Wing Kam Liu

We present a framework for the Convolutional Hierarchical Deep-learning Neural Network (C-HiDeNN) tailored for nonlinear finite element analysis. Building upon the structured foundation of HiDeNN, C-HiDeNN introduces a convolution operator to enhance numerical approximation. A distinctive feature of C-HiDeNN is its higher-order accurate approximation achieved through an expanded set of parameters, such as the polynomial order ‘ p ,’ dilation parameter ‘ a ,’ patch size ‘ s ,’ and nodal position ‘X’. These parameters function as the functional equivalents of weights and biases within each C-HiDeNN patch. In addition, C-HiDeNN can be selectively applied to regions requiring high resolution to adaptively improve local prediction accuracy. To demonstrate the effectiveness of this framework, we provide numerical examples in the context of nonlinear finite element analysis. The results show that our approach achieves significantly higher accuracy than conventional Finite Element Method (FEM) while substantially reducing computational costs.

Metal Laser Powder Bed Fusion (PBF-LB/M) is a leading additive manufacturing technique in which part quality and grain morphology are highly dependent on process parameters. Numerous studies of process variations, such as laser power, scan speed, and spot diameter, have demonstrated that they strongly influence melt pool dynamics; however, the effects of powder layer height and geometric variations remain less well understood. In this article, we focus on variations in powder layer height and part geometry to study their influence on melt pool dynamics. We employed a high-fidelity multiphysics simulation framework based on the open source finite volume method (FVM) solver package `LaserBeamFoam' built on `OpenFOAM' to study the variations in different melt pool metrics -- melt pool depth, width, bead height, overlap depth, overlap width, solidified area, and dilution area. The solver captures coupled phenomena of heat transfer, fluid flow, vaporization, recoil pressure, Marangoni convection, and realistic laser reflection behavior to accurately model the melt pool dynamics. Simulations are performed for different powder layer heights and geometric dimensions for direct comparison with benchmark experiments conducted at the National Institute of Standards and Technology (NIST) in 2025. Quantitative validation against NIST experiment demonstrates excellent agreement in all the melt pool metrics. These results highlight the predictive capability of physics-based PBF-LB models, paving the way for process optimization, defect mitigation, and the integration of simulation into digital twin frameworks for additive manufacturing.

Metal Laser Powder Bed Fusion (PBF-LB/M) is a leading additive manufacturing technique in which part quality and grain morphology are highly dependent on process parameters. Numerous studies of process variations, such as laser power, scan speed, and spot diameter, have demonstrated that they strongly influence melt pool dynamics; however, the effects of powder layer height and geometric variations remain less well understood. In this article, we focus on variations in powder layer height and part geometry to study their influence on melt pool dynamics. We employed a high-fidelity multiphysics simulation framework based on the open source finite volume method (FVM) solver package `LaserBeamFoam' built on `OpenFOAM' to study the variations in different melt pool metrics -- melt pool depth, width, bead height, overlap depth, overlap width, solidified area, and dilution area. The solver captures coupled phenomena of heat transfer, fluid flow, vaporization, recoil pressure, Marangoni convection, and realistic laser reflection behavior to accurately model the melt pool dynamics. Simulations are performed for different powder layer heights and geometric dimensions for direct comparison with benchmark experiments conducted at the National Institute of Standards and Technology (NIST) in 2025. Quantitative validation against NIST experiment demonstrates excellent agreement in all the melt pool metrics. These results highlight the predictive capability of physics-based PBF-LB models, paving the way for process optimization, defect mitigation, and the integration of simulation into digital twin frameworks for additive manufacturing.

