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107 records · Page 6
All sourcesNASA NTRSDOE OSTI
DOE OSTI2024

A face-upwinded spectral element method

Here we present a new high-order accurate discretisation on unstructured meshes of quadrilateral elements. Our Face Upwinded Spectral Element (FUSE) method uses the same node distribution as a high-order continuous Galerkin (CG) method, but with a particular choice of node locations within each element and an upwinded stencil on the face nodes. This results in a number of benefits, including fewer degrees of freedom and straight-forward integration with CG. We present the derivation of the scheme and the analysis of its properties, in particular showing stability using von Neumann analysis. We show numerical evidence for its accuracy and efficiency on multiple classes of problems including convection-dominated flows, Poisson's equation, and the incompressible Navier-Stokes equations.

97 MATHEMATICS AND COMPUTING↗
NASA NTRS1997

Relaxation Revisited: A Fresh Look at Multigrid for Steady Flows

The year 1971 saw the publication of one of the landmark papers in computational aerodynamics, that of Murman and Cole. As with many seminal works, its significance lies not so much in the specific problem that it addressed| small disturbance, plane transonic flow - but in the identification of a general approach to the solution of a technically important and theoretically difficult problem. The key features of Murman and Cole's work were the use of type- dependent differencing to correctly account for the proper domain of dependence of a mixed elliptic/hyperbolic equation, and the introduction of line relaxation to solve the steady flow equation. All subsequent work in transonic potential flows was based on these concepts. Jameson extended Murman and Cole's ideas to the full potential equation with two important contributions. First, he introduced the rotated difference stencil, which generalized the Murman and Cole type-dependent difference operator to general coordinates. Second, he used the interpretation, introduced by Garabedian, of relaxation as an iteration in artificial time to construct stable relaxation schemes, generalizing the original line relaxation method of Reference. The decade of the 1970s saw an explosion of activity in the solution of transonic potential flows, which has been summarized in the review article of Caughey.

Roberts, Thomas W.↗
DOE OSTI2023

Towards a unified nonlocal, peridynamics framework for the coarse-graining of molecular dynamics data with fractures

Molecular dynamics (MD) has served as a powerful tool for designing materials with reduced reliance on laboratory testing. However, the use of MD directly to treat the deformation and failure of materials at the mesoscale is still largely beyond reach. In this work, we propose a learning framework to extract a peridynamics model as a mesoscale continuum surrogate from MD simulated material fracture data sets. Firstly, we develop a novel coarse-graining method, to automatically handle the material fracture and its corresponding discontinuities in the MD displacement data sets. Inspired by the weighted essentially non-oscillatory (WENO) scheme, the key idea lies at an adaptive procedure to automatically choose the locally smoothest stencil, then reconstruct the coarse-grained material displacement field as the piecewise smooth solutions containing discontinuities. Then, based on the coarse-grained MD data, a two-phase optimization-based learning approach is proposed to infer the optimal peridynamics model with damage criterion. In the first phase, we identify the optimal nonlocal kernel function from the data sets without material damage to capture the material stiffness properties. Then, in the second phase, the material damage criterion is learnt as a smoothed step function from the data with fractures. As a result, a peridynamics surrogate is obtained. As a continuum model, our peridynamics surrogate model can be employed in further prediction tasks with different grid resolutions from training, and hence allows for substantial reductions in computational cost compared with MD. We illustrate the efficacy of the proposed approach with several numerical tests for the dynamic crack propagation problem in a single-layer graphene. Our tests show that the proposed data-driven model is robust and generalizable, in the sense that it is capable of modeling the initialization and growth of fractures under discretization and loading settings that are different from the ones used during training.

