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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.

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.

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.

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.