Neural networks and machine learning models for uncertainty quantification suffer from limited scalability and poor reliability compared to their deterministic counterparts. In industry-scale active learning settings, where generating a single high-fidelity simulation may require days or weeks of computation and produce data volumes on the order of gigabytes, they quickly become impractical. This paper proposes a scalable and reliable Bayesian surrogate model, termed the Bayesian Interpolating Neural Network (B-INN). The B-INN combines high-order interpolation theory with tensor decomposition and alternating direction algorithm to enable effective dimensionality reduction without compromising predictive accuracy. We theoretically show that the function space of a B-INN is a subset of that of Gaussian processes, while its Bayesian inference exhibits linear complexity, $\mathcal{O}(N)$, with respect to the number of training samples. Numerical experiments demonstrate that B-INNs can be from 20 times to 10,000 times faster with a robust uncertainty estimation compared to Bayesian neural networks and Gaussian processes. These capabilities make B-INN a practical foundation for uncertainty-driven active learning in large-scale industrial simulations, where computational efficiency and robust uncertainty calibration are paramount.

Neural networks and machine learning models for uncertainty quantification suffer from limited scalability and poor reliability compared to their deterministic counterparts. In industry-scale active learning settings, where generating a single high-fidelity simulation may require days or weeks of computation and produce data volumes on the order of gigabytes, they quickly become impractical. This paper proposes a scalable and reliable Bayesian surrogate model, termed the Bayesian Interpolating Neural Network (B-INN). The B-INN combines high-order interpolation theory with tensor decomposition and alternating direction algorithm to enable effective dimensionality reduction without compromising predictive accuracy. We theoretically show that the function space of a B-INN is a subset of that of Gaussian processes, while its Bayesian inference exhibits linear complexity, $\mathcal{O}(N)$, with respect to the number of training samples. Numerical experiments demonstrate that B-INNs can be from 20 times to 10,000 times faster with a robust uncertainty estimation compared to Bayesian neural networks and Gaussian processes. These capabilities make B-INN a practical foundation for uncertainty-driven active learning in large-scale industrial simulations, where computational efficiency and robust uncertainty calibration are paramount.

Generative artificial intelligence (GAI) plays a fundamental role in high-impact AI-based systems such as SORA and AlphaFold. Currently, GAI shows limited capability in the specialized domains due to data scarcity. In this paper, we develop a continuum mechanics-based theoretical framework to generalize the optimal transport theory from pure mathematics, which can be used to describe the dynamics of data, realizing the generative tasks with a small amount of data. The developed theory is used to solve three typical problem involved in many mechanical designs and engineering applications: at material level, how to generate the stress-strain response outside the range of experimental conditions based on experimentally measured stress-strain data; at structure level, how to generate the temperature-dependent stress fields under the thermal loading; at system level, how to generate the plastic strain fields under transient dynamic loading. Our results show the proposed theory can complete the generation successfully, showing its potential to solve many difficult problems involved in engineering applications, not limited to mechanics problems, such as image generation. The present work shows that mechanics can provide new tools for computer science. The limitation of the proposed theory is also discussed.

Generative artificial intelligence (GAI) plays a fundamental role in high-impact AI-based systems such as SORA and AlphaFold. Currently, GAI shows limited capability in the specialized domains due to data scarcity. In this paper, we develop a continuum mechanics-based theoretical framework to generalize the optimal transport theory from pure mathematics, which can be used to describe the dynamics of data, realizing the generative tasks with a small amount of data. The developed theory is used to solve three typical problem involved in many mechanical designs and engineering applications: at material level, how to generate the stress-strain response outside the range of experimental conditions based on experimentally measured stress-strain data; at structure level, how to generate the temperature-dependent stress fields under the thermal loading; at system level, how to generate the plastic strain fields under transient dynamic loading. Our results show the proposed theory can complete the generation successfully, showing its potential to solve many difficult problems involved in engineering applications, not limited to mechanics problems, such as image generation. The present work shows that mechanics can provide new tools for computer science. The limitation of the proposed theory is also discussed.

