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Challenge 3 of the 2022 NIST additive manufacturing benchmark (AM-Bench) experiments asked modelers to submit predictions for solid cooling rate, liquid cooling rate, time above melt, and melt pool geometry for single and multiple track laser powder bed fusion process using moving lasers. An in-house developed Additive Manufacturing Computational Fluid Dynamics code (AM-CFD) combined with a cylindrical heat source was implemented to accurately predict these experiments. Heuristic heat source calibration was proposed relating volumetric energy density (ψ) based on experiments available in the literature. The parameters of the heat source of the computational model were initially calibrated based on a Higher Order Proper Generalized Decomposition- (HOPGD) based surrogate model. The prediction using the calibrated heat source agreed quantitatively with NIST measurements for different process conditions. A scaling law based on keyhole formation was also utilized in calibrating the parameters of the cylindrical heat source and predicting the challenge experiments. In addition, an improvement on the heat source model was proposed to relate the Volumetric Energy Density (VEDσ) to the melt pool aspect ratio. The model showed further improvement in the prediction of the experimental measurements for the melt pool including cases at higher VEDσ. Overall, it was concluded that the appropriate selection of parameterization scheme along with the heat source model was crucial in the accurate prediction of melt pool geometry and thermal measurements while bypassing the expensive computational simulations that consider increased physics equations.

Dimensionless numbers and scaling laws provide elegant insights into the characteristic properties of physical systems. Classical dimensional analysis and similitude theory fail to identify a set of unique dimensionless numbers for a highly multi-variable system with incomplete governing equations. This paper introduces a mechanistic data-driven approach that embeds the principle of dimensional invariance into a two-level machine learning scheme to automatically discover dominant dimensionless numbers and governing laws (including scaling laws and differential equations) from scarce measurement data. The proposed methodology, called dimensionless learning, is a physics-based dimension reduction technique. It can reduce high-dimensional parameter spaces to descriptions involving only a few physically interpretable dimensionless parameters, greatly simplifying complex process design and system optimization. We demonstrate the algorithm by solving several challenging engineering problems with noisy experimental measurements (not synthetic data) collected from the literature. Examples include turbulent Rayleigh-Bénard convection, vapor depression dynamics in laser melting of metals, and porosity formation in 3D printing. Lastly, we show that the proposed approach can identify dimensionally homogeneous differential equations with dimensionless number(s) by leveraging sparsity-promoting techniques.

In the Air Force Research Laboratory Additive Manufacturing Challenge Series, melted track geometries for a laser powder bed fusion (L-PBF) process of Inconel 625 were used to challenge and validate computational models predicting melting and solidification behavior. The impact of process parameters upon single-track single-layer, multi-track single-layer, and single-track multi-layer L-PBF processes was studied. To accomplish this, a physics-based thermal-fluid model was developed and calibrated using a proper generalized decomposition surrogate model, then compared against the experimental measurements. The thermal-fluid model was enhanced through the usage of an adaptive mesh and residual heat factor (RHF) model, based on the scanning strategy, for improved efficiency and accuracy. It is found that this calibration approach is not only robust and efficient, but it also enables the thermal-fluid model to make predictions which quantitatively agree well with the experimental measurements. The adaptive mesh provides over a 10-times speedup as compared to a uniform mesh. The RHF model improves predictive accuracy by over 60%, particularly near starting and ending points of the melted tracks, which are greatly affected by the thermal behavior of adjacent tracks. Moreover, the thermal-fluid model is shown to potentially predict lack-of-fusion defects and provide insights into the defect generation process in L-PBF.

