase stability (FengGaoZhangGuo 2018), (GludovatzHohenwaterCatoor 2014), (TroparevskyMorrisKent 2015), (FengGaoLeeMathes 2016). 3. Empirical rules (TroparevskyMorrisKent 2015), (ZhangZhouLinChen 2008), (FengGaoLeeMathes 2016), (GaoNgLuLiu 2011). 4. Mesoscale models to predict distribution of phases in the microstructure and their morphologies, as influenced by thermal history and alloy chemistry (RadhakrishnanGorti 2019), (RadhakrishnanGorti 2016). 5. Models for dislocation dynamics and slip band information to help with accurate prediction of stress/life curves. 6. Feature representations useful for characterizing, and distinguishing between, chemical reactions associated with specific corrosion attacks, such as CMAS or calcium sulfate attacks.

(122) Table 8 lists leads towards incorporating physics-based intuition into machine learning algorithms for predicting the material properties of alloys. In reference to Table 8, mechanical nano-twins at low temperature can result in continuous steady strain hardening, which improves both fracture toughness and strength, according to B. Gludovatz et. al. (GludovatzHohenwaterCatoor 2014). By incorporating physics-based models, one can help the ML algorithms to avoid fitting to the input data. The physics-based models can also help with extrapolation into uncharted territories (into subspaces of the parameter space for which there are few or no experimentally obtained data points). The physics-based models can help with developing understanding into complex combinations of material and process-induced imperfections.

(123) 2.1 Thermodynamics: Interaction of ML with Phase Stability Models from CALPHAD

(124) The CALPHAD methodology employs a phenomenological approach to calculate multi-component phase diagrams, based on binary phase information, as shown in FIG. 28, FIG. 29 and FIG. 30. Per FIG. 31 and FIG. 32. CALPHAD enables material designers to estimate the thermodynamic properties of ternary systems (A-B-C combinations), based on (a) thermodynamic

(125) TABLE-US-00008 TABLE 8 Further specifics on application of machine learning to prediction of material properties. Quantity Leads towards Incorporating Physics- ID Predicted Sources of the Input Data Based Intuition 1 Material (MiracleSenkov 2017), X. Q. Chen et. al. can model hardness of Hardness (LyuLeeWangFan 2018) and polycrystalline materials (ChenNiuLiLi 2011) (ZhangZuoTangGao 2014), (TsaiYeh 2014), (Jien-Wei 2006), (YehChenLinChen 2007) 2 Yield Strength (Miracle Senkov 2017), W. A. Curtin et. al. can predict yield strength, (SenkovMiracleChaput 2018), only based on edge or screw dislocation. Can (LyuLeeWangFan 2018), predict yield strength of high-entropy alloys (ZhangZuoTangGao 2014), (LeysonHector Curtin 2012), (RaoVarvenne (ZhangYangLiaw 2012), Woodward 2017), (VarvenneLuque Curtin (DiaoFengDahmen 2017), (TsaiYeh 2016), (LeysonCurtinHector Woodward 2010), 2014), (GludovatzHohenwater (NhringCurtin 2018), (VarvenneLeyson 2017). Catoor 2014), and (YehChenLinGan W. Jiang et. al. have theory to predict strength 2014) and explain why strength higher than conventional metal (JiangZhaoQianSrolovitz 2019). 3 Material (ZhangZuoTangGao 2014), (SarmaDawson 1996), (VanHoutte Delannay Plasticity/ (MiracleSenkov 2017), 2002), (Signorelli BertinettiTurner 2009), (Toth Ductility (DiaoFengDahmen 2017), (TsaiYeh MolinariEstrin 2002), (Boudifa 2014), (GludovatzHohenwater Catoor SaanouniChaboche 2009) 2014), (LiPradeepDeng Raabe 2016). 4 Material (HemphillYuanWang 2012), (Tang Predicting the S/N curve may be difficult; Fatigue YuanTsaiYeh 2015), (Thurston (HemphillYuanWang 2012), GludovatzHohenwater 2017), (Chen (SangidMaierSehitoglu 2011) WangSeifiLewandowski 2018), (ShuklaWangCottonMishra 2018), (LiuNeneFrankSinha 2018), (NeneFrankLiuSinhaMishra 2018), (SeifiLiYongLiawLewandrowski 2015), (GaoYehLiawZhang 2016) 5 Fracture (MiracleSenkov 2017), (Thurston (SinghShetty 1989), Toughness GludovatzHohenwater 2017), J. J. (ShenderovaBrennerOmeltchenko 2000) Lewandowski (SeifiLiYongLiaw Lewandrowski 2015), and (LiLiawGao 2018), (LiZhang 2016), (ZhangZuoTangGao 2014), (GludovatzHohenwaterCatoor 2014) 6 Material Creep (LiWangWuLiaw 2018), (a) Dislocation theory to predict regular (ChenLiXieBrechtl 2018), (smooth) creep behavior (PraveenKim 2018) (PraveenKim 2018) (b) See how stress changes; based on nano- identation (MuthupandiKimNaPark 2017). (WongHellingClark 1988), (LiDasgupta 1993)
database and (b) Gibbs free energy. CALPHAD can also help material designers analyze the phase stability of quad systems (A-B-C-D combinations), based on properties of the binary and ternary systems (Thermo-Calc 2014).

(126) Applications, based on the CALPHAD methodology, such as Thermo-Calc, can tell material designers how stable, or how meta-stable, each phase in the ternary or quad system is. CALPHAD can provide estimates for stability for specific phases in a multi-component systems consisting of up to 20 compositions (TernaryPlot2020).

(127) CALPHAD relies on lots of experimental data, much of which was collected in the fifties or sixties. There are gaps in the CALPHAD database, such as in regards to B2 phase in multi-component systems. The B2 phase that does not appear in binary or ternary systems. As a work-around, the B2 phase may be modeled into binary or ternary systems, even though it is not stable there (only meta-stable).

(128) The methodology of accelerated design and qualification of new alloys can be extended such as to efficiently exploit the configurational entropy in high-entropy alloy systems. The solution thermodynamics of solvent-rich systems is well described by binary interactions. But the HEA systems involve significant contributions from ternary interactions that also need to be quantified for accurate prediction of phase stability. Extension of CALPHAD databases for this purpose can be facilitated by DFT calculations of mixing enthalpies in equi-atomic ternary systems.

(129) The determination of the most appropriate heat-treatment process may also rely on thermodynamics calculations (formation of a 2.sup.nd phase).

(130) 2.2 First-Principle Effects (DFT)

(131) Investigations of alloys from first-principle perspective involves computational modeling at atomic scale, for example of bond energies or phase stability. First-principle studies leverage quantum mechanics calculations, with DFT approximating the Schrodinger equation, to simulate the electronic properties and stability of material candidate from which the most promising leads are confirmed experimentally. Even so, DFT calculations tend to be computationally expensive, often taking hours to days for a single molecular structure, and more accurate results are often associated with lengthier calculations performed at higher-levels of theory. On the other hand, properly trained neural networks can (in theory) yield highly accurate predictions with relatively low computational cost. First-principle (DFT) calculations can be used to validate (sanity check) the prediction outcomes of traditional ML systems, i.e., as a part of a hybrid computational system.

(132) As suggested by FIG. 28, applications, based on the CALPHAD methodology, such as Thermo-Calc, do not carry out first-principle (DFT) calculations explicitly. However, applications based on CALPHAD can accept first-principle data from applications such as VASP. VASP is a computer program for atomic scale materials modelling, e.g. electronic structure calculations and quantum-mechanical molecular dynamics, from first principles. VASP computes an approximate solution to the many-body Schrdinger equation, either within DFT, by solving a Kohn-Sham equations, or within a Hartree-Fock (HF) approximation, by solving Roothaan equations. Hybrid functionals that mix the Hartree-Fock approach with density functional theory are implemented in VASP as well. Furthermore, Green's functions methods and many-body perturbation theory (2nd-order Mller-Plesset) are available in VASP as well (VASP 2020).

(133) 2.3 Empirical Rules

(134) The empirical rules (ZhangZhouLinChen 2008) predict the formation of solid solution phases, based on Delta, which describes the comprehensive effect of the atomic-size difference in multi-component alloy systems, and the mixing enthalpy of a solid solution, H.sub.mix. The empirical rules specify which combinations of (H.sub.mix, Delta) result in formation of a S zone (a zone where only solid solution will form), which combinations result in formation of a S zone (a zone where the a solid solution as a main phase), which combinations result in a B.sub.1 zone, which in a B.sub.2 zone and which in a C zone. The B.sub.2 zone contains Mg and Cu based bulk metallic glasses, while the B.sub.1 zone contains other kinds of bulk metallic glasses, such as Zr. In the C zone, many intermediate phases are expected to form.

(135) In Table 16, we apply the empirical rules to verify the sanity (phase stability) of compositions predicted to yield high tensile strength, on basis of machine learning or regression analysis. Similar to the first-principle (DFT) and thermodynamics (CALPHAD) calculations, the empirical rules can be incorporated into a hybrid computational paradigm, and used to validate (sanity check) the prediction outcomes of traditional ML.

(136) 2.4 Mesoscale Models

(137) The mesoscale models can be used to predict distribution of phases in the microstructure and their morphologies, as influenced by thermal history and alloy chemistry (RadhakrishnanGorti 2019), (RadhakrishnanGorti 2016). Similar to the first-principle calculations, and the empirical rules, this phase information can be used to complement (validate) the prediction outcomes of traditional ML, i.e., as a part of a hybrid computational system.

(138) 2.5 Models Involving Dislocation Dynamics or Slip Band Information

(139) Models Involving Dislocation Dynamics or Slip Band Information may be incorporated into prediction models for fatigue life. The stress/life (S/N) curves tend to be related to crack initiation, which can associated with dislocation dynamics and slip band information.

(140) 2.6 Environmental Resistance (Oxidation, Corrosion or Radiation)

(141) Depending on the chemical reactions involved, and the temperatures at which they occur, one can derive a list of features that properly describe the data, e.g., using canonical component analysis. This can help in terms of developing distinguishing characteristics between CMAS and calcium sulfate (CaSO.sub.4) hot corrosion, with and without the influence of sea salt, and with developing coatings resistant to CMAS and calcium sulfate hot corrosion.

(142) 3. General Approach to Construction of a Physics-Based Model: Application to Prediction of Ultimate Tensile Strength

(143) 3.1 Approach

(144) First, we start out by capturing the physics-based dependencies, per FIG. 18. Assuming a linear model works, each arrow in FIG. 18 represents a coefficient in a matrix. In this case, the parameters at the 1.sup.st level consist of manufacturing, heat treatment, and processing. The parameters at the 2.sup.nd level consist of microstructure, grain size, and defect levels. As depicted in FIG. 18 the dependence relationships still represent a linear model.

(145) Second, the approach assumes constructing an initial, linear regression model with input parameters from a continuous range. The first level of the model can be represented as follows:
z=[micro-structure,grain size,defect level](5)
x=[manufacturing,heat treatment,processing](6)
z=Ax+c.(7)

(146) Now, the second level in FIG. 18 can be written as shown below:
UTS=y=b.sup.Tz+d.(8)

(147) This equation could give rise to an overall linear model of the following type:
UTS=y=b.sup.T(Ax+c)+d=b.sup.TAx+(b.sup.Tc+d)=.sup.Tx+e.(9)

(148) Note if, say, microstructure is a function of other input parameters, then these other input parameters should not be present at the current level, but should be accounted for at a preceding level.

(149) Third, alternatively, one can construct a initial, linear model as follows:
UTS=y=.sup.Tx.sub.B+e,(10)
where, in the case of prediction of the UTS, x.sub.B can be defined as
x.sub.B=[% Al,% Mo,% Nb,% Ti,% V,% Ta,% Zr,% Hf,% Cr].(11)

(150) Fourth, the set of input parameters (x) can be extended, by adding other continuous-valued input parameters, such as temperature. At this point, we have accounted both for the compositions, the temperature, and the continuous-valued input parameters. This arrangement involves a relatively-straight forward extension.

(151) Fifth, one can further extend the set of input parameters (x), by adding categorical inputs, such as for the manufacturing process used. At this point, the problem becomes a mixed-integer optimization problem. Our approach assumes gradual introduction of complexity into the model.