We present JAX-PF, an open-source, GPU-accelerated, and differentiable Phase Field (PF) software package, supporting both explicit and implicit time stepping schemes. Leveraging the modern computing architecture JAX, JAX-PF achieves high performance through array programming and GPU acceleration, delivering ~5x speedup over PRISMS-PF with MPI (24 CPU cores) for systems with ~4.19 million degrees of freedom using explicit schemes, and scaling efficiently with implicit schemes for large-size problems. Furthermore, a key feature of JAX-PF is automatic differentiation (AD), eliminating manual derivations of free-energy functionals and Jacobians. Beyond forward simulations, JAX-PF demonstrates its potential in inverse design by providing sensitivities for gradient-based optimization. We demonstrate, for the first time, the calibration of PF material parameters using AD-based sensitivities, highlighting its capability for high-dimensional inverse problems. By combining efficiency, flexibility, and full differentiability, JAX-PF offers a fast, practical, and integrated tool for forward simulation and inverse design, advancing co-designing of material and manufacturing processes and supporting the goals of the Materials Genome Initiative.

We present JAX-PF, an open-source, GPU-accelerated, and differentiable Phase Field (PF) software package, supporting both explicit and implicit time stepping schemes. Leveraging the modern computing architecture JAX, JAX-PF achieves high performance through array programming and GPU acceleration, delivering ~5x speedup over PRISMS-PF with MPI (24 CPU cores) for systems with ~4.19 million degrees of freedom using explicit schemes, and scaling efficiently with implicit schemes for large-size problems. Furthermore, a key feature of JAX-PF is automatic differentiation (AD), eliminating manual derivations of free-energy functionals and Jacobians. Beyond forward simulations, JAX-PF demonstrates its potential in inverse design by providing sensitivities for gradient-based optimization. We demonstrate, for the first time, the calibration of PF material parameters using AD-based sensitivities, highlighting its capability for high-dimensional inverse problems. By combining efficiency, flexibility, and full differentiability, JAX-PF offers a fast, practical, and integrated tool for forward simulation and inverse design, advancing co-designing of material and manufacturing processes and supporting the goals of the Materials Genome Initiative.

Abstract Data-driven science and computation have advanced immensely to construct complex functional relationships using trainable parameters. However, efficiently discovering interpretable and accurate closed-form expressions from complex dataset remains a challenge. This article presents a novel approach that uses an accurate, frugal, fast, separable, and scalable neural architecture with symbolic regression to discover closed-form expressions from limited observation. The article presents a two-step algorithm with a separability checker embedded in it. The model’s accuracy and efficiency are tested on a canonical benchmark suite, as well as dynamical system discovery, and identification from noisy data. The model generally shows outstanding approximation capability in these benchmarks, producing orders of magnitude smaller errors compared to reference data and traditional symbolic regression. Later, the model is applied to three engineering applications: i) discovering a closed-form fatigue equation, ii) identification of hardness from micro-indentation test data, and iii) discovering an expression for the yield surface with data. In every case, the model outperformed the reference methods used in the literature.

Abstract Spatial heterogeneity in microstructure presents a significant challenge for part qualification in metal additive manufacturing, particularly when relying on physically accurate computational models to replace costly trial-and-error testing. Reliable structural simulation needs a large amount of offline data and several very expensive forward runs to learn the most appropriate material model parameters. This work introduces a systematic framework for accurately calibrating crystal plasticity (CP) material law parameters using limited characterization data. The calibrated CP law is validated through blind predictions of mechanical responses in laser powder-bed fusion Inconel 625 (IN 625) tensile coupons across varying build orientations and strategies. Two surrogate modeling approaches—a higher-order proper generalized decomposition (HOPGD) and a novel interpolating neural network (INN)—are evaluated for their ability to approximate full-field simulations. The study presents an adaptive sampling strategy to efficiently utilize an offline database and outlines a methodology for representing microstructure from sparse characterization inputs. Results demonstrate that the differentiable INN surrogate achieves accurate calibration with significantly reduced data requirements and avoids reliance on computationally expensive genetic algorithms. Both surrogate models exhibit strong predictive performance, and the proposed workflow was instrumental in winning the 2022 NIST Additive Manufacturing Benchmark Challenge.