Challenge 4 of the Air Force Research Laboratory additive manufacturing modeling challenge series asks the participants to predict the grain-average elastic strain tensors of a few specific challenge grains during tensile loading, based on experimental data and extensive characterization of an IN625 test specimen. In this article, we present our strategy and computational methods for tackling this problem. During the competition stage, a characterized microstructural image from the experiment was directly used to predict the mechanical responses of certain challenge grains with a genetic algorithm-based material model identification method. Later, in the post-competition stage, a proper generalized decomposition (PGD)-based reduced order method is introduced for improved material model calibration. This data-driven reduced order method is efficient and can be used to identify complex material model parameters in the broad field of mechanics and materials science. The results in terms of absolute error have been reported for the original prediction and re-calibrated material model. The predictions show that the overall method is capable of handling large-scale computational problems for local response identification. The re-calibrated results and speed-up show promise for using PGD for material model calibration.

Direct representation of material microstructure in a macroscale simulation is prohibitively expensive, if even possible, with current methods. However, the information contained in such a representation is highly desirable for tasks such as material/alloy design and manufacturing process control. In this paper, a mechanistic machine learning framework is developed for fast multiscale analysis of material response and structure performance. The new capabilities stem from three major factors: (1) the use of an unsupervised learning (clustering)-based discretization to achieve significant order reduction at both macroscale and microscale; (2) the generation of a database of interaction tensors among discretized material regions; (3) concurrent multiscale response prediction to solve the mechanistic equations. These factors allow for an orders-of-magnitude decrease in the computational expense compared to FEn, n $$\ge $$ 2. This method provides sufficiently high fidelity and speed to reasonably conduct inverse modeling for the challenging tasks mentioned above.

The hierarchical deep-learning neural network (HiDeNN) is systematically developed through the construction of structured deep neural networks (DNNs) in a hierarchical manner, and a special case of HiDeNN for representing Finite Element Method (or HiDeNN-FEM in short) is established. In HiDeNN-FEM, weights and biases are functions of the nodal positions, hence the training process in HiDeNN-FEM includes the optimization of the nodal coordinates. This is the spirit of r-adaptivity, and it increases both the local and global accuracy of the interpolants. By fixing the number of hidden layers and increasing the number of neurons by training the DNNs, rh-adaptivity can be achieved, which leads to further improvement of the accuracy for the solutions. The generalization of rational functions is achieved by the development of three fundamental building blocks of constructing deep hierarchical neural networks. The three building blocks are linear functions, multiplication, and inversion. With these building blocks, the class of deep learning interpolation functions are demonstrated for interpolation theories such as Lagrange polynomials, NURBS, isogeometric, reproducing kernel particle method, and others. In HiDeNN-FEM, enrichment functions through the multiplication of neurons is equivalent to the enrichment in standard finite element methods, that is, generalized, extended, and partition of unity finite element methods. Numerical examples performed by HiDeNN-FEM exhibit reduced approximation error compared with the standard FEM. Finally, an outlook for the generalized HiDeNN to high-order continuity for multiple dimensions and topology optimizations are illustrated through the hierarchy of the proposed DNNs.

Toolpath choice in metal-based additive manufacturing (AM) affects local thermal environment. We use Hamiltonian paths to systematically enumerate time- and space-continuous toolpaths on example n × n grid geometries. This framework broadens the toolpath design space by establishing a finite and searchable number of AM toolpaths for any discretized geometry. We characterize toolpaths by extracting toolpath internal structures, e.g., the number of corners and pairs of parallel tracks. The enumerated toolpaths serve as an input to thermal simulations to obtain solidification cooling rate statistics, which strongly correlate to the number of internal structures. Hence, toolpath can be linked to microstructural predictions.

In this paper, we derive an analytical expression for reproducing kernel particle method (RKPM) shape functions. Based on this, we propose a necessary and sufficient stability condition for general RKPM in arbitrary function space, and illustrate with degenerate cases. By selecting proper basis vectors and the support of the kernel functions, we demonstrate that the RKPM framework allows generating many kinds of shape functions, including the Lagrangian, B-spline and NURBS shape functions.

