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DOE OSTI2025

Efficient shallow Ritz method for 1D diffusion problems

This paper studies the shallow Ritz method for solving the one-dimensional diffusion problem. It is shown that the shallow Ritz method improves the order of approximation dramatically for non-smooth problems. To realize this optimal or nearly optimal order of the shallow Ritz approximation, we develop a damped block Newton (dBN) method that alternates between updates of the linear and non-linear parameters. Per each iteration, the linear and the non-linear parameters are updated by exact inversion and one step of a modified, damped Newton method applied to a reduced non-linear system, respectively. The computational cost of each dBN iteration is $\mathcal{O}$(n). Starting with the non-linear parameters as a uniform partition of the interval, numerical experiments show that the dBN is capable of efficiently moving mesh points to nearly optimal locations. In conclusion, to improve the efficiency of the dBN further, we propose an adaptive damped block Newton (AdBN) method by combining the dBN with the adaptive neuron enhancement (ANE) method [28].

Diffusion problems↗

DOE OSTI2020

Multiscale modeling of microwave-heated multiphase systems

Modeling of microwave heating of multiphase systems requires resolving the electromagnetic field from the single particle to the entire cavity. We introduce a multiscale methodology for computationally affordable simulations of microwave heating of multiphase systems consisting of one phase dispersed in a continuum phase. The methodology homogenizes the original multiphase system, by taking advantage of the large separation of length scales, to calculate the effective permittivity, effective thermal conductivity, and volumetric power absorbed in each phase. The methodology is rigorously assessed against particle-resolved detailed numerical simulations of model systems. We demonstrate a significant reduction in computational cost while retaining the accuracy of detailed simulations. Lastly, the present methodology enables high throughput exploration of novel multiphase system designs utilizing microwave heating.

37 INORGANIC, ORGANIC, PHYSICAL, AND ANALYTICAL CH↗

DOE OSTI2024

Reduced-order condensed-phase kinetic models for polyethylene, polypropylene and polystyrene thermochemical recycling

Thermochemical recycling of plastic waste (PW) into chemicals and energy vectors requires coupling particle and reactor-scale simulations to accurate condensed phase pyrolysis mechanisms for each constituent. This work proposes a methodology to derive reduced-order condensed-phase kinetic models from validated semi-detailed kinetic mechanisms. Two types of kinetic models are obtained for polyethylene (PE), polypropylene (PP) and polystyrene (PS): reduced semi-detailed models and multi-step fully lumped ones. These families offer different compromises between accuracy and computational cost. The former employ 50–100 gas + liquid species and describe both the radical degradation and the detailed carbon distribution of the products. Conversely, the latter involves 5–10 species per polymer tracking only the main petroleum cuts. The kinetic mechanisms are complemented by the definition of thermochemical properties of gas, liquid, and solid-phase species, accounting for phase-transitions through pseudo-chemical reactions. Model validations are performed by comparison with experimental data and the original semi-detailed mechanisms in terms of mass loss, heat fluxes and product distribution profiles. The resulting CHEMKIN-like condensed-phase models are attached as Supplementary Material and as a GitHub repository. Extending the proposed approach to other polymers and coupling it with existing subsets in the CRECK kinetic framework (e.g., biomass, PVC, PET) offers a powerful tool to model thermochemical recycling of PW and biomass/PW mixtures.

kinetics↗

DOE OSTI2020

Bridging particle and reactor scales in the simulation of biomass fast pyrolysis by coupling particle resolved simulation and coarse grained CFD-DEM

Efficient utilization of biomass is challenging due to the complexity of the feedstock at molecular, particle, and reactor scales. Direct coupling of particle scale model and reactor scale simulation is computationally infeasible. In this work, the results from a particle resolved simulation were used to calibrate the heat transfer and reaction kinetics of particles. A hybrid drag model was used to consider the different fluidization properties of sands and biomass particles. The computation cost was reduced using a coarse-grained Discrete Element Method (DEM). The simulated conversion rate, char yield, tar yield, and pyrolysis gas yield compare well with experiment. The residence time and axial distributions of biomass were also analyzed. This novel multi-scale method provides an efficient and accurate tool for the modeling of biomass pyrolysis reactors.

