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DOE OSTI2015

Numerical Simulation of Big Area Additive Manufacturing (3D Printing) of a Full Size Car

Distortion and buildup of residual stresses are common design problems in structures manufactured using additive manufacturing techniques, popularly known as 3D printing. These problems lead to poor product quality, which often is improved by trial and error, where the sensitivity of different manufacturing configurations to product quality are experimentally determined. With the introduction of Big Area Additive Manufacturing (BAAM), where significant amount of material is deposited per unit time, trial and error philosophy reduces the economical efficiency of the process. In this work, a numerical simulation methodology based on Finite Element Method (FEM), multi-scale damage mechanics and fracture mechanics is introduced to simulate the BAAM process for determining product quality in terms of distortions, material damage and interface fracture due to manufacturing. In addition, this tool can be used to obtain the sensitivities of the manufacturing configurations with respect to product quality in an economically efficient manner compared to the trial and error philosophy. Computational tools providing a fully coupled thermo-mechanical solution of polymer additive manufacturing with chopped fiber reinforced plastics are presented.

42 ENGINEERING↗

DOE OSTI2014

The behavior of transverse waves in nonuniform solar flux tubes. II. Implications for coronal loop seismology

The seismology of coronal loops using observations of damped transverse oscillations in combination with results from theoretical models is a tool to indirectly infer physical parameters in the solar atmospheric plasma. Existing seismology schemes based on approximations of the period and damping time of kink oscillations are often used beyond their theoretical range of applicability. These approximations assume that the variation of density across the loop is confined to a nonuniform layer much thinner than the radius of the loop, but the results of the inversion problem often do not satisfy this preliminary hypothesis. Here, we determine the accuracy of the analytic approximations of the period and damping time, and the impact on seismology estimates when largely nonuniform loops are considered. We find that the accuracy of the approximations when used beyond their range of applicability is strongly affected by the form of the density profile across the loop, that is observationally unknown and so must be arbitrarily imposed as part of the theoretical model. The error associated with the analytic approximations can be larger than 50% even for relatively thin nonuniform layers. This error directly affects the accuracy of approximate seismology estimates compared to actual numerical inversions. In addition, assuming different density profiles can produce noncoincident intervals of the seismic variables in inversions of the same event. The ignorance about the true shape of density variation across the loop is an important source of error that may dispute the reliability of parameters seismically inferred assuming an ad hoc density profile.

79 ASTRONOMY AND ASTROPHYSICS↗

NASA NTRS2002

On Accuracy of Adaptive Grid Methods for Captured Shocks

The accuracy of two grid adaptation strategies, grid redistribution and local grid refinement, is examined by solving the 2-D Euler equations for the supersonic steady flow around a cylinder. Second- and fourth-order linear finite difference shock-capturing schemes, based on the Lax-Friedrichs flux splitting, are used to discretize the governing equations. The grid refinement study shows that for the second-order scheme, neither grid adaptation strategy improves the numerical solution accuracy compared to that calculated on a uniform grid with the same number of grid points. For the fourth-order scheme, the dominant first-order error component is reduced by the grid adaptation, while the design-order error component drastically increases because of the grid nonuniformity. As a result, both grid adaptation techniques improve the numerical solution accuracy only on the coarsest mesh or on very fine grids that are seldom found in practical applications because of the computational cost involved. Similar error behavior has been obtained for the pressure integral across the shock. A simple analysis shows that both grid adaptation strategies are not without penalties in the numerical solution accuracy. Based on these results, a new grid adaptation criterion for captured shocks is proposed.

Yamaleev, Nail K.↗

DOE OSTI2024

Can Error Mitigation Improve Trainability of Noisy Variational Quantum Algorithms?

