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At least 19 records
All sourcesNASA NTRSDOE OSTI
NASA NTRS2004

High Order And High Resolution Methods For a Model CAA Problem

The initial value problem for the first order linear wave equation in one space dimension is treated for two cases with specified initial data and grid, and data from solutions at t = 400 and t = 800 are presented, as prescribed for Problem 1 in Category 1. Results are shown from computations with a sequence of recently developed high order and high resolution methods which combine Hermite interpolation, Cauchy-Kowaleskya recursion for time derivatives, and Taylor series time advancement. These methods have the same order of accuracy in time as in space. Results are shown from methods that range from third to nineteenth order. The stated problems with the prescribed coarse grid can be simulated with errors that are at the level of machine accuracy if the method is sufficiently high order. In addition, the growth of the maximum absolute error out to t = 100,000 is given for simulations with the stated problem data.

Goodrich, John W.↗
DOE OSTI2023

Low-Order Preconditioning for the High-Order Finite Element de Rham Complex

Here, we present a unified framework for constructing spectrally equivalent low-order-refined discretizations for the high-order finite element de Rham complex. This theory covers diffusion problems in H 1 , H(curl), and H(div) and is based on combining a low-order discretization posed on a refined mesh with a high-order basis for Nédélec and Raviart–Thomas elements that makes use of the concept of polynomial histopolation (polynomial fitting using prescribed mean values over certain regions). This spectral equivalence, coupled with algebraic multigrid methods constructed using the low-order discretization, results in highly scalable matrix-free preconditioners for high-order finite element problems in the full de Rham complex. Additionally, a new lowest-order (piecewise constant) preconditioner is developed for high-order interior penalty discontinuous Galerkin (DG) discretizations, for which spectral equivalence results and convergence proofs for algebraic multigrid methods are provided. In all cases, the spectral equivalence results are independent of polynomial degree and mesh size; for DG methods, they are also independent of the penalty parameter. These new solvers are flexible and easy to use; any “black-box” preconditioner for low-order problems can be used to create an effective and efficient preconditioner for the corresponding high-order problem. A number of numerical experiments are presented, based on an implementation in the finite element library MFEM. A range of challenging three-dimensional problems are used to corroborate the theoretical properties and demonstrate the flexibility and scalability of the method.

97 MATHEMATICS AND COMPUTING↗
DOE OSTI2021

Spectral Equivalence of Low-Order Discretizations for High-Order H(curl) and H(div) Spaces

In this paper, we present spectral equivalence results for high-order tensor product edge- and face-based finite elements for the H(curl) and H(div) function spaces. Specifically, we show for certain choices of shape functions that the mass and stiffness matrices of the high-order elements are spectrally equivalent to those for an assembly of low-order elements on the associated Gauss--Lobatto--Legendre mesh. Based on this equivalence, efficient preconditioners can be designed with favorable computational complexity. Numerical results are presented which confirm the theory and demonstrate the benefits of the equivalence results for overlapping Schwarz preconditioners.

97 MATHEMATICS AND COMPUTING↗
DOE OSTI2024

Arbitrary Order Virtual Element Methods for High‐Order Phase‐Field Modeling of Dynamic Fracture

ABSTRACT Accurate modeling of fracture nucleation and propagation in brittle and ductile materials subjected to dynamic loading is important in predicting material damage and failure under extreme conditions. Phase‐field fracture models have garnered a lot of attention in recent years due to their success in representing damage and fracture processes in a wide class of materials and under a variety of loading conditions. Second‐order phase‐field fracture models are by far the most popular among researchers (and increasingly, among practitioners), but fourth‐order models have started to gain broader acceptance since their more recent introduction. The exact solution corresponding to these high‐order phase‐field fracture models has higher regularity. Thus, numerical solutions of the model equations can achieve improved accuracy and higher spatial convergence rates. In this work, we develop a virtual element framework for the high‐order phase‐field model of dynamic fracture. The virtual element method (VEM) can be regarded as a generalization of the classical finite element method. In addition to many other desirable characteristics, the VEM allows computing on polytopal meshes. Here, we use ‐conforming virtual elements and the generalized‐ time integration method for the momentum balance equation, and adopt ‐conforming virtual elements for the high‐order phase‐field equation. We verify our virtual element framework using classical quasi‐static benchmark problems and demonstrate its capabilities with the aid of numerical simulations of dynamic fracture in brittle materials.

