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A multidomain spectral collocation scheme is proposed for the approximation of the two-dimensional Stokes problem. It is shown that the discrete velocity vector field is exactly divergence-free and we prove error estimates both for the velocity and the pressure.
Delaunay-Voronoi mesh systems provide a generalization of the classical rectangular staggered meshes to unstructured meshes. It is shown how such 'covolume' discretizations may be applied to div-curl systems in three dimensions. Error estimates are proved and confirmed by a numerical illustration.
This lecture considers a-posteriori error estimates for the numerical solution of conservation laws with time invariant constraints such as those arising in magnetohydrodynamics (MHD) and gravitational physics. Using standard duality arguments, a-posteriori error estimates for the discontinuous Galerkin finite element method are then presented for MHD with solenoidal constraint. From these estimates, a procedure for adaptive discretization is outlined. A taxonomy of Green's functions for the linearized MHD operator is given which characterizes the domain of dependence for pointwise errors. The extension to other constrained systems such as the Einstein equations of gravitational physics are then considered. Finally, future directions and open problems are discussed.
The development of dynamic mesh capability for turbulent flow simulations using the Streamlined Upwind Petrov-Galerkin (SUPG) discretization is described. The current work extends previous research to include high-order spatial accuracy, including the satisfaction of the discrete geometric conservation law (GCL) on curved elements. Two closely-related schemes are described and the ability of these schemes to satisfy the GCL, while also maintaining temporal accuracy and conservation is assessed. Studies indicate that although one scheme discretizes the time derivative in conservative form, both schemes exhibit temporal conservation errors that decrease according to the expected design order of accuracy. The source of the temporal conservation errors is examined, and it is demonstrated that many finite-volume and finite-element schemes can also be expected to have difficulty strictly satisfying conservation in time. The effects on conservation are examined and, while present in the simulations, are seen to be negligible for the problems considered.
Meshfree discretizations of state-based peridynamic models are attractive due to their ability to naturally describe fracture of general materials. However, two factors conspire to prevent meshfree discretizations of state-based peridynamics from converging to corresponding local solutions as resolution is increased: quadrature error prevents an accurate prediction of bulk mechanics, and the lack of an explicit boundary representation presents challenges when applying traction loads. Herein, we develop a reformulation of the linear peridynamic solid (LPS) model to address these shortcomings, using improved meshfree quadrature, a reformulation of the nonlocal dilatation, and a consistent handling of the nonlocal traction condition to construct a model with rigorous accuracy guarantees. In particular, these improvements are designed to enforce discrete consistency in the presence of evolving fractures, whose a priori unknown location render consistent treatment difficult. In the absence of fracture, when a corresponding classical continuum mechanics model exists, our improvements provide asymptotically compatible convergence to corresponding local solutions, eliminating surface effects and issues with traction loading which have historically plagued peridynamic discretizations. When fracture occurs, our formulation automatically provides a sharp representation of the fracture surface by breaking bonds, avoiding the loss of mass. We provide rigorous error analysis and demonstrate convergence for a number of benchmarks, including manufactured solutions, free-surface, nonhomogeneous traction loading, and composite material problems. Finally, we validate simulations of brittle fracture against a recent experiment of dynamic crack branching in soda-lime glass, providing evidence that the scheme yields accurate predictions for practical engineering problems.
A linearized Euler solver for calculating unsteady flows in turbomachinery blade rows due to both incident gusts and blade motion is presented. The model accounts for blade loading, blade geometry, shock motion, and wake motion. Assuming that the unsteadiness in the flow is small relative to the nonlinear mean solution, the unsteady Euler equations can be linearized about the mean flow. This yields a set of linear variable coefficient equations that describe the small amplitude harmonic motion of the fluid. These linear equations are then discretized on a computational grid and solved using standard numerical techniques. For transonic flows, however, one must use a linear discretization which is a conservative linearization of the non-linear discretized Euler equations to ensure that shock impulse loads are accurately captured. Other important features of this analysis include a continuously deforming grid which eliminates extrapolation errors and hence, increases accuracy, and a new numerically exact, nonreflecting far-field boundary condition treatment based on an eigenanalysis of the discretized equations. Computational results are presented which demonstrate the computational accuracy and efficiency of the method and demonstrate the effectiveness of the deforming grid, far-field nonreflecting boundary conditions, and shock capturing techniques. A comparison of the present unsteady flow predictions to other numerical, semi-analytical, and experimental methods shows excellent agreement. In addition, the linearized Euler method presented requires one or two orders-of-magnitude less computational time than traditional time marching techniques making the present method a viable design tool for aeroelastic analyses.