Variational Quantum Algorithms (VQAs) are often viewed as the best hope for near-term quantum advantage. However, recent studies have shown that noise can severely limit the trainability of VQAs, e.g., by exponentially flattening the cost landscape and suppressing the magnitudes of cost gradients. Error Mitigation (EM) shows promise in reducing the impact of noise on near-term devices. Thus, it is natural to ask whether EM can improve the trainability of VQAs. In this work, we first show that, for a broad class of EM strategies, exponential cost concentration cannot be resolved without committing exponential resources elsewhere. This class of strategies includes as special cases Zero Noise Extrapolation, Virtual Distillation, Probabilistic Error Cancellation, and Clifford Data Regression. Second, we perform analytical and numerical analysis of these EM protocols, and we find that some of them (e.g., Virtual Distillation) can make it harder to resolve cost function values compared to running no EM at all. As a positive result, we do find numerical evidence that Clifford Data Regression (CDR) can aid the training process in certain settings where cost concentration is not too severe. Our results show that care should be taken in applying EM protocols as they can either worsen or not improve trainability. On the other hand, our positive results for CDR highlight the possibility of engineering error mitigation methods to improve trainability.

71 CLASSICAL AND QUANTUM MECHANICS, GENERAL PHYSIC↗

DOE OSTI2014

Communication: Active space decomposition with multiple sites: Density matrix renormalization group algorithm

We extend the active space decomposition method, recently developed by us, to more than two active sites using the density matrix renormalization group algorithm. The fragment wave functions are described by complete or restricted active-space wave functions. Numerical results are shown on a benzene pentamer and a perylene diimide trimer. It is found that the truncation errors in our method decrease almost exponentially with respect to the number of renormalization states M, allowing for numerically exact calculations (to a few μE{sub h} or less) with M = 128 in both cases. This rapid convergence is because the renormalization steps are used only for the interfragment electron correlation.

37 INORGANIC, ORGANIC, PHYSICAL, AND ANALYTICAL CH↗

NASA NTRS1980

UDU/T/ covariance factorization for Kalman filtering

There has been strong motivation to produce numerically stable formulations of the Kalman filter algorithms because it has long been known that the original discrete-time Kalman formulas are numerically unreliable. Numerical instability can be avoided by propagating certain factors of the estimate error covariance matrix rather than the covariance matrix itself. This paper documents filter algorithms that correspond to the covariance factorization P = UDU(T), where U is a unit upper triangular matrix and D is diagonal. Emphasis is on computational efficiency and numerical stability, since these properties are of key importance in real-time filter applications. The history of square-root and U-D covariance filters is reviewed. Simple examples are given to illustrate the numerical inadequacy of the Kalman covariance filter algorithms; these examples show how factorization techniques can give improved computational reliability.

Thornton, C. L.↗

DOE OSTI2015

A heterogeneous stochastic FEM framework for elliptic PDEs

We introduce a new concept of sparsity for the stochastic elliptic operator −div(a(x,ω)∇(⋅)), which reflects the compactness of its inverse operator in the stochastic direction and allows for spatially heterogeneous stochastic structure. This new concept of sparsity motivates a heterogeneous stochastic finite element method (HSFEM) framework for linear elliptic equations, which discretizes the equations using the heterogeneous coupling of spatial basis with local stochastic basis to exploit the local stochastic structure of the solution space. We also provide a sampling method to construct the local stochastic basis for this framework using the randomized range finding techniques. The resulting HSFEM involves two stages and suits the multi-query setting: in the offline stage, the local stochastic structure of the solution space is identified; in the online stage, the equation can be efficiently solved for multiple forcing functions. An online error estimation and correction procedure through Monte Carlo sampling is given. Numerical results for several problems with high dimensional stochastic input are presented to demonstrate the efficiency of the HSFEM in the online stage.

