Engineering Papers⌕ Search

SEARCH · Engineering Papers

Results for “Eikonal equation”

Search indexed NASA NTRS and DOE OSTI research on propulsion, heat transfer, battery materials and energy systems. Follow report and document links to the original sources.

Quote a phrase for an exact phrase match. Source license links do not imply unrestricted reuse.

16 records
All sourcesNASA NTRSDOE OSTI
DOE OSTI2022

A second-order distributed memory parallel fast sweeping method for the Eikonal equation

The Eikonal equation is used to calculate wave propagation and distance fields, and due to its complexity requires numerical treatment for its solution. In this work, we present a second-order distributed memory parallel fast sweeping method. The second-order solution switches on a two-point stencil when two upwind points are available, and reverts to first-order otherwise. In all examples, the second-order method improves the solution over the first-order, allowing for significant savings in memory while achieving the same accuracy. Parallelization over distributed memory saw good weak scaling with optimal convergence. The computational time for second-order was approximately 2.5 times slower than first-order, where the largest amount of mesh points ran on 144 cores (512 GB) was ≈20 billion. The savings in memory from the second-order method combined with the distributed memory algorithm result in the ability to solve problems much larger than are possible with the serial first-order method.

71 CLASSICAL AND QUANTUM MECHANICS, GENERAL PHYSIC↗
DOE OSTI2024

LATTE: Los Alamos TravelTime package based on Eikonal equation

This Fortran code focuses on traveltime computation and tomography based on eikonal equation. Specifically, the package provides three major functionalities: (1) forward modeling of traveltime from single-point or ensemble source based on factorized eikonal equation, (2) adjoint-state first-arrival traveltime tomography based on picked first arrival traveltime using steepest descent, conjugate gradient, or limited-memory BFGS inversion scheme, and (3) adjoint-state joint transmission-reflection tomography based on picked first-arrival and reflection traveltimes. The package applies to forward modeling and tomography based on traveltime in 2D and 3D isotropic regular-grid models. We name this package LATTE – Los Alamos TravelTime package based on Eikonal equation. * The code is for accompanying a journal paper under preparation. The paper will be submitted via LA-UR separately later.

Gao, Kai↗
DOE OSTI2023

Black and gray box learning of amplitude equations: Application to phase field systems

Here, we present a data-driven approach to learning surrogate models for amplitude equations and illustrate its application to interfacial dynamics of phase field systems. In particular, we demonstrate learning effective partial differential equations describing the evolution of phase field interfaces from full phase field data. We illustrate this on a model phase field system, where analytical approximate equations for the dynamics of the phase field interface (a higher-order eikonal equation and its approximation, the Kardar-Parisi-Zhang equation) are known. For this system, we discuss data-driven approaches for the identification of equations that accurately describe the front interface dynamics. When the analytical approximate models mentioned above become inaccurate, as we move beyond the region of validity of the underlying assumptions, the data-driven equations outperform them. In these regimes, going beyond black box identification, we explore different approaches to learning data-driven corrections to the analytically approximate models, leading to effective gray box partial differential equations.

42 ENGINEERING↗
DOE OSTI2025

LATTE: open-source, high-performance traveltime computation, tomography and source location in acoustic and elastic media

Traveltime-based tomography and source location are fundamental approaches for imaging subsurface structures and understanding the spatiotemporal distribution of seismicity from local to global scales. We present an open-source, high-performance framework integrating eikonal equation solvers and adjoint-state theory for traveltime computation, velocity tomography, source location and joint tomography-location in 2-D/3-D acoustic and elastic media. We introduce novel regularization schemes based on total generalized p-variation, structural similarity and multitask machine learning to enhance the fidelity and interpretability of inverted models and source locations. Key features of our implementation also include the ability to leverage both absolute-difference and double-difference traveltime misfits for high-fidelity velocity tomography and source parameter estimation; support for traveltime computation and inversion in diverse 2-D/3-D scenarios with arbitrary source and receiver distributions; and a perturbation-based optimal step-size estimation method to reduce computational costs. In addition, our implementation employs shared-memory and distributed-memory parallelization to provide an efficient solution for traveltime computation, tomography, and source location. In conclusion, we validate the efficacy and accuracy of our approach through multiple synthetic data examples.

