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DOE OSTI2023

Data-driven Whitney forms for structure-preserving control volume analysis

Control volume analysis models physics via the exchange of generalized fluxes between subdomains. Here, we introduce a scientific machine learning framework adopting a partition of unity architecture to identify physically-relevant control volumes, with generalized fluxes between subdomains encoded via Whitney forms. The approach provides a differentiable parameterization of geometry which may be trained in an end-to-end fashion to extract reduced models from full field data while exactly preserving physics. The architecture admits a data-driven finite element exterior calculus allowing discovery of mixed finite element spaces with closed form quadrature rules. An equivalence between Whitney forms and graph networks reveals that the geometric problem of control volume learning is equivalent to an unsupervised graph discovery problem. The framework is developed for manifolds in arbitrary dimension, with examples provided for H(div) problems in $\mathbb{R}$ establishing convergence and structure preservation properties. Finally, we consider a lithium-ion battery problem where we discover a reduced finite element space encoding transport pathways from high-fidelity microstructure resolved simulations. The approach reduces the 5.89M finite element simulation to 136 elements while reproducing pressure to under 0.1% error and preserving conservation.

97 MATHEMATICS AND COMPUTING↗

DOE OSTI2022

Kinetics of particles with short-range interactions

Self-assembly is one of the grand challenges of the 21st century – as the devices and materials we would like to build become too complex or small-scale for top-down manufacturing to be efficient, it is increasingly important to find ways to create these through bottom-up, dynamical approaches. Many particles used in self-assembly have very short-ranged attractive interactions, making simulations expensive or impossible. This proposal develops a set of conceptual and computational tools to study the dynamics of self-assembly for particles with short-ranged interactions, harnessing ideas in differential and computational geometry, and stochastic analysis, to accelerate simulations.

74 ATOMIC AND MOLECULAR PHYSICS↗

DOE OSTI2022

Temperature distribution in a laser-heated diamond anvil cell as described by finite element analysis

Finite element analysis (FEA) is a powerful tool for numerically solving partial differential equations over complex geometries and is thus useful for analyzing heat transport in laser-heated diamond anvil cell (LHDAC) experiments. Our models expand on previously published simulations by calculating the volume-averaged temperatures of both the sample and insulation/pressure media under steady-state heating to determine the thermal pressure of the hot sample. Our goal is to produce an accurate relationship between the measured surface temperature of the absorbing sample and the temperature of the transparent insulating media, which is used to determine thermal pressure but susceptible to steep temperature gradients. We find that in doing so, our FEA models of temperature within the pressure/insulation media can differ from simplified estimates of temperature gradients by more than a factor of 2. We also explore temperature-dependent and temperature-independent thermal conductivity models and find that the volume-averaged temperatures differ by up to a factor of 1.3, forcing the predicted thermal pressures determined to also differ by up to a factor of 1.5 at a temperature of 2000 K at 50 GPa for neon. Higher temperatures exacerbate this difference. We also find that unintentional asymmetric sample insertion and sample heating, which are common in LHDAC experiments, do not have a first-order effect on volume-averaged temperatures. The FEA models, available in both Python and FlexPDE, are versatile across different sample geometries, materials, and heat source laser shapes.

Farah, Frederick↗

DOE OSTI2026

Programmable Phase Selection between Altermagnetic and Noncentrosymmetric Polymorphs of MnTe on InP via Molecular Beam Epitaxy

This dataset contains DFT input and output files supporting the theoretical modeling in the associated publication (ACS Appl. Mater. Interfaces 2026, 18, 15654-15664). The calculations model the interfacial energetics of two MnTe polymorphs — NiAs-MnTe (hexagonal, alpha phase) and ZnS-MnTe (cubic, gamma phase) — on InP(111) substrates with two surface terminations: In-terminated InP(111)A and P-terminated InP(111)B. This gives four interface configurations: NiAs on In-terminated (experimentally observed), NiAs on P-terminated (computed for comparison), ZnS on In-terminated (computed for comparison), and ZnS on P-terminated (experimentally observed). The dataset is organized into four calculation types, each covering all four polymorph/termination combinations: (i) Slabs: Pristine MnTe/InP heterostructure slabs used to compute total energies and interface energy densities (Eint) for all four configurations, as reported in Fig. 6 of the main text. (ii) Disorder: Same slab geometries with a P_Te + Te_P antisite defect pair introduced near the interface, used to assess chemical intermixing effects on interface stability (Fig. S8, SI). (iii) Strain: Pristine slab calculations with in-plane lattice parameters strained by -1% and +1% relative to the InP lattice constant, used to evaluate strain-dependent interface energetics (Fig. S9, SI). (iv) Charge_Density: Single-point calculations on the full heterostructure, the isolated InP slab, and the isolated MnTe slab at fixed geometry, used to compute differential charge density plots showing interfacial charge accumulation and depletion as a function of surface termination (Fig. S10, SI). Each calculation folder contains INCAR, KPOINTS, POSCAR, CONTCAR, OUTCAR, and POTCAR_info.txt (PAW potential information, excluding the full POTCAR due to VASP licensing restrictions). The calculations were performed using VASP 6.4.3 with PBE exchange-correlation, PAW potentials, a Hubbard correction of Ueff = 5 eV on Mn d-states, and A-type AFM spin initialization.