97 MATHEMATICS AND COMPUTING↗
DOE OSTI2023

Preserving Superconvergence of Spectral Elements for Curved Domains via $h$ and $p$-Geometric Refinement

Spectral element methods (SEM), extensions of finite element methods (FEM), have emerged as significant techniques for solving partial differential equations in physics and engineering. SEM can potentially deliver superior accuracy due to the potential superconvergence in nodal solutions for well-shaped tensor-product elements. However, the accuracy of SEM often degrades in complex geometries due to geometric inaccuracies near curved boundaries and the loss of superconvergence with simplicial or non-tensor-product elements. To overcome the first issue, we propose using h- and p-geometric refinement, which refines the mesh near high-curvature regions and increases the degree of geometric basis functions, respectively. We show that when using mixed-element meshes with tensor-product elements in the interior of the domain, curvature-based geometric refinement near boundaries can improve the accuracy of the interior elements by reducing pollution errors and preserving the superconvergence in nodal solutions. To address the second issue, we introduce a post-processing technique using the adaptive extended stencil finite element method (AES-FEM) to recover the accuracy near the curved boundaries. The combination of curvature-based geometric refinement and accurate post-processing offers an effective and easier-to-implement alternative to methods reliant on exact geometries. We demonstrate our techniques by solving the convection-diffusion equation in 2D and show up to two orders of magnitude of improvement in the solution accuracy, even when the elements are poorly shaped near boundaries.

97 MATHEMATICS AND COMPUTING↗
DOE OSTI2022

Learning Optimal Multigrid Smoothers via Neural Networks

Multigrid methods are one of the most efficient techniques for solving large sparse linear systems arising from partial differential equations (PDEs) and graph Laplacians from machine learning applications. One of the key components of multigrid is smoothing, which aims at reducing high-frequency errors on each grid level. However, finding optimal smoothing algorithms is problem-dependent and can impose challenges for many problems. In this paper, we propose an efficient adaptive framework for learning optimized smoothers from operator stencils in the form of convolutional neural networks (CNNs). Here, the CNNs are trained on small-scale problems from a given type of PDEs based on a supervised loss function derived from multigrid convergence theories and can be applied to large-scale problems of the same class of PDEs. Numerical results on anisotropic rotated Laplacian problems and variable coefficient diffusion problems demonstrate improved convergence rates and solution time compared with classical hand-crafted relaxation methods.

97 MATHEMATICS AND COMPUTING↗
DOE OSTI2021

Sparse invariant domain preserving discontinuous Galerkin methods with subcell convex limiting

In this paper, we develop high-order nodal discontinuous Galerkin (DG) methods for hyperbolic conservation laws that satisfy invariant domain preserving properties using subcell flux corrections and convex limiting. These methods are based on a subcell flux corrected transport (FCT) methodology that involves blending a high-order target scheme with a robust, low-order invariant domain preserving method that is obtained using a graph viscosity technique. Furthermore, the new low-order discretizations are based on sparse stencils which do not increase with the polynomial degree of the high-order DG method. As a result, the accuracy of the low-order method does not degrade when used with high-order target methods. The method is applied to both scalar conservation laws, for which the discrete maximum principle is naturally enforced, and to systems of conservation laws such as the Euler equations, for which positivity of density and a minimum principle for specific entropy are enforced. Numerical results are presented on a number of benchmark test cases.

97 MATHEMATICS AND COMPUTING↗
NASA NTRS1995

Explicit and implicit compact high-resolution shock-capturing methods for multidimensional Euler equations 1: Formulation

Two classes of explicit compact high-resolution shock-capturing methods for the multidimensional compressible Euler equations for fluid dynamics are constructed. Some of these schemes can be fourth-order accurate away from discontinuities. For the semi-discrete case their shock-capturing properties are of the total variation diminishing (TVD), total variation bounded (TVB), total variation diminishing in the mean (TVDM), essentially nonoscillatory (ENO), or positive type of scheme for 1-D scalar hyperbolic conservation laws and are positive schemes in more than one dimension. These fourth-order schemes require the same grid stencil as their second-order non-compact cousins. One class does not require the standard matrix inversion or a special numerical boundary condition treatment associated with typical compact schemes. Due to the construction, these schemes can be viewed as approximations to genuinely multidimensional schemes in the sense that they might produce less distortion in spherical type shocks and are more accurate in vortex type flows than schemes based purely on one-dimensional extensions. However, one class has a more desirable high-resolution shock-capturing property and a smaller operation count in 3-D than the other class. The extension of these schemes to coupled nonlinear systems can be accomplished using the Roe approximate Riemann solver, the generalized Steger and Warming flux-vector splitting or the van Leer type flux-vector splitting. Modification to existing high-resolution second- or third-order non-compact shock-capturing computer codes is minimal. High-resolution shock-capturing properties can also be achieved via a variant of the second-order Lax-Friedrichs numerical flux without the use of Riemann solvers for coupled nonlinear systems with comparable operations count to their classical shock-capturing counterparts. The simplest extension to viscous flows can be achieved by using the standard fourth-order compact or non-compact formula for the viscous terms.