71 CLASSICAL AND QUANTUM MECHANICS, GENERAL PHYSIC↗

NASA NTRS2003

System for Centering a Turbofan in a Nacelle During Tests

A feedback position-control system has been developed for maintaining the concentricity of a turbofan with respect to a nacelle during acoustic and flow tests in a wind tunnel. The system is needed for the following reasons: Thermal and thrust loads can displace the fan relative to the nacelle; In the particular test apparatus (see Figure 1), denoted as a rotor-only nacelle (RAN), the struts, vanes, and other stator components of a turbofan engine that ordinarily maintain the required concentricity in the face of thermal and thrust loads are not present; and The struts and stator components are not present because it is necessary to provide a flow path that is acoustically clean in the sense that the measured noise can be attributed to the fan alone. The system is depicted schematically in Figure 2. The nacelle is supported by two struts attached to a two-axis traverse table located outside the wind-tunnel wall. Two servomotors acting through 100:1 gearboxes drive the table along the Y and Z axes, which are perpendicular to the axis of rotation. The Y and Z components of the deviation from concentricity are measured by four laser displacement sensors mounted on the nacelle and aimed at reflective targets on the center body, which is part of the fan assembly. The outputs of the laser displacement sensors are digitized and processed through a personal computer programmed with control software. The control output of the computer commands the servomotors to move the table as needed to restore concentricity. Numerous software and hardware travel limits and alarms are provided to maximize safety. A highly ablative rub strip in the nacelle minimizes the probability of damage in the event that a deviation from concentricity exceeds the radial clearance [<0.004 in. (<0.1 mm)] between the inner surface of the nacelle and the tips of the fan blades. To be able to prevent an excursion in excess of the tip clearance, the system must be accurate enough to control X and Y displacements to within 0.001 in. (.0.025 mm). One characteristic essential to such accuracy is sufficient rigidity in the mechanical components of the system to prevent excitation of vibrations in the strut/ nacelle subsystem. The need for such a high degree of accuracy prompted a comprehensive analysis of sources of measurement and control errors, followed by rigorous design efforts to minimize these errors. As a result, the design of the system incorporates numerous improvements in hardware, software, and operational procedures.

Cunningham, Cameron C.↗

NASA NTRS2002

Local Discontinuous Galerkin Methods for Partial Differential Equations with Higher Order Derivatives

In this paper we review the existing and develop new continuous Galerkin methods for solving time dependent partial differential equations with higher order derivatives in one and multiple space dimensions. We review local discontinuous Galerkin methods for convection diffusion equations involving second derivatives and for KdV type equations involving third derivatives. We then develop new local discontinuous Galerkin methods for the time dependent bi-harmonic type equations involving fourth derivatives, and partial differential equations involving fifth derivatives. For these new methods we present correct interface numerical fluxes and prove L(exp 2) stability for general nonlinear problems. Preliminary numerical examples are shown to illustrate these methods. Finally, we present new results on a post-processing technique, originally designed for methods with good negative-order error estimates, on the local discontinuous Galerkin methods applied to equations with higher derivatives. Numerical experiments show that this technique works as well for the new higher derivative cases, in effectively doubling the rate of convergence with negligible additional computational cost, for linear as well as some nonlinear problems, with a local uniform mesh.

Yan, Jue↗

NASA NTRS1996

Statistical Mechanics and Dynamics of the Outer Solar System.I. The Jupiter/Saturn Zone

We report on numerical simulations designed to understand how the solar system evolved through a winnowing of planetesimals accreeted from the early solar nebula. This sorting process is driven by the energy and angular momentum and continues to the present day. We reconsider the existence and importance of stable niches in the Jupiter/Saturn Zone using greatly improved numerical techniques based on high-order optimized multi-step integration schemes coupled to roundoff error minimizing methods.

numerical simulations planetesimals planet formati↗

DOE OSTI2016

A low-communication, parallel algorithm for solving PDEs based on range decomposition