(152) Sixth, one can introduce nonlinearities (non-linear kernel functions), based on the underlying physics. In case of FIG. 18 [Eq. (9)], this formula can be represented as
UTS=y=.sup.T([micro-structure,grain size,defect level])+e=.sup.T(x)+e.(12)

(153) Here, (.Math.) may represent the sigmoid function, a log(.Math.) function, or an exp(.Math.). This invention assumes that (.Math.) is selected such as to suit the application at hand.

(154) Seventh, in case of the prediction presented in FIG. 24 and FIG. 25, the pre-direction model can be represented as
UTS=y=g(x.sub.B).(13)

(155) Some of the underlying physics may be common for regular alloys and HEAs. Hence, one may develop certain aspects of the model, through the analysis of data available for regular alloys. FIG. 19 and FIG. 20 are presented with this in mind.

(156) Eighth, one can systematically expanding the model, such as to introduce additional complexity.

(157) Ninth, one can systematically retrain the model, as additional complexity is introduced. If the addition of a non-linearity or an input parameter (complexity) does not improve the prediction accuracy of the model, one can analyze why the observed prediction accuracy has not conformed with expectations?

(158) 3.2 Further Theoretical Considerations (Justifications)

(159) Broadly speaking, for the selection of kernel functions based on underlying physics, one should first look at the general shape of the data. Sometimes, the selection may be based on qualitative physics-based insights of an expert. The expert may know what to expect, or if there is too much or too little of a given concentration.

(160) A parametrized description related to the microstructure may be based on the following parameters: Radius asymmetry, difference in atomic radii, number of valence electrons, enthalpy of mixing, ideal entropy of mixing, mean melting temperature, difference in Pauling electro-negativity, electro negativity difference, cohesive energy, first ionization energy, covalence radius, electron affinity, molar volume, etc.

(161) The parametrized description of the defect levels may involve a significant undertaking, as noted above, since there are so many different types to consider. A systematic approach to the parametrized description of defect levels may start with a single case (e.g., powder-bed AM) and a single category of defects (macro-scale, micro-scale or nano-scale defects).

(162) A parametrized description of the grain size may similarly involve a significant undertaking. A systematic approach may exclude metallic glasses (amorphous metals), but instead focus on characterizing the distribution in the size and shape of polycrystalline grains.

(163) The simplest description of the heat treatment process may involve a simple listing by categories. However, such categorical designation may be subject a level of arbitrariness with regards to how the data scatters along the axis listing the categories. In case of hot isostatic pressing, one soon would need to account for the temperature, pressure and duration of the HIP. Similarly, in case of annealing, one would need to account for the temperature and duration of homogenization.

(164) 4. Necessary Step Towards a Physics-Based Model: Characterization of Expected Sources of VariationsApplication to Prediction of Ultimate Tensile Strength

(165) In order to yield highly accurate predictions, one needs to understand the sources causing variations in the property predicted, and properly account for these sources.

(166) 4.1 Expected Dependence of Tensile Strength on Alloy Type

(167) As an example, we expect the mechanical properties of refractory HEAs to differ from those of traditional alloys. For refractory HEAs and transition metals, we expect higher yield strength and lower ductility, compared to other alloys. By increasing the concentration of aluminum (Al) in transition metal type materials, one has a reason to believe the strength will improve (JosephStanford 2017).

(168) 4.2 Expected Dependence of Tensile Strength on Temperature

(169) Usually, when the temperature increases, the ultimate tensile strength tends to decrease. However, we are unaware of a theoretical model describing this relationship. But with that said, FIG. 21 captures empirical results supporting this observation.

(170) 4.3 Expected Dependence of Tensile Strength on Manufacturing Technique

(171) Dependence of the tensile strength on the manufacturing technique employed may depend on specifics of the implementation. Traditionally, arc-melting and spark plasma sintering have been the two main processing techniques employed to fabricate bulk HEAs (GorsseHutchinson 2017). To successfully produce homogeneous bulk HEAs by arc-melting, extensive re-melting and intermittent ingot inversions are required, and powder alloying and refinement (typically via balling milling) is necessary when processing via the SPS route (GorsseHutchinson 2017). The main undesirable feature of the metal additive manufacturing process are the non-equilibrium thermal cycles, consisting of the solid-melting crystallization and solid-remelting recrystallization under fast heating and cooling conditions, which generate anisotropic microstructures and defects.

(172) 4.4 Expected Impact of Grain Size on Yield or Tensile Strength

(173) Grain-boundary strengthening (or Hall-Petch strengthening) is a method of strengthening materials by changing their average grain size (GrainBoundaryStrengthening 2019). It is based on the observation that grain boundaries are insurmountable borders for dislocations and that the number of dislocations within a grain have an effect on how stress builds up in the adjacent grain, which will eventually activate dislocation sources and thus enable deformation in the neighboring grain (GrainBoundaryStrengthening 2019). So by changing grain size, one can influence the number of dislocations piled up at the grain boundary, which impacts the yield and tensile strengths (GrainBoundaryStrengthening 2019). The Hall-Petch relation models the relationship between the yield stress and the grain size as

(174) $\begin{array}{cc}{}_y={}_0+\frac{k_y}{\sqrt{d}}& \left(14\right)\end{array}$

(175) Here, .sub.y is the yield stress, .sub.0 is a materials constant for the starting stress for dislocation movement (or the resistance of the lattice to dislocation motion), k.sub.y is the strengthening coefficient (a constant specific to each material), and d is the average grain diameter (GrainBoundaryStrengthening 2019).

(176) 4.5 Expected Dependence of Endurance Limit on UTS, for Fixed Defect Level and Heat Treatment Process

(177) Upon studying stress states, fracture surfaces, and tensile stress at the fracture-initiation site, one can expect the fatigue-endurance limit to scale in proportion with the UTS, for a fixed defect level and heat-treatment process. Ref. (MenzelDauskardt 2006) cites a study of the stress-life fatigue behavior of a Zr.sub.41.25Ti.sub.13.75Ni.sub.10Cu.sub.12.5Be.sub.22.5 bulk metallic glass using notched cylindrical bars, where the fatigue endurance limit of of the UTS was reported. This result was significantly higher than the value of 1/10 of the fatigue endurance limit previously reported using four-point bend specimens (MenzelDauskardt 2006).

(178) 4.6 Expected Dependence of Endurance Limit on Defect Levels, for Fixed UTS and Heat Treatment Process

(179) Defect levels are here taken to broadly represent microstructural effects. While one expects the increased defect level to exert adversarial impact on the endurance limit, complete characterization of microstructural aspects may involve a significant undertaking. Ref. (LiuGwalaniKomarasamy 2019) reports on the intrinsic role of microstructure on persistent slip bands. Ref (LiuGwalaniKomarasamy 2019) notes that although the nano-sized L1.sub.2 precipitates enhance tensile strength, no improvement in fatigue properties have been observed.

(180) 5. Necessary Step Towards a Physics-Based Model: Characterization of Sources of Observed VariationsApplication to Prediction of Ultimate Tensile Strength

(181) Table 9 shows that one can expect2 variations in the endurance limit, based on defect levels (defect size, density, and type) and raw material purity, for a fixed UTS. This trend suggests that the explicit access to the information on the defect level may be needed in order to accurately predict the endurance limit. The samples in Table 9 were homogenized at 1,000 C. for 6 hour, water quenched, and then cold rolled. For Condition 1, shrinkage pores and macro-segregation remained in some portions. For Conditions 2 and 3, shrinkage pores and macro-segregation were removed before cold rolling.

(182) Table 10 similarly illustrates that one can expect2 variations in the endurance limit, likely caused by variations in the grain size, even for the same microstructure (FCC) and similar process (hot-rolled and heat-treated). Together, Table 9 and Table 10 illustrate that the accurate prediction of the endurance limit is not possible, based on the UTS alone. One also needs to know the defect levels (the defect size, density, and type), the gain size, and even parameters of the heat-treatment process. These observations are consistent with those of (HemphillYuanWang 2012) as well as with those of (TangYuanTsai 2015).

(183) TABLE-US-00009 TABLE 9 For a given composition (Al.sub.0.5CoCrCuFeNi), microstructure, grain size, and process, the endurance limit exhibits a high degree of correlation with the defects reported (HemphillYuanWang 2012), (TangYuanTsai 2015). Tensile Endurance Micro Grain Size Defects Strength Limit Composition structure [um] Process Reported (MPa) (MPa) Al0.5CoCrCuFeNi 2 FCC 2 or 1 mm (matrix annealed + Few defects- 1,344 472 Condition-1 phase or Cu-rich) cold-rolled Al0.5CoCrCuFeNi 2 FCC 2 and 1 mm (matrix annealed + commercial-purity 1,344 382 Condition-2 phase and Cu-rich) cold-rolled raw elements Al0.5CoCrCuFeNi 2 FCC 2 and 1 mm (matrix annealed + high-purity raw 1,344 360 Condition-3 phase and Cu-rich) cold-rolled elements Al0.5CoCrCuFeNi 2 FCC 2 and 1 mm (matrix annealed + High defect level 1,344 270 Condition-1 phase and Cu-rich) cold-rolled

(184) TABLE-US-00010 TABLE 10 Variations in endurance limits for CoCrFeNiMn (the Cantor alloy) (KimHamKimLee 2019), (KashaevVentzke 2019), (ChlupFintovaHadraba 2019), (SuzukiKoyamaHamada 2019). Manufac- Load Tensile Endurance Micro- Grain Size turing Defects Ratio Strength Limit Composition structure [um] Technique Process Reported (R) (MPa) (MPa) CoCrFeNiMn FCC 245.48 Vacuum hot-rolled + Not 0.1 625.6 126 (random solid (avg.) induction heat-treated Specified solution) melting CoCrFeNiMn FCC 250-500 Thermite-type as-sintered 0.1 362 90 CoCrFeNiMn (single phase, 100-300 self-propagating laser beam 0.1 349 90 coarse high-temperature welded grained) synthesis (centrifugal set-up) CoCrFeNiMn FCC 41 Vacuum hot-rolled at 1 585 250 (no sign of induction 1373 K + martensitic melting solution-treated transform.) at 1073 K + water-quenched CoCrFeNiMn Only micro- 0.407 Powder Spark plasma 0.1 N/A 495 structural (median) metallurgy sintering (SPS) (Bending CoCrFeNiMn characteristics 0.628 (ball milling at 1150 C. for 5 0.1 test) 450 Specified (median) process) min
5.1 Comparison Across Compositions, Process Parameters, Defect Levels, and Grain Sizes for a Given UTS Further Explanations of the Scatter
1. UTS1,100 MPa

(185) Variations in the defect level alone can result in 2 variations in the endurance limit for the multi-variate data point, as noted in Table 9. At UTS1,100 MPa, the variations observed can additionally be explained in terms of variations in the grain size, microstructure and composition, per Table 10.

(186) 2. UTS1,340 MPa

(187) At UTS1,340 MPa, the variations observed can similarly be explained in terms of variations in the microstructure, grain size, and processing parameters, per Table 9 and Table 10. One cannot expect the accurate prediction of the fatigue resistance, unless knowing parameters of the heat treatment process and the defect levels, in addition to the UTS.

(188) 5.2 Comparison Across Process Parameters, Defect Levels, Grain Sizes, and UTS for a Given Composition

(189) 1. Comparison for AlCoCrFeNi.sub.2.1

(190) We believe that the increments of the UTS and endurance limit, shown in Table 11, areat least in partcaused by the lower defect level for the AlCoCrFeNi.sub.2.1 cold-rolled and heat-treated eutectic HEA (EHEAw) composition. Since the EHEAw samples were annealed after cold-rolling, the grain sizes are likely similar. In addition to defect structures and grain sizes, the variations may be impacted by persistent slip bands in the micro-structure (FengGaoLeeMathes 2016), (AsmHandbook 1990).

(191) 2. Further Comparison for CoCrFeNiMn

(192) Similar to the case for AlCoCrFeNi.sub.2.1, we believe that the hot-rolled and heat-treated process may have contributed to a little higher UTS and endurance limit for CoCrFeNiMn listed in Table 10 and FIG. 27, compared to the cases of as-sintered or laser-beam welded. But more importantly, we believe that the smaller grain size is resulting in a higher strength and endurance limit, per the Hall-Petch relation (Eq. (14)). The hot rolling and heat treatment may have given rise to smaller grain sizes, compared to sintering or laser beam welding. FIG. 27 demonstrates clear inverse correlation between the endurance limit and the grain size, but positive correlation between the endurance limit and the UTS. For information on the fatigue-crack growth behavior of CoCrFeNiMn, refer to (ThurstonGludovatzHohenwater 2017).