Yee, H. C.↗
NASA NTRS2001

High Order Difference Method for Low Mach Number Aeroacoustics

A high order finite difference method with improved accuracy and stability properties for computational aeroacoustics (CAA) at low Mach numbers is proposed. The Euler equations are split into a conservative and a symmetric non- conservative portion to allow the derivation of a generalized energy estimate. Since the symmetrization is based on entropy variables, that splitting of the flux derivatives is referred to as entropy splitting. Its discretization by high order central differences was found to need less numerical dissipation than conventional conservative schemes. Owing to the large disparity of acoustic and stagnation quantities in low Mach number aeroacoustics, the split Euler equations are formulated in perturbation form. The unknowns are the small changes of the conservative variables with respect to their large stagnation values. All nonlinearities and the conservation form of the conservative portion of the split flux derivatives can be retained, while cancellation errors are avoided with its discretization opposed to the conventional conservative form. The finite difference method is third-order accurate at the boundary and the conventional central sixth-order accurate stencil in the interior. The difference operator satisfies the summation by parts property analogous to the integration by parts in the continuous energy estimate. Thus, strict stability of the difference method follows automatically. Spurious high frequency oscillations are suppressed by a characteristic-based filter similar to but without limiter. The time derivative is approximated by a 4-stage low-storage second-order explicit Runge-Kutta method. The method has been applied to simulate vortex sound at low Mach numbers. We consider the Kirchhoff vortex, which is an elliptical patch of constant vorticity rotating with constant angular frequency in irrotational flow. The acoustic pressure generated by the Kirchhoff vortex is governed by the 2D Helmholtz equation, which can be solved analytically using separation of variables.

Mueller, B.↗
DOE OSTI2024

Graph Analytics on Jellyfish topology

Because large unstructured datasets is important for many science domains, distributed graph analytics is critical to many scientists. Unfortunately, obtaining scaling and performance for irregular communication is challenging because contemporary network interconnects are primarily designed to maximize bandwidths of fixed-neighborhoods large-message exchanges (e.g., stencils). Although there is no consensus on the “best” network topologies for irregular communication, unstructured graph-based interconnects can be more suitable. We analyze three popular graph workloads – clustering, pattern enumeration, and traversal — on comparable networks (in terms of resources and costs) constructed from Jellyfish Random Regular, Dragonfly and Fat tree topologies, varying the routing algorithms. Using packet-level simulations, we demonstrate up to 60% improvement in communication time with Jellyfish due to diversity of the short paths between arbitrary endpoints, which can reduce overall network stalls and congestion.

Graph Analytics, network topology, interconnect, H↗
NASA NTRS2013

Recent Development in the CESE Method for the Solution of the Navier-Stokes Equations Using Unstructured Triangular or Tetrahedral Meshes With High Aspect Ratio

