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All sourcesNASA NTRSDOE OSTI
DOE OSTI2022

A unified framework of stabilized finite element method for solving the Boltzmann transport equation

This paper presents a unified framework of stabilized finite element method for solving the Boltzmann transport equation. Unlike the traditional Petrov-Galerkin finite element method which modifies the test function to construct the stabilization term, we derive the stabilization methods from the standard Galerkin weak form with Sub-grid scale model. The basic idea of this method is to decompose the unknowns into its numerical solution and residual, with an approximation for the residual and embeds it in the Galerkin weak form to yield a stabilized variational formula. Different approximations of the residual lead to different stabilization methods, all the frequently used stabilized methods, including the Streamline Upwinding Petrov-Galerkin (SUPG) method, Galerkin/Least-Square (GLS) method, and Algebraic Sub-Grid Scale (ASGS) method can be obtained from this framework. The similarities and differences of the different approximations are compared in this paper. The numerical results show that the behaviors of the different methods area similar with the same stabilization parameter, and all these stabilized techniques can obtain a correct and stable solution. (authors)

22 GENERAL STUDIES OF NUCLEAR REACTORS↗
DOE OSTI2021

A finite element formulation for deformation twinning induced strain localization in polycrystal magnesium alloys

Deformation twinning induces shear strain localization in hexagonal close-packed crystals and is critical for the material’s ductility and failure. Cracks often occur at twin-twin or twin-grain boundary intersections and propagate along twin bands. However, most crystal plasticity models for deformation twinning are based on a “pseudo-slip” approach and do not capture the localized deformation associated with the formation of each discrete twin band. The few exceptions are discrete twin models that involve very complex numerical algorithms and are often compromised in accuracy due to the numerical convergence. These factors make the discrete twin models hard to adopt. This paper proposes a modification to the conventional finite element weak form, to fully incorporate a twin-induced heterogeneous deformation that does not depend on the “pseudo-slip” assumption. The model starts by splitting the deformation gradient into elastic-slip-twinning components. The twin-induced deformation gradient component is computed separately by solving a microstructural evolution problem and then implemented into finite element weak form by constructing a global “twin-force” vector. The constitutive update (e.g., in the user-defined material subroutine, or UMAT, for ABAQUS) therefore avoids dealing with the twinning and recovers to the form of a regular slip-based crystal plasticity model. The results presented here indicate that the twin-induced strain localization and the associated stress-reversal phenomena near the twin band were naturally captured in the model, which was validated against an in-situ synchrotron X-ray micro-diffraction experiment.

36 MATERIALS SCIENCE↗
DOE OSTI2024

Coarse-graining Hamiltonian systems using WSINDy

Abstract Weak form equation learning and surrogate modeling has proven to be computationally efficient and robust to measurement noise in a wide range of applications including ODE, PDE, and SDE discovery, as well as in coarse-graining applications, such as homogenization and mean-field descriptions of interacting particle systems. In this work we extend this coarse-graining capability to the setting of Hamiltonian dynamics which possess approximate symmetries associated with timescale separation. A smooth $$\varepsilon$$ ε -dependent Hamiltonian vector field $$X_\varepsilon$$ X ε possesses an approximate symmetry if the limiting vector field $$X_0=\lim _{\varepsilon \rightarrow 0}X_\varepsilon$$ X 0 = lim ε → 0 X ε possesses an exact symmetry. Such approximate symmetries often lead to the existence of a Hamiltonian system of reduced dimension that may be used to efficiently capture the dynamics of the symmetry-invariant dependent variables. Deriving such reduced systems, or approximating them numerically, is an ongoing challenge. We demonstrate that WSINDy can successfully identify this reduced Hamiltonian system in the presence of large perturbations imparted in the $$\varepsilon >0$$ ε > 0 regime, while remaining robust to extrinsic noise. This is significant in part due to the nontrivial means by which such systems are derived analytically. WSINDy naturally preserves the Hamiltonian structure by restricting to a trial basis of Hamiltonian vector fields. The methodology is computationally efficient, often requiring only a single trajectory to learn the global reduced Hamiltonian, and avoiding forward solves in the learning process. In this way, we argue that weak-form equation learning is particularly well-suited for Hamiltonian coarse-graining. Using nearly-periodic Hamiltonian systems as a prototypical class of systems with approximate symmetries, we show that WSINDy robustly identifies the correct leading-order system, with dimension reduced by at least two, upon observation of the relevant degrees of freedom. While our main contribution is computational, we also provide a contribution to the literature on averaging theory by proving that first-order averaging at the level of vector fields preserves Hamiltonian structure in nearly-periodic Hamiltonian systems. This provides theoretical justification for our approach as WSINDy’s computations occur at the level of Hamiltonian vector fields. We illustrate the efficacy of our proposed method using physically relevant examples, including coupled oscillator dynamics, the Hénon–Heiles system for stellar motion within a galaxy, and the dynamics of charged particles.

