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All sourcesNASA NTRSDOE OSTI
DOE OSTI2022

Differential geometric approaches to momentum-based formulations for fluids [Slides]

This SAND report documents CIS Late Start LDRD Project 22-0311, "Differential geometric approaches to momentum-based formulations for fluids". The project primarily developed geometric mechanics formulations for momentum-based descriptions of nonrelativistic fluids, utilizing a differential geometry/exterior calculus treatment of momentum and a space+time splitting. Specifically, the full suite of geometric mechanics formulations (variational/Lagrangian, Lie-Poisson Hamiltonian and Curl-Form Hamiltonian) were developed in terms of exterior calculus using vector-bundle valued differential forms. This was done for a fairly general version of semi-direct product theory sufficient to cover a wide range of both neutral and charged fluid models, including compressible Euler, magnetohydrodynamics and Euler-Maxwell. As a secondary goal, this project also explored the connection between geometric mechanics formulations and the more traditional Godunov form (a hyperbolic system of conservation laws). Unfortunately, this stage did not produce anything particularly interesting, due to unforeseen technical difficulties. There are two publications related to this work currently in preparation, and this work will be presented at SIAM CSE 23, at which the PI is organizing a mini-symposium on geometric mechanics formulations and structure-preserving discretizations for fluids. The logical next step is to utilize the exterior calculus based understanding of momentum coupled with geometric mechanics formulations to develop (novel) structure-preserving discretizations of momentum. This is the main subject of a successful FY23 CIS LDRD "Structure-preserving discretizations for momentum-based formulations of fluids".

71 CLASSICAL AND QUANTUM MECHANICS, GENERAL PHYSIC↗
DOE OSTI2020

Quantum simulation of hyperbolic space with circuit quantum electrodynamics: From graphs to geometry

We show how quantum many-body systems on hyperbolic lattices with nearest-neighbor hopping and local interactions can be mapped onto quantum field theories in continuous negatively curved space. Here, the underlying lattices have recently been realized experimentally with superconducting resonators and therefore allow for a table-top quantum simulation of quantum physics in curved background. Our mapping provides a computational tool to determine observables of the discrete system even for large lattices, where exact diagonalization fails. As an application and proof of principle we quantitatively reproduce the ground state energy, spectral gap, and correlation functions of the noninteracting lattice system by means of analytic formulas on the Poincaré disk, and show how conformal symmetry emerges for large lattices. This sets the stage for studying interactions and disorder on hyperbolic graphs in the future. Importantly, our analysis reveals that even relatively small discrete hyperbolic lattices emulate the continuous geometry of negatively curved space, and thus can be used to experimentally resolve fundamental open problems at the interface of interacting many-body systems, quantum field theory in curved space, and quantum gravity.

71 CLASSICAL AND QUANTUM MECHANICS, GENERAL PHYSIC↗
DOE OSTI2026

Neural entropy-stable conservative flux form neural networks for learning hyperbolic conservation laws

We propose a neural entropy-stable conservative flux form neural network (NESCFN) for learning hyperbolic conservation laws and their associated entropy functions directly from solution trajectories, without requiring any predefined numerical discretization. While recent neural network architectures have successfully integrated classical numerical principles into learned models, most rely on prior knowledge of the governing equations or assume a fixed discretization. Our approach removes this dependency by embedding entropy-stable design principles into the learning process itself, enabling the discovery of physically consistent dynamics in a fully data-driven setting. By jointly learning both the flux function and a corresponding entropy, NESCFN promotes conservation and entropy dissipation, which is critical for long-term stability and fidelity in the system of hyperbolic conservation laws. Furthermore, numerical results demonstrate that the method achieves stability and conservation over extended time horizons and accurately captures shock propagation speeds, even without oracle access to future-time solution profiles in the training data.

Conservative flux form↗
DOE OSTI2020

On differentiable local bounds preserving stabilization for Euler equations

This work presents the design of nonlinear stabilization techniques for the finite element discretization of Euler equations in both steady and transient form. Implicit time integration is used in the case of the transient form. A differentiable local bounds preserving method has been developed, which combines a Rusanov artificial diffusion operator and a differentiable shock detector. Nonlinear stabilization schemes are usually stiff and highly nonlinear. This issue is mitigated by the differentiability properties of the proposed method. Moreover, in order to further improve the nonlinear convergence, we also propose a continuation method for a subset of the stabilization parameters. The resulting method has been successfully applied to steady and transient problems with complex shock patterns. Numerical experiments show that it is able to provide sharp and well resolved shocks. Furthermore, the importance of the differentiability is assessed by comparing the new scheme with its non-differentiable counterpart. Numerical experiments suggest that, for up to moderate nonlinear tolerances, the method exhibits improved robustness and nonlinear convergence behavior for steady problems. Additionally, in the case of transient problem, we also observe a reduction in the computational cost.