42 ENGINEERING↗
DOE OSTI2023

End-to-end GPU acceleration of low-order-refined preconditioning for high-order finite element discretizations

In this article, we present algorithms and implementations for the end-to-end GPU acceleration of matrix-free low-order-refined preconditioning of high-order finite element problems. The methods described here allow for the construction of effective preconditioners for high-order problems with optimal memory usage and computational complexity. The preconditioners are based on the construction of a spectrally equivalent low-order discretization on a refined mesh, which is then amenable to, for example, algebraic multigrid preconditioning. The constants of equivalence are independent of mesh size and polynomial degree. For vector finite element problems in H(curl) and H(div) (e.g., for electromagnetic or radiation diffusion problems), a specially constructed interpolation–histopolation basis is used to ensure fast convergence. Detailed performance studies are carried out to analyze the efficiency of the GPU algorithms. The kernel throughput of each of the main algorithmic components is measured, and the strong and weak parallel scalability of the methods is demonstrated. The different relative weighting and significance of the algorithmic components on GPUs and CPUs is discussed. Results on problems involving adaptively refined nonconforming meshes are shown, and the use of the preconditioners on a large-scale magnetic diffusion problem using all spaces of the finite element de Rham complex is illustrated.

97 MATHEMATICS AND COMPUTING↗
NASA NTRS2000

A Low Order and a High Order Solution for a Converging-Diverging Nozzle Problem

The Computational Aeroacoustics Workshops on Benchmark problems are conducted in order to generate solutions with a variety of methods for problems that are chosen both to be representative of significant and relevant acoustic phenomena. It is generally recognized that CAA applications require the resolution in time and in space of solution details over a significantly broader range of scales than typical computational fluid dynamics applications. These requirements are forcing the development of new methodologies for CAA applications. This paper presents numerical results from a third and a seventh order algorithm for the propagation of an acoustic signal through a converging-diverging nozzle. These two algorithms are members of a new family of high accuracy methods that have the same order accuracy in both space and time, and are an extension of previous work for linearized Euler equations to fully nonlinear time dependent problems. The simulations are all with the fully nonlinear quasi-1D Euler equations for the total solution, which includes both the fluid dynamics and the acoustics. The acoustic solution is obtained from the time dependent nonlinear solution by subtracting the steady solution.

Goodrich, John W.↗
NASA NTRS1989

Low-order design and high-order simulation of active closed-loop control for aerospace structures under construction

Partially constructed/assembled structures in space are complicated enough but their dynamics will also be operating in closed-loop with feedback controllers. The dynamics of such structures are modeled by large-scale finite element models. The model dimension L is extremely large (approximately 10,000) while the numbers of actuators (M) and sensors (P) are small. The model parameters M(sub m) mass matrix, D(sub o) damping matrix, and K(sub o) stiffness matrix, are all symmetric and sparse (banded). Thus simulation of open-loop structure models of very large dimension can be accomplished by special integration techniques for sparse matrices. The problem of simulation of closed-loop control of such structures is complicated by the addition of controllers. Simulation of closed-loop controlled structures is an essential part of the controller design and evaluation process. Current research in the following areas is presented: high-order simulation of actively controlled aerospace structures; low-order controller design and SCI compensation for unmodeled dynamics; prediction of closed-loop stability using asymptotic eigenvalue series; and flexible robot manipulator control experiment.

Balas, Mark J.↗
DOE OSTI2022

Highly ordered graphite (HOPG) to hexagonal diamond (lonsdaleite) phase transition observed on picosecond time scales using ultrafast x-ray diffraction

Here, the response of rapidly compressed highly oriented pyrolytic graphite (HOPG) normal to its basal plane was investigated at a pressure of ~80 GPa. Ultrafast x-ray diffraction using ~100 fs pulses at the Materials Under Extreme Conditions sector of the Linac Coherent Light Source was used to probe the changes in crystal structure resulting from picosecond timescale compression at laser drive energies ranging from 2.5 to 250 mJ. A phase transformation from HOPG to a highly textured hexagonal diamond structure is observed at the highest energy, followed by relaxation to a still highly oriented, but distorted graphite structure following release. We observe the formation of a highly oriented lonsdaleite within 20 ps, subsequent to compression. This suggests that a diffusionless martensitic mechanism may play a fundamental role in phase transition, as speculated in an early work on this system, and more recent static studies of diamonds formed in impact events.

58 GEOSCIENCES↗
NASA NTRS2004

Relaxation and Preconditioning for High Order Discontinuous Galerkin Methods with Applications to Aeroacoustics and High Speed Flows

This project is about the investigation of the development of the discontinuous Galerkin finite element methods, for general geometry and triangulations, for solving convection dominated problems, with applications to aeroacoustics. Other related issues in high order WENO finite difference and finite volume methods have also been investigated. methods are two classes of high order, high resolution methods suitable for convection dominated simulations with possible discontinuous or sharp gradient solutions. In [18], we first review these two classes of methods, pointing out their similarities and differences in algorithm formulation, theoretical properties, implementation issues, applicability, and relative advantages. We then present some quantitative comparisons of the third order finite volume WENO methods and discontinuous Galerkin methods for a series of test problems to assess their relative merits in accuracy and CPU timing. In [3], we review the development of the Runge-Kutta discontinuous Galerkin (RKDG) methods for non-linear convection-dominated problems. These robust and accurate methods have made their way into the main stream of computational fluid dynamics and are quickly finding use in a wide variety of applications. They combine a special class of Runge-Kutta time discretizations, that allows the method to be non-linearly stable regardless of its accuracy, with a finite element space discretization by discontinuous approximations, that incorporates the ideas of numerical fluxes and slope limiters coined during the remarkable development of the high-resolution finite difference and finite volume schemes. The resulting RKDG methods are stable, high-order accurate, and highly parallelizable schemes that can easily handle complicated geometries and boundary conditions. We review the theoretical and algorithmic aspects of these methods and show several applications including nonlinear conservation laws, the compressible and incompressible Navier-Stokes equations, and Hamilton-Jacobi-like equations.