Dimensionless numbers and scaling laws provide elegant insights into the characteristic properties of physical systems. Classical dimensional analysis and similitude theory fail to identify a set of unique dimensionless numbers for a highly-multivariable system with incomplete governing equations. In this study, we embed the principle of dimensional invariance into a two-level machine learning scheme to automatically discover dominant and unique dimensionless numbers and scaling laws from data. The proposed methodology, called dimensionless learning, can be treated as a physics-based dimension reduction. It can reduce high-dimensional parameter spaces into descriptions involving just a few physically-interpretable dimensionless parameters, which significantly simplifies the process design and optimization of the system. We demonstrate the algorithm by solving several challenging engineering problems with noisy experimental measurements (not synthetic data) collected from the literature. The examples include turbulent Rayleigh-Bénard convection, vapor depression dynamics in laser melting of metals, and porosity formation in 3D printing. We also show that the proposed approach can identify dimensionally homogeneous differential equations with minimal parameters by leveraging sparsity-promoting techniques.

To design microstructure and microhardness in the additive manufacturing (AM) of nickel (Ni)-based superalloys, the present work develops a novel data-driven approach that combines physics-based models, experimental measurements, and a data-mining method. The simulation is based on a computational thermal-fluid dynamics (CtFD) model, which can obtain thermal behavior, solidification parameters such as cooling rate, and the dilution of solidified clad. Based on the computed thermal information, dendrite arm spacing and microhardness are estimated using well-tested mechanistic models. Experimental microstructure and microhardness are determined and compared with the simulated values for validation. To visualize process–structure–properties (PSPs) linkages, the simulation and experimental datasets are input to a data-mining model—a self-organizing map (SOM). The design windows of the process parameters under multiple objectives can be obtained from the visualized maps. The proposed approaches can be utilized in AM and other data-intensive processes. Data-driven linkages between process, structure, and properties have the potential to benefit online process monitoring control in order to derive an ideal microstructure and mechanical properties. Keywords: Additive manufacturing, Data science, Multiphysics modeling, Self-organizing map, Microstructure, Microhardness, Ni-based superalloy

Accurate and efficient modeling of microstructural interaction and evolution for prediction of the macroscopic behavior of materials is important for material design and manufacturing process control . This paper approaches this challenge with a reduced-order method called self-consistent clustering analysis (SCA). It is reformulated for general elasto-viscoplastic materials under large deformation . The accuracy and efficiency for predicting overall mechanical response of polycrystalline materials is demonstrated with a comparison to traditional full-field solution methods such as finite element analysis and the fast Fourier transform . It is shown that the reduced-order method enables fast prediction of microstructure–property relationships with quantified variation. The utility of the method is demonstrated by conducting a concurrent multiscale simulation of a large-deformation manufacturing process with sub-grain spatial resolution while maintaining reasonable computational expense. This method could be used for microstructure-sensitive properties design as well as process parameters optimization.

Recently proposed clustering-based methods considerably reduce numerical cost for homogenizing heterogeneous materials, while maintaining the accuracy of mechanical property predictions in an online stage. In such an algorithm, however, the calculation of interaction tensors consumes much of the total computing time. We introduce a new method that expedites the interaction tensors calculation, thereby enhancing the clustering-based methods. We first cast a cubic/rectangular coarse grid over the representative volume element. Using analytical expressions for the integral of the Green’s functions, we then calculate interaction tensors on the coarse grid. Finally, the desired interaction tensors on the clusters are approximated based on composition ratios. Moreover, in virtual clustering analysis, we derive the Lippmann–Schwinger equation for finite strain problems. Numerical tests in two and three space dimensions verify the efficiency and accuracy of the proposed method.