42 ENGINEERING↗
DOE OSTI2021

Second-order invariant domain preserving approximation of the compressible Navier–Stokes equations

Here, we present a fully discrete approximation technique for the compressible Navier–Stokes equations that is second-order accurate in time and space, semi-implicit, and guaranteed to be invariant domain preserving. The restriction on the time step is the standard hyperbolic CFL condition, i.e. τ ≲ O ( h ) / V where V is some reference velocity scale and h the typical meshsize.

42 ENGINEERING↗
DOE OSTI2022

High Order Strong Stability Preserving MultiDerivative Implicit and IMEX Runge--Kutta Methods with Asymptotic Preserving Properties

In this article we present a class of high order unconditionally strong stability preserving (SSP) implicit two-derivative Runge--Kutta schemes and SSP implicit-explicit (IMEX) multi-derivative Runge--Kutta schemes where the time-step restriction is independent of the stiff term. The unconditional SSP property for a method of order $p>2$ is unique among SSP methods and depends on a backward-in-time assumption on the derivative of the operator. We show that this backward derivative condition is satisfied in many relevant cases where SSP IMEX schemes are desired. We devise unconditionally SSP implicit Runge--Kutta schemes of order up to $p=4$ and IMEX Runge--Kutta schemes of order up to $p=3$. For the multiderivative IMEX schemes, we also derive and present the order conditions, which have not appeared previously. The unconditional SSP condition ensures that these methods are positivity preserving, and we present sufficient conditions under which such methods are also asymptotic preserving when applied to a range of problems, including a hyperbolic relaxation system, the Broadwell model, and the Bhatnagar--Gross--Krook kinetic equation. We present numerical results to support the theoretical results on a variety of problems.

97 MATHEMATICS AND COMPUTING↗
DOE OSTI2026

A sweeping positivity-preserving high-order finite difference WENO scheme for Euler equations

We develop a simple, high-order, conservative and robust positivity-preserving sweeping procedure for the density and the nonlinear pressure function in the compressible Euler equations. Using the scaling limiter in Zhang and Shu (J Comput Phys 229:3091–3120, 2010), we obtain a non-trivial extension of the scalar sweeping technique in Liu et al. (J Sci Comput 73:1028–1071, 2017) for the positivity of pressure. The sweeping procedure developed in this paper is a post-processing technique, which can be applied to any concave functions of the conserved variables in hyperbolic conservation law systems. Thus, it has applications beyond the Euler equations. This procedure preserves positivity and conservation of physical quantities without destroying the accuracy of the underlying scheme. The algorithm works for general schemes including finite difference, finite volume and discontinuous Galerkin methods; however, in this paper we focus on finite difference weighted essentially non-oscillatory (WENO) methods. As a result, we provide numerical tests of the fifth-order finite difference WENO scheme to demonstrate the accuracy and robustness of the technique.

Compressible Euler equations↗
DOE OSTI2021

Simulations of plasmas and fluids using anti-symmetric models

ALMA (Anti-symmetric, Large-Moment, Accelerated) is a fast, flexible, and scalable toolkit designed to solve hyperbolic conservation law systems in hybrid supercomputers. Here this manuscript describes the theoretical background and implementation of ALMA, which uses the anti-symmetric formulation of fluids to obtain simple, robust, and easily paralellizable code. Practical GPU acceleration is realized on entire applications with an overall gain factor of 2 to 4. ALMA also provides a parallel, GPU accelerated sparse solver based on geometric multigrid, capable of diagonalizing linear systems with 239 unknowns. Here we demonstrate ALMA's scaling and performance in petascale supercomputers and use standard fluid models to verify the overall approach with canonical benchmark problems.

71 CLASSICAL AND QUANTUM MECHANICS, GENERAL PHYSIC↗
DOE OSTI2021

Spontaneous emission dynamics of Eu 3+ ions coupled to hyperbolic metamaterials

We show that sub-wavelength nanostructured systems with tunable electromagnetic properties, such as hyperbolic metamaterials (HMMs), provide a useful platform to tailor spontaneous emission processes. Here, we investigate a system comprising Eu 3+ (NO 3 ) 3 ·6H 2 O nanocrystals on an HMM structure featuring a hexagonal array of Ag nanowires in a porous Al2O 3 matrix. The HMM-coupled Eu 3+ ions exhibit up to a 2.4 fold increase in their decay rate, accompanied by an enhancement of the emission rate of the 5 D 0 → 7 F 2 transition. Using finite-difference time-domain modeling, we corroborate these observations with the increase in the photonic density of states seen by the Eu 3+ ions in the proximity of the HMM. Our results indicate that HMMs can serve as a valuable tool to control the emission from weak transitions and, hence, hint at a route toward more practical applications of rare-earth ions in nanoscale optoelectronics and quantum devices.