Shu, Chi-Wang↗
DOE OSTI2022

Electron Tomography and Machine Learning for Understanding the Highly Ordered Structure of Leafhopper Brochosomes

Insects known as leafhoppers (Hemiptera: Cicadellidae) produce hierarchically structured nanoparticles known as brochosomes that are exuded and applied to the insect cuticle, thereby providing camouflage and anti-wetting properties to aid insect survival. Although the physical properties of brochosomes are thought to depend on the leafhopper species, the structure–function relationships governing brochosome behavior are not fully understood. Brochosomes have complex hierarchical structures and morphological heterogeneity across species, due to which a multimodal characterization approach is required to effectively elucidate their nanoscale structure and properties. In this work, we study the structural and mechanical properties of brochosomes using a combination of atomic force microscopy (AFM), electron microscopy (EM), electron tomography, and machine learning (ML)-based quantification of large and complex scanning electron microscopy (SEM) image data sets. This suite of techniques allows for the characterization of internal and external brochosome structures, and ML-based image analysis methods of large data sets reveal correlations in the structure across several leafhopper species. Our results show that brochosomes are relatively rigid hollow spheres with characteristic dimensions and morphologies that depend on leafhopper species. Nanomechanical mapping AFM is used to determine a characteristic compression modulus for brochosomes on the order of 1–3 GPa, which is consistent with crystalline proteins. Altogether, this work provides an improved understanding of the structural and mechanical properties of leafhopper brochosomes using a new set of ML-based image classification tools that can be broadly applied to nanostructured biological materials.

Chemical structure↗
DOE OSTI2024

Enhancement of harmonic generation by an intense driving laser with high-order waveguide modes in a high-pressure gas-filled hollow waveguide

High-order harmonics have been widely used as reliable tabletop coherent radiation sources recently, but their applications have often been limited by the available pulse energy. Here, we report that by using an overdriven intense laser in a long waveguide with high-pressure gas, phase matching can be achieved in three distinct “regimes”. In the third regime, favorable phase matching is achieved at near-axis positions to enhance harmonic yields. Our results are supported by a full theoretical analysis, and we demonstrate that coupling of the driving laser with the high-order waveguide modes (instead of the fundamental mode used in most prior experiments) is responsible for achieving phase matching. Furthermore, we establish that this phase matching (and harmonic enhancement) is robust, and a scaling relation is derived for the necessary waveguide and gas parameters, allowing our predictions to be tested immediately in any laboratory today.

Li, Baochang [Nanjing University of Science and Te↗
NASA NTRS1981

A method for obtaining reduced-order control laws for high-order systems using optimization techniques

A method of synthesizing reduced-order optimal feedback control laws for a high-order system is developed. A nonlinear programming algorithm is employed to search for the control law design variables that minimize a performance index defined by a weighted sum of mean-square steady-state responses and control inputs. An analogy with the linear quadractic Gaussian solution is utilized to select a set of design variables and their initial values. To improve the stability margins of the system, an input-noise adjustment procedure is used in the design algorithm. The method is applied to the synthesis of an active flutter-suppression control law for a wind tunnel model of an aeroelastic wing. The reduced-order controller is compared with the corresponding full-order controller and found to provide nearly optimal performance. The performance of the present method appeared to be superior to that of two other control law order-reduction methods. It is concluded that by using the present algorithm, nearly optimal low-order control laws with good stability margins can be synthesized.

Mukhopadhyay, V.↗
DOE OSTI2025

Cavity-enhanced high-order harmonic generation for high-performance time-resolved photoemission experiments

Recent breakthroughs in high repetition-rate extreme ultraviolet (XUV) light sources and photoelectron analyzers have led to dramatic advances in the performance of time-resolved photoemission experiments. In this perspective article, we discuss the application of cavity-enhanced high-order harmonic generation, with repetition rates exceeding 10 MHz, to photoemission experiments using advanced endstations incorporating time-of-flight photoelectron analyzers. We discuss recent results, perspective on future areas for further technological improvement, and the wide array of science enabled by ultrafast XUV photoemission experiments, now in a qualitatively new regime.

Allison, Thomas K. (ORCID:0000000302898830)↗