A new data-driven method that generalizes experimentally measured and/or computational generated data sets under different loading paths to build three dimensional nonlinear elastic material law with objectivity under arbitrary loadings using neural networks is proposed. The proposed approach is first demonstrated by exploiting the concept of representative volume element (RVE) in the principal strain and stress spaces to numerically generate the data. A computational data-training algorithm on the generalization of these principal space data to three dimensional objective isotropic material laws subjected to arbitrary deformation is given. To validate these data-driven derived material laws, large deformation and buckling analysis of nonlinear elastic solids with reference material models and engineering structure with microstructure are performed. Numerical experiments show that only seven sets of data under different stress loading paths on RVEs are required to reach reasonable accuracy. The requirements for constitutive law such as objectivity are preserved approximately. The consistent tangent modulus is also derived. The proposed approach also shows a great potential to obtain the material law between different scales in the multiscale analysis by pure data.

Mechanical science and engineering can use machine learning. However, data sets have remained relatively scarce; fortunately, known governing equations can supplement these data. This paper summarizes and generalizes three reduced order methods: self-consistent clustering analysis, virtual clustering analysis, and FEM-clustering analysis. These approaches have two-stage structures: unsupervised learning facilitates model complexity reduction and mechanistic equations provide predictions. These predictions define databases appropriate for training neural networks. The feed forward neural network solves forward problems, e.g., replacing constitutive laws or homogenization routines. The convolutional neural network solves inverse problems or is a classifier, e.g., extracting boundary conditions or determining if damage occurs. We will explain how these networks are applied, then provide a practical exercise: topology optimization of a structure (a) with non-linear elastic material behavior and (b) under a microstructural damage constraint. This results in microstructure-sensitive designs with computational effort only slightly more than for a conventional linear elastic analysis.

In order to represent the functionally graded properties of interphase, a multi-coated micromechanical model is developed. Based on elliptic shell integration of Green’s function, the strain disturbance in each phase is obtained. According to computational investigation of this model, the outer layer of the interphase does not bring in strain disturbance within the inner ones. To this end, a sequential computational homogenization method is proposed. The inhomogeneities are added sequentially from outside to inside. The temporary effective modulus on each stage is obtained by the Self Consistency Scheme. Then the effective modulus of the overall composites are fitted with a Mori–Tanaka estimation for practical applications. The effectiveness of present method is verified by the results of “2 + 1” and “3 + 1” models in prior researches and finite element simulations. Finally, the influence of thickness and stiffness of interphase on the composites’ effective modulus are investigated.

Process−property relationships in additive manufacturing (AM) play critical roles in process control and rapid certification. In laser-based directed energy deposition, powder mass flow into the melt pool influences the cooling behavior and properties of a built part. This study develops predictive computational models that provide the microhardness of AM components processed with miscible dissimilar alloys, and then investigates the influence of varying process parameters on properties in experiments and modeling. Experimentally-determined clad dilution and microhardness results of Ni-based superalloy Inconel 718 clads deposited onto 1045 carbon steel substrates are compared to the values from a computational thermo-fluid dynamics (CtFD) model. The numerical model considers the fluidic mechanisms of molten metal during powder deposition and the resulting transient melt pool geometry changes. The model also handles the change in thermo-physical properties caused by the composition mixture between the powder and substrate materials in the melt pool. Based on the computed temperature and velocity distributions in the melt pool, cooling rate, dilution of the melt pool and microhardenss are evaluated. The capability to predict thermal histories in such models is calibrated and validated with experimental thermal imaging and microstructures of additive manufactured clads. In addition, the roles of cooling rate and alloy composition on the microhardness are examined. The results show that variation in microhardness is dominated by composition mixture between the powder and substrate materials, rather than cooling behavior or dendrite arm spacing at liquid-solid interface in laser deposited Inconel 718 on AISI 1045 carbon steel.