36 MATERIALS SCIENCE↗

DOE OSTI2023

Four lectures on Euler integrals

These lecture notes provide a self-contained introduction to Euler integrals, which are frequently encountered in applications. In particle physics, they arise as Feynman integrals or string amplitudes. Our four selected topics demonstrate the diverse mathematical techniques involved in the study of Euler integrals, including polyhedral geometry, very affine varieties, differential equations, and computational algebra.

Matsubara-Heo, Saiei-Jaeyeong↗

DOE OSTI2021

Challenge Solutions for a Significant Water Cut Reduction History Match on Heavy Oil Polymer EOR

Milne Point Field is conducting the first-ever active polymer flood pilot program designed to enhance oil recovery (EOR) of heavy oils on the Alaska North Slope (ANS). The experimental field program consists of two pairs of horizontal injection and production wells deployed in an isolated fault block in the Schrader Bluff reservoir. The EOR method combines polymer flooding and low salinity water flooding to improve recovery. Prior average water cut reduction in polymer EOR fields ordinarily ranges between 10% to 15% for most light oil and heavy oil reservoirs. When comparing other field applications, the long-lasting (24 months) and much lower water cut poses a challenge for numerical history matching when applying typical simulation methods. This abstract discusses two practical new approaches used for history matching of the long-lasting ultralow water cut. These new applications include 1) multiple relative permeability curves with J-function incorporation with low endpoints of relative permeability of water (Krw). This approach integrated the permeability differential adjustment in the geometries of horizontal and vertical directions based on geological review. 2) A specific permeability channel strips along with the horizontal or vertical planar assumption. The current history matching results indicate success in achieving an excellent agreement with the field observed low water-cut.

Namie, S.↗

DOE OSTI2022

Thermal model for time-domain thermoreflectance experiments in a laser-flash geometry

Time-domain thermoreflectance (TDTR) is a well-established pump–probe method for measuring thermal conductivity and interface conductance of multilayers. Interpreting signals in a TDTR experiment requires a thermal model. In standard front/front TDTR experiments, both pump and probe beams typically irradiate the surface of a multilayer. As a result, existing thermal models for interpreting thermoreflectance experiments assume that the pump and probe beams both interact with the surface layer. Here, we present a frequency-domain solution to the heat-diffusion equation of a multilayer in response to nonhomogeneous laser heating. This model allows analysis of experiments where the pump and probe beams irradiate opposite sides of a multilayer. We call such a geometry a front/back experiment to differentiate such experiments from standard TDTR experiments. As an example, we consider a 60nm amorphous Si film. We consider how signals differ in a front/front vs front/back geometry and compare thermal model predictions to experimental data.

Peng, Wanyue↗

DOE OSTI2025

Morphology Effects on Free Energies of Proton-Coupled Electron Transfer in Polyoxotungstates

Polyoxotungstates have previously been established to facilitate the hydrogenation of small molecule substrates via hydrogen atom transfer from reactive hydroxyl groups formed at the assembly surface. Understanding structure−function relationships that dictate the thermochemistry and kinetics of protoncoupled electron transfer is key to controlling this chemistry. In this work, we combine comprehensive electrochemical experiments and density functional theory calculations to address how different polyoxotungstate morphologies, specifically W 6 O 19 −2 , W 10 O 32 −4 , SiW 12 O 40 −4 , and P 2 W 18 O 62 −6 , affect the bond dissociation free energies of surface hydroxides (BDFE(O−H)) formed upon reduction of the assembly in acidic media. Our results reveal increasing hydroxide bond strengths with increasing cluster size, and that anisotropic cluster geometries result in substantial thermodynamic differentiation of H-binding sites. We demonstrate an excellent agreement between theory and experiments on the reported BDFE(O−H) values and, importantly, we elucidate how cluster size and shape affect electronic properties (local charges and frontier molecular orbitals), giving rise to sites with increased preference for hydrogen binding, demonstrated in higher BDFE(O−H). Overall, this work aids the understanding and design of polyoxometalates exhibiting surface sites with tailored interaction strengths.

