Patent Publication Number: US-2022219239-A1

Title: Methods of designing printed metallic materials

Description:
CROSS-REFERENCE TO RELATED APPLICATIONS 
     This application claims the benefit of U.S. provisional patent application No. 63/090,882, filed on Oct. 13, 2020, and entitled “METHODS OF DESIGNING PRINTED METALLIC MATERIALS,” the disclosure of which is expressly incorporated herein by reference in its entirety. 
    
    
     STATEMENT REGARDING GOVERNMENT SUPPORT 
     This invention was made with government support under sponsor award W911NF-18-1-0278, awarded by the Army Research Office and under sponsor award 1846676, awarded by the National Science Foundation. The government has certain rights in the invention. 
    
    
     BACKGROUND 
     In additive manufacturing (AM) (also known as 3D printing), a three-dimensional object is created by incrementally adding material (for example, a metal or metal alloy) to the object. AM technologies rely on the correct configuration of AM parameters to prevent defects. Selecting the correct AM parameters can be challenging. Some of these challenges can be attributed to the fact that existing commercial raw materials used in AM may have been originally developed for other manufacturing processes such as casting, forging, and machining. When attempting to process these materials using AM, the materials undergo different—and sometimes more complex—physical transformations during the AM process, which can result in defects, microstructural inconsistencies, and high degrees of variability. 
     Developing new alloys for AM involves yet another set of challenges. For example, there is no standard systematic procedure to determine the processing recipes (or parameters) for these new materials. One possible approach is through brute force which can include exhaustive, exploration of the entire parameter space. However, brute force methods can be undesirable due to the amount of time and materials required. 
     Therefore, additional methods of selecting the AM process parameters that address these and other challenges are needed. 
     SUMMARY 
     To improve manufacturing techniques for additive manufacturing and to overcome the limitations of conventional design techniques, systems, methods and devices are disclosed which use models to determine desirable additive manufacturing process parameters. 
     In one aspect, the present disclosure relates to a method for determining processing parameters for an alloy. In one embodiment, the method includes performing a simulation of melt pool temperature and melt pool geometries for an alloy at a plurality of combinations of a laser speed parameter and a laser power parameter, creating an initial printability map based on the laser speed parameter and the laser power parameter based on the simulation of melt pool temperature and melt pool geometries, defining, within the printability map, one or more regions of the printability map that correspond to one or more manufacturing defects, sampling the printability map to determine a plurality of samples within the printability map, where each sample includes a value of the laser speed parameter and a value of the laser power parameter, performing a set of single-track experiments corresponding to the plurality of samples, calibrating the printability map based on the set of single-track experiments to create a revised printability map, generating a plurality of hatch spacing contours defining a spacing between adjacent beads in a three-dimensional printed part, and adding the plurality of hatch spacing contours to the revised printability map to create a final printability map, where the final printability map represents a printability characteristic of the alloy at a plurality of combinations of laser speed, laser power, and hatch spacing. 
     In one embodiment, the manufacturing defects comprise keyholing, balling, and lack of fusion. 
     In one embodiment, the method includes fabricating a bulk sample of the alloy, measuring a porosity value of the bulk sample, and identifying an optimal combination of processing parameters based on the porosity value of the bulk sample. 
     In one embodiment, the method includes revising the final printability map based on evaluating the bulk sample of the alloy for porosity/density and mechanical properties. 
     In one embodiment, the method includes sampling the printability map to generate a set of processing parameter values for the laser speed parameter and the laser power parameter, fabricating a plurality of sample parts based on each of the set of processing parameter values for the laser speed parameter and the laser power parameter, measuring a material property of each of the plurality of sample parts to generate a plurality of material property data points, and performing an optimization of the material property based on the material property data points. 
     In one embodiment, the material property is tensile strength. 
     In one embodiment, the step of calibrating the printability map includes performing a Bayesian calibration. 
     In one embodiment, the hatch spacing contours are based on a geometric criterion, where the geometric criterion defines the maximum value of hatch spacing that allows for complete fusion within and between layers of beads of the three-dimensional printed part. 
     In one embodiment, the final printability map is used to set one or more printer parameters of an additive manufacturing printer. 
     In one embodiment, sampling the printability map includes defining a grid within the printability map and sampling each point of the grid, where each point in the grid includes a value of the laser speed parameter and a value of the laser power parameter. 
     In one embodiment, the calibration of the printability map is validated by calculating an absolute prediction error for one or more regions of the printability map. 
     In one embodiment, sampling the printability map includes defining one or more regions of the printability map, and, for each region in the printability map, selecting a sampling technique from a plurality of sampling techniques and sampling the region using the sampling technique. 
     In one embodiment, the plurality of sampling techniques includes a grid based sampling technique or a Latin hypercube sampling technique. 
     In one embodiment, the plurality of sampling techniques includes orthogonal array sampling or central composite design sampling technique. 
     In one embodiment, the simulation of melt pool temperature and melt pool geometries is an Eagar-Tsai (E-T) simulation. 
     In one embodiment, defining the printability map includes comparing the melt pool temperature and melt pool geometries to a plurality of threshold ratios, where the threshold ratios represent thresholds at which defects are predicted to occur. 
     In one embodiment, the threshold ratios are adjusted based on the plurality of single track experiments. 
     In one aspect the present disclosure relates to a system for additive manufacturing. In one embodiment, the system includes an additive manufacturing printer; a processor; and a memory coupled to the processor; where the memory stores instructions which when executed by the processor cause the system to: perform a simulation of melt pool temperature and melt pool geometries for an alloy at a plurality of combinations of a laser speed parameter and a laser power parameter; create an initial printability map based on the laser speed parameter and the laser power parameter based on the simulation of melt pool temperature and melt pool geometries; define, within the printability map, one or more regions of the printability map that correspond to one or more manufacturing defects; sample the printability map to determine a plurality of samples within the printability map, where each sample includes a value of the laser speed parameter and a value of the laser power parameter; print, using the additive manufacturing printer, a plurality of sample tracks corresponding to the plurality of samples; perform a set of single-track experiments on the plurality of sample tracks to corresponding to the plurality of samples; calibrate the printability map based on the set of single-track experiments to create a revised printability map; generate a plurality of hatch spacing contours defining a spacing between adjacent beads in a three-dimensional printed part; add the plurality of hatch spacing contours to the revised printability map to create a final printability map, where the final printability map represents a printability characteristic of the alloy at a plurality of combinations of laser speed, laser power, and hatch spacing; print, using additive manufacturing printer, a part using a combination of laser speed, laser power, and hatch spacing selected from the plurality of combinations of laser speed, laser power, and hatch spacing of the printability map. 
     In one embodiment, the combination of laser speed, laser power and hatch spacing is selected from a region of the printability map that does not correspond to any of the one or more manufacturing defects. 
     In one embodiment, the additive manufacturing printer is a laser powder bed fusion printer configured for metal additive manufacturing processes. 
     It should be understood that the above-described subject matter may also be implemented as a computer-controlled apparatus, a computer process, a computing system, or an article of manufacture, such as a computer-readable storage medium. 
     Other systems, methods, features and/or advantages will be or may become apparent to one with skill in the art upon examination of the following drawings and detailed description. It is intended that all such additional systems, methods, features and/or advantages be included within this description and be protected by the accompanying claims. 
    