In the NIST additive manufacturing benchmark (AM-Bench) experiments, melt pool geometry, cooling rates, surface topography, and dendritic microstructure in laser melted Inconel 625 were used to challenge and validate computational models of the melting and solidification process. To this end, three thermal models incorporating different physics are compared with the experimental data. It is identified that the heat convection enhanced by the thermocapillary flow inside the melt pool and heat loss caused by vaporization play pivotal roles to guarantee the accuracy of the predictions, and thus should be considered in the thermal model. Neglecting fluid flow and vaporization leads to nearly 100% difference in cooling rate during solidification, and 20% difference in cooling rate after solidification from the results. With the most accurate thermal model, surface topographies of the melt tracks are predicted and quantitatively analyzed. Using the Kurz-Fisher model, the primary dendrite arm spacing is predicted based on the thermal gradient and solidification rate predictions, while elemental segregation is predicted using the Scheil-Gulliver model and a non-equilibrium solidification model. Additionally, it is shown that increasing scan speed inhibits elemental microsegregation.

Phase-field models for fracture problems have attracted considerable attention in recent years, which are capable of tracking the discontinuities numerically, and also produce complex crack patterns in many applications. In this paper, a phase-field model for a general nonlinearly elastic material is proposed using a novel additive decomposition of strain energy. This decomposition has two parts: one is principal stretch related and the other solely composed of volumetric deformation, which accounts for different behaviors of fracture in tension and compression. We construct the Lagrangian by integrating the split energies and the separation energy from phase-field approximation for discrete cracks. A coupled system of equations is also derived that governs the deformation of the body and the evolution of phase field. The capability and performance of the proposed model are demonstrated in several representative examples. Our results show that the predicted fracture surfaces are in good agreement with experimental observations. Compared with the previous models in which the energy is simply split into the isochoric and volumetric parts, the present model is numerically more robust and effective in simulating sharp cracks. The present model can also aid researchers to control the degree of tension–compression asymmetry in the nonlinear regime of deformation, which can be naturally extended to simulate the fracture of the rubber-like materials with tension–compression asymmetry.

This work elucidates the impacts of model construction choices on turbulence characteristics and solution fidelity in the simulation of coupled freestream and porous turbulent flows. A freestream-porewater interface is modeled numerically as a matrix of regularly spaced spheres submerged in a surrounding flow. Simulations are conducted to solve the continuity and momentum equations via Large Eddy Simulation (LES) using the Control Volume Finite Element Method (CVFEM) on an unstructured, surface-conforming mesh, and simulated flow fields are compared with experimental results. Key parameters are identified, allowing for model creation recommendations. A mesh refinement study is performed, and characteristic required mesh sizes in both the bed and the freestream are identified that achieve a good trade-off between accuracy and efficiency. Additionally, it is shown that similar to wall-bounded flows, the computational domain for a coupled freestream and porous flow must be sufficiently large to capture the relevant largest-sized eddies and to avoid the spanwise locking of flow structures; such structures may affect the flow field in the pores as well as in the freestream. Dimensions of 7.5H × 3.5H × H, where H is the freestream height, are found to give satisfactory comparison with experimental results for the cases studied. Finally, it is found that the wall-adapting local eddy-viscosity (WALE) turbulence closure scheme is better able to model the fluid velocity in the problem domain compared with the Smagorinsky model. Failure to select the proper turbulence closure model or domain size leads to a misrepresentation of the turbulent structures. Because of the strong coupling between the porewater flow and the freestream, these modeling errors propagate in both flow regions.

This year marks the eightieth anniversary of the invention of the finite element method (FEM). FEM has become the computational workhorse for engineering design analysis and scientific modeling of a wide range of physical processes, including material and structural mechanics, fluid flow and heat conduction, various biological processes for medical diagnosis and surgery planning, electromagnetics and semi-conductor circuit and chip design and analysis, additive manufacturing, i.e. virtually every conceivable problem that can be described by partial differential equations (PDEs). FEM has fundamentally revolutionized the way we do scientific modeling and engineering design, ranging from automobiles, aircraft, marine structures, bridges, highways, and high-rise buildings. Associated with the development of finite element methods has been the concurrent development of an engineering science discipline called computational mechanics, or computational science and engineering. In this paper, we present a historical perspective on the developments of finite element methods mainly focusing on its applications and related developments in solid and structural mechanics, with limited discussions to other fields in which it has made significant impact, such as fluid mechanics, heat transfer, and fluid-structure interaction. To have a complete storyline, we divide the development of the finite element method into four time periods: I. (1941-1965) Early years of FEM; II. (1966-1991) Golden age of FEM; III. (1992-2017) Large scale, industrial applications of FEM and development of material modeling, and IV (2018-) the state-of-the-art FEM technology for the current and future eras of FEM research. Note that this paper may not strictly follow the chronological order of FEM developments, because often time these developments were interwoven across different time periods.