72 PHYSICS OF ELEMENTARY PARTICLES AND FIELDS↗
DOE OSTI2021

Linear embedding of nonlinear dynamical systems and prospects for efficient quantum algorithms

The simulation of large nonlinear dynamical systems, including systems generated by discretization of hyperbolic partial differential equations, can be computationally demanding. Such systems are important in both fluid and kinetic computational plasma physics. This motivates exploring whether a future error-corrected quantum computer could perform these simulations more efficiently than any classical computer. In this work, we describe a method for mapping any finite nonlinear dynamical system to an infinite linear dynamical system (embedding) and detail three specific cases of this method that correspond to previously studied mappings. Then we explore an approach for approximating the resulting infinite linear system with finite linear systems (truncation). Using a number of qubits only logarithmic in the number of variables of the nonlinear system, a quantum computer could simulate truncated systems to approximate output quantities if the nonlinearity is sufficiently weak. Other aspects of the computational efficiency of the three detailed embedding strategies are also discussed.

97 MATHEMATICS AND COMPUTING↗
DOE OSTI2022

A 1D–0D–3D coupled model for simulating blood flow and transport processes in breast tissue

In this work, we present mixed dimensional models for simulating blood flow and transport processes in breast tissue and the vascular tree supplying it. These processes are considered, to start from the aortic inlet to the capillaries and tissue of the breast. Large variations in biophysical properties and flow conditions exist in this system necessitating the use of different flow models for different geometries and flow regimes. In total, we consider four different model types. First, a system of 1D nonlinear hyperbolic partial differential equations (PDEs) is considered to simulate blood flow in larger arteries with highly elastic vessel walls. Second, we assign 1D linearized hyperbolic PDEs to model the smaller arteries with stiffer vessel walls. The third model type consists of ODE systems (0D models). It is used to model the arterioles and peripheral circulation. Finally, homogenized 3D porous media models are considered to simulate flow and transport in capillaries and tissue within the breast volume. Sink terms are used to account for the influence of the venous and lymphatic systems. Combining the four model types, we obtain two different 1D–0D–3D coupled models for simulating blood flow and transport processes: The first model results in a fully coupled 1D–0D–3D model covering the complete path from the aorta to the breast combining a generic arterial network with a patient specific breast network and geometry. The second model is a reduced one based on the separation of the generic and patient specific parts. The information from a calibrated fully coupled model is used as inflow condition for the patient specific sub-model allowing a significant computational cost reduction. Several numerical experiments are conducted to calibrate the generic model parameters and to demonstrate realistic flow simulations compared to existing data on blood flow in the human breast and vascular system. Moreover, we use two different breast vasculature and tissue data sets to illustrate the robustness of our reduced sub-model approach.

multiscale blood flow models↗
DOE OSTI2025

Adaptive Uncertainty Quantification for Stochastic Hyperbolic Conservation Laws

Here, we propose a predictor-corrector adaptive method for the study of hyperbolic partial differential equations (PDEs) under uncertainty. Constructed around the framework of stochastic finite volume (SFV) methods, our approach circumvents sampling schemes or simulation ensembles while also preserving fundamental properties, in particular hyperbolicity of the resulting systems and conservation of the discrete solutions. Furthermore, we augment the existing SFV theory with a priori convergence results for statistical quantities, in particular push-forward densities, which we demonstrate through numerical experiments. By linking refinement indicators to regions of the physical and stochastic spaces, we drive anisotropic refinements of the discretizations, introducing new degrees of freedom where deemed profitable. To illustrate our proposed method, we consider a series of numerical examples for nonlinear hyperbolic PDEs based on Burgers’ and Euler’s equations.