    
     
       BRIEF DESCRIPTION OF THE DRAWINGS 
       The accompanying drawings, which are incorporated in and constitute a part of this specification, illustrate embodiments and together with the description, serve to explain the principles of the methods and systems: 
         FIG. 1  illustrates a flowchart of a method of determining processing parameters for an alloy, according to an implementation described herein. 
         FIG. 2  illustrates an example of a printability map that has been divided into regions and sampled. 
         FIGS. 3A-3B  illustrate examples of maps of absolute prediction error for melt pool dimensions compared to markers representing single track experiments.  FIG. 3A  illustrates a map of absolute prediction error for melt pool depth, and  FIG. 3B  illustrates a map of absolute prediction error for melt pool width. 
         FIGS. 4A-4B  illustrate examples of a printability map, according to one implementation of the present disclosure. In  FIGS. 4A-4B , the white regions represent combinations of print parameters without the modeled manufacturing defects.  FIG. 4A  is an illustration of a printability map that has been revised based on a number of single-track experiments, which are denoted as a series of “x” and circle marks on the printability map.  FIG. 4B  illustrates a finalized printability map with hatch spacing contours, where hatch spacing is another important print parameter. 
         FIGS. 5A-5D  illustrate examples of single track samples. The samples include a good track ( 5 A), a lack of fusion track ( 5 B), a keyholing defect track ( 5 C), and a balling defect track ( 5 D). 
         FIG. 6  illustrates an exemplary computer that may comprise all or a portion of the system for determining gradient paths for compositionally graded alloys, or a control system for multi-material printers; conversely, any portion or portions of the computer illustrated in  FIG. 3  may comprise all or a portion of the system for determining gradient paths for compositionally graded alloys, or a control system for multi-material printers; conversely. 
     
    
    