Computational science and engineering are shifting toward data-centric, optimization-based, and self-correcting solvers with artificial intelligence. This transition faces challenges such as low accuracy with sparse data, poor scalability, and high computational cost in complex system design. This work introduces Interpolating Neural Network (INN)-a network architecture blending interpolation theory and tensor decomposition. INN significantly reduces computational effort and memory requirements while maintaining high accuracy. Thus, it outperforms traditional partial differential equation (PDE) solvers, machine learning (ML) models, and physics-informed neural networks (PINNs). It also efficiently handles sparse data and enables dynamic updates of nonlinear activation. Demonstrated in metal additive manufacturing, INN rapidly constructs an accurate surrogate model of Laser Powder Bed Fusion (L-PBF) heat transfer simulation. It achieves sub-10-micrometer resolution for a 10 mm path in under 15 minutes on a single GPU, which is 5-8 orders of magnitude faster than competing ML models. This offers a new perspective for addressing challenges in computational science and engineering.

In this paper, we propose a time-marching multi-level Variational Multiscale-Tensor Decomposition (VMS-TD) algorithm to solve the heat equation with a moving heat source model that arises from additive manufacturing. First, we take a second-order centered difference for time semi-discretization. The temperature field is decomposed according to multiple space resolution levels, each represented by the TD method. Then we adopt the VMS formulation [T.J.R. Hughes, G.R. Feijoo, L. Mazzei, J.B. Quincy. Comput. Methods Appl. Mech. Engrg. 166:3-24 (1998)] for the resulting elliptic problem to obtain a Galerkin weak form, and design VMS-TD algorithm to solve it. Furthermore, to comply with the TD solution scheme, special inter-scale data transfers are made at the scale interface and moving fine-scale subdomains. Numerical results demonstrate that the multi-level VMS-TD algorithm is much more efficient than the fully resolved TD algorithm, let alone traditional direct numerical simulation methods such as finite difference or finite element analysis. Compared with the well-known multi-grid methods or more recent GO-MELT framework [J.P. Leonor, G.J. Wagner. Comput. Methods Appl. Mech. Engrg, 426:116977 (2024)], the three-level VMS-TD uses much smaller degrees of freedom to reach accurate results. A multi-time-scale extension of VMS-TD algorithm is also proposed.

Software development has entered a new era where large language models (LLMs) now serve as general-purpose reasoning engines, enabling natural language interaction and transformative applications across diverse domains. This paradigm is now extending into computer-aided engineering (CAE). Recent applications of LLMs in CAE have successfully automated routine tasks, including CAD model generation and FEM simulations. Nevertheless, these contributions, which primarily serve to reduce manual labor, are often insufficient for addressing the significant computational challenges posed by large-scale, high-dimensional systems. To this aim, we first introduce the concept of LLM-empowered CAE agent, where LLMs act as autonomous collaborators that plan, execute, and adapt CAE workflows. Then, we propose an LLM-empowered CAE agent for data-free model order reduction (MOR), a powerful yet underused approach for ultra-fast large-scale parametric analysis due to the intrusive nature and labor-intensive redevelopment of solvers. LLMs can alleviate this barrier by automating derivations, code restructuring, and implementation, making intrusive MOR both practical and broadly accessible. To demonstrate feasibility, we present an LLM-empowered CAE agent for solving ultra-large-scale space-parameter-time (S-P-T) physical problems using Tensor-decomposition-based A Priori Surrogates (TAPS). Our results show that natural language prompts describing parametric partial differential equations (PDEs) can be translated into efficient solver implementations, substantially reducing human effort while producing high-fidelity reduced-order models. Moreover, LLMs can synthesize novel MOR solvers for unseen cases such as nonlinear and high-dimensional parametric problems based on their internal knowledge base. This highlights the potential of LLMs to establish the foundation for next-generation CAE systems.