37 INORGANIC, ORGANIC, PHYSICAL, AND ANALYTICAL CH↗

DOE OSTI2025

A simplified CFD approach for modeling mass transport in catalytic open-cell foams

A simplified macroscopic CFD approach is presented to model mass transport including chemical reactions in washcoated open-cell foams. The foam is treated as a porous medium. Species conversion during chemical reactions is modeled using appropriate source terms based on reaction rate expressions and modified to account for the mass transport resistances occurring at the fluid-washcoat interfaces and within the washcoat layers. As example, the catalytic CO oxidation over platinum is studied. The simulation results show good agreement with experimental data from literature. A parametric study on washcoat parameters, such as thickness, tortuosity, porosity, and size, is carried out. Increasing the washcoat thickness from 5 to 100 µm or decreasing the tortuosity to porosity ratio from 5 to 20 decreases the CO conversion by 10 %. The proposed model is found to be reliable and has the advantage of lower computational cost, making it a suitable tool for foam-based catalytic reactor design.

37 INORGANIC, ORGANIC, PHYSICAL, AND ANALYTICAL CH↗

DOE OSTI2023

Linearization errors in discrete goal-oriented error estimation

This paper is concerned with goal-oriented a posteriori error estimation for nonlinear functionals in the context of nonlinear variational problems solved with continuous Galerkin finite element discretizations. A two-level, or discrete, adjoint-based approach for error estimation is considered. The traditional method to derive an error estimate in this context requires linearizing both the nonlinear variational form and the nonlinear functional of interest which introduces linearization errors into the error estimate. In this paper, we investigate these linearization errors. In particular, we develop a novel discrete goal-oriented error estimate that accounts for traditionally neglected nonlinear terms at the expense of greater computational cost. We demonstrate how this error estimate can be used to drive mesh adaptivity. Here, we show that accounting for linearization errors in the error estimate can improve its effectivity for several nonlinear model problems and quantities of interest. We also demonstrate that an adaptive strategy based on the newly proposed estimate can lead to more accurate approximations of the nonlinear functional with fewer degrees of freedom when compared to uniform refinement and traditional adjoint-based approaches.

42 ENGINEERING↗

DOE OSTI2023

A subdivision-stabilized B-spline mixed material point method

Subjected to external loadings, polymeric materials, e.g., biological tissues, hydrogels, and elastomers, may undergo extreme, nearly incompressible, (self-)contact deformations. For numerical modeling employing mesh-based techniques such as the finite element method (FEM), these deformations pose significant challenges due to large distortions in the deformed geometry, accuracy issues stemming from volumetric locking effects, and increased computational cost from complex contact searches. As an alternative to mesh-based methods, the material point method (MPM), a continuum-based particle technique, is gaining attention for its ability to handle extreme distortions and capture no-slip contact without added cost. For nearly incompressible material behaviors, while mixed formulations can address locking effects by treating displacements and pressure as independent fields, they can suffer from numerical instabilities close to the incompressibility limit due to the violation of the inf-sup condition, leading to inaccurate nodal pressure solutions. Here we propose an efficient and stable mixed B-spline material point method with highest achievable regularity for quasi-compressible polymeric materials. Using the two-scale relation of B-splines, we introduce a subdivision-stabilization for the two-field mixed MPM and obtain numerically stable, oscillation-free nodal solutions with equal-order interpolations with optimal regularity. Building on the Eulerian-Lagrangian nature of MPM, a previously-converged solution framework is adopted to mitigate issues related to cell-crossing and numerical fracture artifact present in standard MPM. We assess the stability and accuracy of the developed mixed MPM at large deformations for soft materials through the benchmark Cook’s membrane problem. Additionally, we test the robustness of the proposed MPM by modeling several examples, including the compression and indentation of a circular block into a quasi-compressible substrate and the twisting deformation of a rectangular block. The findings demonstrate the MPM’s capabilities for modeling practical soft material applications.