     DETAILED DESCRIPTION 
     Before the present methods and systems are disclosed and described, it is to be understood that the methods and systems are not limited to specific synthetic methods, specific components, or to particular compositions. It is also to be understood that the terminology used herein is for the purpose of describing particular embodiments only and is not intended to be limiting. 
     As used in the specification and the appended claims, the singular forms “a,” “an” and “the” include plural referents unless the context clearly dictates otherwise. Ranges may be expressed herein as from “about” one particular value, and/or to “about” another particular value. When such a range is expressed, another embodiment includes from the one particular value and/or to the other particular value. Similarly, when values are expressed as approximations, by use of the antecedent “about,” it will be understood that the particular value forms another embodiment. It will be further understood that the endpoints of each of the ranges are significant both in relation to the other endpoint, and independently of the other endpoint. 
     “Optional” or “optionally” means that the subsequently described event or circumstance may or may not occur, and that the description includes instances where said event or circumstance occurs and instances where it does not. 
     Throughout the description and claims of this specification, the word “comprise” and variations of the word, such as “comprising” and “comprises,” means “including but not limited to,” and is not intended to exclude, for example, other additives, components, integers or steps. “Exemplary” means “an example of” and is not intended to convey an indication of a preferred or ideal embodiment. “Such as” is not used in a restrictive sense, but for explanatory purposes. 
     Throughout this application, various publications may be referenced. The disclosures of these publications in their entireties are hereby incorporated by reference into this application in order to more fully describe the state of the art to which the methods and systems pertain. Some references, which may include various patents, patent applications, and publications, are cited in a reference list and discussed in the disclosure provided herein. The citation and/or discussion of such references is provided merely to clarify the description of the present disclosure and is not an admission that any such reference is “prior art” to any aspects of the present disclosure described herein. 
     Disclosed are components that can be used to perform the disclosed methods and systems. These and other components are disclosed herein, and it is understood that when combinations, subsets, interactions, groups, etc. of these components are disclosed that while specific reference of each various individual and collective combinations and permutation of these may not be explicitly disclosed, each is specifically contemplated and described herein, for all methods and systems. This applies to all aspects of this application including, but not limited to, steps in disclosed methods. Thus, if there are a variety of additional steps that can be performed it is understood that each of these additional steps can be performed with any specific embodiment or combination of embodiments of the disclosed methods. 
     The present methods and systems may be understood more readily by reference to the following detailed description of preferred embodiments and the Examples included therein and to the Figures and their previous and following description. 
     As will be appreciated by one skilled in the art, the methods and systems may take the form of an entirely hardware embodiment, an entirely software embodiment, or an embodiment combining software and hardware aspects. Furthermore, the methods and systems may take the form of a computer program product on a computer-readable storage medium having computer-readable program instructions (e.g., computer software) embodied in the storage medium. More particularly, the present methods and systems may take the form of web-implemented computer software. Any suitable computer-readable storage medium may be utilized including hard disks, CD-ROMs, optical storage devices, or magnetic storage devices. 
     Embodiments of the methods and systems are described below with reference to block diagrams and flowchart illustrations of methods, systems, apparatuses and computer program products. It will be understood that each block of the block diagrams and flowchart illustrations, and combinations of blocks in the block diagrams and flowchart illustrations, respectively, can be implemented by computer program instructions. These computer program instructions may be loaded onto a general-purpose computer, special purpose computer, or other programmable data processing apparatus to produce a machine, such that the instructions which execute on the computer or other programmable data processing apparatus create a means for implementing the functions specified in the flowchart block or blocks. 
     These computer program instructions may also be stored in a computer-readable memory that can direct a computer or other programmable data processing apparatus to function in a particular manner, such that the instructions stored in the computer-readable memory produce an article of manufacture including computer-readable instructions for implementing the function specified in the flowchart block or blocks. The computer program instructions may also be loaded onto a computer or other programmable data processing apparatus to cause a series of operational steps to be performed on the computer or other programmable apparatus to produce a computer-implemented process such that the instructions that execute on the computer or other programmable apparatus provide steps for implementing the functions specified in the flowchart block or blocks. 
     Accordingly, blocks of the block diagrams and flowchart illustrations support combinations of means for performing the specified functions, combinations of steps for performing the specified functions and program instruction means for performing the specified functions. It will also be understood that each block of the block diagrams and flowchart illustrations, and combinations of blocks in the block diagrams and flowchart illustrations, can be implemented by special purpose hardware-based computer systems that perform the specified functions or steps, or combinations of special purpose hardware and computer instructions. 
     Throughout the present disclosure, the terms “optimal,” “optimum” and “optimally” are used to refer to the results of a mathematical optimization problem. It should be understood that the mathematical optimization results described herein are not intended to be limiting, and that different optimization techniques, path planning techniques, constraints, and results are contemplated. 