(193) TABLE-US-00011 TABLE 11 Variations in endurance limits for as-cast eutectic HEA (EHEAc) and cold- rolled and heat-treated eutectic HEAs (ShuklaWangCottonMishra 2018). Tensile Endurance Micro- Grain Size Defects Strength Limit Composition Structure [um] Process Reported (MPa) (MPa) AlCoCrFeNi2.1 FCC + BCC Not as-cast Not 1,057 374 EHEAc Specified Specified AlCoCrFeNi2.1 FCC + BCC Not cold-rolled + Not 1,340 466 EHEAw Specified heat-treated Specified
5.3 Comparison for 4340 Steel

(194) Since both 4340 steel samples were heat treated, we expect similar defect levels. Judging from Table 9-Table 11, the difference in the tensile strength and endurance limit, listed in Table 12, likely is caused by the difference in the grain size, and possiblydepending on the annealing temperature (not specified in (GorsseHutchinson 2017))microstructure (phases).

(195) TABLE-US-00012 TABLE 12 Variations in endurance limits for 4340 steels (BrownHoMindlin 1979). Tensile Endurance Micro- Grain Size Defects Strength Limit Composition structure [um] Process Reported (MPa) (MPa) 4340 Steel Not Not Quenched & Not 1,260 335 Specified Specified tempered at 538 C. Specified 4340 Steel Annealed 745 170
6. Example 1: Prediction of Fatigue Endurance Limit

(196) The overall methodology for predicting the fatigue resistance is presented in FIG. 22. This is a two-stage prediction process, where we first determine the tensile strength, given an input combination, and then determine the fatigue resistance, given the tensile strength and the inputs.

(197) FIG. 26 compares the high-cycle fatigue properties of HEAs to those of conventional alloys. The HEAs seem to generally result in higher UTS and endurance limits, compared to conventional alloys. Still, there is a lot of scatter in the data. But despite the scatter, the endurance limit seems primarily correlated with the UTS. Hence, by identifying compositions with larger UTS, one can expect greater fatigue resistance. A key to accurate prediction entails understanding, and properly accounting for (explaining), the sources of variations in the data.

(198) The scatter in the data is caused by input sources, such as defect levels, process parameters, or grain sizes, which are not accounted for in the prediction model, or accounted for in the prediction model, but not available at the time of prediction.

(199) The endurance limits in FIG. 26 represent multi-dimensional data points, which in addition to the UTS exhibit dependence on the grain size, process parameters, and defect characteristics.

(200) 6.1 Formulation of the Input Combinations

(201) Our intent is to capture in the input sources that contribute to variations in the fatigue resistance (to variations in the output). In this invention, we like to model the input combination as
input combination=(composition,heat treatment process,defect level).(15)

(202) Here defects are defined broadly such as to include inhomogenities, impurities, and unwanted features.

(203) While we are primarily looking for compositions yielding attractive fatigue resistance, we will see (from Table 9) that the defect level also significantly impacts the fatigue resistance. Hence, the accurate prediction of the fatigue resistance (endurance limit) may be impossible, without knowing the defect information. Similarly, we preferably would like to determine the heat-treatment process that yields the least scatter in the fatigue resistance observed.

(204) 6.2 Estimating the Tensile Strength, Given an Input Combination

(205) Step 1 in FIG. 22 summarizes the overall approach. If the UTS corresponding to a given input combination is known, we can apply a simple look-up. If the UTS corresponding to a given input combination is not known, we can apply prediction (interpolation or extrapolation), based on the nearest neighbors.

(206) FIG. 18 captures a general model of physical dependencies for prediction of the UTS. This model is a generalization of the input sources modeled in Eq. (15). Capturing of the physical dependencies helps greatly in terms of the incorporation of a priori knowledge, derived from the underlying physics, and in terms of making the most of theusually limitedinput data available.

(207) 6.3 Arriving at the Fatigue Resistance, Given the Tensile Strength and the Remaining Inputs

(208) Step 2 in FIG. 22 summarizes the overall approach, for one embodiment of the invention. Here, we model the fatigue resistance (the fatigue endurance limit) as
endurance limit=(UTS,heat treatment process,defect level)(16)

(209) If the endurance limit corresponding to a given input combination is known, we can apply simple look-up. If the endurance limit corresponding to a given input combination is not known, we can predict the endurance limit, on basis of the estimated
UTS(composition,heat treatment process,defect level)(17)
as shown in FIG. 26.

(210) The significant variations observed in FIG. 26 reinforce the need for the access to the relevant input data, for accurate prediction. To obtain the prediction accuracy within the measurement error (10%-20%), or within the accuracy limits on fatigue life expected for given parts, one may need to know quite a bit more about the input (configurational) parameters. An accurate prediction model seems to have the form
endurance limit=(UTS,process,defect(process),grain(process),microstructure(process),T . . . )(18)
where
UTS=UTS(composition,heat treatment process,defect level(process),grain size,T).(19)
7. Example 2: Identification of Compositions Yielding High Tensile Strength
7.1 Review of the Original Data SetRational for Restricting Analysis to Room-Temperature Data

(211) As illustrated in FIG. 21, the ultimate tensile strength exhibits the significant dependence on the temperature. While all compositions in FIG. 21 contained a BCC phase, and were subjected to some type of annealing, the tensile strength at 1,000 C. can be .sup.th (12%) to .sup.rd (33%) of the tensile strength at room temperature. With this in mind, and to maintain consistency across compositions, we elected only to apply our optimization framework to tensile strengths at room temperature. Our original data set, listed in Table 13, contains some 24 compositions that yield relatively-high UTS at room temperature. We derived two feature sets, hereafter referred to as A and B from the original data set in Table 13:
Feature Vector A=x.sub.A=[% Al,% Mo,% Nb,% Ti,% V,% Ta,% Zr,% Hf](20)
Feature Vector B=x.sub.B=[% Al,% Mo,% Nb,% Ti,% V,% Ta,% Zr,% Hf,% Cr].(21)

(212) We have available nineteen (19) instances of the feature vector, A, and twenty two (22), of the feature vector, B. While the set of input data may seem small, we will show that it suffices for the meaningful prediction, provided that a suitable optimization technique is selected.

(213) TABLE-US-00013 TABLE 13 Compositions from the original and enhanced databases yielding the high UTS at room temperature (25 C.). Compositions No. 1 - No. 36 were all fabricated using arc melting. Micro- Select Process Example No. Composition structure Process Specifics UTS (Data Set) 1 Al.sub.0.25NbTaTiZr BCC + B2 HIP + anneal HIP: 2 hr., 1400 C. 1,830 MPa A, B, C Original 2 Al.sub.0.2MoTaTiV BCC As-cast N/A. Remelted a few x 1,249 MPa A, B, C Database 3 Al.sub.0.3NbTa.sub.0.8Ti.sub.1.4V.sub.0.2Zr.sub.1.3 BCC HIP + anneal HIP: 2 hr., 1200 C. 2,061 MPa A, B, C 4 Al.sub.0.3NbTaTi.sub.1.4Zr.sub.1.3 2 BCC HIP + anneal HIP: 2 hr., 1200 C. 2,054 MPa A, B, C 5 Al.sub.0.4Hf.sub.0.6NbTaTiZr BCC HIP + anneal HIP: 2 hr., 1200 C. 2,269 MPa A, B, C 6 Al.sub.0.5Mo.sub.0.5NbTa.sub.0.5TiZr BCC + B2 HIP + anneal HIP: 2 hr., 1400 C. 2,460 MPa A, B, C 7 Al.sub.0.5NbTa.sub.0.8Ti.sub.1.5V.sub.0.2Zr 2 BCC HIP + anneal HIP: 2 hr., 1200 C. 2,105 MPa A, B, C 8 Al.sub.0.6MoTaTiV BCC As-cast N/A. Remelted a few x 1,033 MPa A, B, C 9 AlCr.sub.0.5NbTiV BCC Annealed Homog: 24 hr., 1200C 1,430 MPa B, C 10 AlCrNbTiV BCC + Laves Annealed Homog: 24 hr., 1200C 1,570 MPa B, C 11 AlMo.sub.0.5NbTa.sub.0.5TiZr BCC + B2 HIP + anneal HIP: 2 hr., 1400 C. 2,370 MPa A, B, C 12 AlNb.sub.1.5Ta.sub.0.5Ti.sub.1.5Zr.sub.0.5 BCC HIP + anneal HIP: 2 hr., 1400 C. 1,367 MPa A, B, C 13 AlNbTa.sub.0.5TiZr.sub.0.5 B2 HIP + anneal HIP: 2 hr., 1400 C. 1,357 MPa A, B, C 14 AlNbTiV BCC Annealed Homog: 24 hr., 1200 C. 1,280 MPa A, B, C 15 AlNbTiVZr B2 + AI3Zr5 + Annealed Homog: 24 hr., 1200 C 1,675 MPa A, B, C Laves 99.9 + % purities; 16 AlNbTiVZr.sub.0.1 B2 + AI3Zr5 Annealed Prior to annealing, the 1,395 MPa A, B, C 17 AlNbTiVZr.sub.0.25 B2 + AI3Zr5 Annealed samples were 1,480 MPa A, B, C 18 AlNbTiVZr.sub.1.5 B2 + AI3Zr5 + Annealed incapsulated in 1,550 MPa A, B, C Laves vacuumed (10.sup.2 Torr) quartz tubes. 19 CrHfNbTiZr BCC + Laves Annealed 973 K for 600 sec 1,908 MPa B, C 20 Hf.sub.0.5Mo.sub.0.5NbTiZr BCC As-cast N/A. Melt 5 times 1,538 MPa A, B, C 21 MoTaTiV BCC As-cast N/A. Remelted a few x 1,454 MPa A, B, C 22 HfNbTaTiZr BCC Cold roll + 1373 K anneal in 1,095 MPa A, B, C anneal He atmos. for 5 hr 23 CrMo.sub.0.5NbTa.sub.0.5TiZr 2BCC + FCC HIP + anneal 1723 K/207MN/m.sup.2/3 hr 2,046 MPa C Enhanced 24 MoNbTaV BCC As-cast N/A. Remelted a few x 2,400 MPa C Database 25 CrNbTiVZr BCC + Laves HIP + anneal HIP at 1473 K & 1,725 MPa C 26 CrNbTiZr BCC + Laves HIP + anneal 207 MPa for 2 hr. 1,575 MPa C 27 HfNbTiVZr BCC + Unknow As-cast N/A. Remelted a few x 1,463 MPa C 28 MoNbTiV.sub.0.25Zr BCC As-cast N/A. Remelted a few x 3,893 MPa C 29 MoNbTiV.sub.0.5Zr BCC As-cast N/A. Remelted a few x 3,307 MPa C 30 MoNbTiV.sub.0.75Zr BCC As-cast N/A. Remelted a few x 3,929 MPa C 31 MoNbTiV.sub.1.5Zr 2 BCC As-cast N/A. Remelted a few x 3,300 MPa C 32 MoNbTiV.sub.2Zr 2 BCC As-cast N/A. Remelted a few x 3,176 MPa C 33 MoNbTiV.sub.3Zr 2 BCC As-cast N/A. Remelted a few x 2,508 MPa C 34 MoNbTiVZr BCC As-cast N/A. Remelted a few x 3,828 MPa C 35 MoNbTiZr BCC As-cast N/A. Remelted a few x 3,450 MPa C 36 MoTaTiV BCC As-cast N/A. Remelted a few x 1,454 MPa C 37 Al Purity: 99.99% [44] 45 MPa C 38 Mo Annealed [45] 324 MPa C 39 Nb Annealed [45] 275 MPa C 40 Ti Purity 99.9% [44] 235 MPa C 41 V Cold rolled [44] 828 MPa C 42 Ta Cold worked [45] 900 MPa C 43 Zr Typical [45] 330 MPa C 44 Hf Typical [45] 485 MPa C 45 Cr As-swaged [44] 413 MPa C
7.2 Analysis of Variations in UTS for the Pure Elements Selection of a Suitable Prediction Model

(214) In order to develop insight into the causes of variations in tensile strengths for the pure elements comprising feature vectors A and B, and for the identification of a model for predicting compositions yielding high tensile strengths, and presumably attractive fatigue resistance, we present FIG. 19 and FIG. 20. FIG. 19 shows that processing conditions and purity can contribute to variations in tensile strengths of Al of 3 and of 4 in the tensile strength of Co. FIG. 20 similarly illustrates that processing methods have significant influence on the tensile strength of V and Cr. For the Vanadium, the variations in the tensile strength are 2, and for the Chromium, we are looking at variations of up to 5 (!) This trend suggests that an accurate model for predicting the tensile strength may indeed have the form of Eq. (19). But for relative comparison of tensile strengths across compositions, for the same heat-treatment process and defect levels, and at fixed (room) temperature, the model
UTS=UTS(composition)(22)
may suffice. For the prediction presented in FIG. 36 and the experimental verification outlined in FIG. 37, we employ the prediction model of Eq. (UTS=UTS(composition)(22).
7.3 Selection of a Suitable Optimization Technique

(215) Given the small size of the data set in Table 13, it suffices to say that we are not ready for traditional ML models. Models, such as artificial neural networks, decision trees, support vector machines, Bayesian networks, or genetic algorithms, tend to be effective in organizing and extracting complex patterns from large sets of data, as noted above. But for the application and limited data set at hand, it makes sense to select a simple linear-prediction model, multi-variate linear regression, to begin with, and build from there. As suggested by Agrawal et. al. (AgrawalDeshpande 2014), changing the method may not change the results that much. According to FIG. 5 and Table 2 in (AgrawalDeshpande 2014), the linear regression yields R.sup.2 of 0.963, when predicting the fatigue strength of the stainless steel, compared to R.sup.2 of 0.972 for the artificial neural networks.