58 GEOSCIENCES↗
DOE OSTI2023

A Fast Butterfly-Compressed Hadamard–Babich Integrator for High-Frequency Helmholtz Equations in Inhomogeneous Media with Arbitrary Sources

Here we present a butterfly-compressed representation of the Hadamard-Babich (HB) ansatz for the Green's function of the high-frequency Helmholtz equation in smooth inhomogeneous media. For a computational domain discretized with Nv discretization cells, the proposed algorithm first solves and tabulates the phase and HB coefficients via eikonal and transport equations with observation points and point sources located at the Chebyshev nodes using a set of much coarser computation grids, and then butterfly compresses the resulting HB interactions from all Nv cell centers to each other. The overall CPU time and memory requirement scale as O(Nv log2 Nv) for any bounded two-dimensional (2D) domains with arbitrary excitation sources. A direct extension of this scheme to bounded 3D domains yields an O(Nv4/3) CPU complexity, which can be further reduced to quasi-linear complexities with proposed remedies. The scheme can also efficiently handle scattering problems involving inclusions in inhomogeneous media. Although the current construction of our HB integrator does not accommodate caustics, the resulting HB integrator itself can be applied to certain sources, such as concave-shaped sources, to produce caustic effects. Compared to finite-difference frequency domain methods, the proposed HB integrator is free of numerical dispersion and requires fewer discretization points per wavelength. As a result, it can solve wave propagation problems well beyond the capability of existing solvers. Remarkably, the proposed scheme can accurately model wave propagation in 2D domains with 640 wavelengths per direction and in 3D domains with 54 wavelengths per direction on a state-of-the-art supercomputer at Lawrence Berkeley National Laboratory.

Hadamard--Babich ansatz↗
DOE OSTI2024

Not all that is β0 is β-function: the DGLAP resummation and the running coupling in NLO JIMWLK

Abstract We reanalyze the origin of the large transverse logarithms associated with the QCD one loopβfunction coefficient in the NLO JIMWLK Hamiltonian. We show that some of these terms are not associated with the running of the QCD coupling constant but rather with the DGLAP evolution. The DGLAP-like resummation of these logarithms is mandatory within the JIMWLK Hamiltonian, as long as the color correlation length in the projectile is larger than that in the target. This regime in fact covers the whole range of rapidities at which JIMWLK evolution is supposed to be applicable. We derive the RG equation that resums these logarithms to all orders inα s in the JIMWLK Hamiltonian. This is a nonlinear equation for the eikonal scattering matrixS(x). We solve this equation, and perform the DGLAP resummation in two simple cases: the dilute limit, where both the projectile and the target are far from saturation, and the saturated regime, where the target correlation length also determines its saturation momentum.

Physics↗
DOE OSTI2026

Unpolarized GPDs at small x and non-zero skewness

We study the small-x asymptotics of unpolarized generalized parton distributions (GPDs) and generalized transverse momentum distributions (GTMDs). Unlike the previous works in the literature, we consider the case of non-zero (but small) skewness while allowing for non-linear contributions to the evolution equations. We first show that unpolarized GPDs and GTMDs at small x are related to the eikonal dipole amplitude N, whose small-x evolution is given by the BK/JIMWLK evolution equations, and to the odderon amplitude $\mathscr{O}$, whose evolution is also known in the literature. We then show that the effect of non-zero skewness ξ ≠ 0 is to modify the value of the evolution parameter (rapidity) in the arguments for the dipole amplitudes N and $\mathscr{O}$ from Y = ln(1/x) to Y = ln min{1/|x|, 1/|ξ|}.

Generalized parton distributions (GPDs)↗
DOE OSTI2024

Quasiclassical Gluon Fields and Low’s Soft Theorem at Small Momentum-Fraction x

In the high-energy limit, soft gluons can be approximately described by quasiclassical gluon fields. It is well known that the gluon field is a pure gauge field on the transverse plane at eikonal order. We derived the complete next-to-eikonal order solutions of the classical Yang-Mills equations for soft gluons in the dense nuclear regime. Utilizing these solutions, it is shown that Low’s soft theorem at small x can be obtained by considering off-diagonal matrix elements of quasiclassical chromoelectric field between single-gluon states in the dilute regime. We further propose on extending Low’s soft theorem at small x to incorporate the effects of gluon saturation in the dense regime.