     Embodiments of the present disclosure relate to systems and methods for determining a printability map for materials. As a non-limiting example, embodiments of the present disclosure can be used as a unified framework to determine printability maps for a given newly developed material in laser powder bed fusion (LPBF) metal AM processes. Throughout the present disclosure, “a printability map” can refer to windows of processing parameters space within which parts free of macroscopic defects can be produced. Non-limiting examples of processing parameters that can be used in embodiments of the present disclosure include: laser power, P [W], scan speed, V [m/s], and hatch spacing, h [μm]. Non-limiting examples of macroscopic defects that can be mitigated are porosities, cracks, and delamination. These defects can be linked to three common phenomena that may occur during LBPF AM (laser powder bed fusion additive manufacturing): keyholing, lack of fusion, and balling. Again, it should be understood that these defects are only provided as non-limiting examples and that the present disclosure can be used to design materials free of other types of defect. 
     Methods are disclosed herein that integrate physics-based modeling, experimental characterization, and statistical uncertainty quantification (UQ), to construct a printability map for a desired alloy in a systematic and accelerated fashion. Additionally, implementations described herein can use analytical models. Embodiments of the present disclosure include a framework that can be summarized as follows: through integrating physics-based modeling, experimental characterization, and statistical uncertainty quantification (UQ), a printability map can be constructed in a systematic and accelerated fashion. The modeling step can include an analytical model that is accessible to users, eliminating the need for proprietary computational codes. Furthermore, the analytical model can be computationally tractable which enhances the accelerated aspect of the proposed framework. In some embodiments, the model is an analytical and relatively low fidelity model; and the method can include performing an uncertainty quantification UQ) step. In addition to constructing a printability map for a given new material, embodiments of the present disclosure can guide the microstructural and mechanical characterization of the specimens that are printed according to that map. 
     Second, for the modeling step implementations described herein can use computationally traceable analytical models combined with an uncertainty quantification step. Further, implementations described herein can be used to characterize both the microstructure and mechanical attributes of the printability map. 
       FIG. 1  illustrates a method  100  for determining processing parameters for an alloy, according to one implementation described herein. In step  102 , a thermal model is used to simulate melt pool temperature and geometry of the target alloy. In the method  100  shown in  FIG. 1 , the Eagar-Tsai (E-T) analytical thermal model is used, but the use of other types of thermal model to determine melt pool temperature and geometry is contemplated. The analysis of the melt pool geometry can include analyzing information about the melt pool width and depth by identifying temperature contours in the material. Additionally, information including the additive manufacturing (AM) processing parameters can be input into the thermal model as part of performing the simulation step  102 . 
     Defining different regions within the scan speed-laser power space can be correlated with single track melt pool characteristics. For example, the melt pool width, W, and melt pool depth, D, of a single track can be used to determine the potential for defect formation in a printed part. To achieve this, a prediction of the melt pool geometry as a function of the AM processing parameters and material properties in establishing a processing parameter space of producing fully dense parts can be used. This can be done over a wide range of processing parameter combination (i.e. a parameter sweep). Some embodiments of the present disclosure can use a computationally inexpensive analytical thermal model, Eagar-Tsai (E-T) for this task. The E-T model can model welding processes as a traveling heat source with Gaussian profile over a semi-infinite flat plate and calculate the temperature distribution across the plate. The temperature distribution can be used to calculate the melt pool width and depth by identifying melting temperature contours. The E-T model can be used to approximate LPBF AM processes, for example LPBF AM processes according to embodiments of the present disclosure. In some embodiments, the E-T model can exclude phenomena such as phase transformations (e.g. melting and boiling), convective currents within the melt pool as well as liquid-laser interactions resulting in keyhole formation and also considers that thermophysical properties are temperature-independent. Furthermore, an uncertainty quantification (or Bayesian calibration) step can adjust model predictions such that they agree with experiments as described later in this section. Alternatively, in some embodiments of the present disclosure, other models can be used to attain better fidelity with respect to the physics of the process. 
     The E-T model can take the LPBF processing parameters as inputs. Non-limiting examples of inputs that can be used in embodiments of the present disclosure include laser power, scan speed, and Gaussian-distributed laser beam diameter, and the use of other inputs and processing parameters is contemplated by the present disclosure. The Gaussian-distributed beam diameter which is the diameter corresponding to four standard deviations of the Gaussian profile of the beam. The model can also take thermo-physical material properties as inputs, including thermal conductivity ‘k’ [W/(m·K)], specific heat capacity ‘c’ [K/(kg·K)], bulk density ‘ρ’ [kg/m 3 ], melting temperature ‘T m ’ [K], boiling temperature ‘T b ’ [K]. The absorptivity ‘A’ [0-1] can provide a measure of the effectiveness in the laser-material energy transfer. Two sources of uncertainty in model predictions are: (1) uncertainty of the thermo-physical material parameters due to the unknown material properties of a newly developed material, and (2) model uncertainty or model discrepancy. Model uncertainty can also originate from missing physics or simplifying assumptions in the model, e.g., the temperature-independent material properties, the semi-infinite plate, and solid substrate surface in contrast to a powder bed as can be the case with LPBF. To account for these and other sources of uncertainty and increase the accuracy of predictions, a Bayesian statistical calibration can be used in some embodiments of the present disclosure. This can estimate the values of uncertain parameters that can make model predictions agree with experiments, and also estimate the discrepancy function through a Gaussian-process approximation to account for deviations between model predictions and experimental observations. 
     Defining different regions within the scan speed-laser power space can be correlated with single-track melt pool characteristics. In particular, the melt pool width, W, and melt pool depth, D, of a single-track can be used to determine the potential for defect formation in a printed part. The melt pool geometry can be predicted as a function of the AM processing parameters and material properties in establishing a processing parameter space of producing fully dense parts. Furthermore, it can be desirable to perform these predictions over a wide range of processing parameter combinations (i.e. a parameter sweep). 