anions↗

DOE OSTI2024

Preserving Superconvergence of Spectral Elements for Curved Domains [Slides]

Finite Element Methods (FEM) and Spectral Element Methods (SEM) are crucial for solving partial differential equations (PDEs) on complex geometries. SEM offers superior accuracy due to potential superconvergence for simple domains. Challenges persist for domains with curved boundaries, restricting SEM’s advantages in real-world applications. A proposed solution is the introduction of a novel strategy to enhance accuracy and maintain superconvergence of SEM in curved domains. The strategy includes a mesh-generation procedure with geometrically refined elements near curved boundaries and a post-processing phase using the Adaptive Extended Stencil Finite Element Method (AES-FEM). The method, named AES-FEM post-processed Spectral Element Method (ApSEM), aligns the accuracy of non-tensor-product elements with superconvergent spectral elements.

97 MATHEMATICS AND COMPUTING↗

DOE OSTI2024

On the Geometry of the Near-core Magnetic Field in Massive Stars

It is well-known that the cores of massive stars sustain a stellar dynamo with a complex magnetic field configuration. However, the same cannot be said for the field's strength and geometry at the convective–radiative boundary, which are crucial when performing asteroseismic inference. In this Letter, we present 3D magnetohydrodynamic (MHD) simulations of a 7 M ⊙ mid-main-sequence star, with particular attention given to the convective–radiative boundary in the near-core region. Our simulations reveal that the toroidal magnetic field is significantly stronger than the poloidal field in this region, contrary to recent assumptions. Moreover, the rotational shear layer, also important for asteroseismic inference, is specifically confined within the extent of the Brunt–Väisälä frequency peak. These results, which are based on the inferred properties of HD 43317, have widespread implications for asteroseismic studies of rotation, mixing, and magnetism in stars. While we expect our results to be broadly applicable across stars with similar Brunt–Väisälä frequency profiles and stellar masses, we also expect the MHD parameters (e.g., Re m ) and the initial stellar rotation rate to impact the geometry of the field and differential rotation at the convective–radiative interface.

79 ASTRONOMY AND ASTROPHYSICS↗

DOE OSTI2024

Hadamard products and BPS networks

We study examples of fourth-order Picard-Fuchs operators that are Hadamard products of two second-order Picard-Fuchs operators. Each second-order Picard-Fuchs operator is associated with a family of elliptic curves, and the Hadamard product computes period integrals on the fibred product of the two elliptic surfaces. We construct 3-cycles on this geometry as the union of 2-cycles in the fibre over contours on the base. We then use the special Lagrangian condition to constrain the contours on the base. This leads to a construction that is reminiscent of spectral networks and exponential networks that have previously appeared in string theory literature.

72 PHYSICS OF ELEMENTARY PARTICLES AND FIELDS↗

DOE OSTI2026

High-Order Mesh r-Adaptivity with Tangential Relaxation and Guaranteed Mesh Validity

High-order meshes are crucial for achieving optimal convergence rates in curvilinear domains, preserving symmetry, and aligning with key flow features in moving mesh simulations [1], but their quality is challenging to control. In prior work, we have developed techniques based on Target-Matrix Optimization Paradigm (TMOP) to adapt a given high-order mesh to the geometry and solution of the partial differential equation (PDE) [2, 3]. Here, we extend this framework to address two key gaps in the literature for highorder mesh 𝑟-adaptivity. First, we introduce tangential relaxation on curved surfaces using solely the discrete mesh representation, eliminating the need for access to underlying geometry (e.g., CAD model). Second, we ensure a continuously positive Jacobian determinant throughout the domain. This determinant positivity is essential for using the high-order mesh resulting from 𝑟-adaptivity with arbitrary quadrature schemes in simulations. The proposed approach is demonstrated to be robust using a variety of numerical experiments.

Mathematics and Computing↗

DOE OSTI2020

Open associahedra and scattering forms

We continue the study of open associahedra associated with bi-color scattering amplitudes initiated in ref. [1]. We focus on the facet geometries of the open associahedra, uncovering many new phenomena such as fiber-product geometries. We then provide novel recursion procedures for calculating the canonical form of open associahedra, generalizing recursion relations for bounded polytopes to unbounded polytopes.