71 CLASSICAL AND QUANTUM MECHANICS, GENERAL PHYSIC↗
DOE OSTI2025

Systematic study of the validity of the eikonal model including uncertainties

Nuclear reactions at intermediate beam energies are often interpreted using the eikonal model. In the analysis of complex reaction probes, where few-body reaction methods are needed, the eikonal method may be used as an efficient way for describing the fragment-target reaction process. In this work, we perform a systematic study to test the validity of the eikonal approximation for nucleon-nucleus reactions. We also quantify uncertainties due to the nucleon optical potential on reaction observables. We inspect the validity of the eikonal model and its semiclassical correction by comparing it to exact solutions (obtained from solving the optical-model equation with a finite-differences method) for a wide range of reactions. We also study the effect of relativistic corrections, both kinematic and dynamic, by effectively incorporating the relativistic effects at intermediate energies. The uncertainties from a Bayesian global optical potential (KDUQ) are propagated to the observables of interest. Our study includes neutron and proton reactions on 27 Al , 40 Ca , 90 Zr , and 208 Pb , for a wide range of energies 𝐸 lab = 0–400 MeV. We calculate neutron-total cross sections (elastic and reactions) as well as proton-absorption cross sections as a function of beam energy, using the eikonal model, the eikonal model with a semiclassical correction, and the exact solution. Here, we also compute angular distributions for the methods above. Our results show that for the proton-absorption cross section, the eikonal model can be used down to around 60 MeV and the semiclassical correction extends its use to 30 MeV. However, the validity of the eikonal model for the neutron-total cross section only goes down to ≈120 MeV, a range extended to ≈ 50 MeV when using the semiclassical correction. We find the semiclassical correction to the eikonal model to be less effective in describing the angular distributions. The 1⁢𝜎 uncertainty intervals on the observables we studied is less than 5% for most of the energies considered, but increases rapidly for higher energies, namely energies outside the range of KDUQ (𝐸 lab > 200MeV).

Cluster models↗
DOE OSTI2025

Quasiclassical evaluation of gluon saturation induced helicity effects

At sub-eikonal order in the high-energy limit, helicity dependences are generally contained in the transverse components of the gluon field. In the gluon saturation regime, novel helicity effects arise from the nonlinear interaction between the eikonal-order longitudinal gluon field and the sub-eikonal-order transverse gluon field. We derive this saturation induced helicity-dependent field both by solving the classical Yang-Mills equations and through direct diagrammatic calculations. Our analysis shows that the saturation induced helicity effect is intrinsically a two-particle (or multiparticle) correlation effect, rather than a single-particle distribution effect. Furthermore, we evaluate this effect in the context of double-spin asymmetry for incoherent diffractive dijet production in longitudinally polarized electron-nucleus collisions, using a simplified helicity-extended McLerran-Venugopalan model. We find that the saturation induced helicity effect further suppresses the back-to-back peak in the dijet azimuthal angle correlation, with a magnitude comparable to that of the direct helicity effect. This gluon saturation induced helicity effect may offer a novel avenue to probe gluon saturation in polarized collisions. Published by the American Physical Society 2025

Li, Ming (ORCID:0000000195509691)↗
DOE OSTI2022

T-odd leading-twist quark TMDs at small x

We study the small-x asymptotics of the flavor non-singlet T-odd leading-twist quark transverse momentum dependent parton distributions (TMDs), the Sivers and Boer-Mulders functions. While the leading eikonal small-x asymptotics of the quark Sivers function is given by the spin-dependent odderon [1, 2], we are interested in revisiting the sub-eikonal correction considered by us earlier in [3]. We first simplify the expressions for both TMDs at small Bjorken x and then construct small-x evolution equations for the resulting operators in the large-N c limit, with N c the number of quark colors. For both TMDs, the evolution equations resum all powers of the double-logarithmic parameter α s ln 2 (1/x), where α s is the strong coupling constant, which is assumed to be small. Solving these evolution equations numerically (for the Sivers function) and analytically (for the Boer-Mulders function) we arrive at the following leading small-x asymptotics of these TMDs at large N c :$$ {\displaystyle \begin{array}{l}{f}_{1T}^{\perp NS}\left(x\ll 1,{k}_T^2\right)={C}_O\left(x,{k}_T^2\right)\frac{1}{x}+{C}_1\left(x,{k}_T^2\right){\left(\frac{1}{x}\right)}^{3.4\sqrt{\frac{\alpha_s{N}_c}{4\pi }}}\\ {}{h}_1^{\perp \textrm{NS}}\left(x\ll 1,{k}_T^2\right)=C\left(x,{k}_T^2\right){\left(\frac{1}{x}\right)}^{-1}.\end{array}} $$ The functions C O (x,$ {k}_T^2 $), C 1 (x,$ {k}_T^2 $), and C(x,$ {k}_T^2 $) can be readily obtained in our formalism: they are mildly x-dependent and do not strongly affect the power-of-x asymptotics shown above. The function C O , along with the 1/x factor, arises from the odderon exchange. For the sub-eikonal contribution to the quark Sivers function (the term with C 1 ), our result shown above supersedes the one obtained in [3] due to the new contributions identified recently in [4].