This work proposes a new, low-communication algorithm for solving PDEs on massively parallel computers. The range decomposition (RD) algorithm exposes coarse-grain parallelism by applying nested iteration and adaptive mesh refinement locally before performing a global communication step. Just a few such steps are observed to be sufficient to obtain accuracy within a small multiple of discretization error. The target applications are petascale and exascale machines, where hierarchical parallelism is required and traditional parallel numerical PDE communication patterns are costly because of message latency. The RD algorithm uses a partition of unity to equally distribute the error, and thus, the work. The computational advantages of this approach are that the decomposed problems can be solved in parallel without any communication until the partitioned solutions are summed. This offers potential advantages in the paradigm of expensive communication but very cheap computation. This paper introduces the method and explains the details of the communication step. Two performance models are developed, showing that the latency cost associated with a traditional parallel implementation of nested iteration is proportional to log(P) 2 , whereas the RD method reduces the communication latency to log(P), while maintaining similar bandwidth costs. Numerical results for two problems, Laplace and advection diffusion, demonstrate the enhanced performance, and a heuristic argument explains why the method converges quickly.

97 MATHEMATICS AND COMPUTING↗

DOE OSTI2018

On the convergence and convergence order of finite volume gradient schemes for oblique derivative boundary value problems

This work is an improvement of the previous note (Bradji in: Fuhrmann et al., Finite volumes for complex applications VII–methods, theoretical aspects. Proceedings of the FVCA 7, Berlin, 2014) which dealt with the convergence analysis of a finite volume scheme for the Poisson’s equation with a linear oblique derivative boundary condition. The formulation of the finite volume scheme given in Bradji (in: Fuhrmann et al., Finite volumes for complex applications VII–methods, theoretical aspects. Proceedings of the FVCA 7, Berlin, 2014) involves the discrete gradient introduced recently in Eymard et al. (IMA J Numer Anal 30(4):1009–1043, 2010). In this paper, we consider the convergence analysis of finite volume schemes involving the discrete gradient of Eymard et al. (IMA J Numer Anal 30(4):1009–1043, 2010) for elliptic and parabolic equations with linear oblique derivative boundary conditions. Linear oblique derivative boundary conditions arise for instance in the study of the motion of water in a canal, cf. Lesnic (Commun Numer Methods Eng 23(12):1071–1080, 2007). We derive error estimates in several norms which allow us to get error estimates for the approximations of the exact solutions and its first derivatives. In particular, we provide an error estimate between the gradient of the exact solutions and the discrete gradient of the approximate solutions. Convergence of the family of finite volume approximate solutions towards the exact solution under weak regularity assumption is also investigated. In the case of parabolic equations with oblique derivative boundary conditions, we develop a new discrete a priori estimate result. The proof of this result is based on the use of a discrete mean Poincaré–Wirtinger inequality. Thanks to the stated a priori estimate and to a comparison with an appropriately chosen auxiliary finite volume scheme, we derive the convergence results. This work can be viewed as a continuation of the previous work (Bradji and Gallouët in Int J Finite Vol 3(2):1–35, 2006) where a convergence analysis for a finite volume scheme, based on the admissible mesh of Eymard et al. (In: Ciarlet and Lions, Handbook of numerical analysis, North-Holland, Amsterdam, 2000), for the Poisson’s equation with a linear oblique derivative boundary conditions is given. The obtained convergence results do not require any relation between the mesh sizes of the spatial and time discretizations. Some numerical tests are presented for both elliptic and parabolic equations. In particular, we present three methods to compute the discrete solution.

97 MATHEMATICS AND COMPUTING↗

NASA NTRS1982

The effects of transmitter/receiver clock time-base instability on coherent communication system performance

The purpose of this paper is to present a model and associated analysis of the deleterious effects which both transmitter and receiver data clock time-base instability have on bit-error probability. The types of time-base instability modeled and analyzed include what is commonly referred to as bit jitter (epoch jitter), bit-jitter rate (time interval jitter), and data asymmetry. The bit-error probability is determined as a function of the statistical characteristics of the time-base instability, the data-transition density, the signal-to-noise ratio in the bit-rate bandwidth and the mechanizations of both the data-bit detector and bit synchronizer. The latter is manifested through the bit synchronizer's natural frequency and loop-damping parameters. Numerical results which indicate the sensitivity of the system bit-error rate to various parameters of the model are provided.