97 MATHEMATICS AND COMPUTING↗
DOE OSTI2026

Weak-charge form-factor determination at the electron-ion collider

Determining the weak charge form factor, 𝐹 𝑊 ⁡(𝑄 2 ), of nuclei over a continuous range of momentum transfers, 0 ≲ 𝑄 2 ≲ 0.1 GeV 2 , is essential for mapping out the distribution of neutrons in nuclei. The neutron density distribution has significant implications for a broad range of areas, including studies of nuclear structure, neutron stars, and physics beyond the Standard Model. Currently, our knowledge of 𝐹 𝑊 ⁡(𝑄 2 ) comes primarily from fixed target experiments that measure the parity-violating asymmetry in coherent elastic electron-ion scattering. Fixed target experiments, such as CREX and PREX-1,2, have provided high-precision weak charge form factor extractions for the 48 Ca and 208 Pb nuclei, respectively. However, a major limitation of fixed target experiments is that they each provide data only at a single value of 𝑄 2 . With the proposed electron-ion collider (EIC) on the horizon, we explore its potential to impact the determination of the weak charge form factor. While it cannot compete with the precision of fixed target experiments, it can provide data over a wide and continuous range of 𝑄 2 values, and for a wide variety of nuclei. We show that with data corresponding to an integrated luminosity of ℒ ∼ 500/𝐴 fb −1 , where 𝐴 is the nucleus atomic weight, the EIC can significantly impact constraints by lifting degeneracies in theoretical models of the neutron density distribution. Ensuring EIC detector coverage at low 𝑄 2 and large negative pseudorapidities will be essential for such 𝐹 𝑊 ⁡(𝑄 2 ) measurements.

72 PHYSICS OF ELEMENTARY PARTICLES AND FIELDS↗
DOE OSTI2019

Coherent elastic neutrino-nucleus scattering on Ar 40 from first principles

Coherent elastic neutrino scattering on the 40 Ar nucleus is computed with coupled-cluster theory based on nuclear Hamiltonians inspired by effective field theories of quantum chromodynamics. Our approach is validated by calculating the charge form factor and comparing it to data from electron scattering. We make predictions for the weak form factor, the neutron radius, and the neutron skin and estimate systematic uncertainties. The neutron-skin thickness of 40 Ar is consistent with results from density-functional theory. Precision measurements from coherent elastic neutrino-nucleus scattering could potentially be used to extract these observables and help to constrain nuclear models.

39 ≤ A ≤ 58↗
DOE OSTI2021

Generalized Boozer coordinates: A natural coordinate system for quasisymmetry

We prove the existence of a straight-field-line coordinate system we call generalized Boozer coordinates. This coordinate system exists for magnetic fields with nested toroidal flux surfaces provided ∮ d l / B   ( j · ∇ ψ ) = 0 , where symbols have their usual meaning, and the integral is taken along closed magnetic field lines. All quasisymmetric fields, regardless of their associated form of equilibria, must satisfy this condition. This coordinate system presents itself as a convenient form to describe general quasisymmetric configurations and their properties. Insight can be gained analytically into the difference between strong and weak forms of quasisymmetry, as well as axisymmetry, and the interaction of quasisymmetry with different forms of equilibria.

70 PLASMA PHYSICS AND FUSION TECHNOLOGY↗
DOE OSTI2025

Formation of Water Networks on Anionic Perylene

We present infrared photodissociation spectra of hydrated perylene anion clusters with up to four water molecules, as well as electronic structure calculations based on density functional theory. Water molecules form weak hydrogen bonds to the π system of the perylene anion. For clusters with more than one water molecule, water−water hydrogen bonds are formed, which generally appear to be stronger than water−π hydrogen bonds, especially for the trihydrate and tetrahydrate. The resulting water networks exist as water subclusters on the surface of the carbon frame of perylene. Here, we observe temperature-dependent dynamic effects, which highlight large amplitude motions of the water network and the shallowness of the potential energy surfaces governing the structures of these clusters.

