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All sources NASA NTRS DOE 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↗

NASA NTRS2002

A Discontinuous Galerkin Discretization of the Eikonal Equation on Curved Piecewise Isoparametric Triangulated Manifolds

This viewgraph presentation provides information on optimizing the travel distance between two points on a curved surface. The presentation addresses the single source shortest path problem, fast algorithms for estimating the eikonal equation, fast schemes and barrier theorems, and the discontinuous Galerkin method, including hyperbolic causality, finite element method, scalars, and marching the discontinuous Galerkin Eikonal approximation.

Barth, TIm↗

NASA NTRS1971

On the integrability cases of the equation of motion for a satellite in an axially symmetric gravitational field.

Investigation of two cases of integrability of a second-order differential equation describing the projection of an axisymmetric satellite orbit on to a plane perpendicular to the rotation axis. It is demonstrated that for these two cases the integration can be carried out either by quadratures or reduced to a first-order differential equation. Analytical and physical properties are expressed, and it is shown that the equation can be derived from the classical plane eikonal equation of geometric optics.

Ghaffari, A.↗

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↗

NASA NTRS1997

Numerical Schemes for the Hamilton-Jacobi and Level Set Equations on Triangulated Domains

Borrowing from techniques developed for conservation law equations, numerical schemes which discretize the Hamilton-Jacobi (H-J), level set, and Eikonal equations on triangulated domains are presented. The first scheme is a provably monotone discretization for certain forms of the H-J equations. Unfortunately, the basic scheme lacks proper Lipschitz continuity of the numerical Hamiltonian. By employing a virtual edge flipping technique, Lipschitz continuity of the numerical flux is restored on acute triangulations. Next, schemes are introduced and developed based on the weaker concept of positive coefficient approximations for homogeneous Hamiltonians. These schemes possess a discrete maximum principle on arbitrary triangulations and naturally exhibit proper Lipschitz continuity of the numerical Hamiltonian. Finally, a class of Petrov-Galerkin approximations are considered. These schemes are stabilized via a least-squares bilinear form. The Petrov-Galerkin schemes do not possess a discrete maximum principle but generalize to high order accuracy.

Barth, Timothy J.↗

NASA NTRS1992

Wave diffraction in weak cosmic-ray-modified shocks

Weakly multidirectional, long-wavelength cosmic-ray-modified shocks are studied via multiple scale perturbation techniques. The effects of diffraction are discussed in terms of Green's function solutions of the linearized 1 + 3D Burgers and 1 + 3D KdVB equations, and also in terms of solutions with singular Dirac delta initial distributions. The solutions show a monotonic decrease of the wave-front curvature with increasing time owing to the effects of wave diffraction. The shape of the wave surface is discussed in terms of solutions S to the wave eikonal equation corresponding to singular initial conditions. For the fast magnetosonic wave propagating in the positive x-direction, the wave phase surface S = 0 has elliptic cross sections with the planes x = constant and has a convex paraboloidal shape. Plane-wave solutions of the 1 + 3D KdVB equation are discussed.

Webb, G. M.↗

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↗

NASA NTRS1978

Turbulence in planetary occultations. I - A theoretical formulation

The Eikonal equation of geometrical optics and a weak scattering wave-optical formulation are separately used to calculate first- and second-order effects of turbulence on radio propagation through an atmosphere with uniformly varying average refractive index combined with a random component due to turbulence. The extent to which perturbations in bending angle and Doppler frequency affects atmospheric profiles of temperature and pressure derived from Doppler measurements is considered. The determination of the power spectra of turbulence effects, the contribution of different terms, and the use of the spectra for studying planetary atmospheres are discussed.

Haugstad, B. S.↗

NASA NTRS2004

Computations of Wall Distances Based on Differential Equations

The use of differential equations such as Eikonal, Hamilton-Jacobi and Poisson for the economical calculation of the nearest wall distance d, which is needed by some turbulence models, is explored. Modifications that could palliate some turbulence-modeling anomalies are also discussed. Economy is of especial value for deforming/adaptive grid problems. For these, ideally, d is repeatedly computed. It is shown that the Eikonal and Hamilton-Jacobi equations can be easy to implement when written in implicit (or iterated) advection and advection-diffusion equation analogous forms, respectively. These, like the Poisson Laplacian term, are commonly occurring in CFD solvers, allowing the re-use of efficient algorithms and code components. The use of the NASA CFL3D CFD program to solve the implicit Eikonal and Hamilton-Jacobi equations is explored. The re-formulated d equations are easy to implement, and are found to have robust convergence. For accurate Eikonal solutions, upwind metric differences are required. The Poisson approach is also found effective, and easiest to implement. Modified distances are not found to affect global outputs such as lift and drag significantly, at least in common situations such as airfoil flows.