Scoliosis is a 3D deformation of the spinal column, characterized by a lateral deviation of the spine, accompanied by axial rotation of the vertebrae. Adolescent Idiopathic Scoliosis (AIS), is the most common type, affecting children between ages 8 to 18 when bone growth is at its maximum rate. The selection of the most appropriate treatment options is based on the surgeon’s experience. So, developing a clinically validated patient-specific model of the spine would aid surgeons in understanding AIS in early stages and propose an efficient method of treatment for the individual patient. This project steps include: Developing a clinically validated patient-specific Reduced Order Finite Element Model (ROFEM) of the spine, predicting AIS progression using data mining and proposing a method of treatment. First we implement FE synergistically with bio-mechanical information, image processing and data science techniques to improve predictive ability. Initial geometry of the spine will be extracted from the x-ray images from different planes and imported to FEM software to generate the spine model and perform analysis. A RO model is developed based on the detailed spinal FEM. Next, a neural network is used to predict the spinal curvature. The ability to predict the severity of AIS will have an immense impact on the treatment of AIS-affected children. Access to a predictive and patient-specific model will enable the physicians to have a better understanding of spinal curvature progression. Consequently, the physicians will be able to educate families, choose the most appropriate treatment option and asses for surgical intervention.

Metal additive manufacturing provides remarkable flexibility in geometry and component design, but localized heating/cooling heterogeneity leads to spatial variations of as-built mechanical properties, significantly complicating the materials design process. To this end, we develop a mechanistic data-driven framework integrating wavelet transforms and convolutional neural networks to predict location-dependent mechanical properties over fabricated parts based on process-induced temperature sequences, i.e., thermal histories. The framework enables multiresolution analysis and importance analysis to reveal dominant mechanistic features underlying the additive manufacturing process, such as critical temperature ranges and fundamental thermal frequencies. We systematically compare the developed approach with other machine learning methods. The results demonstrate that the developed approach achieves reasonably good predictive capability using a small amount of noisy experimental data. It provides a concrete foundation for a revolutionary methodology that predicts spatial and temporal evolution of mechanical properties leveraging domain-specific knowledge and cutting-edge machine and deep learning technologies.

Scoliosis, an abnormal curvature of the human spinal column, is characterized by a lateral deviation of the spine, accompanied by axial rotation of the vertebrae. Adolescent Idiopathic Scoliosis (AIS) is the most common type, affecting children between ages 8 to 18 when bone growth is at its maximum rate. We propose a mechanistic machine learning algorithm in order to study patient-specific AIS curve progression, which is associated with the bone growth and other genetic and environmental factors. Two different frameworks are used to analyse and predict curve progression, one with implementing clinical data extracted from 2D X-ray images and the other one with incorporating both clinical data and physical equations governing the non-uniform bone growth. The physical equations governing bone growth are affiliated with calculating all stress components at each region. The stress values are evaluated through a surrogate finite element simulation and a bone growth model on a detailed patient-specific geometry of the human spine. We also propose a patient-specific framework to generate the volumetric model of human spine which is partitioned into different tissues for both vertebra and intervertebral disc. It is shown that implementing physical equations governing bone growth into the prediction framework will notably improve the prediction results as compared to only using clinical data for prediction. In addition, we can predict curve progression at ages outside the range of training samples.