(216) Our intent is to start out with the statistical (linear) regression analysis, and account for the underlying sources of (input) variations. We intend to then expand the model, and add non-linearities, based on the underlying physics, and as necessitated by the application at hand and the data available.

(217) 7.4 Setting Up the Optimization Problem

(218) 1. Multi-Variate Linear Regression

(219) When applying the linear regression, we solve a constrained optimization problem of the form

(220) $\begin{array}{cc}{\min }_x{.Math.\mathrm{Bx}-y.Math.}_2^2\mathrm{such}\mathrm{that}\begin{array}{c}0x_{i}100\\ {.Math.}_i{x}_i=100\end{array}.& \left(23\right)\end{array}$

(221) Here, y represents a vector of tensile-strength values, but B the training set of compositions [a stacked version of x vectors, derived from Table 13]. To solve this constrained optimization problem, one can use a function from Matlab or Octave called lsqlin(.Math.).

(222) 2. Quadratic Regression with Diagonal Matrix (for Comparison)

(223) When applying the quadratic regression, we model the UTS (y) as
y=x.sup.TAx+b.sup.Tx+c.(24)

(224) Assuming a general A matrix and a 9-element feature vector (x.sub.B), this model consists of
9+99+1=91 parameters.(25)

(225) Using an unconstrained model with 91 parameters to fit to the data sets in Table 13 does not make sense, since the number of model parameters greatly exceeds the number of data points. Hence, we should be able to fit the model perfectly to the data. In order to reign in the model complexity, we restrict the A matrix to a diagonal form. In this case, the model consists of
9+9+1=19 parameters,(26)
i.e., fewer parameters than listed for Data Set C in Table 13.

(226) In the case of a diagonal A matrix,

(227) $\begin{array}{cc}A=\left[\begin{array}{cccc}{a}_{11}& 0& \mathrm{.Math.}& 0\\ 0& a_{22}& \mathrm{.Math.}& 0\\ 0& 0& \mathrm{.Math.}& 0\\ \mathrm{.Math.}& & & \mathrm{.Math.}\\ 0& 0& \mathrm{.Math.}& a_{\mathrm{mm}}\end{array}\right],& \left(27\right)\end{array}$
the quadratic regression can be cast as an enhanced version of standard linear regression. To accomplish this process, we write the training set, as shown below:

(228) $\begin{array}{cc}{y}_1=a_{11}{x}_1\left(1\right)x_1\left(1\right)+a_{22}{x}_1\left(2\right)x_1\left(2\right)+a_{33}{x}_1\left(3\right)x_1\left(3\right)+\mathrm{.Math.}+a_{99}{x}_1\left(9\right)x_1\left(9\right)+b^T{x}_1+c{y}_2=a_{11}{x}_2\left(1\right)x_2\left(1\right)+a_{22}{x}_2\left(2\right)x_2\left(2\right)+a_{33}{x}_2\left(3\right)x_2\left(3\right)+\mathrm{.Math.}+a_{99}{x}_2\left(9\right)x_2\left(9\right)+b^T{x}_2+c{y}_3=a_{11}{x}_3\left(1\right)x_3\left(1\right)+a_{22}{x}_3\left(2\right)x_3\left(2\right)+a_{33}{x}_3\left(3\right)x_3\left(3\right)+\mathrm{.Math.}+a_{99}{x}_3\left(9\right)x_3\left(9\right)+b^T{x}_3+c\mathrm{.Math.}{y}_N=a_{11}{x}_N\left(1\right)x_N\left(1\right)+a_{22}{x}_N\left(2\right)x_N\left(2\right)+a_{33}{x}_N\left(3\right)x_N\left(3\right)+\mathrm{.Math.}+a_{99}{x}_N\left(9\right)x_N\left(9\right)+b^T{x}_N+c.& \left(28\right)\end{array}$

(229) We then rearrange the terms as follows:

(230) $\begin{array}{cc}{y}_1=c+x_1\left(1\right)x_1\left(1\right)a_{11}+x_1\left(2\right)x_1\left(2\right)a_{22}+x_1\left(3\right)x_1\left(3\right)a_{33}+\mathrm{.Math.}+x_1\left(9\right)x_1\left(9\right)a_{99}+x_1^{T}b{y}_2=c+x_2\left(1\right)x_2\left(1\right)a_{11}+x_2\left(2\right)x_2\left(2\right)a_{22}+x_2\left(3\right)x_2\left(3\right)a_{33}+\mathrm{.Math.}+x_2\left(9\right)x_2\left(9\right)a_{99}+x_2^{T}b{y}_3=c+x_3\left(1\right)x_3\left(1\right)a_{11}+x_2\left(3\right)x_2\left(3\right)a_{22}+x_3\left(3\right)x_3\left(3\right)a_{33}+\mathrm{.Math.}+x_3\left(9\right)x_3\left(9\right)a_{99}+x_3^{T}b\mathrm{.Math.}{y}_N=c+x_N\left(1\right)x_N\left(1\right)a_{11}+x_N\left(2\right)x_N\left(2\right)a_{22}+x_N\left(3\right)x_N\left(3\right)a_{33}+\mathrm{.Math.}+x_N\left(9\right)x_N\left(9\right)a_{99}+x_N^{T}b.& \left(29\right)\end{array}$

(231) This process results in the linear system

(232) $\begin{array}{cc}\left[\begin{array}{c}{y}_1\\ y_2\\ y_3\\ \mathrm{.Math.}\\ y_N\end{array}\right]=\left[\begin{array}{cccccc}1& x_1\left(1\right)x_1\left(1\right)& x_1\left(2\right)x_1\left(2\right)& \mathrm{.Math.}& x_1\left(9\right)x_1\left(9\right)& x_1^T\\ 1& x_2\left(1\right)x_2\left(1\right)& x_2\left(2\right)x_2\left(2\right)& \mathrm{.Math.}& x_2\left(9\right)x_2\left(9\right)& x_2^T\\ 1& x_3\left(1\right)x_3\left(1\right)& x_3\left(2\right)x_3\left(2\right)& \mathrm{.Math.}& x_3\left(9\right)x_3\left(9\right)& x_3^T\\ 1& \mathrm{.Math.}& \mathrm{.Math.}& & \mathrm{.Math.}& \mathrm{.Math.}\\ 1& x_N\left(1\right)x_N\left(1\right)& x_N\left(2\right)x_N\left(2\right)& \mathrm{.Math.}& x_N\left(9\right)x_N\left(9\right)& x_N^T\end{array}\right]\left[\begin{array}{c}{a}_{11}^C\\ a_{22}\\ a_{33}\\ \mathrm{.Math.}\\ a_{99}\\ b\end{array}\right].& \left(30\right)\\ \stackrel{\_}{y}=\tilde{x}\stackrel{\_}{b}.& \left(31\right)\end{array}$

(233) The least-squared solution of Eq. (30) can now be obtained in closed form as.
{circumflex over (b)}=({tilde over (X)}.sup.T{tilde over (X)}).sup.1{tilde over (X)}.sup.T{tilde over (y)}(32)
7.5 Prediction of Composition Yielding Higher UTS, and Presumably More Attractive Fatigue Resistance, than Previously Observed Based on Data Sets A and B

(234) FIG. 23 and FIG. 24 illustrate the process of predicting compositions yielding higher UTS, and presumably higher fatigue resistance, based on Data Sets A and B. The prediction process consists of three main steps:

(235) First, we identify the composition with the highest measured UTS, which in the case of Data Sets A and B is Al.sub.0.5Mo.sub.0.5NbTa.sub.0.5TiZr, with the measured UTS of 2,460 MPa.

(236) Second, we decrease the concentrations corresponding to negative (or small, positive) values of the a vector. In case of FIG. 23 and FIG. 24 this results in decreasing the concentrations of Ti and Hf. But the concentration of Hf in Al.sub.0.5Mo.sub.0.5NbTa.sub.0.5TiZr is already 0.0%, so the Hf cannot be decreased further.

(237) Third, we increase the concentration of elements corresponding to the largest values of the weighting vector, a. These elements exhibit the largest correlation with (or contributions to) the UTS observed. Hence, by increasing these elements, one can expect the largest relative increase in the UTS. In an effort to maximize the UTS, this process results in increasing the concentrations of Nb and Zr (and for Data Set B, the concentration of Cr).

(238) Both Data Sets A and B give rise to the same, predicted composition (Al.sub.0.5Mo.sub.0.5Nb.sub.1.5Ta.sub.0.5Zr.sub.1.5). This consistency suggests that the prediction algorithm may be somewhat immune to minor variations or redundancy (or even discrepancy) in the input data.

(239) Next, we assess the predictive capability of the regression model. The top two figures in FIG. 36 show the predicted UTS as a function of the measured UTS. Here we are applying the same data points from Table 13 (19 and 22, respectively) for training and testing. Even so, we are looking at R.sup.2=0.592 and normalized standard deviation per data point of 15.0 MPa for Data Set A. Similarly, for Data Set B, we are looking at R.sup.2=0.591 and normalized standard deviation of 12.2 MPa. We attribute the spread observed to the fact we are obtaining the data from the open literature, and that variations primarily in the process, but also the microstructure, shown in Table 13, are not accounted for in the prediction model in Eq. (22). FIG. 19 and FIG. 20 clearly indicate that the heat-treatment process applied can significantly impact the UTS observed.

(240) 7.6 Towards Understanding What is Causing Limitations of the ModelAnalysis of Variance (Outliers)

(241) For the purpose of confirming the conjecture about variance in the post-processing (heat-treatment process) applied being a major cause of the variance observed in FIG. 36, we pick a few outliers for further analysis. We are in particular interested in identifying (analyzing) the processing applied to the outliers. Table 14 and Table 15 summarize our analysis of the annotated outliers from FIG. 36. Our anticipated outcome can be characterized as follows:

(242) For outliers above the red lines in FIG. 36 corresponding to
Measured(UTS)<Predicted(UTS),(32)
we expected poor processing (no heat-treatment process, or a cheap process) to be applied.

(243) Here, the other compositions, esp. the ones with good heat treatments, impact the overall weighting (the a vector) such as to improve the overall prediction of the UTS for this data point.

(244) But for outliers below the red lines in FIG. 36, corresponding to
Measured(UTS)>Predicted(UTS),(33)
we thought good processing might have been applied. Here, the other compositions, esp. the ones with poor heat treatment, impact the overall weighting (the a vector) such as to degrade the overall prediction of the UTS for this data point.

(245) The results from Table 14 indeed serve to confirm our conjecture: The outlier, Al.sub.0.4Hf.sub.0.6NbTaTiZr, from FIG. 36 falls below the red line, and has a reasonably-good heat treatment applied (HIP for 2 hours at. 1,200 C.). The other outlier from FIG. 36, Hf.sub.0.5Mo.sub.0.5NbTiZr, shows up quite a bit above the red line, and has no heat-treatment process applied.