     For example, in some implementations described herein, an E-T model can take the laser bed powder fusion (LPBF) processing parameters of laser power, scan speed, and Gaussian-distributed laser beam diameter as inputs. The Gaussian-distributed beam diameter is the diameter corresponding to four standard deviations of the Gaussian profile of the beam. The E-T model can also take thermo-physical material properties as inputs, including thermal conductivity ‘k’ [W/(m·K)], specific heat capacity ‘c’ [K/(kg·K)], bulk density ‘ρ’ [kg/m 3 ], melting temperature ‘Tm’ [K], boiling temperature ‘Tb’ The absorptivity ‘A’ can provide a measure of the effectiveness in the laser-material energy transfer. 
     Again referring to  FIG. 1 , in step  104  of the method  100 , an initial printability map is generated based on the thermal model. The initial printability map can be based on the thermal model used in step  102  and defect criteria. As shown in  104 , the initial printability map is a  2 D map that includes combinations of laser power (y-axis) and laser speed (x-axis). The initial printability map includes regions where defects are predicted. In the implementation shown in  FIG. 1 , the defects are Keyholing, Balling, and Lack of Fusion, although the present disclosure contemplates the analysis of other types of defects. The region in the center of the chart depicted in step  104  (“Good Tracks”) represents combinations of processing parameters that are predicted to result in fabricated parts that are free of defects. 
     The printability map can define regions within the processing parameter space that that correspond to different phenomena (also referred to in the present disclosure as “modes”) that occur during LPBF. In some embodiments of the present disclosure, defining regions within the processing parameter space can include reducing the parameter space from a theoretically infinite space in the positive quadrant to a finite space. 
     Upper and lower bounds on the laser scan speed and laser power can be established. The upper bound on the laser speed, V max , can be set to the maximum attainable speed by the laser optics on the AM system while the lower bound, V min , can be set to an arbitrarily small value (e.g. 0.05 m/s) slightly above the theoretical minimum (i.e. zero). Using an arbitrarily small V min  can be used in models like the E-T model that can specify a moving heat source. The upper bound on the laser power, Pm ax , can be set as the maximum power attainable by the AM system (i.e. a limitation or parameter of the AM machine). The lower bound on the laser power, P min , can be set as the minimum laser power that will cause melting at a speed of V min . This value can be computed using the E-T model. 
     The E-T model is used to further reduce this space into sub-regions corresponding to phenomena that result in porosity; namely lack of fusion, keyholing, and balling. Examples of these phenomena are depicted in  FIG. 5 . Lack of fusion can occur when the melt pool depth is smaller than powder layer thickness, t, due to an insufficient amount of laser energy being deposited into the powder bed. 
     The lack of fusion boundary line can be plotted as the line passing through speed-power combinations that result in a melt pool depth that is equal to the layer thickness t. Large laser energy density can lead to the development of vapor cavities resulting from the recoil pressure associated with the rapid evaporation of the molten liquid. This can cause the laser beam to “drill” into the material to a larger depth than is the case during the general conduction mode. This can ultimately result in the collapse of the cavity, leaving voids known as keyholing porosity. 
     The balling effect is observed at high laser power and scan speed combinations as the melt pool form into droplets (as opposed to a continuous weld track) due to Plateau-Rayleigh capillary instability. As a non-limiting example, thresholds for plotting lack of fusion, keyholing and balling boundaries are set as D≤t, W/D≤1.5 and L/W≥2.3, respectively. These threshold ratios are derived from empirical observations, physical principles, and geometric considerations. Therefore it is contemplated by the present disclosure that other threshold ratios can be calculated, estimated, or determined according to material properties, melt pool characteristics, risk tolerance, and any other factor. Furthermore, these initial values of the ratios can be revised after experimental measurements. The region of the printability map that is not labeled with a specific defect-causing phenomenon can be considered to be a good region for printing nearly full density parts. It should be understood that these defects, the causes of these defects, and the thresholds identified for these defects are intended only as non-limiting examples, and that the use of other defects and associated thresholds are contemplated by the present disclosure. 
     In step  106 , single-track experiments are performed based on the initial printability map. The single-track experiments can include measurements of melt pool depth and melt pool width. Some of the thermo-physical properties may not be known at the time of running the simulation and can be estimated based on domain expertise or on uncertain values reported in the literature. When finalizing the printability map, these uncertainties can be quantified and accounted for such that model predictions are in agreement with experimental observations. This process can include statistical model calibration, described herein. Single-track experiments can be used to obtain the experimental observations that can be required for calibrating the printability map. The printability map can be sampled, for example by grid sampling. The sampling selects different laser scan speed and laser power combinations within the parameter space. Other sampling strategies are contemplated by the present disclosure. Non-limiting examples of other sampling techniques that can be used include Latin hypercube sampling (LHS), orthogonal array sampling, and central composite design. 
     The initial printability map in steps  102   104  can be constructed based on E-T model simulations. In some embodiments of the present disclosure, the parameters and models can include uncertainties, for example, the value of some of the thermo-physical properties can be unknown at the time of running the simulation, or based on estimates which include uncertainty. Embodiments of the present disclosure can quantify and account for these uncertainties so that model predictions are in agreement with experimental observations (e.g., by statistical model calibration). Single track experiments can be conducted to obtain the experimental observations needed for calibration. Sampling techniques (e.g., grid sampling) can be used to select different laser scan speed-power combinations within the parameter space. Again, as a non-limiting example, 60 processing parameter combinations can be used to cover a finite model space. The present disclosure also contemplates the use of other sampling strategies, such as Latin hypercube sampling (LHS), orthogonal array sampling, or central composite design. 