97 MATHEMATICS AND COMPUTING↗
DOE OSTI2022

ZnO-Au x Cu 1-x Alloy and ZnO-Au x Al 1-x Alloy Vertically Aligned Nanocomposites for Low-Loss Plasmonic Metamaterials

Hyperbolic metamaterials are a class of materials exhibiting anisotropic dielectric function owing to the morphology of the nanostructures. In these structures, one direction behaves as a metal, and the orthogonal direction behaves as a dielectric material. Applications include subdiffraction imaging and hyperlenses. However, key limiting factors include energy losses of noble metals and challenging fabrication methods. In this work, self-assembled plasmonic metamaterials consisting of anisotropic nanoalloy pillars embedded into the ZnO matrix are developed using a seed-layer approach. Alloys of Au x Al 1-x or Au x Cu 1-x are explored due to their lower losses and higher stability. Optical and microstructural properties were explored. The ZnO-Au x Cu 1-x system demonstrated excellent epitaxial quality and optical properties compared with the ZnO-Au x Al 1-x system. Both nanocomposite systems demonstrate plasmonic resonance, hyperbolic dispersion, low losses, and epsilon-near-zero permittivity, making them promising candidates towards direct photonic integration.

36 MATERIALS SCIENCE↗
DOE OSTI2024

Stochastic finite volume method for uncertainty quantification of transient flow in gas pipeline networks

We develop a weakly intrusive framework to simulate the propagation of uncertainty in solutions of generic hyperbolic partial differential equation systems on graph-connected domains with nodal coupling and boundary conditions. The method is based on the Stochastic Finite Volume (SFV) approach and can be applied for uncertainty quantification (UQ) of the dynamical state of fluid flow over actuated transport networks. The numerical scheme has specific advantages for modeling intertemporal uncertainty in time-varying boundary parameters, which cannot be characterized by strict upper and lower (interval) bounds. We describe the scheme for a single pipe, and then formulate the controlled junction Riemann problem (JRP) that enables the extension to general network structures. In conclusion, we demonstrate the method's capabilities and performance characteristics using a standard benchmark test network.

97 MATHEMATICS AND COMPUTING↗
DOE OSTI2020

In situ compression artifact removal in scientific data using deep transfer learning and experience replay

The massive amount of data produced during simulation on high-performance computers has grown exponentially over the past decade, exacerbating the need for streaming compression and decompression methods for efficient storage and transfer of this data---key to realizing the full potential of large-scale computational science. Lossy compression approaches such as JPEG when applied to scientific simulation data realized as a stream of images can achieve good compression rates but at the cost of introducing compression artifacts and loss of information. This paper develops a unified framework for in situ compression artifact removal in which the fully convolutional neural network architectures are combined with scalable training, transfer learning, and experience replay to achieve superior accuracy and efficiency while significantly decreasing the storage footprint as compared with the traditional optimization-based approaches. We demonstrate the proposed approach and compare it with compressed sensing postprocessing and other baseline deep learning models using climate simulations and nuclear reactor simulations, both of which are driven by hyperbolic partial differential equations. Our approach when applied to remove the compression artifacts on the JPEG-compressed nuclear reactor simulation data (using a transfer-trained model that was pretrained on the climate simulation data and updated incrementally as the nuclear reactor simulation progressed), achieved a significant improvement---mean peak signal-to-noise ratio of 42.438 as compared with 27.725 obtained with the compressed sensing approach.

97 MATHEMATICS AND COMPUTING↗
DOE OSTI2024

Shearless effective barriers to chaotic transport induced by even twin islands in nontwist systems

For several decades now it has been known that systems with shearless invariant tori, nontwist Hamiltonian systems, possess barriers to chaotic transport. These barriers are resilient to breakage under perturbation and therefore regions where they occur are natural places to look for barriers to transport. Here we describe a kind of effective barrier that persists after the shearless torus is broken. Because phenomena are generic, for convenience we study the standard nontwist map (SNM), an area-preserving map that violates the twist condition locally in the phase space. The barrier occurs in nontwist systems when twin even period islands are present, which happens for a broad range of parameter values in the SNM. With a phase space composed of regular and irregular orbits, the movement of chaotic trajectories is hampered by the existence of shearless curves, total barriers, and a network of partial barriers formed by the stable and unstable manifolds of the hyperbolic points. Being a degenerate system, the SNM has twin islands and, consequently, twin hyperbolic points. We show that the structures formed by the manifolds intrinsically depend on period parity of the twin islands. For this even scenario the structure that we call a torus free barrier occurs because the manifolds of different hyperbolic points form an intricate chain atop a dipole configuration and the transport of chaotic trajectories through the chain becomes a rare event. This structure impacts the emergence of transport, the escape basin for chaotic trajectories, the transport mechanism, and the chaotic saddle. The case of odd periodic orbits is different: we find for this case the emergence of transport immediately after the breakup of the last invariant curve, and this leads to a scenario of higher transport, with intricate escape basin boundary and a chaotic saddle with nonuniformly distributed points.

classical mechanics↗