72 PHYSICS OF ELEMENTARY PARTICLES AND FIELDS↗

DOE OSTI2024

Cosmology meets cohomology

The cosmological polytope and bootstrap programs have revealed interesting connections between positive geometries, modern on-shell methods and bootstrap principles studied in the amplitudes community with the wavefunction of the Universe in toy models of FRW cosmologies. To compute these FRW correlators, one often faces integrals that are too difficult to evaluate by direct integration. Borrowing from the Feynman integral community, the method of (canonical) differential equations provides an efficient alternative for evaluating these integrals. Moreover, we further develop our geometric understanding of these integrals by describing the associated relative twisted cohomology. Leveraging recent progress in our understanding of relative twisted cohomology in the Feynman integral community, we give an algorithm to predict the basis size and simplify the computation of the differential equations satisfied by FRW correlators.

72 PHYSICS OF ELEMENTARY PARTICLES AND FIELDS↗

DOE OSTI2021

Generalising G 2 geometry: involutivity, moment maps and moduli

We analyse the geometry of generic Minkowski N = 1, D = 4 flux compactifications in string theory, the default backgrounds for string model building. In M-theory they are the natural string theoretic extensions of G2 holonomy manifolds. In type II theories, they extend the notion of Calabi-Yau geometry and include the class of flux backgrounds based on generalised complex structures first considered by Graña et al. (GMPT). Using E 7(7) × $\mathbb{R}$ + generalised geometry we show that these compactifications are characterised by an SU(7) ⊂ E 7(7) structure defining an involutive subbundle of the generalised tangent space, and with a vanishing moment map, corresponding to the action of the diffeomorphism and gauge symmetries of the theory. The Kähler potential on the space of structures defines a natural extension of Hitchin’s G 2 functional. Using this framework we are able to count, for the first time, the massless scalar moduli of GMPT solutions in terms of generalised geometry cohomology groups. It also provides an intriguing new perspective on the existence of G 2 manifolds, suggesting possible connections to Geometrical Invariant Theory and stability.

71 CLASSICAL AND QUANTUM MECHANICS, GENERAL PHYSIC↗

DOE OSTI2020

Integrable deformation of $\mathbb{C}$P n and generalised Kähler geometry

We build on the results of [1] for generalised frame fields on generalised quotient spaces and study integrable deformations for CP n . In particular we show how, when the target space of the Principal Chiral Model is a complex projective space, a two-parameter deformation can be introduced in principle. The second parameter can however be removed via a diffeomorphism, which we construct explicitly, in accordance with the results stemming from a thorough integrability analysis we carry out. We also elucidate how the deformed target space can be seen as an instance of generalised Kähler, or equivalently bi-Hermitian, geometry. In this respect, we find the generic form of the pure spinors for CP n and the explicit expression for the generalised Kähler potential for n = 1, 2.

71 CLASSICAL AND QUANTUM MECHANICS, GENERAL PHYSIC↗

DOE OSTI2025

cymyc: $\underline{C}$alabi-$\underline{Y}$au $\underline{M}$etrics, $\underline{Y}$ukawas, and $\underline{C}$urvature

We introduce cymyc, a high-performance Python library for numerical investigation of the geometry of a large class of string compactification manifolds and their associated moduli spaces. We develop a well-defined geometric ansatz to numerically model tensor fields of arbitrary degree on a large class of Calabi-Yau manifolds. cymyc includes a machine learning component which incorporates this ansatz to model tensor fields of interest on these spaces by finding an approximate solution to the system of partial differential equations they should satisfy.

differential and algebraic geometry↗

DOE OSTI2022

Analyses of internal structures and defects in materials using physics-informed neural networks

Characterizing internal structures and defects in materials is a challenging task, often requiring solutions to inverse problems with unknown topology, geometry, material properties, and nonlinear deformation. Here, we present a general framework based on physics-informed neural networks for identifying unknown geometric and material parameters. By using a mesh-free method, we parameterize the geometry of the material using a differentiable and trainable method that can identify multiple structural features. We validate this approach for materials with internal voids/inclusions using constitutive models that encompass the spectrum of linear elasticity, hyperelasticity, and plasticity. We predict the size, shape, and location of the internal void/inclusion as well as the elastic modulus of the inclusion. Our general framework can be applied to other inverse problems in different applications that involve unknown material properties and highly deformable geometries, targeting material characterization, quality assurance, and structural design.

36 MATERIALS SCIENCE↗