Process‐induced defects are inherent to additively manufactured parts and significantly influence the performance of printed materials. This article introduces a predictive performance modeling framework for defect analysis in fused filament fabrication (FFF)‐printed materials, leveraging a mechanistic data science‐based reduced‐order modeling (ROM) approach. The proposed model addresses process‐induced variabilities by quantifying defects across micro‐ and mesoscales and integrating them into part‐scale performance predictions. Employing mechanistic ROM enables concurrent multiscale modeling of FFF‐printed materials, explicitly accounting for local microstructures and defects. A thermoelastic extension of the mechanistic ROM is introduced to evaluate the thermal residual stresses developed during the FFF process. Applying the mechanistic ROM framework, defects in FFF‐printed polylactic acid (PLA) and PLA with short carbon fiber (PLA/SCF) composites are analyzed to establish a multiscale model for predicting the mechanical performance of tensile and three‐point bending specimens. The results highlight that accurate material performance prediction relies on capturing process‐induced defects, with the mechanistic ROM successfully simulating behavior across both local and part‐scale domains. The proposed defect analysis and modeling approach can be extended to other additive manufacturing processes to offer valuable insights into microstructure‐guided material design.

Data-driven science and computation have advanced immensely to construct complex functional relationships using trainable parameters. However, efficiently discovering interpretable and accurate closed-form expressions from complex dataset remains a challenge. The article presents a novel approach called Explainable Hierarchical Deep Learning Neural Networks or Ex-HiDeNN that uses an accurate, frugal, fast, separable, and scalable neural architecture with symbolic regression to discover closed-form expressions from limited observation. The article presents the two-step Ex-HiDeNN algorithm with a separability checker embedded in it. The accuracy and efficiency of Ex-HiDeNN are tested on several benchmark problems, including discerning a dynamical system from data, and the outcomes are reported. Ex-HiDeNN generally shows outstanding approximation capability in these benchmarks, producing orders of magnitude smaller errors compared to reference data and traditional symbolic regression. Later, Ex-HiDeNN is applied to three engineering applications: a) discovering a closed-form fatigue equation, b) identification of hardness from micro-indentation test data, and c) discovering the expression for the yield surface with data. In every case, Ex-HiDeNN outperformed the reference methods used in the literature. The proposed method is built upon the foundation and published works of the authors on Hierarchical Deep Learning Neural Network (HiDeNN) and Convolutional HiDeNN. The article also provides a clear idea about the current limitations and future extensions of Ex-HiDeNN.

Abstract Additive manufacturing has enabled the fabrication of functionally graded materials (FGMs), such as compositionally graded alloys (CGAs), offering unprecedented flexibility in structural design. CGAs hold significant potential for thermal-elastic applications, yet existing design methods often overlook temperature-dependent material properties due to the complexity of coupled physics, design-dependent temperature fields, and local constraints. To address these challenges, we propose a topology optimization (TO) framework that concurrently designs geometry and graded material composition while accounting for temperature-dependent material behaviors and nonlinear thermal analysis. Our method employs a radial basis function (RBF)-based interpolation scheme to model material properties as functions of both temperature and material composition. Additionally, we leverage automatic differentiation and adjoint sensitivity analysis for computational efficiency and extensibility to GPU acceleration. Numerical examples demonstrate the effectiveness of our approach, underscoring (1) the critical role of temperature-dependent material properties in thermal-elastic structure optimization and (2) the benefits of continuous material grading in enhancing structural performance.

A data-free, predictive scientific AI model, Tensor-decomposition-based A Priori Surrogate (TAPS), is proposed for tackling ultra large-scale engineering simulations with significant speedup, memory savings, and storage gain. TAPS can effectively obtain surrogate models for high-dimensional parametric problems with equivalent zetta-scale ($10^{21}$) degrees of freedom (DoFs). TAPS achieves this by directly obtaining reduced-order models through solving governing equations with multiple independent variables such as spatial coordinates, parameters, and time. The paper first introduces an AI-enhanced finite element-type interpolation function called convolution hierarchical deep-learning neural network (C-HiDeNN) with tensor decomposition (TD). Subsequently, the generalized space-parameter-time Galerkin weak form and the corresponding matrix form are derived. Through the choice of TAPS hyperparameters, an arbitrary convergence rate can be achieved. To show the capabilities of this framework, TAPS is then used to simulate a large-scale additive manufacturing process as an example and achieves around 1,370x speedup, 14.8x memory savings, and 955x storage gain compared to the finite difference method with $3.46$ billion spatial degrees of freedom (DoFs). As a result, the TAPS framework opens a new avenue for many challenging ultra large-scale engineering problems, such as additive manufacturing and integrated circuit design, among others.