36 MATERIALS SCIENCE↗

DOE OSTI2024

Divide and conquer: Learning chaotic dynamical systems with multistep penalty neural ordinary differential equations

Forecasting high-dimensional dynamical systems is a fundamental challenge in various fields, such as geosciences and engineering. Neural Ordinary Differential Equations (NODEs), which combine the power of neural networks and numerical solvers, have emerged as a promising algorithm for forecasting complex nonlinear dynamical systems. However, classical techniques used for NODE training are ineffective for learning chaotic dynamical systems. In this work, we propose a novel NODE-training approach that allows for robust learning of chaotic dynamical systems. Here, our method addresses the challenges of non-convexity and exploding gradients associated with underlying chaotic dynamics. Training data trajectories from such systems are split into multiple, non-overlapping time windows. In addition to the deviation from the training data, the optimization loss term further penalizes the discontinuities of the predicted trajectory between the time windows. The window size is selected based on the fastest Lyapunov time scale of the system. Multi-step penalty(MP) method is first demonstrated on Lorenz equation, to illustrate how it improves the loss landscape and thereby accelerates the optimization convergence. MP method can optimize chaotic systems in a manner similar to least-squares shadowing with significantly lower computational costs. Our proposed algorithm, denoted the Multistep Penalty NODE, is applied to chaotic systems such as the Kuramoto-Sivashinsky equation, the two-dimensional Kolmogorov flow, and ERA5 reanalysis data for the atmosphere. It is observed that MP-NODE provide viable performance for such chaotic systems, not only for short-term trajectory predictions but also for invariant statistics that are hallmarks of the chaotic nature of these dynamics.

Chaotic dynamical systems↗

DOE OSTI2025

Generalizable machine learning potentials for quantum-accurate predictions of non-equilibrium behavior in 2D materials

Machine learning interatomic potentials (ML-IAPs) are emerging as transformative tools in materials modeling, promising quantum-level accuracy at a fraction of the computational cost. However, their ability to generalize beyond equilibrium configurations and to reliably capture defect- and temperature-driven behavior remains underexplored. Here, we develop and benchmark two state-of-the-art ML-IAPs, Spectral Neighbor Analysis Potential (SNAP) and Allegro, on a comprehensive dataset for monolayer MoSe₂. Using density functional theory (DFT) as the reference, we evaluate their performance in capturing stress–strain behavior, phase transition energetics, defect evolution, edge stability, and fracture toughness. Allegro, a deep equivariant neural network potential, surpasses both SNAP and the classical Tersoff potential in accuracy, efficiency, and transferability. Importantly, both ML potentials accurately reproduce experimental fracture measurements and ab initio predictions of inversion domain formation—phenomena well beyond their training sets. Our findings establish ML-IAPs as viable replacements for traditional force fields in the study of non-equilibrium mechanical phenomena, enabling large-scale, high-fidelity simulations in 2D materials and beyond. In conclusion, this work provides a broadly applicable framework for data-driven modeling of structural and functional transformations under extreme conditions.

2D materials↗

DOE OSTI2025

Personalized and uncertainty-aware coronary hemodynamics simulations: From Bayesian estimation to improved multi-fidelity uncertainty quantification

Non-invasive simulations of coronary hemodynamics have improved clinical risk stratification and treatment outcomes for coronary artery disease, compared to relying on anatomical imaging alone. However, simulations typically use empirical approaches to distribute total coronary flow amongst the arteries in the coronary tree, which ignores patient variability, the presence of disease, and other clinical factors. Further, uncertainty in the clinical data often remains unaccounted for in the modeling pipeline. We present an end-to-end uncertainty-aware pipeline to (1) personalize coronary flow simulations by incorporating vessel-specific coronary flows as well as cardiac function; and (2) predict clinical and biomechanical quantities of interest with improved precision, while accounting for uncertainty in the clinical data. We assimilate patient-specific measurements of myocardial blood flow from clinical CT myocardial perfusion imaging to estimate branch-specific coronary artery flows. Simulated noise in the clinical data is used to estimate the joint posterior distributions of the model parameters using adaptive Markov Chain Monte Carlo sampling. Additionally, the posterior predictive distribution for the relevant quantities of interest is determined using a new approach combining multi-fidelity Monte Carlo estimation with non-linear, data-driven dimensionality reduction. This leads to improved correlations between high- and low-fidelity model outputs. Our framework accurately recapitulates clinically measured cardiac function as well as branch-specific coronary flows under measurement noise uncertainty. We observe substantial reductions in confidence intervals for estimated quantities of interest compared to single-fidelity Monte Carlo estimation and state-of-the-art multi-fidelity Monte Carlo methods. This holds especially true for quantities of interest that showed limited correlation between the low- and high-fidelity model predictions. In addition, the proposed multi-fidelity Monte Carlo estimators are significantly cheaper to compute than traditional estimators, under a specified confidence level or variance. The proposed pipeline for personalized and uncertainty-aware predictions of coronary hemodynamics is based on routine clinical measurements and recently developed techniques for CT myocardial perfusion imaging. The proposed pipeline offers significant improvements in precision and reduction in computational cost.