(246) The results from Table 15 further serve to confirm our conjecture: The outlier, AlCrNbTiV, from FIG. 36 falls below the red line, and has a reasonably-good heat treatment applied (annealed for 24 hours at 1,200 C.). The other outlier from FIG. 36, MoNbTiZr, shows up somewhat above the red line, and has no heat-treatment process applied. [For full disclosure, the compositions within the green circle had no heat-treatment applied, but yet appeared beneath the red line. This trend may have had to do with the fact that none of the compositions with highest UT Shad heat-treatments applied.]

(247) Overall, these observations strengthen our belief in that the prediction accuracy, measured in terms of R.sup.2 and the standard deviation normalized per data point, is primarily limited by the quality of (variance in) the input data. These limitations in the prediction accuracy are consistent with the variations observed in Table 13, FIG. 19 and FIG. 20. The observations further underline the importance of selecting an optimization technique suitable for the application at hand and data available, and suggest that multi-variate regression is indeed suitable for analysis of the tensile strength.

(248) The methodology presented here is not specific to the tensile strength. Comparison, such as Eq. (32) and Eq. (33), can be presented as a part of outlier analysis for other quantities of interest, as long as combinations of predicted and measured values are available.

(249) TABLE-US-00014 TABLE 14 Analysis of .properties, in particular heat treatment properties, corresponding to two of the annotated outliers in FIG. 36. Outlier Al.sub.0.4Hf.sub.0.6NbTaTiZr Hf.sub.0.5Mo.sub.0.5NbTiZr Process HIP + anneal As-cast Process HIP: 2 hr., N/A. Re-melt Specific 1,200 C. 5 times UTS.sub.meas 2,269 MPa 1,538 MPa UTS.sub.predict 1,657 MPa 2,700 MPa Above or Below Above Below Red Line in FIG. 36? Expectation Since UTS.sub.meas > UTS.sub.predict, Since UTS.sub.meas < UTS.sub.predict, we expect good processing we expect poor processing applied applied Observation Indeed, here a reasonably Indeed, here no heat treatment good heat treatment process process was applied has been applied

(250) TABLE-US-00015 TABLE 15 Analysis of .properties, in particular heat treatment properties, corresponding to other two of the annotated outliers in FIG. 36. Outlier AlCrNbTiV MoNbTiZr Process Annealed As-cast Process Homogenized for N/A. Remelted Specific 24 hr. at 1,200 C. a few times. UTS.sub.meas 1,570 MPa 3,450 MPa UTS.sub.predict 1,043 MPa 3,615 MPa Above or Below Above Below Red Line in FIG. 36? Expectation Since UTS.sub.meas > UTS.sub.predict, Since UTS.sub.meas < UTS.sub.predict, we expect good processing we expect poor processing applied applied Observation Indeed, here a reasonably Indeed, here no heat treatment good heat treatment process process was applied has been applied
7.7 Assessing the Need for a More Sophisticated Prediction Model Further Analysis of Data Set C
1. Criterion for Assessment of Suitability of the Linear Regression, Upon Addition of New Data

(251) The results for Data Sets A and B in FIG. 36 show that one can predict the UTS, using multi-variate linear regression, but that the variance is somewhat large, for reasons stated above.

(252) Further assessment of the suitability of the linear regression can be obtained, by looking at the error margins, upon the addition of new data: If the variances decrease (improve), upon the addition of new data, then linear regression is a good model. If the variance stays approximately the same, then the linear regression is a questionable technique. But if the variance increases (deteriorates), upon the addition of new data, then we may need to look for another method.

(253) 2. Prediction of Compositions with Higher UTS than Previously Observed, Based on Data Set C

(254) FIG. 25 illustrates the process of predicting compositions yielding higher UTS, and presumably higher fatigue resistance, than previously observed, based on Data Set C from Table 13. The prediction process consists of the same three, main steps, as the prediction in FIG. 23 and FIG. 24. The 4,847.1 MPa tensile strength for the predicted composition, MoNbZr, is 918 MPa higher than the highest UTS that has been previously observed (which is 3,929 MPa, observed for MoNbTiV.sub.0.75Zr).

(255) If desired, one can include the tensile strength of the pure elements comprising Feature Vector B, such as to introduce natural barriers, whichtogether with the measured compositionscan limit the extent of the extrapolation. The nine (9) pure elements in feature vector B provide references, which one can compare against, during the extrapolation.

(256) When expanding the data set, we considered also expanding the feature vector, by adding Co, Fe, and Ni. But eventually, we decided against it. While expanding the feature vector would have allowed us to introduce at least forty (40) additional compositions, and the addition of Cu would have allowed us to introduce several new compositions on top of that, all of these compositions corresponded to lower UTS than the highest UTS observed in Table 13. Out of these additional compositions considered, the UTS, which came closest to the composition in Table 13 with the highest UTS (MoNbTiV.sub.0.25Zr with UTS of 3,893 MPa), was AlCoCrFeNi, which exhibited UTS of 3,531 MPa. Out of the additional compositions considered, a dozen or so exhibited UTS in the range of 2,500-3,200 MPa. But very few measured at higher UTS.

(257) 3. Suitability of Linear Regression for Data Set C

(258) To the bottom left, FIG. 36 presents results from applying the multi-variate linear regression to Data Set C. Based on the figure, and R.sup.2=0.77, it does seem that the data points are more or less following linear regression. But the results also may not be conclusive; still more data may be necessary. Although quadratic dependence is not apparent in the figure, such (weak?) dependence may still be present in the data.

(259) When comparing the accuracy of the fit for the linear regression for Data Sets B and C, we notice that the normalized standard deviation per data point has decreased somewhat (from 12.0 MPa to 11.1 MPa). Applying the criterion above, this trend suggests that linear regression is a reasonably-good technique. Linear regression seems to provide the reasonably-good description of the data.

(260) 4. Suitability of Quadratic Regression for Data Set C

(261) To the bottom right, FIG. 36 presents results from applying quadratic regression, with the diagonal A matrix per Eqs. (24) and (27), to Data Set C. Judging from the figure, R.sup.2=0.94, and the normalized standard deviation of only 5.6 MPa per data point, it may seem that the prediction accuracy is limited by the prediction method more so than the variance in the inputs. Note though that, given limitations in the availability of the input data, we are using Data Sets A, B, and C (the same data sets) both for training and testing. Note also that the linear regression involves 10 model parameters, but the quadratic regression 19, per Eq. (26). Hence, the improvement in the accuracy of the fit, in case of the quadratic regression, is accomplished in part by fitting data to the 9 new model parameters. While quadratic regression with a diagonal matrix may indeed be a good technique for the application at hand, proper characterization of the relative merits of linear vs. quadratic regression will require separate data sets for training and testing. Further, although we may have sufficient data for a technique more sophisticated than linear regression (e.g., quadratic regression), we still believe the overall accuracy is primarily limited by variations in the inputs. This assessment is motivated by FIG. 19 and FIG. 20.

(262) 7.8 Verifying Feasibility of the Predicted Compositions Empirical Rules

(263) Table 16 captures the outcomes from applying the empirical rules of (ZhangZhou 2008), (FengGaoLeeMathes 2016) to the formation of the predicted compositions, Al.sub.0.5Mo.sub.0.5Nb.sub.1.5Ta.sub.0.5Zr.sub.1.5 and MoNbZr.

(264) TABLE-US-00016 TABLE 16 Assessment of viability of the predicted compositions (Al.sub.0.5Mo.sub.0.5Nb.sub.1.5 Ta.sub.0.5Zr.sub.1.5 and MoNbZr) through the application of empirical rules (ZhangZhou 2008). Alloy Al.sub.0.5Mo.sub.0.5NbTa.sub.0.5TiZr Al.sub.0.5Mo.sub.0.5Nb.sub.1.5Ta.sub.0.5Zr.sub.1.5 MoNbTiV.sub.0.75Zr MoNbZr .sub.r (%) 4.41 4.89 5.65 5.56 H.sub.mix (kJ/mol) 10.52 10.17 2.70 3.56 S.sub.mix (J/K/mol) 14.43 12.18 13.33 9.13 3.16 2.89 11.79 6.66
1. Expected Properties of Al.sub.0.5Mo.sub.0.5Nb.sub.0.5Ta.sub.0.5Zr.sub.1.5, Based on the Empirical Rules

(265) By comparing the calculated parameters for the atomic difference, .sub.r, and the enthalpy of mixing, H.sub.mix, from Table 16 to FIG. 2 from (ZhangZhou 2008), one can see that a solid solution will likely form both in the predicted composition, Al.sub.0.5Mo.sub.0.5Nb.sub.0.5Ta.sub.0.5Zr.sub.1.5, and in the reference composition, Al.sub.0.5Mo.sub.0.5NbTa.sub.0.5TiZr. However, the predicted and reference compositions fall near the boundary between S and S regions in FIG. 2 from (ZhangZhou 2008). This trend suggests a small amount of the ordered solid solution precipitates may also form as a minor phase. We expect Al.sub.0.5Mo.sub.0.5Nb.sub.1.5Ta.sub.0.5Zr.sub.1.5 to be a stable composition with two types of phases.

(266) 2. Expected Properties of MoNbZr, Based on the Empirical Rules

(267) Again, by comparing the calculated parameters for the atomic difference, Sr, and the enthalpy of mixing, H.sub.mix, from Table 16 to FIG. 2 from (ZhangZhou 2008), one notices that the predicted composition, MoNbZr, sits in the middle of the S region in FIG. 2, which suggests the composition has high chance of forming a solid solution main phase with ordered solid solution precipitates.

(268) 7.9 Experimental Verification of Predicted CompositionsAl.sub.0.5Mo.sub.0.5Nb.sub.1.5Ta.sub.0.5Zr.sub.1.5 and MoNbZr

(269) FIG. 37 summarizes results from the experimental verification of the strength of the predicted composition, Al.sub.0.5Mo.sub.0.5Nb.sub.1.5Ta.sub.0.5Zr.sub.1.5, in comparison to the reference, Al.sub.0.5Mo.sub.0.5NbTa.sub.0.5TiZr. We conclude from the experimental results in FIG. 37 that the candidate composition indeed exhibits higher UTS than the reference, hence confirming the outcome of our prediction. Coupons for both the predicted and reference compositions were prepared, using arc melting and compression testing, for as-cast samples, and with no heat treatment applied. The casted samples had the initial diameter of a quarter inch (6.27 mm) based on the mold used. The diameter of the samples was then decreased by grinding the cylindrical surfaces to diameter of 4-5 mm, in order to obtain higher stress, and ultimately break the samples, using a mechanical compression testing machine with maximum force of 15,000 lbs. From the compression tests, it turned out that the predicted composition had relatively higher yield strength and fracture strength than the reference, as shown in FIG. 37. But both compositions were very brittle. Reminiscent of the trade-off between strength and ductility, this came as no surprise.

(270) 8. Example 3: Prediction of Fatigue Life (Stress Life or Strain Life) and Crack Growth

(271) 8.1 Prediction of Fatigue Life Remaining (S/N Curve), in Presence of No Cracks

(272) 1. Objective

(273) For the triplet
(process,stress applied,cycles to failure),(34)
the goal is to accurately infer the process parameter from the combination
(stress applied,cycles to failure),(35)
in order to properly differentiate between process categories.

(274) Similarly, we can look to estimate cycles to failure (fatigue life) from the combination (process, stress applied), or stress applied (fatigue strength) from the combination (process, cycles to failure).

(275) 2. Method

(276) The method, presented in FIG. 36, is analogous to the method used for prediction of the endurance limit (see Eq. (16)-Eq. (19)). The main difference is that here we are taking the stress applied as an additional input parameter. The method will be essentially the same as for predicting the endurance limit, except here we are doing separate prediction at each stress level.

(277) 3. Metric Used to Measure Success:

(278) We will characterize quality in terms of the variance of the predictor or the MSE.

(279) The variance in the system output, A y, for the system model in Eq. (1), is in part determined by the variance in the system input, A k, and in part by the model. In case of independent inputs, the variance in the system output can be modeled as

(280) 0$\begin{array}{cc}{y}_i={.Math.}_{j=1}^p\frac{f}{x_j}{x}_j.& \left(36\right)\end{array}$
4. Expected Results

(281) Defects are easier to introduce with AM. Hence, we expect more scatter in S/N data for AM than for casting.