The global total-f gyrokinetic particle-in-cell code XGC, used to study transport in magnetic fusion plasmas or to couple with a core gyrokinetic code while functioning as an edge gyrokinetic code, implements a five-dimensional continuum grid to perform the dissipative operations, such as plasma collisions, or to exchange the particle distribution function information with a core code. To transfer the distribution function between marker particles and a rectangular two-dimensional velocity-space grid, XGC employs a bilinear mapping. The conservation of particle density and momentum is accurate enough in this bilinear operation, but the error in the particle energy conservation can become undesirably large and cause non-negligible numerical heating in a steep edge pedestal. In the present work we update XGC to use a novel mapping technique, based on the calculation of a pseudo-inverse, to exactly preserve moments up to the order of the discretization space. Here we describe the details of the implementation and we demonstrate the reduced interpolation error for a tokamak test plasma using first- and second-order elements with the pseudo-inverse method and comparing with the bilinear mapping.
Many iterative parallel-in-time algorithms have been shown to be highly efficient for diffusion-dominated partial differential equations (PDEs) but are inefficient or even divergent when applied to advection-dominated PDEs. We consider the application of the multigrid reduction-in-time (MGRIT) algorithm to linear advection PDEs. Here, the key to efficient time integration with this method is using a coarse-grid operator that provides a sufficiently accurate approximation to the so-called ideal coarse-grid operator. For certain classes of semi-Lagrangian discretizations, we present a novel semi-Lagrangian-based coarse-grid operator that leads to fast and scalable multilevel time integration of linear advection PDEs. The coarse-grid operator is composed of a semi-Lagrangian discretization followed by a correction term, with the correction designed so that the leading-order truncation error of the composite operator is approximately equal to that of the ideal coarse-grid operator. Parallel results show substantial speed-ups over sequential time integration for variable-wave-speed advection problems in one and two spatial dimensions, and using high-order discretizations up to order five. The proposed approach establishes the first practical method that provides small and scalable MGRIT iteration counts for advection problems.
Numerical sensitivity to 3 different Voronoi seeding methods is investigated for Large Eddy Simulations (LES). A second order accurate, non-dissipative finite volume discretization is used to systematically investigate the effects of different polyhedral Voronoi mesh types using a sequence of three canonical problems with increasing complexity. First, inviscid isentropic vortex propagation is studied to demonstrate the substantial reduction of errors for rhombic dodecahedron and truncated octahedral cell types over Cartesian hexagonal cells of identical spacing. Furthermore, the reduction of errors induced at cell-size transitions (grid-coarsening interfaces) due to Lloyd smoothing iterations is quantified. It is shown that by utilizing an appropriate viscous flux discretization, a constant coefficient subgrid scale model is sufficient for non-linear stability at the 2:1 cell-size transitions on polyhedral grids, although some further error reduction does occur when smoothing is utilized. Next, forced homogeneous isotropic turbulence at an asymptotically large Reynolds number is studied to demonstrate the non-dissipative character of the inviscid flux discretization, and the non-linear robustness and accuracy offered by the viscous flux discretization using a subgrid scale model for all three Voronoi grid types. Finally,Wall-Modeled Large Eddy Simulations (WMLES) are performed to study the high-lift aerodynamics on the McDonnell Douglas 30P30N multi-element airfoil at two distinct grid levels and for two distinct Voronoi cell types. The formulation is shown to predict the aerodynamic loading with high accuracy at all angles of attack when sufficient resolution is reached, and the hexagonal prism grid topology, while computationally more expensive, has higher effective resolution compared to the Cartesian grid topology with the same spacing.