In the multidimensional CESE development, triangles and tetrahedra turn out to be the most natural building blocks for 2D and 3D spatial meshes. As such the CESE method is compatible with the simplest unstructured meshes and thus can be easily applied to solve problems with complex geometries. However, because the method uses space-time staggered stencils, solution decoupling may become a real nuisance in applications involving unstructured meshes. In this paper we will describe a simple and general remedy which, according to numerical experiments, has removed any possibility of solution decoupling. Moreover, in a real-world viscous flow simulation near a solid wall, one often encounters a case where a boundary with high curvature or sharp corner is surrounded by triangular/tetrahedral meshes of extremely high aspect ratio (up to 106). For such an extreme case, the spatial projection of a space-time compounded conservation element constructed using the original CESE design may become highly concave and thus its centroid (referred to as a spatial solution point) may lie far outside of the spatial projection. It could even be embedded beyond a solid wall boundary and causes serious numerical difficulties. In this paper we will also present a new procedure for constructing conservation elements and solution elements which effectively overcomes the difficulties associated with the original design. Another difficulty issue which was addressed more recently is the wellknown fact that accuracy of gradient computations involving triangular/tetrahedral grids deteriorates rapidly as the aspect ratio of grid cells increases. The root cause of this difficulty was clearly identified and several remedies to overcome it were found through a rigorous mathematical analysis. However, because of the length of the current paper and the complexity of mathematics involved, this new work will be presented in another paper.

Chang, Sin-Chung↗
DOE OSTI2021

Nonlinear convergence in contact mechanics: Immersed boundary finite volume

In this report we present an immersed boundary finite volume (IBM) method for simulating quasistatic contact mechanics of linearly elastic domains at small strains. In IBM, all external boundaries and internal contacts of an object are represented by embedded surfaces inside a Cartesian mesh, which need not conform to the grid lines. The contact constraints consist of the non-penetrability condition and Coulomb’s friction law, which are discretized using special interpolation stencils and enforced via penalty parameters. The resulting nonlinear system depends on displacement unknowns only. To solve it, we use the Newton method but find that it diverges frequently. To understand the divergence pattern, we analyze a simplified 2-cell problem and show that the global convergence of Newton cannot be ensured for any choice of penalty parameters. We thus propose a modified Newton solver, which guarantees convergence for the 2-cell problem and is numerically verified to converge for all the challenging simulations considered herein. While both 1 st - and 2 nd -order variants of IBM, in displacement unknowns, are proposed, the modified Newton solver applies only to the 1 st -order variant.

42 ENGINEERING↗
NASA NTRS2017

Geometrically Flexible and Efficient Flow Analysis of High Speed Vehicles Via Domain Decomposition, Part 1: Unstructured-Grid Solver for High Speed Flows

The ability to solve the equations governing the hypersonic turbulent flow of a real gas on unstructured grids using a spatially-elliptic, 2nd-order accurate, cell-centered, finite-volume method has been recently implemented in the VULCAN-CFD code. This paper describes the key numerical methods and techniques that were found to be required to robustly obtain accurate solutions to hypersonic flows on non-hex-dominant unstructured grids. The methods and techniques described include: an augmented stencil, weighted linear least squares, cell-average gradient method, a robust multidimensional cell-average gradient-limiter process that is consistent with the augmented stencil of the cell-average gradient method and a cell-face gradient method that contains a cell skewness sensitive damping term derived using hyperbolic diffusion based concepts. A data-parallel matrix-based symmetric Gauss-Seidel point-implicit scheme, used to solve the governing equations, is described and shown to be more robust and efficient than a matrix-free alternative. In addition, a y+ adaptive turbulent wall boundary condition methodology is presented. This boundary condition methodology is deigned to automatically switch between a solve-to-the-wall and a wall-matching-function boundary condition based on the local y+ of the 1st cell center off the wall. The aforementioned methods and techniques are then applied to a series of hypersonic and supersonic turbulent flat plate unit tests to examine the efficiency, robustness and convergence behavior of the implicit scheme and to determine the ability of the solve-to-the-wall and y+ adaptive turbulent wall boundary conditions to reproduce the turbulent law-of-the-wall. Finally, the thermally perfect, chemically frozen, Mach 7.8 turbulent flow of air through a scramjet flow-path is computed and compared with experimental data to demonstrate the robustness, accuracy and convergence behavior of the unstructured-grid solver for a realistic 3-D geometry on a non-hex-dominant grid.