     As a non-limiting example 10 mm-long single tracks spaced  1 mm apart from one another can be printed as samples. The powder layer thickness can be set to the 80 th  percentile of the powder size distribution (known as d 80 ). Characterization of the single tracks using microscopy can be conducted to measure melt pool width and depth. Scanning electron microscope (SEM) images of the single-track tops can be used to measure the melt pool width at locations (e.g. 9 locations) along the track and the average of these measurements can be taken as the melt pool width. Optical microscope (OM) images of 3 melt pool cross sections can be used to measure melt pool depth after sectioning, polishing, and etching. Again, the instruments, number of samples, and measurement techniques described herein are intended only as non-limiting examples and the use of different types of samples, and different methods of measuring and processing the samples, is contemplated by the present disclosure. 
       FIG. 2  depicts experimentally characterized single tracks shown in an example printability map  200 . In some embodiments of the present disclosure, the printability map  200  can be divided into regions  202   204   206  as shown in  FIG. 2 . The printability map shown in  FIG. 2  includes three regions  202   204   206 , but the present disclosure contemplates that the printability map can be divided into any number of regions  202   204   206 . Different sampling techniques can be used to sample the space in each region  202   204   206 , and/or some regions  202   204   206  may use the same sampling techniques as other regions. As a non-limiting example, the printability map shown in  FIG. 2  is divided into three regions  202   204   206 , where the first region  202  is sampled using Latin hypercube sampling, the second region  204  is sampled using a grid, and the third region  206  is sampled using a grid. The nodes of balling criterion line and lack of fusion criterion line of Pm ax  boundary (p 1  and p 2 ) were used to draw lines perpendicular to Pmim boundary at point p 3  and point p 4 . Then these two lines  210   212 , p 1 -p 3  and p 2 -p 4 , split the finite space into different regions. The first line  210  runs between p 1  and p 3 , and the second line  212  runs between p 2  and p 4 . It should be understood that the spacing and shape of the lines shown is intended only as a non-limiting example, and that the present disclosure contemplates that other regions  202   204   206  can be divided by different lines selected using different criteria. 
     Different sizes of single-tracks are possible. According to a non-limiting example, 10 mm-long single-tracks spaced 1 mm apart from one another can be printed. The powder layer thickness can be set to the 80th percentile of the powder size distribution (commonly known as d80). Characterization of the single-tracks using microscopy can be conducted to measure melt pool width and depth. Scanning electron microscope (SEM) images of the single-track tops can be used to measure the melt pool width at locations along the track and the average of these measurements can be selected as the melt pool width. However, methods of measuring or calculating the melt pool width are contemplated. Optical microscope (OM) images of melt pool cross sections can be used to measure melt pool depth after sectioning, polishing, and etching. Again, different methods of viewing, measuring, and calculating the melt pool depth are contemplated, and these values of powder layer thickness, depth, and length are intended as non-limiting examples. 
     Based on the experiments performed in step  106 , the model can be calibrated in step  108  to produce a revised printability map, as shown in step  110 . The calibration performed in step  108  can be a Bayesian calibration, or a calibration performed based on uncertainty quantification, or any other calibration that can adjust the model predictions to agree with the experiments performed in step  106 . The present disclosure contemplates that steps  102 - 108  may be repeated as necessary to update the statistical model. 
     According to one implementation described herein, the calibration is performed using a Bayesian procedure. The Bayesian procedure for calibrating the model can include constructing a Gaussian process surrogate model of the original E-T model. The surrogate model can be computationally less expensive than the original E-T model and can be used to generate sufficiently large numbers of simulations for performing calibration. The surrogate model is developed based on simulations first generated from the E-T model according to a Latin hypercube sampling strategy. a Gaussian process is fit to these simulations. Simulations generated through the surrogate model with experimental observations obtained from single-track experiments to calibrate the model parameters and estimate model uncertainty. Three input model parameters can be identified as calibration parameters: thermal conductivity ‘k’, specific heat capacity ‘c’ and absorptivity ‘A’. When developing the surrogate model, a range of values can be selected for each calibration parameter including its prior estimation value in order for the surrogate model to be valid for many or all of the possible calibration parameter values. For example, the estimation of absorptivity for Nickel Titanium alloy (NiTi) is 0.56. Choices of ‘A’ that are generated can be used to train the surrogate model. The mean of the posterior distribution of ‘A’ can be derived through statistical model calibration as its calibrated parameter value. To test the accuracy of the calibrated model, the absolute prediction error for melt pool width and depth across the processing space can be calculated.  FIGS. 3A and 3B  illustrate shaded maps  300   350  of absolute prediction error for melt pool depth and melt pool width, respectively. Markers  302  illustrate single track experiments. The absolute prediction error can be used by embodiments of the present disclosure to determine if an adequate number of single track experiments have been performed. In some embodiments of the present disclosure, performing additional single track experiments in a region (e.g. the regions  202   204   206  shown in  FIG. 2 ) can reduce the absolute prediction error in that region. 
     In the non-limiting example shown in  FIGS. 3A-B , the processing parameter combinations with linear energy density (EL) less than 300 J/m have larger prediction errors. This can indicate that, in this non-limiting example, more single-track experiments with EL&lt;300 J/m are can be used to acquire missing information. The mean absolute percentage error (MAPE) can also be calculated in embodiments of the present disclosure, and the MAPE for the width predictions and depth predictions corresponding to  FIGS. 3A-B  were determined as 3.6% and 4.05%. This can be an acceptable MAPE in some embodiments of the present disclosure. 
     In some embodiments of the present disclosure, statistical model calibration can involve combining experimental observations y E  of the real process run at some values of control inputs x, and model simulations y S  to calibrate unknown model parameters θ, and estimate model uncertainty (systematic bias between model predictions and experiments due to missing physics, a discrepancy function δ(x), and experimental measurement errors ε). This can be described by the following equation: 
         yE ( x )= yS ( x,  θ)+δ( x )+ε( x )   (1)
 