Numerical sensitivity to 3 different Voronoi seeding methods is investigated for Large Eddy Simulations (LES). A second order accurate, non-dissipative finite volume discretization is used to systematically investigate the effects of different polyhedral Voronoi mesh types using a sequence of three canonical problems with increasing complexity. First, inviscid isentropic vortex propagation is studied to demonstrate the substantial reduction of errors for rhombic dodecahedron and truncated octahedral cell types over Cartesian hexagonal cells of identical spacing. Furthermore, the reduction of errors induced at cell-size transitions (grid-coarsening interfaces) due to Lloyd smoothing iterations is quantified. It is shown that by utilizing an appropriate viscous flux discretization, a constant coefficient subgrid scale model is sufficient for non-linear stability at the 2:1 cell-size transitions on polyhedral grids, although some further error reduction does occur when smoothing is utilized. Next, forced homogeneous isotropic turbulence at an asymptotically large Reynolds number is studied to demonstrate the non-dissipative character of the inviscid flux discretization, and the non-linear robustness and accuracy offered by the viscous flux discretization using a subgrid scale model for all three Voronoi grid types. Finally, Wall-Modeled Large Eddy Simulations (WMLES) are performed to study the high-lift aerodynamics on the McDonnell Douglas 30P30N multi-element airfoil at two distinct grid levels and for two distinct Voronoi cell types. The formulation is shown to predict the aerodynamic loading with high accuracy at all angles of attack when sufficient resolution is reached, and the hexagonal prism grid topology, while computationally more expensive, has higher effective resolution compared to the Cartesian grid topology with the same spacing.
Active learning sequentially selects the best instance for labeling by optimizing an acquisition function to enhance data/label efficiency. The selection can be either from a discrete instance set (pool-based scenario) or a continuous instance space (query synthesis scenario). In this work, we study both active learning scenarios for Gaussian Process Classification (GPC). The existing active learning strategies that maximize the Estimated Error Reduction (EER) aim at reducing the classification error after training with the new acquired instance in a onestep-look-ahead manner. The computation of EER-based acquisition functions is typically prohibitive as it requires retraining the GPC with every new query. Moreover, as the EER is not smooth, it can not be combined with gradient-based optimization techniques to efficiently explore the continuous instance space for query synthesis. To overcome these critical limitations, we develop computationally efficient algorithms for EER-based active learning with GPC. Further, we derive the joint predictive distribution of label pairs as a one-dimensional integral, as a result of which the computation of the acquisition function avoids retraining the GPC for each query, remarkably reducing the computational overhead. We also derive the gradient chain rule to efficiently calculate the gradient of the acquisition function, which leads to the first query synthesis active learning algorithm implementing EER-based strategies. Our experiments clearly demonstrate the computational efficiency of the proposed algorithms. We also benchmark our algorithms on both synthetic and real-world datasets, which show superior performance in terms of sampling efficiency compared to the existing state-of-the-art algorithms.
The effects of instrumentation accuracy and configuration on estimation error are studied for the small expendable-tether deployment system (SEDS) using a continuous-discrete extended Kalman filter (CDEKF) state estimator. A twelfth order model that incorporates the rigid body modes of the tether as well as the satellite attitude dynamics is developed. Simulation results using the model and the estimator indicate that the originally planned instrumentation package could not estimate the state vector adequately. Recommendations are made and results presented that reduce the estimation error by adding instruments and increasing selected measurement accuracies.
In This report, the central challenge in building a quantum computer is error correction. Unlike classical bits, which are susceptible to only one type of error, quantum bits (“qubits”) are susceptible to two types of error, corresponding to flips of the qubit state about the X- and Z-directions. While the Heisenberg Uncertainty Principle precludes simultaneous monitoring of X- and Z-flips on a single qubit, it is possible to encode quantum information in large arrays of entangled qubits that enable accurate monitoring of all errors in the system, provided the error rate is low. Another crucial requirement is that errors cannot be correlated. Here, we characterize a superconducting multiqubit circuit and find that charge fluctuations are highly correlated on a length scale over 600 µm; moreover, discrete charge jumps are accompanied by a strong transient suppression of qubit energy relaxation time across the millimeter-scale chip. The resulting correlated errors are explained in terms of the charging event and phonon-mediated quasiparticle poisoning associated with absorption of gamma rays and cosmic-ray muons in the qubit substrate. Robust quantum error correction will require the development of mitigation strategies to protect multiqubit arrays from correlated errors due to particle impacts.
A method is presented for performing color or grayscale image compression that eliminates redundant and invisible image components. The image compression uses a Discrete Cosine Transform (DCT) and each DCT coefficient yielded by the transform is quantized by an entry in a quantization matrix which determines the perceived image quality and the bit rate of the image being compressed. The quantization matrix comprises visual masking by luminance and contrast technique all resulting in a minimum perceptual error for any given bit rate, or minimum bit rate for a given perceptual error.