White, Jeffery A.↗
DOE OSTI2026

A variational mimetic finite difference method for elliptic interface problems on non-matching polytopal meshes with geometric interface inconsistencies

A new variational mimetic finite difference method for elliptic interface problems with perfect and imperfect thermal contacts on non-matching polytopal meshes with geometric interface inconsistencies is developed and analyzed theoretically and numerically. The method is defined on multiple non-matching submeshes with gaps and overlaps along their interfaces. The discrete equations are derived from a minimization problem for the augmented Dirichlet functional. For a perfect thermal contact, the functional uses a modified mimetic gradient with extended stencil which couples unknowns from both sides of an interface, as well as penalty terms to enforce weak continuity of temperature across the interface. The method leads to a symmetric positive definite matrix for any scaling of the penalty terms. For an imperfect thermal contact, the Dirichlet functional is supplemented with a quadratic jump term along the interface related to the interface thermal resistance. We prove that the method conserves the total heat flux across each interface. In conclusion, the obtained results are verified with numerical experiments showing convergence in the discrete L 2 and L ∞ norms.

97 MATHEMATICS AND COMPUTING↗
NASA NTRS1995

Solutions of the benchmark problems by the dispersion-relation-preserving scheme

The 7-point stencil Dispersion-Relation-Preserving scheme of Tam and Webb is used to solve all the six categories of the CAA benchmark problems. The purpose is to show that the scheme is capable of solving linear, as well as nonlinear aeroacoustics problems accurately. Nonlinearities, inevitably, lead to the generation of spurious short wave length numerical waves. Often, these spurious waves would overwhelm the entire numerical solution. In this work, the spurious waves are removed by the addition of artificial selective damping terms to the discretized equations. Category 3 problems are for testing radiation and outflow boundary conditions. In solving these problems, the radiation and outflow boundary conditions of Tam and Webb are used. These conditions are derived from the asymptotic solutions of the linearized Euler equations. Category 4 problems involved solid walls. Here, the wall boundary conditions for high-order schemes of Tam and Dong are employed. These conditions require the use of one ghost value per boundary point per physical boundary condition. In the second problem of this category, the governing equations, when written in cylindrical coordinates, are singular along the axis of the radial coordinate. The proper boundary conditions at the axis are derived by applying the limiting process of r approaches 0 to the governing equations. The Category 5 problem deals with the numerical noise issue. In the present approach, the time-independent mean flow solution is computed first. Once the residual drops to the machine noise level, the incident sound wave is turned on gradually. The solution is marched in time until a time-periodic state is reached. No exact solution is known for the Category 6 problem. Because of this, the problem is formulated in two totally different ways, first as a scattering problem then as a direct simulation problem. There is good agreement between the two numerical solutions. This offers confidence in the computed results. Both formulations are solved as initial value problems. As such, no Kutta condition is required at the trailing edge of the airfoil.

Tam, Christopher K. W.↗
NASA NTRS2000

High Order Discontinuous Gelerkin Methods for Convection Dominated Problems with Application to Aeroacoustics