Deep learning has been extensively employed as a powerful function approximator for modeling physics-based problems described by partial differential equations (PDEs). Despite their popularity, standard deep learning models often demand prohibitively large computational resources and yield limited accuracy when scaling to large-scale, high-dimensional physical problems. Their black-box nature further hinders the application in industrial problems where interpretability and high precision are critical. To overcome these challenges, this paper introduces Interpolating Neural Network-Tensor Decomposition (INN-TD), a scalable and interpretable framework that has the merits of both machine learning and finite element methods for modeling large-scale physical systems. By integrating locally supported interpolation functions from finite element into the network architecture, INN-TD achieves a sparse learning structure with enhanced accuracy, faster training/solving speed, and reduced memory footprint. This makes it particularly effective for tackling large-scale high-dimensional parametric PDEs in training, solving, and inverse optimization tasks in physical problems where high precision is required.

A common trend in simulation-driven engineering applications is the ever-increasing size and complexity of the problem, where classical numerical methods typically suffer from significant computational time and huge memory cost. Methods based on artificial intelligence have been extensively investigated to accelerate partial differential equations (PDE) solvers using data-driven surrogates. However, most data-driven surrogates require an extremely large amount of training data. In this paper, we propose the Convolutional Hierarchical Deep Learning Neural Network-Tensor Decomposition (C-HiDeNN-TD) method, which can directly obtain surrogate models by solving large-scale space-time PDE without generating any offline training data. We compare the performance of the proposed method against classical numerical methods for extremely large-scale systems.

A seamless integration of neural networks with Isogeometric Analysis (IGA) was first introduced in [1] under the name of Hierarchical Deep-learning Neural Network (HiDeNN) and has systematically evolved into Isogeometric Convolution HiDeNN (in short, C-IGA) [2]. C-IGA achieves higher order approximations without increasing the degree of freedom. Due to the Kronecker delta property of C-IGA shape functions, one can refine the mesh in the physical domain like standard finite element method (FEM) while maintaining the exact geometrical mapping of IGA. In this article, C-IGA theory is generalized for multi-CAD-patch systems with a mathematical investigation of the compatibility conditions at patch interfaces and convergence of error estimates. Two compatibility conditions (nodal compatibility and G^0 (i.e., global C^0) compatibility) are presented and validated through numerical examples.

Artificial intelligence (AI) has revolutionized software development, shifting from task-specific codes (Software 1.0) to neural network-based approaches (Software 2.0). However, applying this transition in engineering software presents challenges, including low surrogate model accuracy, the curse of dimensionality in inverse design, and rising complexity in physical simulations. We introduce an interpolating neural network (INN), grounded in interpolation theory and tensor decomposition, to realize Engineering Software 2.0 by advancing data training, partial differential equation solving, and parameter calibration. INN offers orders of magnitude fewer trainable/solvable parameters for comparable model accuracy than traditional multi-layer perceptron (MLP) or physics-informed neural networks (PINN). Demonstrated in metal additive manufacturing, INN rapidly constructs an accurate surrogate model of Laser Powder Bed Fusion (L-PBF) heat transfer simulation, achieving sub-10-micrometer resolution for a 10 mm path in under 15 minutes on a single GPU. This makes a transformative step forward across all domains essential to engineering software.