37 INORGANIC, ORGANIC, PHYSICAL, AND ANALYTICAL CH↗
DOE OSTI2021

Accurate Determination of the Neutron Skin Thickness of Pb 208 through Parity-Violation in Electron Scattering

We report a precision measurement of the parity-violating asymmetry A PV in the elastic scattering of longitudinally polarized electrons from 208 Pb. We measure A PV = 550 ± 16 (stat) ±8 (syst) parts per billion, leading to an extraction of the neutral weak form factor F W (Q 2 = 0.00616 GeV 2 ) = 0.368 ± 0.013. Combined with our previous measurement, the extracted neutron skin thickness is R n -R p = 0.283 ± 0.071 fm. The result also yields the first significant direct measurement of the interior weak density of 208 Pb: ρ$^0_W$ = -0.0796 ± 0.0036(exp) ± 0.0013(theo) fm -3 leading to the interior baryon density ρ$^0_b$ = 0.1480 ± 0.0036(exp) ± 0.0013(theo) fm -3 . Finally, the measurement accurately constrains the density dependence of the symmetry energy of nuclear matter near saturation density, with implications for the size and composition of neutron stars.

73 NUCLEAR PHYSICS AND RADIATION PHYSICS↗
DOE OSTI2023

Precise Measurement of the Neutron Skin Thicknesses of 208Pb and 48Ca

Despite great leaps of development in nuclear physics over the past century, we still do not have a comprehensive understanding of nuclear structure. This is mainly due to the lack of precise knowledge of the neutron distribution inside nucleus. Electromagnetic probes are ineffective in probing neutrons, unlike their charged partners, the protons. Therefore, it is of great importance to constrain the neutron distributions experimentally. Heavy nuclei have more neutrons than protons in order to balance the repulsion between protons. In such neutron-rich nuclei, the extra neutrons are pushed out to the surface by the nuclear symmetry energy, forming the so-called ?neutron skin?. The neutron skin can be probed with a well-established experimental technique ? parity-violating electron scattering (PVES). Using the scattering of longitudinally polarized electrons by an unpolarized target, PREX-II and CREX measure the small parity-violating asymmetry in cross sections. Employing electrons with opposite helicities the weak form factor, the neutron distribution and the neutron skin thickness of the target nucleus are extracted. With excellent beam qualities and dedicated instrumentation at Jefferson Lab, the asymmetry measurements are statistics-limited. We report the results of these two high-precision measurements and their implications on broad topics, from the nuclear structures to the neutron stars.

Zhang, Weibin↗
DOE OSTI2025

A higher-order finite-element implementation of the nonlinear Fokker–Planck collision operator for charged particle collisions in a low density plasma

Collisions between particles in a low density plasma are described by the Fokker–Planck collision operator. In applications, this nonlinear integro-differential operator is often approximated by linearised or ad-hoc model operators due to computational cost and complexity. In this work, we present an implementation of the nonlinear Fokker–Planck collision operator written in terms of Rosenbluth potentials in the Rosenbluth–MacDonald–Judd (RMJ) form. The Rosenbluth potentials may be obtained either by direct integration or by solving partial differential equations (PDEs) similar to Poisson's equation: we optimise for performance and scalability by using sparse matrices to solve the relevant PDEs. We represent the distribution function using a tensor-product continuous-Galerkin finite-element representation and we derive and describe the implementation of the weak form of the collision operator. We present tests demonstrating a successful implementation using an explicit time integrator and we comment on the speed and accuracy of the operator. Finally, we speculate on the potential for applications in the current and next generation of kinetic plasma models.

70 PLASMA PHYSICS AND FUSION TECHNOLOGY↗
DOE OSTI2024

Numerical integration in the virtual element method with the scaled boundary cubature scheme

Abstract The virtual element method (VEM) is a stabilized Galerkin method on meshes that consist of arbitrary (convex and nonconvex) polygonal and polyhedral elements. A crucial ingredient in the implementation of low‐ and high‐order VEM is the numerical integration of monomials and nonpolynomial functions over such elements. In this article, we apply the recently proposed scaled boundary cubature (SBC) scheme to compute the weak form integrals in various virtual element formulations over polygonal and polyhedral meshes. In doing so, we demonstrate the flexibility of the approach and the accuracy that it delivers on a broad suite of boundary‐value problems in 2D and 3D over polytopes with affine faces as well as on elements with curved boundaries. In addition, the use of the SBC scheme is exemplified in an enriched Poisson formulation of the VEM in which weakly singular functions are required to be integrated. This study establishes the SBC method as a simple, accurate and efficient integration scheme for use in the VEM.