Tucker, Paul G.↗

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↗

NASA NTRS1988

Eikonal solutions to optical model coupled-channel equations

Methods of solution are presented for the Eikonal form of the nucleus-nucleus coupled-channel scattering amplitudes. Analytic solutions are obtained for the second-order optical potential for elastic scattering. A numerical comparison is made between the first and second order optical model solutions for elastic and inelastic scattering of H-1 and He-4 on C-12. The effects of bound-state excitations on total and reaction cross sections are also estimated.

Cucinotta, Francis A.↗

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↗

NASA NTRS2003

Transport Equation Based Wall Distance Computations Aimed at Flows With Time-Dependent Geometry

Eikonal, Hamilton-Jacobi and Poisson equations can be used for economical nearest wall distance computation and modification. Economical computations may be especially useful for aeroelastic and adaptive grid problems for which the grid deforms, and the nearest wall distance needs to be repeatedly computed. Modifications are directed at remedying turbulence model defects. For complex grid structures, implementation of the Eikonal and Hamilton-Jacobi approaches is not straightforward. This prohibits their use in industrial CFD solvers. However, both the Eikonal and Hamilton-Jacobi equations can be written in advection and advection-diffusion forms, respectively. These, like the Poisson s Laplacian, are commonly occurring industrial CFD solver elements. Use of the NASA CFL3D code to solve the Eikonal and Hamilton-Jacobi equations in advective-based forms is explored. The advection-based distance equations are found to have robust convergence. Geometries studied include single and two element airfoils, wing body and double delta configurations along with a complex electronics system. It is shown that for Eikonal accuracy, upwind metric differences are required. The Poisson approach is found effective and, since it does not require offset metric evaluations, easiest to implement. The sensitivity of flow solutions to wall distance assumptions is explored. Generally, results are not greatly affected by wall distance traits.

Tucker, Paul G.↗

NASA NTRS2003

Transport Equation Based Wall Distance Computations Aimed at Flows With Time-Dependent Geometry

Eikonal, Hamilton-Jacobi and Poisson equations can be used for economical nearest wall distance computation and modification. Economical computations may be especially useful for aeroelastic and adaptive grid problems for which the grid deforms, and the nearest wall distance needs to be repeatedly computed. Modifications are directed at remedying turbulence model defects. For complex grid structures, implementation of the Eikonal and Hamilton-Jacobi approaches is not straightforward. This prohibits their use in industrial CFD solvers. However, both the Eikonal and Hamilton-Jacobi equations can be written in advection and advection-diffusion forms, respectively. These, like the Poisson's Laplacian, are commonly occurring industrial CFD solver elements. Use of the NASA CFL3D code to solve the Eikonal and Hamilton-Jacobi equations in advective-based forms is explored. The advection-based distance equations are found to have robust convergence. Geometries studied include single and two element airfoils, wing body and double delta configurations along with a complex electronics system. It is shown that for Eikonal accuracy, upwind metric differences are required. The Poisson approach is found effective and, since it does not require offset metric evaluations, easiest to implement. The sensitivity of flow solutions to wall distance assumptions is explored. Generally, results are not greatly affected by wall distance traits.

Tucker, Paul G.↗

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)↗

NASA NTRS1995

Comparison of exact solution with Eikonal approximation for elastic heavy ion scattering

A first-order optical potential is used to calculate the total and absorption cross sections for nucleus-nucleus scattering. The differential cross section is calculated by using a partial-wave expansion of the Lippmann-Schwinger equation in momentum space. The results are compared with solutions in the Eikonal approximation for the equivalent potential and with experimental data in the energy range from 25A to 1000A MeV.

Dubey, Rajendra R.↗

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↗