(282) 8.2 Prediction of Fatigue Life Remaining, Given a Crack

(283) Fatigue life, defined in terms of the number of cycles, N, in the presence of cracks, is usually estimated as

(284) $\begin{array}{cc}N={}_{a_0}^{a_f}\left(\frac{\mathrm{dN}}{\mathrm{da}}\right)\mathrm{da}.& \left(37\right)\end{array}$

(285) According to the NASGRO equation by Forman and Mettu (FormanMettu 1992)

(286) $\begin{array}{cc}\frac{\mathrm{da}}{\mathrm{dN}}={\left(C*F\right)}^{-1}*K^{-m}*\frac{{\left(1-\frac{K_{\max }}{K_C}\right)}^q}{{\left(1-\frac{K_{\mathrm{th}}}{K}\right)}^p}={\left(C*F\right)}^{-1}*K^{p-m}{K}_C^{-q}\frac{{\left(K_C-K_{\max }\right)}^q}{{\left(K-K_{\mathrm{th}}\right)}^p}.& \left(38\right)\end{array}$
Here, C represents a crack growth constant, F a crack velocity factor, AK a stress intensity factor range and K.sub.max a maximum stress intensity factor. Furthermore, m denotes Paris exponent and K.sub.th threshold of stress intensity factor range for crack propagation.

(287) By applying ML to estimating the number of cycles, N, directly, one may avoid error magnification that otherwise could occur during the integration process, due to over- or under-fitting.

(288) 9 Example 4: Towards Corrosion Resistant CoatingsAnalysis of CMAS and Calcium Sulfate Attacks

(289) 9.1 Overall Picture

(290) This invention can utilize literature materials data or experimental data to develop models or algorithms for machine learning (ML) that will detect data patterns and characteristic trends, learn from the accumulated data, and evolve distinguishing characteristics between calcium-magnesium-alumino-silicate attack (CMAS) and calcium sulfate (CaSO4) hot corrosion, with and without the influence of sea salt, in order to develop resistant coatings to CMAS and calcium sulfate hot corrosion. FIG. 33 presents corrosion data analytics in context with the overall effort to identify high-risk areas for deposit build-up in gas turbines.

(291) We seek to link structure and chemistry to observed reaction mechanisms to accelerate materials design for corrosive environments.

(292) The intent is to develop sophisticated physics-based prediction models from experimental data.

(293) The deposition model of (KulkarniEPRI 2020) provides valuable information on where deposits are likely to take place inside a gas turbine. The deposition model incorporates the particulate characteristics and component design conditions to identify high risk areas for deposit buildup for actual components. The deposition model covers the cases of hot corrosion for land-based, air-borne or sea water applications of gas turbines. In sea water, sodium chloride or sodium sulfate tend to be prevalent, and the corrosion reactions may occur at lower temperature. But the deposition model still applies (with adjusted input values).

(294) 9.2 Essence of CMAS and Calcium Sulfate Corrosion Attacks

(295) The CMAS attacks the ceramic (top coating) first, and then attacks the metal side. For CMAS, reaction with TBC is the only thing we consider. The calcium sulfate (CaSO4), on the other hand, soaks into the ceramics (top coating), and then attacks the bond coat surface and the base alloy (the super-alloy). For the calcium sulfate, there is less interaction with the TBC, but more with the base alloys.

(296) 9.3 Specifics of Interaction of CMAS with the Thermal Barrier Coating

(297) CMAS consists a combination of SiO.sub.2, CaO, Mg, Al.sub.2O.sub.3 and FeO. For information on relative concentration of these constituents between different types of CMAS simulated sand, engine deposits, average earth's crust, Saudi sand, airport runway sand, Mt. St. Helen's volcanic ash, Eyjafjallajokull volcanic ash, subbituminous fly ash or bituminous fly ash, refer to (LeviHutchinson 2012).

(298) CMAS degradation is both thermochemical and thermomechanical to the thermal barrier coatings (TBCs), as shown in FIG. 34. Molten CMAS (1,150-1,240 C.) penetrates the TBC pores and freezes a given depth within the TBC. Above 1250 C., CMAS deposits can infiltrate and significantly dissolve YSZ top coatings via thermochemical interactions. Upon cooling, zirconia can reprecipitate with a spherical morphology and a composition that depended on the local melt chemistry. Upon cooling, the CMAS can destabilize the YSZ top coating through a detrimental phase transformation (t.fwdarw.t.fwdarw.f+m).

(299) Calcium oxide (CaO) is known to react with chromium contained in MCrAlY (M=Ni, Co) alloys and nickel-based superalloys to form a low-melting (1,100 C.) calcium chromate. The reactivity of gamma-NiAl and gamma-Ni-based NiCoCrAlY alloys with CaO at 1,100 C. produced multi-layer scales of Al.sub.2O.sub.3 and calcium aluminates (xCaO-yAl2O3). Increasing alloy chromium content only enhances corrosion severity. The reaction of two-phase beta-gamma MCrAlY alloys with CaO progressed according to two distinct mechanisms: 1. During the initial stage, formation of a liquid calcium chromate led to the rapid consumption of the Cr-rich gamma-phase. The extent of degradation was particularly important for a single-phase gamma-composition, and was significantly reduced by increasing the alloy beta fraction. 2. In the subsequent stage, a continuous Al.sub.2O.sub.3 layer was established at the base of the scale, which led to a much lower oxidation rate. Additions of Al.sub.2O.sub.3 or SiO.sub.2 decreased the CaO reactivity due to the formation of aluminates or silicates.

(300) Upon cooling, the glass and reaction product phases solidify and the void structure that is utilized to reduce thermal conductivity and provide the strain compliance is lost leading to TBC delamination, as shown in FIG. 34.

(301) 9.4 Specifics of Interaction of Calcium Sulfate with the Thermal Barrier Coating and Base Alloy

(302) Due to relatively short history, modeling of CaSO.sub.4 is in part based on analogy with sodium sulfate (Na.sub.2SO.sub.4).

(303) Early research has shown that CaSO.sub.4 tends to attack yttria, destabilizing zirconia-based TBCs.

(304) CoNiCrAlY in both as-sprayed and preoxidized condition suffered a significant damage by CaSO.sub.4 deposits via a basic fluxing mechanism that yielded CaCrO.sub.4 and CaAl.sub.2O.sub.4.

(305) TABLE-US-00017 TABLE 17 Characterization of the reaction space for CMAS and calcium sulfate attacks, with and without sea salt. Siemens has a few sites where they have seen CaS0.sub.4 attacks, and they have done the reaction studies for those super-alloys. Attack/ Deposit Environment Reactions Temperature CMAS Natural/Air CaSO.sub.4 + 2H.sub.2O = CaSO.sub.4 Dehydration at 150 C. (Calcium- (without sea salt) CaCO.sub.3 = CaO + CO.sub.2 ~800 C. magnesium- CaSO.sub.4 = CaSO.sub.3 + 1/2O.sub.2 ~1200 C. alumino- CaSO.sub.3 = CaO + SO.sub.2 silicate; Reaction with TBC CaOMgOAl2O3SiO2Y2O3ZrO.sub.2 is the key system CaOMgOAl.sub.2O.sub.3SiO.sub.2) (7YSZ) Prototypical salt: Fluxing process (Type I) ~900 C. Na.sub.2SO.sub.4 (T.sub.melt = 881 C.) Salt-component processes (Type II) ~700 C. Calcium Natural/Air CaSO.sub.42H.sub.2O = CaO + SO.sub.3 + 2 H.sub.2O ~1220 C. Sulfate Sodium sulfate Natural/Air 2 NaCl + SO.sub.2 + O.sub.2 = Na.sub.2SO.sub.4 + Cl.sub.2 ~900 C.
9.5 Overview of the Reaction Space

(306) The reaction space for CMAS, calcium sulfate and sodium sulfate (Na.sub.2SO.sub.4) deposition is summarized in Table 17. We are including sodium sulfate for historic reference. Research into hot corrosion and its preventive measures in gas turbine engines has mostly focused on sodium sulfate since the early 1950s.

(307) 9.6 List of Features Jointly Characterizing CMAS and Calcium Sulfate Corrosion Attacks

(308) Table 18 summarizes a unified list of features characterizing CMAS and calcium sulfate corrosion attacks, both in air and sea water. In sea water, sodium chloride or sodium sulfate tend to be prevalent, and the corrosion reactions may occur at lower temperature. But the same feature list still applies (with adjusted input values).

(309) TABLE-US-00018 TABLE 18 Unified feature list for analysis of CMAS and calcium sulfate hot corrosion, both covering attacks in air and sea water. I/O Parameter Note Inputs Composition of CMAS Applicable to CMAS attacks Composition of TBC Applicable to CMAS and CaSO.sub.4 attacks Melting/solidification Applicable to CMAS attacks temperature Modulus of CMAS Applicable to CMAS attacks Modulus of TBC Applicable to CMAS attacks Particle size Applicable to CMAS attacks Surface temperature Applicable to CMAS attacks Wall shear velocity Applicable to CMAS attacks CaSO.sub.4 concentration Applicable to CaSO.sub.4 attacks Dew point Applicable to CaSO.sub.4 attacks Base alloy composition Applicable to CaSO.sub.4 attacks Air-fuel ratio Applicable to CaSO.sub.4 attacks Surface pressure Applicable to CaSO.sub.4 attacks Time Applicable to CMAS and CaSO.sub.4 attacks Outputs Chemical reactions Applicable to CMAS attacks Particle temperature Applicable to CMAS attacks Particle velocity Applicable to CMAS attacks Coating modulus after Applicable to CMAS attacks infiltration Deposit build-up rate Applicable to CMAS attacks Weight change Applicable to CaSO.sub.4 attacks Metal loss Applicable to CaSO.sub.4 attacks Depth of attack Applicable to CaSO.sub.4 attacks
9.7 Canonical Component Analysis for Deriving Distinguishing Characteristics Between CMAS and Calcium Sulfate Hot Corrosion Attacks

(310) In order to evolve distinguishing characteristics between CMAS and calcium sulfate hot corrosion, with or without the influence of sea salt, in order to develop resistant coatings to CMAS and calcium sulfate hot corrosion, we apply canonical component analysis, as qualitatively shown in FIG. 35.

(311) As an alternative to canonical component analysis, one also can conduct correlation between the input and output features, using regression analysis, as shown in (SteingrimssonJonesKisialiou 2018).

(312) 9.8 Joint Optimization

(313) To accurately identify a (TBC, base alloy) combination that is likely both protect against a CMAS and calcium sulfate attack, assuming the CMAS protection is measured through deposit build-up rate, but the calcium sulfate protection through weight change, metal loss and depth of attack, we apply joint optimization.

(314) We optimize a weighted objective function of the form
Objective=w.sub.1 deposit_build_up_rate+w.sub.2 weight_change+w.sub.3 metal_loss+w.sub.4 depth_of_attack.(39)

Example 5: Joint Optimization of Material Strength and Ductility

(315) For operating turbines or other energy conversion devices at higher temperature, and with improved efficiency, there is need for materials yielding good strength at higher temperature, without sacrificing ductility at room temperature. The design goals involve joint optimization: 5. Achieving the required material strength at higher temperature. 6. Improving room temperature ductility (decreasing the brittle-to-ductile transition below room temperature). 7. Offering acceptable oxidation resistance.

(316) In principle, there are two, primary routes for formulating, such joint optimization problems: 1. Through maximization of a joint objective function, one accounting both for strength and ductility. 2. Through maximization of an objective function only capturing the strength, but where the ductility is accounted for in the constraints.
7.3 Specific Approach to Prediction of Employing Statistical ModelingPrediction of Fatigue Life of Additively Manufactured Components
1. Statistical Modeling Compared and Contrasted to Physics-Based Modeling

(317) For certain applications, the selection of the prediction model may be based on a probabilistic, not physics-based, derivation. The assumption of a constant failure rate, i.e., of 1/ representing the time to failure, results in an exponential function (a standard Weibull distribution).

(318) At times, one may assume independence between events, and invoke the Central Limit Theorem of statistics. In case of alloy design, the parameters may tend to be inter-related, and hence, we may not be able to invoke the Central Limit Theorem. One may need to derive dependent distributions, based on the dependency relations established (per FIG. 18).