Bayesian parameter estimation↗

DOE OSTI2021

A two-equation soot-in-flamelet modeling approach applied under Spray A conditions

Soot production (including formation and oxidation) is studied in the transient, high-pressure and turbulent n-dodecane Spray A flames from the Engine Combustion Network (ECN) using computational fluid dynamics (CFD) simulations. A two-equation soot-in-flamelet modeling approach is applied within the framework of the Unsteady Flamelet Progress Variable (UFPV) model and results are validated against experimental data. Equations for soot mass fraction and soot number density derived in the mixture fraction space are solved in the context of detailed flamelet calculations. Source terms for the different steps in the soot chemistry are tabulated and incorporated in the flamelet manifold. For the reference condition, the modeling approach based on the tabulated flamelet manifold reduces the computational cost of a CFD calculation by approximately 40 times compared to a non-tabulated well-mixed (WM) modeling approach. The soot-in-flamelet approach is then extended to study the effect of ambient oxygen concentration, ambient mixture composition and ambient temperature on soot production. Results show that the modeling approach is able to capture the experimental trends for the soot volume fraction (SVF) with good quantitative agreement, especially in the soot ramp-up region.

37 INORGANIC, ORGANIC, PHYSICAL, AND ANALYTICAL CH↗

DOE OSTI2023

A co-kurtosis PCA based dimensionality reduction with nonlinear reconstruction using neural networks

For turbulent reacting flow systems, identification of low-dimensional representations of the thermo-chemical state space is vitally important, primarily to significantly reduce the computational cost of device-scale simulations. Principal component analysis (PCA), and its variants, are a widely employed class of methods. Recently, an alternative technique that focuses on higher-order statistical interactions, co-kurtosis PCA (CoK-PCA), has been shown to effectively provide a low-dimensional representation by capturing the stiff chemical dynamics associated with spatiotemporally localized reaction zones. While its effectiveness has only been demonstrated based on a priori analyses with linear reconstruction, in this work, we employ nonlinear techniques to reconstruct the full thermo-chemical state and evaluate the efficacy of CoK-PCA compared to PCA. Specifically, we combine a CoK-PCA-/PCA-based dimensionality reduction (encoding) with an artificial neural network (ANN) based reconstruction (decoding) and examine, a priori, the reconstruction errors of the thermo-chemical state. In addition, we evaluate the errors in species production rates and heat release rates, which are nonlinear functions of the reconstructed state, as a measure of the overall accuracy of the dimensionality reduction technique. We employ four datasets to assess CoK-PCA/PCA coupled with ANN-based reconstruction: zero-dimensional (homogeneous) reactor for autoignition of an ethylene/air mixture that has conventional single-stage ignition kinetics, a dimethyl ether (DME)/air mixture which has two-stage (low and high temperature) ignition kinetics, a one-dimensional freely propagating premixed ethylene/air laminar flame, and a two-dimensional dataset representing turbulent autoignition of ethanol in a homogeneous charge compression ignition (HCCI) engine. Finally, results from the analyses demonstrate the robustness of the CoK-PCA based low-dimensional manifold with ANN reconstruction in accurately capturing the data, specifically from the reaction zones.

37 INORGANIC, ORGANIC, PHYSICAL, AND ANALYTICAL CH↗

DOE OSTI2026

Time-dependent-bases with local CUR decomposition method for accelerating turbulent combustion simulations