A collaborative team of researchers from fields of Computational Fluid Dynamics (CFD), fluid physics, computer architectures, and computer science and knowledge engineering have begun work on a prototype system that addresses several of industry's concerns in using NASA-developed CFD codes as part of the design cycle. A major problem exists in the application of CFD technologies within the aeronautics design cycle due primarily to misunderstandings in the ranges of applicability of the various solver codes or turbulence models. Features that arise during the CFD solution process need to be discriminated and recognized as actual flow features with physical support in the geometry and flow conditions of the problem being solved, or as numerical or non-physical errors arising from mis-application of solver code and its parameters, gridding strategies, or discretization. interpolations. The fundamental concept is to develop an intelligent computational system that can accept the engineer's definition of the problem and construct an optimal CFD solution. To do this requires capturing both the knowledge of how to apply the various CFD tools and how to adapt the application of those tools to flow structures as they evolve during the flow simulation. Embedded within this adaptive system approach is the additional desire to automatically identify and quantify the quality of resolution of the pertinent flow structures, be they genuine or error-induced, and then to adjust the solution strategy accordingly. This paper discusses the status of that prototyping effort.
Concentrating solar power (CSP) plants depend on thousands of heliostats whose reflectance declines as dust accumulates. Operators routinely measure reflectance to estimate soiling and, in turn, inform cleaning schedules, but the value of collecting more frequent or more accurate data has not been formally quantified. This study introduces a Monte Carlo discrete event simulation framework that integrates stochastic models of soiling, weather, and measurement error with a dynamic cleaning dispatch policy to estimate annual energy production and operations costs. Applied to two representative central-receiver field configurations, the results show that both the frequency and accuracy of reflectance measurements can meaningfully impact plant performance. In both case studies, reducing measurement intervals yields significant returns, with the energy gains greatly exceeding the cost of more frequent data collection. The simulation framework serves as a decision-support tool for CSP operators, allowing them to input site-specific soiling conditions, measurement accuracy, and survey frequency to evaluate the tradeoffs between data collection cost and energy recovery, and to identify measurement strategies that maximize plant profit.
ABSTRACT Since its formulation by Sir Isaac Newton, the problem of solving the equations of motion for three bodies under their own gravitational force has remained practically unsolved. Currently, the solution for a given initialization can only be found by performing laborious iterative calculations that have unpredictable and potentially infinite computational cost, due to the system’s chaotic nature. We show that an ensemble of converged solutions for the planar chaotic three-body problem obtained using an arbitrarily precise numerical integrator can be used to train a deep artificial neural network (ANN) that, over a bounded time interval, provides accurate solutions at a fixed computational cost and up to 100 million times faster than the numerical integrator. In addition, we demonstrate the importance of training an ANN using converged solutions from an arbitrary precise integrator, relative to solutions computed by a conventional fixed precision integrator, which can introduce errors in the training data, due to numerical round-off and time discretization, that are learned by the ANN. Our results provide evidence that, for computationally challenging regions of phase space, a trained ANN can replace existing numerical solvers, enabling fast and scalable simulations of many-body systems to shed light on outstanding phenomena such as the formation of black hole binary systems or the origin of the core collapse in dense star clusters.
The generalized winding number (GWN) is a scalar field that supports robust containment queries on curved geometry, including non-watertight, overlapping, and nested boundary representations. While queries can be easily parallelized over samples, direct evaluation on parametric curves and surfaces remains costly for large and complex models. Fast, state-of-the-art GWN approaches leverage a spatial index to approximate the GWN, typically coupled with a Taylor expansion which approximates the GWN contribution for far clusters of geometric primitives. However, such methods operate only on discrete inputs such as triangle meshes and point clouds, and would introduce containment errors near boundaries if applied to curved input. We extend support for fast GWN evaluation over arbitrary collections of NURBS curves in 2D and trimmed NURBS patches in 3D via a Bounding Volume Hierarchy that stores efficiently precomputed moment data in the hierarchy nodes. When querying the hierarchy, approximations for far clusters are used alongside direct evaluation for nearby NURBS primitives, achieving sub-linear complexity while preserving the geometric features in the vicinity of the query point. Central to our performance improvements is an adaptive subdivision strategy for NURBS primitives during a preprocessing phase, creating better spatial partitions while retaining the same accuracy for containment decisions as a direct evaluation. We demonstrate the performance and accuracy of our approach across a large collection of 2D and 3D datasets.