(319) 2. Statistical Fatigue Life Model for Analytical Representation of Stress/Life (S/N) Curves

(320) A commonly used analytical representation of S/N curves, like the ones presented in FIG. 38, is given by (ChernNandwana 2019), (HemphillYuanWang 2012):
N()=c.sup.d.(40)

(321) Here, represents the applied stress, N() is the expected cycles to failure at the stress a, and c and d are positive material parameters. Taking the logarithm of the S/N relation given by Eq. (39) yields (ChernNandwana 2019), (HemphillYuanWang 2012):
log[N()]=a+b*log(),(41)
where a=log(c) and b=d. Eq. (40) describes the relationship between the mean-log of the fatigue life and the applied stress. In order to account for scattering observed in the fatigue-life experiments, a regression model is formulated by adding a random error term (ChernNandwana 2019), (HemphillYuanWang 2012):
log[N.sub.ij]=.sub.i(.sub.ij)+.sub.ij=a.sub.i+b.sub.i*log(.sub.ij)+.sub.ij(42)

(322) Here, i indexes the process categories available, but j the data points available for each process category. N.sub.ij is the j-th data point under condition i collected at the stress .sub.ij . . . .sub.ij is a random error term, which is assumed to follow normal distribution with mean zero and standard deviation, s.sub.i. .sub.i () is the mean (also the median) logarithm transformed fatigue life at stress, , under condition i. The collected fatigue life data is denoted by {N.sub.ij,.sub.ij,.sub.ij)}, where is a runout indicator, defined as =1 for a failure observation and =0 for a runout. The likelihood function for the observed fatigue life data under condition i is then given by (ChernNandwana 2019), (HemphillYuanWang 2012):

(323) $\begin{array}{cc}{L}_i\left(a_i,b_i,s_i\right)={.Math.}_{j=1}^{m_i}{\left[\left(\frac{\log \left(N_{\mathrm{ij}}\right)-a_i-b_i*\log \left({}_{\mathrm{ij}}\right)}{s_i}\right)\right]}^{{}_{\mathrm{ij}}}{\left[1-\left(\frac{\log \left(N_{\mathrm{ij}}\right)-a_i-b_i*\log \left({}_{\mathrm{ij}}\right)}{s_i}\right)\right]}^{-1-{}_{\mathrm{ij}}}& \left(43\right)\end{array}$
where (.Math.) and (.Math.) represent the probability density function and the cumulative distribution function of the standardized normal distribution, respectively.
3. Augmentations of the Statistical Fatigue Life Model

(324) The Statistical Fatigue Life Model can be augmented such as to include additional input parameters.

(325) In case of N independent events, the probability of failure, P(fail), can be formulated using Poisson distribution:
P(fail)=.sub.i=1.sup.NP.sub.i(fail)=.sub.i=1.sup.Nf.sub.i exp(c.sub.i)=.sub.1*.sub.2* . . . .sub.N*exp(.sub.i=1.sup.Nc.sub.i).(44)

(326) With this in mind, it makes sense to model the augmented version of the Statistical Fatigue Life model (Eq. (39)) as
N(,p.sub.1,p.sub.2,p.sub.3, . . . ,p.sub.N)=.sub.1(p.sub.1,p.sub.2,p.sub.3, . . . ,p.sub.N).sup..sup.2.sup.(p.sup.1.sup.,p.sup.2.sup.,p.sup.3.sup., . . . ,p.sup.N.sup.).(45)

(327) Here, p.sub.1, p.sub.2, p.sub.3, . . . , p.sub.N model the input parameters impacting the fatigue life of AM metallic components. For a specific parameter selection, refer to Table 1. We are assuming that multiple effects cause failure and that these effects are close to independent. The function .sub.1(.Math.) models a prior knowledge and the function .sub.2(.Math.) conditional probabilities. For dependent events, one can apply a Bayesian model, with the same definition of .sub.1(.Math.) and .sub.2(.Math.).

(328) In case of independent events, it makes sense to apply the direct linear regression to assess .sub.1(.Math.) and .sub.2(.Math.). But in case of coupled failure modes, .sub.1(.Math.) and .sub.2(.Math.) may consist of complex Bayesian functions. We may not know these functions, and one may be able to apply regression analysis, but these functions still may be hard to derive. So this is where ML comes in. One can apply neural networks or support vector machines to effectively deduce these functions from the data. Even if 100-200 parameters impact the fatigue life of AM metallic components, this is still a relatively small set by the standards of ML.

(329) The model of Eq. (44) should be able to predict the time to failure with better accuracy, for reasons similar to the quadratic regression model in FIG. 36 providing accuracy superior to that of the linear regression. The main objective is to quantify the accuracy and reduce the error bars (improve the accuracy of the fatigue prediction).

(330) One of the primary advantages of the model in Eq. (44) involves the ability to (a) determine the top factors that contribute to fatigue life, and (b) provide feedback, through sensitivity analysis. By estimating

(331) $\begin{array}{cc}\frac{N\left(,p_1,p_2,p_3,\mathrm{.Math.},p_N\right)}{p_i},i=1,2,3,\mathrm{.Math.},N,& \left(46\right)\end{array}$
one can assess the contribution of the input parameter, i, to the fatigue life. Such feedback may yield significant, tangible benefits. The dominant factors contributing to the reliability may involve something in manufacturing. Maybe it is something involving the material properties. Maybe the smaller gain size will improve the fatigue life. Our model may be able to quantify for how long to expose the laser, at which temperature, with which grain size, and translate into fatigue life. Such information can be of great value, and may lead to iterative refinements. The fatigue-prediction toolset can advise on, say, how to change a given material property. Once the property has been changed, and new data generated, the data can be fed back into the model and the impact assessed.

(332) It is our understanding that small variations in the atomic % of Sulphur or Carbon can have significant impacts on fatigue life of the stainless steel.

(333) 4. Review of Advantages of Machine Learning for Prediction of Fatigue Life of AM Components

(334) ML can help address (avoid) inaccuracies in fitting traditional models to real-world fatigue data. Traditional models are prone to under- or over-fitting, and can lead to error magnification during integration.

(335) ML can help in terms of accounting for all the sources that can impact fatigue life of AM components. As noted above, it has been reported that over 100 different process parameters can affect the fatigue life of AM components (ChernNandwana 2019). Traditionally, parametric models have been designed to account for key sources, but not for all the sources.

(336) Existing software tools cannot predict the material properties or account for AM.

(337) Given the Statistical Fatigue Life model, or its augmented version, one can play with the input parameters such as to maximize fatigue life (optimize for fatigue life), and provide feedback on manufacturing process. The invention can provide feedback on the impact that variations in input parameters have on fatigue life.

(338) 7.4 Prediction for an Intelligent AM System

(339) Key steps will include systematically changing the key parameters (the laser power and travel speed during deposition, powder feed rate and increment, number of laser tracks for each patch and overlap value, repeat times and laser power during re-melting, the powder size, powder shape, powder distribution and purity), as shown in FIG. 39.

(340) Factors that make AM, and in particular laser powder bed fusion AM, such a challenging manufacturing process are: 1. The smallness of the laser processing volume and rapid melt time when compared to the final part size and build time respectively and the associated process variabilities that result from them. 2. The intrinsic variability of all the powder bed physical (mass, heat capacity, thermal conductivity, emissivity, reflectivity) and chemical (composition, oxidation state, wetting angle) properties that compound to the above-mentioned process variabilities. 3. The large power densities required to process the powder bed and the associated large heating rates and thermal gradients, which when combined with the above-mentioned variability makes it difficult to control the microstructure of the processed volume. 4. The chaotic nature of the AM process that results from combining the small spatial and temporal scales described above with the high energy densities required for melting the powder, which makes it difficult to reliably predict the process trajectory in the multi-parameter process space before the build process starts and virtually impossible to control it in real time. 5. The large number of process parameters (in some cases over 100) that can affect the outcome of the AM process and make it almost impossible to model with physics-based models. 6. The non-symmetric deposition of the processing energy that results from rastering a single laser beam over the powder bed which leads to non-uniform heating/cooling rates, thermal gradients, residual stresses and part defects and distortion.

(341) Most of these challenges can be alleviated by better controlling and distributing the laser energy at and around the melt pool area and/or the processing part surface area combined with real-time monitoring of the same area or beyond and by intelligently linking the laser energy control parameters with the process monitoring sensors to learn and adapt to the continuously evolving environment. Distributing the process energy intelligently at and around the melt pool would help reduce the process variability, the powder bed physical property variability, the heating/cooling rates and the thermal gradients. For example, it might be desirable to pre-heat the powder ahead of the melt-pool without melting it, to reduce the heating rates and thermal gradients later during melting. Doing so might allow processing the powder faster and reducing the build time while at the same time reducing evaporative recoils, ejecta and denudation effects (which induce defects in the final part). Monitoring the temperature profile around the melt-pool area could be used to adjust the distributed laser energy control parameters (power levels and distribution) in real time in a system where the temperature profile is directly linked to the heating source control parameters via an Al processor. Similar improvements could be achieved by intelligently distributing the laser processing energy over the entire part surface while monitoring the temperature evolution over the same area.

(342) The intelligent AM system links the actuators controlling the laser energy distribution over the powder bed with the sensors that monitor the temperature distribution and/or other relevant process parameters over the powder bed using a real-time Al controller.

(343) Specifically, the intelligent AM system employs a multi-beam strategy to customize the melt pool temperature and the energy distribution over the powder bed, as shown in FIG. 40.

(344) 7.5 Other ApplicationsPrediction of Properties Beyond Alloy CompositionsComposites

(345) Our approach entails establishing correspondence with existing research work for HEAs. The data analytics and optimization techniques would be extended to next-generation multi-functional composites.

(346) FIG. 3 presents the primary sources impacting the material properties of HEAs to matrix composites.

(347) FIG. 13 shows how prediction of material properties of HEAs can be extended to matrix composites.

(348) One can formulate models capturing the underlying physical dependencies, similar can FIG. 18.

(349) Similar to the case of alloy design, one can start with simple models and build up from there (look to capture the underlying physics and correspondingly introduce non-linearities in the models, for improved accuracy).

(350) In further correspondence with the case of alloy design, one can select an analysis technique, suitable for the application at hand and for the data available.

(351) 8. Verification, Validation and Reporting

(352) 8.1 Verifying Predictive Capability of ML Algorithms

(353) Verification of the predictive capability of the ML algorithms is primarily based on comparison with experimental results, as shown in FIG. 37. One can print materials in relevant geometries from down-selected compositions predicted by the ML algorithm, such as materials predicted to perform well in elevated temperature creep environments. Comparison of predicted material properties to the observed properties will enable assessment of the predictive capabilities of the machine learning algorithms.

(354) 8.2 Analysis of OutliersVerification Addressing Implications of Incomplete Data Sets

(355) Table 14 and Table 15 illustrate how outliers, resulting from a limited data set (FIG. 36) can be analyzed.

(356) 8.3 Approach to Rapid Screening

(357) For rapid screening (high throughput experiments), one may emphasize properties that do not depend heavily on the microstructure.

(358) 8.4 Uncertainty Quantification

(359) We assume the uncertainty quantification is consistent with similar activities within NIST or CALPHAD.

(360) NIST has some interesting projects for AI/ML data extraction and uncertainty prediction, such as Al self-quality assurance using learning curves in feedback loops and CALPHAD Uncertainty (NistAiUncertainty 2019).

(361) 8.5 Reporting

(362) The reporting mechanism is integrated into the user interface listed above.

(363) In one embodiment of the invention, reporting of results predicted is based on the traditional Model-View-Controller paradigm (Steingrimsson 2017).

(364) 9. System Integration (Case of Embedded Implementation)

(365) 9.1 General Considerations

(366) The primary use case for an embedded implementation involves a plugin to or web service for ICME or Product Lifecycle Management tools, such as Thermo-Calc or Siemens STAR CCM+. Thermo-Calc, which was created using the CALPHAD methodology, comes with a diffusion module, called DICTRA. DICTRA is an add-on package for accurate simulation of diffusion controlled reactions in multi-component alloy systems. Thermo-Calc also comes with a SDK and a publicly available Thermo-Calc Application Programming Interface (TC-API), consisting of a library of C functions.

(367) An alternative use case for an embedded implementation involves a plugin or add-on to, or even web service for, toolboxes (or libraries) for machine learning, artificial intelligence or data analytics, including the TensorFlow package (TensorFlow 2020) or scikit-learn (SciKit-Learn 2020) The physics-based models may be incorporated as add-ons to open-source, off-the-shelf toolboxes (libraries) for machine learning, artificial intelligence or data analytics, to provide physical insight as unexplored sections of the composition space are navigated.