This paper presents a proper generalized decomposition (PGD) based reduced-order model of hierarchical deep-learning neural networks (HiDeNN). The proposed HiDeNN-PGD method keeps both advantages of HiDeNN and PGD methods. The automatic mesh adaptivity makes the HiDeNN-PGD more accurate than the finite element method (FEM) and conventional PGD, using a fraction of the FEM degrees of freedom. The accuracy and convergence of the method have been studied theoretically and numerically, with a comparison to different methods, including FEM, PGD, HiDeNN and Deep Neural Networks. In addition, we theoretically showed that the PGD converges to FEM at increasing modes, and the PGD error is a direct sum of the FEM error and the mode reduction error. The proposed HiDeNN-PGD performs high accuracy with orders of magnitude fewer degrees of freedom, which shows a high potential to achieve fast computations with a high level of accuracy for large-size engineering problems., Comment: 35 pages, 12 figures

Thermal residual stress is identified as one of the major reasons of stress concentration in material’s microstructures which initiates failure in the microstructure. Considering the details of microstructural features (inclusions shape, size, and distri-bution) can provide better understanding of the thermal residual stress developed in the materials due to the temperature change. In this paper, we have extended the self-consistent clustering analysis (SCA) method for efficient and accurate modeling of thermal residual stress for thermoelastic heterogeneous materials. The governing equations of the thermoelasticity has been implemented through a eigenstrain problem and solved using a Fast Fourier Transform (FFT) based solution scheme and later extended for SCA formulation. The thermoelastic formulation of SCA method has been verified for a single inclusion (homogeneous and inhomoge-neous) problem with the analytical solution of Eshelby and the FFT-based solution for multiple clusters. An example problem with multiple inhomogeneous inclusions has been solved for different temperatures and the residual stress developed has been analyzed. Results show that the thermoelastic formulation of SCA can have order of hundred times speed up compare to the traditional FFT-based solution scheme. The proposed methodology can be implemented in thermomechanical problems and provide efficient multiscale capabilities for prediction of thermal residual stress.

Design of additively manufactured metallic parts requires computational models that can predict the mechanical response of the parts considering the microstructural, manufacturing, and operating conditions. This article documents our response to Air Force Research Laboratory (AFRL) Additive Manufacturing Modeling Challenge 3, which asks the participants to predict the mechanical response of tensile coupons of IN625 as function of microstructure and manufacturing conditions. A representative volume element (RVE) approach was coupled with a crystal plasticity material model, solved within the fast Fourier transformation (FFT) framework for mechanics, to address the challenge. During the competition, material model calibration proved to be a challenge, prompting the introduction in this manuscript of an advanced material model identification method using proper generalized decomposition (PGD). Finally, a mechanistic reduced order method called self-consistent clustering analysis (SCA) is shown as a possible alternative to the FFT method for solving these problems. Apart from presenting the response analysis, some physical interpretation and assumptions associated with the modeling are discussed.

Metal three-dimensional (3D) printing includes a vast number of operation and material parameters with complex dependencies, which significantly complicates process optimization, materials development, and real-time monitoring and control. We leverage ultrahigh-speed synchrotron X-ray imaging and high-fidelity multiphysics modeling to identify simple yet universal scaling laws for keyhole stability and porosity in metal 3D printing. The laws apply broadly and remain accurate for different materials, processing conditions, and printing machines. We define a dimensionless number, the Keyhole number, to predict aspect ratio of a keyhole and the morphological transition from stable at low Keyhole number to chaotic at high Keyhole number. Furthermore, we discover inherent correlation between keyhole stability and porosity formation in metal 3D printing. By reducing the dimensions of the formulation of these challenging problems, the compact scaling laws will aid process optimization and defect elimination during metal 3D printing, and potentially lead to a quantitative predictive framework., Identifying scaling laws in metal 3D printing is key to process optimization and materials development. Here the authors report scaling laws to quantify correlation between process parameters, keyhole stability and pore formation by high-speed synchrotron X-ray imaging and multiphysics modeling.