Chin, Eric B.↗
DOE OSTI2019

Electroweak probes of ground state densities

Background: Elastic electron scattering has been used for decades to paint the most accurate picture of the proton distribution in atomic nuclei. This stands in stark contrast to the neutron distribution that is traditionally probed using hadronic reactions that are hindered by large uncertainties in the reaction mechanism. Spurred by new experimental developments, it is now possible to gain valuable insights into the neutron distribution using exclusively electroweak probes. Purpose: In this work, we aim to assess the information content and complementarity of the following three electroweak experiments in constraining the neutron distribution of atomic nuclei: (a) parity-violating elastic electron scattering, (b) coherent elastic neutrino-nucleus scattering, and (c) elastic electron scattering of unstable nuclei. Methods: Relativistic mean-field models informed by the properties of finite nuclei and neutron stars are used to compute ground state densities and form factors of a variety of nuclei. All the models follow the same fitting protocol, except for the assumed—and presently unknown—value of the neutron skin thickness of 208 Pb. This enables one to tune the density dependence of the symmetry energy without compromising the success in reproducing well known physical observables. Results: We found that the ongoing PREX-II and upcoming CREX campaigns at Jefferson Lab will play a vital role in constraining the weak form factor of xenon and argon, liquid noble gases that are used for the detection of both neutrinos and dark matter particles. Conclusions: Remarkable new advances in experimental physics have opened a new window into ground state densities of atomic nuclei using solely electroweak probes. The diversity and versatility of these experiments reveal powerful correlations that impose important nuclear structure constraints. In turn, these constraints provide quantitative theoretical uncertainties that are instrumental in searches for new physics and insights into the behavior of dense matter.

73 NUCLEAR PHYSICS AND RADIATION PHYSICS↗
DOE OSTI2025

Inference of phase field fracture models

The phase field approach to modeling fracture uses a diffuse damage field to represent cracks. This representation mollifies singularities that arise in computations with sharp interface models and some of the resultant difficulties in the mathematical and numerical treatment of fracture. Phase field fracture models have proven effective at representing crack propagation, branching, and merging. Specific formulations, beginning with brittle fracture, have also been shown to converge to classical solutions. Extensions to cover the range of material failure, including ductile and cohesive fracture, lead to an array of possible models. There exists a large body of literature focusing on this class of models and on the impact of model form on the predicted crack evolution. However, there have not been systematic studies into how optimal models may be chosen. Here, we take a first step in this direction by developing formal methods for identification of the best parsimonious model of phase field fracture given full-field data on the damage and deformation fields. We consider some of the main models that have been used for the degradation of elastic response due to damage and its propagation. Our approach builds upon Variational System Identification (VSI), a weak form variant of the Sparse Identification of Nonlinear Dynamics (SINDy). Furthermore, in this first communication we focus on synthetically generated data but we also consider central issues associated with the use of experimental full-field data, such as data sparsity and noise.

71 CLASSICAL AND QUANTUM MECHANICS, GENERAL PHYSIC↗
DOE OSTI2020

N -Arylimide Molecular Balances: A Comprehensive Platform for Studying Aromatic Interactions in Solution

Noncovalent interactions of aromatic surfaces play a key role in many biological processes and in determining the properties and utility of synthetic materials, sensors, and catalysts. However, the study of aromatic interactions has been challenging because these interactions are usually very weak and their trends are modulated by many factors such as structural, electronic, steric, and solvent effects. Recently, N-arylimide molecular balances have emerged as highly versatile and effective platforms for studying aromatic interactions in solution. These molecular balances can accurately measure weak noncovalent interactions in solution via their influence on the folded–unfolded conformational equilibrium. The structure (i.e., size, shape, π-conjugation, and substitution) and nature (i.e., element, charge, and polarity) of the π-surfaces and interacting groups can be readily varied, enabling the study of a wide range of aromatic interactions. These include aromatic stacking, heterocyclic aromatic stacking, and alkyl–π, chalcogen–π, silver–π, halogen–π, substituent–π, and solvent–π interactions. The ability to measure a diverse array of aromatic interactions within a single model system provides a unique perspective and insights as the interaction energies, stability trends, and solvent effects for different types of interactions can be directly compared. Some broad conclusions that have emerged from this comprehensive analysis include: (1) The strongest aromatic interactions involve groups with positive charges such as pyridinium and metal ions which interact with the electrostatically negative π-face of the aromatic surface via cation–π or metal–π interactions. Attractive electrostatic interactions can also form between aromatic surfaces and groups with partial positive charges. (2) Electrostatic interactions involving aromatic surfaces can be switched from repulsive to attractive using electron-withdrawing substituents or heterocycles. These electrostatic trends appear to span many types of aromatic interactions involving a polar group interacting with a π-surface such as halogen–π, chalcogen–π, and carbonyl–π. (3) Nonpolar groups form weak but measurable stabilizing interactions with aromatic surfaces in organic solvents due to favorable dispersion and/or solvophobic effects. Furthermore, a good predictor of the interaction strength is provided by the change in solvent-accessible surface area. (4) Solvent effects modulate the aromatic interactions in the forms of solvophobic effects and competitive solvation, which can be modeled using solvent cohesion density and specific solvent–solute interactions.