Here, this study presents a novel reduced-order modeling framework, Time-Dependent Bases with Local CUR decomposition (TDB-L-CUR), designed to efficiently and accurately approximate the species transport equations in reacting flow simulations. The method extends the existing TDB-CUR approach for chemically reacting flows (Jung et al. Comput. Methods Appl. Mech. Engrg. 437 (2025) 117758), which leverages matrix decomposition techniques to form a global-in-space, time-dependent low-dimensional manifold. While TDB-CUR performs well in homogeneous systems, it may be less well-suited to spatially heterogeneous systems such as turbulent flames, where higher-rank approximations are typically required. The proposed TDB-L-CUR framework introduces two methodological extensions to the baseline approach. First, it applies unsupervised clustering to partition the physical domain into distinct regions, enabling spatially localized manifold construction, thereby reducing the rank required for the reduced-order representation. Second, it incorporates a computational singular perturbation (CSP)-based scheme for identifying and penalizing fast species, allowing for spatio-temporally adaptive mitigation of chemical stiffness. The proposed framework is validated on a hierarchy of test cases, including a one-dimensional premixed flame, a two-dimensional nonpremixed ignition case with vortex interaction, and a three-dimensional turbulent premixed flame. TDB-L-CUR significantly improves accuracy over TDB-CUR while further reducing computational cost. The fully on-the-fly formulation of TDB-L-CUR (i.e., requiring no offline training or prior knowledge) makes it a robust and scalable tool for reduced-order modeling of reactive flows.

Local manifold↗

DOE OSTI2020

An automated cluster surface scanning method for exploring reaction paths on metal-cluster surfaces

Metal-cluster surfaces present a wide variety of unique coordination environments. This complexity makes it difficult to manually probe the surface reactivity of such clusters. Here, we present a simple and automated method to systematically discover reaction pathways on cluster surfaces, based on the automated cluster surface scanning (ACSS) technique for mapping out potential energy surfaces. We showcase our method on 55-atom icosahedral Cu and Ag clusters, where we determine the activation energies of four elementary steps common in heterogeneous catalysis – hydrogen recombination (H* + H* → H 2 * + *), oxygen recombination (O* + O* → O 2 * + *), water formation (OH* + H* → H 2 O* + *), and CO oxidation (CO* + O* → CO 2 * + *) – with density functional theory calculations (DFT-PBE + D3). We show that the ACSS method requires significantly less human effort than the established manually performed climbing-image nudged elastic band (MP + CI-NEB) technique and locates transition states with comparable accuracy (root-mean-squared error of 0.10 eV) and similar computational cost. Rigorous sampling of the potential energy surface with the ACSS method allows one to locate all lowest-energy reaction pathways obtained via the MP+CI-NEB approach, as well as alternative pathways that one may have missed with the MP+CI-NEB approach due to the many possible pathways available on these clusters. The accuracy and efficiency afforded by the ACSS method could enable high-throughput exploration of the diverse reactivity of metal clusters.

36 MATERIALS SCIENCE↗

DOE OSTI2020

Rapid generation of optimal generalized Monkhorst-Pack grids

In this report computational modeling of the properties of crystalline materials has become an increasingly important aspect of materials research, consuming hundreds of millions of CPU-hours at scientific computing centers around the world each year, if not more. A routine operation in such calculations is the evaluation of integrals over the Brillouin zone. We have previously demonstrated that performing such integrals using generalized Monkhorst-Pack k-point grids can roughly double the speed of these calculations relative to the widely-used traditional Monkhorst-Pack grids. However the generation of optimal generalized Monkhorst-Pack grids is not implemented in most software packages due to the computational cost and difficulty of identifying the best grids. To address this problem, we present new algorithms that allow rapid generation of optimal generalized Monkhorst-Pack grids on the fly. We demonstrate that the grids generated by these algorithms are on average significantly more efficient than those generated using existing algorithms across a range of grid densities. For grids that correspond to a real-space supercell with at least 50 Å between lattice points, which is sufficient to converge density functional theory calculations within 1 meV/atom for nearly all materials, our algorithm finds optimized grids in an average of 0.19 s on a single processing core. To facilitate the widespread adoption of this approach, we present new open-source tools including a library designed for integration with third-party software packages.

36 MATERIALS SCIENCE↗

DOE OSTI2021

Saddle point sampling using scaled normal coordinates

The sampling of saddle points (SPs) on a potential energy surface (PES) is critical for describing the dynamics and transport properties of materials. In this paper we propose a saddle point search (SPS) method that enables efficient sampling of practically meaningful SPs around a local minimum by converting the PES to the scaled normal coordinates (SNCs), together with the dimer method. We find that the pathway leading to a SP using SNCs is highly efficient and is independent of the system size. This results in the significantly increased SPS efficiency and the enhanced possibility of a complete catalog of the SPs. In addition, we perform SPSs using the SNCs for the diffusion of a vacancy and a dumbbell in body-centered cubic Fe and demonstrate (1) the required number of system force evaluations for a SPS decreases by at least an order of magnitude, (2) the effects of the system size on the number of force evaluations are greatly decreased, and (3) the effects of the number of atoms involved in the SPS on the probability of finding practically meaningful SPs is mostly eliminated. We also discuss the computational cost of introducing the SNCs. It should be highlighted that the SNCs could also be applied to other minimum-mode following methods, demonstrating the general versatility of the proposed method.