     In some embodiments of the present disclosure, calibration can be conducted using a Bayesian procedure. A Gaussian process surrogate model of the original E-T model can be constructed. This surrogate model can be computationally less expensive than the original E-T model and is needed to generate sufficiently large numbers of simulations needed to conduct calibration. As a non-limiting example, to develop the surrogate model, 1000 simulations can be generated from the E-T model according to an LHS (Latin Hypercube Sampling strategy. Next, a Gaussian process is fit to these simulations. Simulations generated through that surrogate model with experimental observations obtained from single track experiments to calibrate the model parameters and estimate model uncertainty. Three input model parameters are identified as calibration parameters since model simulations tend to be sensitive to them: thermal conductivity ‘k’, specific heat capacity ‘c’ and absorptivity ‘A’. a range of values need to be selected for each calibration parameter for the surrogate model including its prior estimation value. As a non-limiting example, the estimation of absorptivity for Nickel Titanium alloy (NiTi) can be given as 0.56. 100 choices of ‘A’ can be generated from (0, 1) and used to train the surrogate model. Then the mean of the posterior distribution of ‘A’ can be derived through statistical model calibration as its calibrated parameter value. 
     The calibrated E-T surrogate model can now be used to revise the defect boundaries in the initial printability map resulting in a revised printability map (e.g. the printability map of step  110 ). 
     In step  110 , a revised printability map can be created. The revised printability map can include regions corresponding to defects, as well as a region or regions where no defects are predicted. As shown in step  110 , some regions of the printability map represent defects (keyholing, balling, and lack of fusion) a region, near the center of the graph shown in  110 , corresponds to a “GOOD” region (i.e. region of the printability map where the three defects are not predicted to occur). In step  112 , a finalized printability map is generated including contour lines. The contour lines represent the maximum hatch spacing (the distance between two adjacent passes of the laser beam within the same layer) for the laser power and laser speed combinations shown in the revised printability map. These contour lines can be calculated based on geometric criteria relating melt pool depth to the known layer thickness of the powder layers. The contour lines in the final printability map can be used to show how the range of valid (i.e. predicted to be defect-free) laser power and laser speed parameters changes for different values of hatch spacing.  FIG. 4A  depicts a revised printability map  200  including “x” marks representing experimental results. As shown in  FIG. 4A , different shading can represent different regions of the printability map  400 . The different regions of the printability map can correspond to different types of defects, or to measures of the defects. As a non-limiting example, in the printability map  400  shown in  FIG. 4A , keyholing is represented by three regions  402 ,  404 ,  406 , where each of the three regions can represent a different minimum depth. In the non-limiting example shown in  FIG. 4A , a keyhole region with a depth greater than or equal to width/1.2 is shown as a region  402 , and a depth greater than or equal to width/1.5 is shown as another region  404 . Another keyhole region  406  is shown where the depth is greater than width/2.0. Similarly, a lack of fusion where depth is less than or equal to thickness, can be shown as another shaded region  408 . And a balling region  410  can be shown as another shaded region. A region without any predicted defects (i.e. a “good region”) can be shown using a region with a different shade, or as a region without shading  412 , as shown in  FIG. 4A . Experimental information can be included in the printability map, or overlaid on the printability map. As shown in  FIG. 4A , dots representing samples to print, or samples that have been printed are overlaid on the printability map. These samples can be used to confirm or validate the accuracy of the printability map  400 . 
       FIG. 4B  depicts a final printability map  450  including contour lines  452  representing different hatch spacings, given in micrometers (μm). Similar to the printability map depicted in  FIG. 4A , the printability map  250  in  FIG. 4B  depicts a keyhole region (depth greater than or equal to Width/1.2)  402 , balling region  410 , and lack of fusion region (depth less than or equal to thickness)  408 . It should be understood that the defect ratios and thresholds are provided only as examples, and that other defect measurements/thresholds are contemplated by the present disclosure. In particular, defect thresholds can vary based on the processes and materials used. It should be understood that the thresholds given throughout the present disclosure (e.g. the threshold that the depth is greater than the width/1.2) are intended only as non-limiting examples. Example illustrations of SEM data illustrating Balling, Keyholing, and Lack of fusion, as compared to a “good” track are illustrated in  FIGS. 5A-D . These examples of balling, keyholing, and lack of fusion can correspond to the shapes of those defects in embodiments of the present disclosure (e.g. the single track samples referred to above with reference to  FIG. 1 ). The illustrations show an example of a “good track” ( 5 A), a lack of fusion track ( 5 B), a keyholing defect track ( 5 C), and a balling defect track ( 5 D). It should be understood that the illustrations in  FIG. 5A-5D  are intended only as non-limiting examples, and that embodiments of the present disclosure can include different definitions or criteria for assessing whether these or other defects are present. 
     In implementations of the present disclosure, the maximum hatch spacing can be computed. Hatch spacing can be defined ash, the distance between two adjacent passes of the laser beam within the same layer. For example, a geometric criterion can be used to compute the maximum value for h that allows for full fusion within and between layers for a given melt pool width, melt pool depth, and layer thickness. Maximum hatch spacing h max  can be calculated as shown in equation 1, below. In equation 1, W represents the melt pool width, D represents the melt pool depth, and t represents the layer thickness. 
     