37 INORGANIC, ORGANIC, PHYSICAL, AND ANALYTICAL CH↗
DOE OSTI2023

A conservative Galerkin solver for the quasilinear diffusion model in magnetized plasmas

We propose a conservative Galerkin scheme for the quasilinear model in three-dimensional momentum space and three-dimensional spectral space, with cylindrical symmetry. We construct an unconditionally conservative weak form and use a discretization that preserves conservation properties independent of the wave emission probability. The discrete operators, combined with a consistent quadrature rule, preserve all the conservation laws rigorously. The proposed scheme is quite general: it works for both relativistic and non-relativistic systems, for both magnetized and unmagnetized plasmas, and even for problems with time-dependent dispersion relations. We represent the particle distribution by continuous basis functions and use discontinuous basis functions for the wave spectral energy density, which enables the application of a positivity-preserving technique. We adopt the marching simplex algorithm, designed initially for computer graphics, for numerical integration on the resonance manifold. Furthermore, the numerical examples with a bump-on-tail initial configuration show how the unstable waves produce strong momentum space diffusion.

70 PLASMA PHYSICS AND FUSION TECHNOLOGY↗
DOE OSTI2020

An adaptive space-time phase field formulation for dynamic fracture of brittle shells based on LR NURBS

Here, we present an adaptive space-time phase field formulation for dynamic fracture of brittle shells. Their deformation is characterized by the Kirchhoff–Love thin shell theory using a curvilinear surface description. All kinematical objects are defined on the shell’s mid-plane. The evolution equation for the phase field is determined by the minimization of an energy functional based on Griffith’s theory of brittle fracture. Membrane and bending contributions to the fracture process are modeled separately and a thickness integration is established for the latter. The coupled system consists of two nonlinear fourth-order PDEs and all quantities are defined on an evolving two-dimensional manifold. Since the weak form requires C 1 -continuity, isogeometric shape functions are used. The mesh is adaptively refined based on the phase field using Locally Refinable (LR) NURBS. Time is discretized based on a generalized-α method using adaptive time-stepping, and the discretized coupled system is solved with a monolithic Newton–Raphson scheme. Finally, the interaction between surface deformation and crack evolution is demonstrated by several numerical examples showing dynamic crack propagation and branching.

42 ENGINEERING↗
DOE OSTI2024

A Particle Method for the Multispecies Landau Equation

Abstract The multispecies Landau collision operator describes the two-particle, small scattering angle or grazing collisions in a plasma made up of different species of particles such as electrons and ions. Recently, a structure preserving deterministic particle method (Carrillo et al. in J. Comput. Phys. 7:100066, 2020) has been developed for the single species spatially homogeneous Landau equation. This method relies on a regularization of the Landau collision operator so that an approximate solution, which is a linear combination of Dirac delta distributions, is well-defined. Based on a weak form of the regularized Landau equation, the time dependent locations of the Dirac delta functions satisfy a system of ordinary differential equations. In this work, we extend this particle method to the multispecies case, and examine its conservation of mass, momentum, and energy, and decay of entropy properties. We show that the equilibrium distribution of the regularized multispecies Landau equation is a Maxwellian distribution, and state a critical condition on the regularization parameters that guarantees a species independent equilibrium temperature. A convergence study comparing an exact multispecies Bobylev-Krook-Wu (BKW) solution to the particle solution shows approximately 2nd order accuracy. Important physical properties such as conservation, decay of entropy, and equilibrium distribution of the particle method are demonstrated with several numerical examples.

Mathematics↗
DOE OSTI2020

Eulerian thermo-mechanical simulations of heterogeneous solid propellants using an approximate projection method

We present a mathematical model that describes thermo-mechanical deformations and thermal gradients on the unsteady burning of a heterogeneous solid propellant. A scaling study shows that the deformations in solid at combustion timescales can be treated as quasi-static. The resulting thermo-mechanical formulation is formulated on a Cartesian grid and makes use of a weak form of Chorin-type projection method to deal with large difference in shear modulus of constituent materials. Further, a one-dimensional verification study is carried out by comparing numerical simulations with those of an analytical model. In addition, convergence studies for a two-dimensional propellant sandwich configuration are presented for the stress, velocity, and reference map components. Finally, simulations are carried out for a two-dimensional random propellant pack and the time-averaged burn rate and strains are reported.

42 ENGINEERING↗