73 NUCLEAR PHYSICS AND RADIATION PHYSICS↗
DOE OSTI2022

Quark and gluon helicity evolution at small x: revised and updated

We revisit the problem of small Bjorken-x evolution of the gluon and flavor-singlet quark helicity distributions in the shock wave (s-channel) formalism. Earlier works on the subject in the same framework resulted in an evolution equation for the gluon field-strength F 12 and quark “axial current” $\overline{\psi}\gamma$ + γ 5 ψ operators (sandwiched between the appropriate light-cone Wilson lines) in the double-logarithmic approximation (summing powers of α s ln 2 (1/x) with α s the strong coupling constant). In this work, we observe that an important mixing of the above operators with another gluon operator, ${}_D{}^{\leftarrow i} {}_D{}^{i}$, also sandwiched between the light-cone Wilson lines (with the repeated transverse index i = 1, 2 summed over), was missing in the previous works. This operator has the physical meaning of the sub-eikonal (covariant) phase: its contribution to helicity evolution is shown to be proportional to another sub-eikonal operator, ${}_D{}^{i} {}_D{}^{\leftarrow i}$, which is related to the Jaffe-Manohar polarized gluon distribution. In this work we include this new operator into small-x helicity evolution, and construct novel evolution equations mixing all three operators (${}_D{}^{i} {}_D{}^{\leftarrow i}$, F 12 , and $\overline{\psi}\gamma$ + γ 5 ψ), generalizing the results of . We also construct closed double-logarithmic evolution equations in the large-N c and large-N c &N f limits, with N c and N f the numbers of quark colors and flavors, respectively. Solving the large-N c equations numerically we obtain the following small-x asymptotics of the quark and gluon helicity distributions ΔΣ and ΔG, along with the g 1 structure function, $$ \Delta \Sigma \left(x,{Q}^2\right)\sim \Delta G\left(x,{Q}^2\right)\sim {g}_1\left(x,{Q}^2\right)\sim {\left(\frac{1}{x}\right)}^{3.66\sqrt{\frac{\alpha_s{N}_c}{2\pi }}} $$ in complete agreement with the earlier work by Bartels, Ermolaev and Ryskin.

73 NUCLEAR PHYSICS AND RADIATION PHYSICS↗
DOE OSTI2026

Back-to-back dijet production in DIS at arbitrary Bjorken x: TMD gluon distributions to twist-3 accuracy

We derive the gluon transverse-momentum-dependent (TMD) operator structure of back-to-back\\\\r\\\\nquark–antiquark dijet production in deep inelastic scattering at arbitrary Bjorken-x to twist-3 ac\\\\r\\\\ncuracy. Working at leading order in the strong coupling and in the kinematic regime where the\\\\r\\\\ntransverse momentum imbalance of the jets is much smaller than their individual transverse mo\\\\r\\\\nmenta, we perform a systematic gradient expansion of the quark propagator in a background gluon\\\\r\\\\nfield. This expansion organizes multiple interactions with the target in terms of longitudinal Wilson\\\\r\\\\nlines and gauge-invariant field-strength insertions, yielding a TMD description valid beyond the\\\\r\\\\nstrict high-energy eikonal (x → 0) approximation. We obtain explicit cross sections for longitudi\\\\r\\\\nnally and transversely polarized virtual photons, identifying all contributing gluon TMD operators\\\\r\\\\nup to twist-3, including structures involving F+−, Fij, and three-gluon correlators. The full lon\\\\r\\\\ngitudinal phase eixP+z− associated with Bjorken-x is retained throughout. In the small-x limit,\\\\r\\\\nour results reproduce the known sub-eikonal expressions obtained in the Color Glass Condensate\\\\r\\\\nframework, establishing a direct connection between the general-x TMD expansion and high-energy\\\\r\\\\nfactorization. We further reduce the operator basis using equations of motion, minimizing the num\\\\r\\\\nber of independent nonperturbative matrix elements entering the cross section. This work provides\\\\r\\\\na systematic foundation for extending TMD analyses of dijet production beyond leading twist, es\\\\r\\\\ntablishing a unified operator framework valid at arbitrary Bjorken-x that smoothly interpolates\\\\r\\\\nbetween moderate- and small-x descriptions of gluon TMDs.