A representative volume element (RVE) is a reasonably small unit of microstructure that can be simulated to obtain the same effective properties as the entire microstructure sample. Finite element (FE) simulation of RVEs, as opposed to much larger samples, saves computational expenses, especially in multiscale modeling. Therefore, it is desirable to have a framework that determines the RVE size prior to FE simulations. Existing methods select the RVE size based on when the FE-simulated properties of samples of increasing sizes converge with insignificant statistical variations, with the drawback being that many samples must be simulated. We propose a simulation-free alternative that determines the RVE size based only on a micrograph. The approach utilizes a machine learning model trained to implicitly characterize the stochastic nature of the input micrograph. The underlying rationale is to view RVE size as the smallest moving window size for which the stochastic nature of the microstructure within the window is stationary as the window moves across a large micrograph. For this purpose, we adapt a recently developed Fisher score-based framework for microstructure nonstationarity monitoring. Because the resulting RVE size is based solely on the micrograph and does not involve any FE simulation of specific properties, it constitutes an RVE for any property of interest that solely depends on the microstructure characteristics. Through numerical experiments of simple and complex microstructures, we validate our approach and show that our selected RVE sizes are consistent with when the chosen FE-simulated properties converge.

Abstract 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 A dditive M anufacturing C omputational F luid D ynamics code (AM-CFD) combined with a cylindrical heat source is implemented to accurately predict these experiments. Heuristic heat source calibration is proposed relating volumetric energy density (ψ) based on experiments available in the literature. The parameters of the heat source of the computational model are initially calibrated based on a Higher Order Proper Generalized Decomposition- (HOPGD) based surrogate model. The prediction using the calibrated heat source agrees quantitatively with NIST measurements for different process conditions (laser spot diameter, laser power, and scan speed). A scaling law based on keyhole formation is 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 is proposed to relate the Volumetric Energy Density (VED σ ) to the melt pool aspect ratio. The model shows further improvement in the prediction of the experimental measurements for the melt pool, including cases at higher VED σ . Overall, it is concluded that the appropriate selection of laser heat source parameterization scheme along with the heat source model is crucial in the accurate prediction of melt pool geometry and thermal measurements while bypassing the expensive computational simulations that consider increased physics equations.

A digital twin (DT) is a virtual representation of physical process, products and/or systems that requires a high-fidelity computational model for continuous update through the integration of sensor data and user input. In the context of laser powder bed fusion (LPBF) additive manufacturing, a digital twin of the manufacturing process can offer predictions for the produced parts, diagnostics for manufacturing defects, as well as control capabilities. This paper introduces a parameterized physics-based digital twin (PPB-DT) for the statistical predictions of LPBF metal additive manufacturing process. We accomplish this by creating a high-fidelity computational model that accurately represents the melt pool phenomena and subsequently calibrating and validating it through controlled experiments. In PPB-DT, a mechanistic reduced-order method-driven stochastic calibration process is introduced, which enables the statistical predictions of the melt pool geometries and the identification of defects such as lack-of-fusion porosity and surface roughness, specifically for diagnostic applications. Leveraging data derived from this physics-based model and experiments, we have trained a machine learning-based digital twin (PPB-ML-DT) model for predicting, monitoring, and controlling melt pool geometries. These proposed digital twin models can be employed for predictions, control, optimization, and quality assurance within the LPBF process, ultimately expediting product development and certification in LPBF-based metal additive manufacturing.

The localized stress and strain field simulation results are critical for understanding the mechanical properties of materials, such as strength and toughness. However, applying off-the-shelf machine learning or deep learning methods to a digitized microstructure restricts the image samples to be of a fixed size and also lacks interpretability. Additionally, existing methods that utilize deep learning models to solve boundary value problems require retraining the model for each set of boundary conditions. To address these limitations, we propose a customized Pixel-Wise Convolutional Neural Network (PWCNN) to make fast predictions of stress and strain fields pixel-by-pixel under different loading conditions and for a wide range of composite microstructures of any size (e.g., much larger or smaller than the sample on which the PWCNN is trained). Through numerical experiments, we show that our PWCNN model serves as an alternative approach to numerical solution methods, such as finite element analysis, but is computationally more efficient, and the prediction errors on the test microstructure are around 5%. Moreover, we also propose an interpretable machine learning framework to facilitate the scientific discovery of why certain microstructures have better or worse performance than others, which has important implications in the design of composite microstructures in advanced manufacturing.