       
         
           
             
               
                 
                   
                     h 
                     max 
                   
                   = 
                   
                     W 
                     ⁢ 
                     
                       
                         1 
                         - 
                         
                           t 
                           
                             t 
                             + 
                             D 
                           
                         
                       
                     
                   
                 
               
               
                 
                   ( 
                   1 
                   ) 
                 
               
             
           
         
       
     
     The values of h max  for different values of W, t, and D, can be used to calculate maximum hatch spacing contours in the final printability map. Other equations and methods for determining maximum hatch spacing are contemplated by the present disclosure. 
     Based on the final printability map, three-dimensional parts can be created based on AM processes. The information in the printability map (laser speed, laser power, and hatch spacing) can correspond to processing parameters on AM machines, including commercially available AM machines such as selective laser melting (SLM) and direct metal laser sintering (DMLS) machines. Therefore, the techniques described with reference to  FIG. 1 , steps  102 ,  104 ,  106 ,  108 ,  110 , and  112  as well as  FIGS. 4A-4B  can be used to determine calibration settings that reduce or eliminate defects in AM for the specified alloy. 
     Additional steps can be performed to further calibrate the printability map, improve the performance of the model, and perform additional desired optimization steps. With reference to  FIG. 1 , in step  114 , bulk samples (i.e. “bulk coupons”) can be fabricated based on the final printability map. These bulk samples can be used to evaluate the porosity/density of parts made with different parameter combinations. This information can be used to further refine the final printability map. The evaluation of parameters other than porosity in step  114  is contemplated by the present disclosure. 
     Again referring to  FIG. 1 , in step  116  the properties of the bulk samples can be analyzed. For example, the evaluation of the bulk samples can include measurements of tensile strength and ductility. By evaluating the tensile strength of multiple bulk samples, an optimization step can be performed to determine the valid print parameters that result in the optimal tensile strength. For example, processing parameters that resulted in samples with highest density (&gt;99% of theoretical density) can be selected to print mechanical test samples. As a general guideline, 8-15 processing parameter combinations for porosity coupons and 4-5 parameter combinations for mechanical test specimens are recommended. Optionally, the samples printed in step  116  can be printed with print parameters that are predicted to be free of defects based on steps  102 - 114 . However, these numbers of processing parameter combinations are intended only as non-limiting examples. The optimization of properties other than tensile strength, as well as the optimization of more than one bulk sample property, is contemplated by the present disclosure. 
       FIG. 6  illustrates an exemplary computer that may comprise all or a portion of a system for generating printability maps for AM. Conversely, any portion or portions of the computer illustrated in  FIG. 6  may comprise all or part of the system for generating printability maps for AM. As used herein, “computer” may include a plurality of computers. The computers may include one or more hardware components such as, for example, a processor  1021 , a random-access memory (RAM) module  1022 , a read-only memory (ROM) module  1023 , a storage  1024 , a database  1025 , one or more input/output (I/O) devices  1026 , and an interface  1027 . Alternatively, and/or additionally, the computer may include one or more software components such as, for example, a computer-readable medium including computer executable instructions for performing a method associated with the exemplary embodiments such as, for example, an algorithm for determining a property profile gradient. It is contemplated that one or more of the hardware components listed above may be implemented using software. For example, storage  1024  may include a software partition associated with one or more other hardware components. It is understood that the components listed above are exemplary only and not intended to be limiting. 
     Processor  1021  may include one or more processors, each configured to execute instructions and process data to perform one or more functions associated with a computer for controlling a system (e.g., a system for generating printability maps for AM) and/or receiving and/or processing and/or transmitting data associated with electrical sensors. Processor  1021  may be communicatively coupled to RAM  1022 , ROM  1023 , storage  1024 , database  1025 , I/O devices  1026 , and interface  1027 . Processor  1021  may be configured to execute sequences of computer program instructions to perform various processes. The computer program instructions may be loaded into RAM  1022  for execution by processor  1021 . 
     RAM  1022  and ROM  1023  may each include one or more devices for storing information associated with operation of processor  1021 . For example, ROM  1023  may include a memory device configured to access and store information associated with the computer, including information for identifying, initializing, and monitoring the operation of one or more components and subsystems. RAM  1022  may include a memory device for storing data associated with one or more operations of processor  1021 . For example, ROM  1023  may load instructions into RAM  1022  for execution by processor  1021 . 
     Storage  1024  may include any type of mass storage device configured to store information that processor  1021  may need to perform processes consistent with the disclosed embodiments. For example, storage  1024  may include one or more magnetic and/or optical disk devices, such as hard drives, CD-ROMs, DVD-ROMs, or any other type of mass media device. 
     Database  1025  may include one or more software and/or hardware components that cooperate to store, organize, sort, filter, and/or arrange data used by the computer and/or processor  1021 . For example, database  1025  may store data related to the plurality of thrust coefficients. The database may also contain data and instructions associated with computer-executable instructions for controlling a system (e.g., an multi-material printer) and/or receiving and/or processing and/or transmitting data associated with a network of sensor nodes used to measure water quality. It is contemplated that database  1025  may store additional and/or different information than that listed above. 
     I/O devices  1026  may include one or more components configured to communicate information with a user associated with computer. For example, I/O devices may include a console with an integrated keyboard and mouse to allow a user to maintain a database of digital images, results of the analysis of the digital images, metrics, and the like. I/O devices  1026  may also include a display including a graphical user interface (GUI) for outputting information on a monitor. I/O devices  1026  may also include peripheral devices such as, for example, a printer, a user-accessible disk drive (e.g., a USB port, a floppy, CD-ROM, or DVD-ROM drive, etc.) to allow a user to input data stored on a portable media device, a microphone, a speaker system, or any other suitable type of interface device. 
     Interface  1027  may include one or more components configured to transmit and receive data via a communication network, such as the Internet, a local area network, a workstation peer-to-peer network, a direct link network, a wireless network, or any other suitable communication platform. For example, interface  1027  may include one or more modulators, demodulators, multiplexers, demultiplexers, network communication devices, wireless devices, antennas, modems, radios, receivers, transmitters, transceivers, and any other type of device configured to enable data communication via a wired or wireless communication network. 
     The figures illustrate the architecture, functionality, and operation of possible implementations of systems, methods and computer program products according to various implementations of the present invention. In this regard, each block of a flowchart or block diagrams may represent a module, segment, or portion of code, which comprises one or more executable instructions for implementing the specified logical function(s). It should also be noted that, in some alternative implementations, the functions noted in the block may occur out of the order noted in the figures. For example, two blocks shown in succession may, in fact, be executed substantially concurrently, or the blocks may sometimes be executed in the reverse order, depending upon the functionality involved. It will also be noted that each block of the block diagrams and/or flowchart illustration, and combinations of blocks in the block diagrams and/or flowchart illustration, can be implemented by special purpose hardware-based systems that perform the specified functions or acts, or combinations of special purpose hardware and computer instructions. 
     The corresponding structures, materials, acts, and equivalents of all means or step plus function elements in the claims below are intended to include any structure, material, or act for performing the function in combination with other claimed elements as specifically claimed. The description of the present invention has been presented for purposes of illustration and description, but is not intended to be exhaustive or limited to the invention in the form disclosed. Many modifications and variations will be apparent to those of ordinary skill in the art without departing from the scope and spirit of the invention. The implementation was chosen and described in order to best explain the principles of the invention and the practical application, and to enable others of ordinary skill in the art to understand the invention for various implementations with various modifications as are suited to the particular use contemplated. 
     Any combination of one or more computer readable medium(s) may be used to implement the systems and methods described herein above. The computer readable medium may be a computer readable signal medium or a computer readable storage medium. A computer readable storage medium may be, for example, but not limited to, an electronic, magnetic, optical, electromagnetic, infrared, or semiconductor system, apparatus, or device, or any suitable combination of the foregoing. More specific examples (a non-exhaustive list) of the computer readable storage medium would include the following: an electrical connection having one or more wires, a portable computer diskette, a hard disk, a random access memory (RAM), a read-only memory (ROM), an erasable programmable read-only memory (EPROM or Flash memory), an optical fiber, a portable compact disc read-only memory (CD-ROM), an optical storage device, a magnetic storage device, or any suitable combination of the foregoing. In the context of this document, a computer readable storage medium may be any tangible medium that can contain, or store a program for use by or in connection with an instruction execution system, apparatus, or device. 
     Program code embodied on a computer readable medium may be transmitted using any appropriate medium, including but not limited to wireless, wireline, optical fiber cable, RF, etc., or any suitable combination of the foregoing. 
     Computer program code for carrying out operations for aspects of the present invention may be written in any combination of one or more programming languages, including an object-oriented programming language such as Java, Smalltalk, C++or the like and conventional procedural programming languages, such as the “C” programming language or similar programming languages. The program code may execute entirely on the user&#39;s computer, partly on the user&#39;s computer, as a stand-alone software package, partly on the user&#39;s computer and partly on a remote computer or entirely on the remote computer or server. In the latter scenario, the remote computer may be connected to the user&#39;s computer through any type of network, including a local area network (LAN) or a wide area network (WAN), or the connection may be made to an external computer (for example, through the Internet using an Internet Service Provider). 
     While the methods and systems have been described in connection with preferred embodiments and specific examples, it is not intended that the scope be limited to the particular embodiments set forth, as the embodiments herein are intended in all respects to be illustrative rather than restrictive. 
     Unless otherwise expressly stated, it is in no way intended that any method set forth herein be construed as requiring that its steps be performed in a specific order. Accordingly, where a method claim does not actually recite an order to be followed by its steps or it is not otherwise specifically stated in the claims or descriptions that the steps are to be limited to a specific order, it is no way intended that an order be inferred, in any respect. This holds for any possible non-express basis for interpretation, including: matters of logic with respect to arrangement of steps or operational flow; plain meaning derived from grammatical organization or punctuation; the number or type of embodiments described in the specification. 
     It will be apparent to those skilled in the art that various modifications and variations can be made without departing from the scope or spirit. Other embodiments will be apparent to those skilled in the art from consideration of the specification and practice disclosed herein. It is intended that the specification and examples be considered as exemplary only, with a true scope and spirit being indicated by the following claims.