73 NUCLEAR PHYSICS AND RADIATION PHYSICS↗
DOE OSTI2025

Precise measurement of nuclear interaction cross sections towards neutron-skin determination with R 3 B

The R 3 B (Reactions with Relativistic Radioactive Beams) experiment as a major instrument of the NUSTAR collaboration for the research facility FAIR in Darmstadt is designed for kinematically complete studies of reactions with high-energy radioactive beams. Part of the broad physics program of R 3 B is to constrain the asymmetry term in the nuclear equation-of-state and hence improve the description of highly asymmetric nuclear matter (e.g., in neutron stars). For a precise determination of the neutron-skin thickness – an observable which is directly correlated with the symmetry energy in theoretical calculations – by measuring absolute fragmentation cross sections, it is essential to quantify the uncertainty and challenge the reaction model under stable conditions. During the successful FAIR Phase-0 campaign of R 3 B, we precisely measured the energy dependence of total interaction cross sections in 12 C + 12 C collisions, for a direct comparison with calculations based on the eikonal reaction theory.

Ponnath, L. (ORCID:0000000286742624)↗
DOE OSTI2023

Scattering in black hole backgrounds and higher-spin amplitudes. Part I

The scattering of massless waves of helicity |$h$| = $0$, $\frac{1}{2}$, $1$ in Schwarzschild and Kerr backgrounds is revisited in the long-wavelength regime. Using a novel description of such backgrounds in terms of gravitating massive particles, we compute classical wave scattering in terms of 2 → 2 QFT amplitudes in flat space, to all orders in spin. The results are Newman-Penrose amplitudes which are in direct correspondence with solutions of the Regge-Wheeler/Teukolsky equation. By introducing a precise prescription for the point-particle limit, in Part I of this work we show how both agree for $h$ = 0 at finite values of the scattering angle and arbitrary spin orientation. Associated classical observables such as the scattering cross sections, wave polarizations and time delay are studied at all orders in spin. The effect of the spin of the black hole on the polarization and helicity of the waves is found in agreement with previous analysis at linear order in spin. In the particular limit of small scattering angle, we argue that wave scattering admits a universal, point-particle description determined by the eikonal approximation. We show how our results recover the scattering eikonal phase with spin up to second post-Minkowskian order, and match it to the effective action of null geodesics in a Kerr background. Using this correspondence we derive classical observables such as polar and equatorial scattering angles. This study serves as a preceding analysis to Part II, where the Gravitational Wave ($h$ = 2) case will be studied in detail.

79 ASTRONOMY AND ASTROPHYSICS↗
DOE OSTI2024

Quantum field theory, worldline theory, and spin magnitude change in orbital evolution

A previous paper [Z. Bern et al., Binary dynamics through the fifth power of spin at 𝑂⁡(𝐺 2 ), Phys. Rev. Lett. 130, 201402 (2023)] identified a puzzle stemming from the amplitudes-based approach to spinning bodies in general relativity: additional Wilson coefficients appear compared to current worldline approaches to conservative dynamics of generic astrophysical objects, including neutron stars. In this paper we clarify the nature of analogous Wilson coefficients in the simpler theory of electrodynamics. We analyze the original field-theory construction, identifying definite-spin states some of which have negative norms, and relating the additional Wilson coefficients in the classical theory to transitions between different quantum spin states. We produce a new version of the theory which also has additional Wilson coefficients, but no negative-norm states. We match, through 𝒪⁡(𝛼 2 ) and 𝒪⁡(𝑆 2 ), the Compton amplitudes of these field theories with those of a modified worldline theory with extra degrees of freedom introduced by releasing the spin supplementary condition. We build an effective two-body Hamiltonian that matches the impulse and spin kick of the modified field theory and of the worldline theory, displaying additional Wilson coefficients compared to standard worldline approaches. The results are then compactly expressed in terms of an eikonal formula. Our key conclusion is that, contrary to standard approaches, while the magnitude of the spin tensor is still conserved, the magnitude of the spin vector can change under conserved Hamiltonian dynamics and this change is governed by the additional Wilson coefficients. For specific values of Wilson coefficients the results are equivalent to those from a definite spin obeying the spin supplementary condition, but for generic values they are physically inequivalent. These results warrant detailed studies of the corresponding issues in general relativity.

classical black holes↗