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. On the analysis side, we have studied the efficient and stable discontinuous Galerkin framework for small second derivative terms, for example in Navier-Stokes equations, and also for related equations such as the Hamilton-Jacobi equations. This is a truly local discontinuous formulation where derivatives are considered as new variables. On the applied side, we have implemented and tested the efficiency of different approaches numerically. Related issues in high order ENO and WENO finite difference methods and spectral methods have also been investigated. Jointly with Hu, we have presented a discontinuous Galerkin finite element method for solving the nonlinear Hamilton-Jacobi equations. This method is based on the RungeKutta discontinuous Galerkin finite element method for solving conservation laws. The method has the flexibility of treating complicated geometry by using arbitrary triangulation, can achieve high order accuracy with a local, compact stencil, and are suited for efficient parallel implementation. One and two dimensional numerical examples are given to illustrate the capability of the method. Jointly with Hu, we have constructed third and fourth order WENO schemes on two dimensional unstructured meshes (triangles) in the finite volume formulation. The third order schemes are based on a combination of linear polynomials with nonlinear weights, and the fourth order schemes are based on combination of quadratic polynomials with nonlinear weights. We have addressed several difficult issues associated with high order WENO schemes on unstructured mesh, including the choice of linear and nonlinear weights, what to do with negative weights, etc. Numerical examples are shown to demonstrate the accuracies and robustness of the methods for shock calculations. Jointly with P. Montarnal, we have used a recently developed energy relaxation theory by Coquel and Perthame and high order weighted essentially non-oscillatory (WENO) schemes to simulate the Euler equations of real gas. The main idea is an energy decomposition under the form epsilon = epsilon(sub 1) + epsilon(sub 2), where epsilon(sub 1) is associated with a simpler pressure law (gamma)-law in this paper) and the nonlinear deviation epsilon(sub 2) is convected with the flow. A relaxation process is performed for each time step to ensure that the original pressure law is satisfied. The necessary characteristic decomposition for the high order WENO schemes is performed on the characteristic fields based on the epsilon(sub l) gamma-law. The algorithm only calls for the original pressure law once per grid point per time step, without the need to compute its derivatives or any Riemann solvers. Both one and two dimensional numerical examples are shown to illustrate the effectiveness of this approach.

Shu, Chi-Wang↗
DOE OSTI2024

Lattice Green’s Functions for High-Order Finite Difference Stencils

Lattice Green's Functions (LGFs) are fundamental solutions to discretized linear operators, and as such they are a useful tool for solving discretized elliptic PDEs on domains that are unbounded in one or more directions. The majority of existing numerical solvers that make use of LGFs rely on a second-order discretization and operate on domains with free-space boundary conditions in all directions. Under these conditions, fast expansion methods are available that enable precomputation of 2D or 3D LGFs in linear time, avoiding the need for brute-force multi-dimensional quadrature of numerically unstable integrals. Here we focus on higher-order discretizations of the Laplace operator on domains with more general boundary conditions, by (1) providing an algorithm for fast and accurate evaluation of the LGFs associated with high-order dimension-split centered finite differences on unbounded domains, and (2) deriving closed-form expressions for the LGFs associated with both dimension-split and Mehrstellen discretizations on domains with one unbounded dimension. Through numerical experiments we demonstrate that these techniques provide LGF evaluations with near machine-precision accuracy, and that the resulting LGFs allow for numerically consistent solutions to high-order discretizations of the Poisson's equation on fully or partially unbounded 3D domains.

97 MATHEMATICS AND COMPUTING↗
DOE OSTI2022

Scalable and Energy-Efficient Methods for Interactive Exploration of Scientific Data

The main scientific contributions of this project are the following novel concepts for multidimensional arrays: shape-based similarity join (SIGMOD 2016), incremental view maintenance (SIGMOD 2017), user-defined stencil functions (HPDC 2017), and distributed caching for in-situ processing (SSDBM 2018). Building on our collaboration with the astrophysics group at LBNL, we applied these techniques to the data generated in the Palomar Transient Factory (PTF) astronomical survey. They played a pivotal role in the first-ever observation of a neutron star merger, which produces gravitational waves and turns out to be the origin of heavy elements, including gold. This has lead to a Science magazine article that has received extensive media coverage on ACM TechNews, Slashdot, FiveThirtyEight, and Quanta Magazine, among others. Additionally, two other articles detailing related aspects of the same discovery have been published in the Astrophysical Journal Letters journal. These publications have more than 3,000 citations according to Google Scholar (as of February 2022). This cross-disciplinary collaboration provided very good opportunities to apply database techniques to real-life scientific problems. The fact that they facilitated major discoveries in astrophysics proves the importance of our research. In addition to the work on multidimensional array databases, this project has also developed stochastic gradient descent (SGD) optimization algorithms for training large scale machine learning models, methods for querying in-situ data, and a database query optimizer based on sketch synopses.

79 ASTRONOMY AND ASTROPHYSICS↗