(368) 9.2 Specific Example: Prediction of Fatigue Life of Additively Manufactured Components

(369) 9.1.1.1 Input

(370) The input consists of finite-element data files from tools, such as Abaqus, ANSYS, NX Nastran and Altair OptiStruct.

(371) 9.1.1.2 Desired Output

(372) The toolset is expected to calculate (estimate) realistic fatigue lives, accounting for AM.

(373) 9.1.1.3 API to Abaqus

(374) Abaqus returns output database (.odb) files, which are binary files, but can be imported into our plugin, and decoded, using the Abaqus C++ API (Abaqus 2019).

(375) 9.1.1.4 API to ANSYS

(376) ANSYS returns results (.rst) files, which are also binary files, and which can be imported into our plugin, using source code, such as available through ParaView or OpenFOAM (ParaView 2019), (OpenFOAM 2019).

(377) 9.1.1.5 API to NX Nastran and Through FEmap

(378) FEmap is an advanced engineering simulation application for creating, editing and importing/re-using mesh-centric FE analysis models of complex products or systems. You can combine FEmap with a wide variety of CAD systems and finite-element solvers, including NX Nastran. The FEmap API is an OLC/COM based programming interface that supports object oriented programming. There are a number of codes that can call FEmap through the API (Visual Basic, Excel, Word, Access, C or C++) (PredictiveEngineering 2019).

(379) 9.1.1.6 Case of Altair OptiStruct

(380) OptiStruct in HyperWorks supports scripts for reading in .fem files (AltairHyperWorks 2019).

(381) 10. How to Make the Invention

(382) 10.1 User Interface

(383) In case of a prediction engine employing statistical modeling, such as for predicting the fatigue life of additively manufactured components, the preferred embodiment of the invention assumes the design of an efficient user interface, one capable of effectively guiding the user through (effectively helping the user specify) the multiple parameters impacting the fatigue life of additively manufactured components.

(384) 10.2 Data Base System

(385) Using matlab .dat files or .xlsx or .csv files from Excel available, one can populate a test SQL data base with material records resembling the ones from (GorsseNguyenSenkovMiracle 2018). In one embodiment of the invention, Ref. (GorsseNguyenSenkovMiracle 2018) may provide a good starting point for a classification scheme: Universe.fwdarw.Family.fwdarw.Class.fwdarw.Member.fwdarw.Attributes.fwdarw.Material Records.
The structure of the data in (GorsseNguyenSenkovMiracle 2018) seems to map well to relational databases.

(386) In one embodiment of the invention, the feature extraction may resemble the Citrination platform for materials science data (LingAntonoBajaj 2018), (O'MaraMeredigMichel 2016). The Citrination platform automatically parses chemical formulas and alloy compositions, calculating over ninety different features based on the elemental properties (e.g., ionization energy, melting temperature and the number of valence electronics).

(387) 10.3 Prediction Logic

(388) The prediction logic receives primitives from the prediction engine and passes to the reporting and validation module. The primitives include the quantity predicted, such as the feature list.

(389) 10.4 Prediction Engine

(390) In one embodiment of the invention, the procedure for constructing the prediction engine may consist of the following steps: 1. One can start out by populating a sample SQL database with pertinent materials data, such as from (ChenWangSeifi Lewandowski 2018). (HemphillYuanWang 2012) (TangYuanTsaiYeh 2015), (Thurston GludovatzHohenwater 2017) (ShuklaWangCottonMishra 2018) (SeifiLiYongLiaw Lewandrowski 2015) (LiuKomarasamyGwalani 2019) or (JiaoSimKomarasamyMishra 2018). 2. One can normalize the data according to (1) and derive features in accordance with Table 7. 3. One can train the prediction model on pertinent materials data, such as from ((ChenWangSeifi Lewandowski 2018) (HemphillYuanWang 2012), (TangYuanTsaiYeh 2015), (Thurston GludovatzHohenwater 2017). (ShuklaWangCottonMishra 2018), (SeifiLiYongLiaw Lewandrowski 2015) (LiuKomarasamyGwalani 2019) and (JiaoSimKomarasamyMishra 2018)), and expand from there. 1. Forward prediction can be based on approaches such as outlined in FIG. 13 and FIG. 14. 2. Inverse prediction can be based on approaches, such as outlined in FIG. 15-FIG. 17.
10.5 Verification, Validation and Reporting

(391) One can report verification and validation results in the form of scatter plots and error histograms similar to (Ling AntonoBajaj 2018) and (AgrawalDeshpande 2014).

(392) 11. How to Use the Invention

(393) 1. For Accelerating the Development of New Materials (Alloys or Composites)

(394) The machine learning prediction framework is presented as a dual-use technology, intended both for military and civilian use. Direct dual use applications are expected to include a wide range of commercial applications, e.g., within the aerospace, marine, automotive, and oil and gas industries.

(395) Tools for accelerating the new material discovery and optimizing AM processes for HEAs may benefit DoD Warfare Centers and Production Facilities. Dual-use applications may include aircrafts, land vehicles, ships, submarines and materials processing entities.

(396) In case of an embedded implementation of the prediction engine, host applications used by alloy designers may include ICME tools, such as Thermo-Calc, DICTRA or the Pandat Software from CompuTherm, CFD tools, or tools for crack growth analysis (NASGRO, FE-SAFE, nCode DesignLife, AFGROW).

(397) 2. For Design of RHEAs for Use in Energy Conversion or Propelling Systems with Increased Conversion Efficiency

(398) The invention addresses development of useful inverse design representations, and advanced physics-based metallurgical models, enabling the identification of HEAs suitable for energy conversion devices, such as for compressor blades of land-based gas turbines operating with ultrahigh efficiency.

(399) The invention can be used for design of refractory HEAs for application in land-based or air-borne gas turbines, or for application in propeller systems, such as used by the Navy, Air Force or the Army, operating with superior conversion efficiency, compared to conventional designs.

(400) Stage one turbine blades are currently cast from nickel-base (Ni-base) superalloys, and due to material-property limitations, the combustor firing temperatures are capped at 1,450 C., which correlates to a carbon conversion efficiency of 62%. In order to achieve CCE in excess of 65%, a 300 C. increase in the combustor firing temperature is required. In materials terms, this requirement places a demand on stage one blade alloys to increase material capability by 200 C. (cooling and coating is expected to account for 100 C. difference).

(401) 3. For Design of RHEAs for Use in Energy Conversion or Propelling Systems Resulting in Reduced Cost of Manufacturing

(402) The invention helps advance the manufacture of new energy materials leading to improved performance and to lower manufacturing cost. The cost reduction can be achieved in part by redesigning the turbine blades such as to take advantage of the superior yield strength of the RHEAs, per FIG. 8, for the purpose of reducing the mass. Additional cost reductions are expected for air-borne turbines, where lighter turbine blades contribute to lighter vessels, and improved fuel economy.

(403) Further, according to Siemens Energy, the high temperature materials capability of HEAs, combined with AM processing, enables a 30-50% reduction in product deployment time, leading to earlier realization of higher engine efficiency improvements.

(404) 4. For Design of RHEAs for Use in Automotive Applications

(405) The invention can be used to design and develop advanced, lightweight materials for use in advanced internal combustion engines or emission systems. These will require structural materials, to withstand environments that exhibit temperatures (and pressures) well above conventional technologies. To successfully implement such advanced vehicle technologies, materials are required that can maintain their original material properties over a vehicle's lifespan.

(406) ML and AM methods to design and fabricate lightweight HEAs for motor vehicle fuel efficiency applications. The AM and ML techniques have an enormous potential to produce lightweight materials that have superior corrosion and fatigue resistance as compared to conventional alloys. Furthermore, the combined use of the AI and ML techniques have the potential to reduce product development times from 15 to within 5 years. This potential can be made possible in part by taking advantage of the physics-based metallurgical predictions that are inherent in the models.

(407) 5. For Developing Coatings Protecting Against Corrosion Attacks, Esp. CMAS or Calcium Sulfate, in Air or Sea Water

(408) The invention can be used in conjunction with applications for Product Lifecycle Management, such as Siemens STAR CCM+, to develop coatings resistant of CMAS or calcium sulfate corrosion attacks.

(409) 6. For Predicting Properties (Performance) of Additively Manufactured Parts

(410) The invention is capable of optimizing the process parameters for additively manufactured components to achieve enhanced performance, including enhanced fatigue performance, such as for aircraft components.

(411) 7. For Estimating the Fatigue Life of Additively Manufactured Components

(412) The invention can be used to estimate the fatigue life of an AM component, such as from a rotor of an aircraft (or any aircraft). One can use the invention to predict the stress/life curves for components of interest. From estimates of the fatigue life of a given component of interest, one can look to estimate service life of an entire subsystem or even of an entire aircraft.

(413) 8. For Deriving Optimal AM Processes

(414) The invention (software application) can also accelerate and optimize the process of additively manufacturing alloys, such as high-entropy alloys. In terms of the fabrication of the high-entropy alloys, the invention can help optimize the selection of the laser power and travel speed during deposition, the powder distribution, shape, feed rate and increment, the number of laser tracks for each patch and overlap value, as well as the repeat times and laser power during re-melting.

(415) The invention can help reliably predict the optimal trajectory in the multidimensional process parameter space due to the inherent spatiotemporal variability in the process parameter and the chaotic nature of the AM process. Despite the continued progress in AM technologies, AM parts still require several trial and error runs with post-processing treatments and machining to optimize the build, reduce defects and residual stresses, and meet tolerances. AM still lacks a stable process that can produce consistent, defect-free parts on a first time basis due to our inability to reliably predict the optimal trajectory.

(416) 9. For Intelligent AM Systems

(417) Machine learning or Al can help with better distributing, monitoring, and controlling the processing energy in a laser metal powder bed fusion AM systems, for purposes of real-time process monitoring and control towards producing high-quality, defect-free AM parts with build periods comparable to or shorter than present ones. Using an Al controller (ML, deep neural network or neuromorphic processor), one can monitor the temperature distribution and/or other relevant process parameters over the powder bed.

(418) 10. As a Plugin to ICME Tools

(419) In one embodiment of the invention, the prediction engine can be employed as a plugin to ICME tools, such as Thermo-Calc.

(420) 11. As a Plugin to Tools for Crack Growth Analysis

(421) In another embodiment of the invention, the prediction engine can be employed as a plugin to tools for crack growth analysis, such as NASGRO, FE-SAFE, nCode DesignLife or AFGROW.

(422) 12. As a Plugin to Toolboxes (Libraries) for Machine Learning, Artificial Intelligence or Data Analytics

(423) As noted above, the physics-based models may be incorporated as add-ons to open-source, off-the-shelf toolboxes (libraries) for machine learning, artificial intelligence or data analytics, to provide physical insight as unexplored sections of the composition space are navigated.

(424) 13. As a Plugin or Web Service for Tools for Product Lifecycle Management

(425) The prediction engine can be deployed as a plugin or web service for tools for Product Lifecycle Management, such as Siemens STAR CCM+.

(426) 14. For Design of CMCs or PMCs for High-Temperature Aerospace Applications

(427) There is an ever-changing, constant need for designing novel composite materials for aerospace applications that offer a broader gamut of multi-functionality (sensing, electrical and thermal properties, desired interface characteristics, adhesion, energy storage/harvesting, low density, etc.) and structural stability (mechanical behavior, stiffness or compliance, fracture toughness, high temperature stability, minimal physical aging, etc.) with an eventual goal of minimizing costs and maximizing operational performance and efficiency. Experimentally, this design space is often explored via building upon previous reported literature towards synthesizing and characterizing state-of-the-art composite materials for various applications. Hence, there is opportunity to formalize modeling methods and data analytics, machine learning or artificial intelligence frameworks to investigate and understand structure-property-performance relationships in multi-functional PMCs or CMCs to (a) complement experimental efforts towards better appreciation of molecular origins of structure-property-performance relationships; and (b) facilitate accelerated materials design for next-generation multi-functional composites, geared towards aerospace applications.

(428) In the gas turbine industry, CMCs are particular attractive, since they have the potential to replace nickel based super-alloys in various hot section components.

12. Further Examples of the Invention

(429) Thus, it will be appreciated by those skilled in the art that the present invention is not restricted to the particular preferred embodiments described with reference to the drawings, and that variations may be made therein without departing from the scope of the invention.