36 MATERIALS SCIENCE↗

DOE OSTI2024

Interpretable, extensible linear and symbolic regression models for charge density prediction using a hierarchy of many-body correlation descriptors

Here, density functional theory (DFT) is routinely used to make electronic structure predictions for high-throughput screening of materials and molecules for technologically relevant areas, like the identification of better catalysts, electronic materials, and drug discovery. However, the DFT formalism is limited by (a) its poor (quadratic-to-quartic) scaling, and (b) the need to perform repeated eigenvalue computations of the electronic Hamiltonian as part of its self-consistent field (SCF) iteration procedure to obtain the converged ground state electron density, ρ (r). Approaches that directly predict ρ (r) of a structure with high accuracy can accelerate conventional SCF calculations and can also be used in linearly scaling methods such as orbital-free DFT. To this end, we present a procedure to predict the ground state electron density of molecular and periodic three-dimensional systems directly from the atomic structure with a particular emphasis on physical interpretability. In our framework, ρ (r) is modeled using many-body correlation descriptors that accurately capture the effects of local atomic arrangements in the neighborhood of a grid point. Our use of a linear regression scheme to fit to charge density data enables transparent analysis of the relative contributions of various types of local atomic correlations. By systematically including increasingly complex correlations, our model is shown to accurately predict ρ (r) for a variety of chemically and electronically diverse systems — amorphous Ge, Al(001) slab, crystalline Ga 2 O 3 , molecular benzene, and polyethylene. We then demonstrate a symbolic regression-based protocol to construct easily computable, interpretable features from lower-order correlations that significantly improves our electron density predictions with effectively no increase in the computational cost.

71 CLASSICAL AND QUANTUM MECHANICS, GENERAL PHYSIC↗

DOE OSTI2025

On the numerical sensitivity of cellular automata grain structure predictions to large thermal gradients and cooling rates

Cellular automata (CA) models of as-solidified grain structure, originally developed and applied to casting, have become a common means of predicting grain structure resulting from Additive Manufacturing (AM) processes. The majority of these models are based on the decentered octahedron approach, which attempts to correct for the effect of grid anisotropy on the prediction of competitive solidification of dendritic grains. However, AM solidification occurs under cooling rates ($\dot{T}$) and thermal gradients (G) that are orders of magnitude larger than those encountered in casting, and no systematic investigation on the effect of the CA model cell size (Δx) and time step (Δt) on AM microstructure predictions has been performed. Here, in this study, such an investigation is first performed via simulation of individual grains of various crystallographic orientations with a fixed, unidirectional G, showing that CA prediction of the steady-state undercooling matched the expected values based on the interfacial response function at small G and deviated from the expected values at large G. Simulation of competitive growth of multiple grains showed a weakening of the predicted texture as G and Δx became large. Simulation of solidification under AM conditions, where G and $\dot{T}$ vary spatially across the melt pools, showed that not only does grain selection weaken and deviate from expectations at large Δx, but grains with crystallographic $\langle$100$\rangle$ aligned with the grid directions are more adversely affected by the temperature field discontinuities than grains with other crystallographic orientations. Despite the fact that the exact grain competition results depended on Δt, the overall texture development was notably less sensitive to Δt than Δx, provided that a reasonable value of Δt is selected based on the ratio of Δx to the maximum local solidification velocity in the simulation domain. Finally, from the directional solidification and AM simulation results, an analysis of computational cost compared to simulation resolution is performed based on an equation derived to quantify the relatively inaccuracy in grain selection based on the model and temperature field inputs. From this analysis, it is concluded that there is a need for algorithmic improvements to improve CA grain competition accuracy for large G processing conditions as sufficiently small Δx to resolve the necessary competition is intractable for many AM processing conditions.

36 MATERIALS SCIENCE↗