Chie, C. M.↗

NASA NTRS1973

Heat conduction errors and time lag in cryogenic thermometer installations

Installation practices are recommended that will increase rate of heat exchange between the thermometric sensing element and the cryogenic fluid, in addition to bringing about a reduction in the rate of undesired heat transfer to higher temperature objects. Formulas and numerical data are given that help to estimate the magnitude of heat conduction errors and of time lag in response.

Warshawsky, I.↗

DOE OSTI2018

Degenerate variational integrators for magnetic field line flow and guiding center trajectories

Symplectic integrators offer many benefits for numerically approximating solutions to Hamiltonian differential equations, including bounded energy error and the preservation of invariant sets. Two important Hamiltonian systems encountered in plasma physics—the flow of magnetic field lines and the guiding center motion of magnetized charged particles—resist symplectic integration by conventional means because the dynamics are most naturally formulated in non-canonical coordinates. New algorithms were recently developed using the variational integration formalism; however, those integrators were found to admit parasitic mode instabilities due to their multistep character. This work eliminates the multistep character, and therefore the parasitic mode instabilities via an adaptation of the variational integration formalism that we deem “degenerate variational integration.” Both the magnetic field line and guiding center Lagrangians are degenerate in the sense that the resultant Euler-Lagrange equations are systems of first-order ordinary differential equations. We show that retaining the same degree of degeneracy when constructing discrete Lagrangians yields one-step variational integrators preserving a non-canonical symplectic structure. Lastly, numerical examples demonstrate the benefits of the new algorithms, including superior stability relative to the existing variational integrators for these systems and superior qualitative behavior relative to non-conservative algorithms.

70 PLASMA PHYSICS AND FUSION TECHNOLOGY↗

DOE OSTI2019

Space–Time Least-Squares Petrov–Galerkin Projection for Nonlinear Model Reduction

This work proposes a space--time least-squares Petrov--Galerkin (ST-LSPG) projection method for model reduction of nonlinear dynamical systems. In contrast to typical nonlinear model-reduction methods that first apply (Petrov--)Galerkin projection in the spatial dimension and subsequently apply time integration to numerically resolve the resulting low-dimensional dynamical system, the proposed method applies projection in space and time simultaneously. To accomplish this, the method first introduces a low-dimensional space--time trial subspace, which can be obtained by computing tensor decompositions of state-snapshot data. The method then computes discrete-optimal approximations in this space--time trial subspace by minimizing the residual arising after time discretization over all space and time in a weighted $\ell^2$-norm. This norm can be defined to enable complexity reduction (i.e., hyper-reduction) in time, which leads to space--time collocation and space--time Gauss--Newton with Approximated Tensors (GNAT) variants of the ST-LSPG method. Advantages of the approach relative to typical spatial-projection-based nonlinear model reduction methods such as Galerkin projection and least-squares Petrov--Galerkin projection include a reduction of both the spatial and temporal dimensions of the dynamical system, and a priori error bounds that bound the solution error by the best space--time approximation error and whose stability constants exhibit slower growth in time. Numerical examples performed on model problems in fluid dynamics demonstrate the ability of the method to generate orders-of-magnitude computational savings relative to spatial-projection-based reduced-order models without sacrificing accuracy for a fixed spatio-temporal discretization.

97 MATHEMATICS AND COMPUTING↗

NASA NTRS1992

Error analysis applied to several inversion techniques used for the retrieval of middle atmospheric constituents from limb-scanning MM-wave spectroscopic measurements

The formal retrieval error analysis of Rodgers (1990) allows the quantitative determination of such retrieval properties as measurement error sensitivity, resolution, and inversion bias. This technique was applied to five numerical inversion techniques and two nonlinear iterative techniques used for the retrieval of middle atmospheric constituent concentrations from limb-scanning millimeter-wave spectroscopic measurements. It is found that the iterative methods have better vertical resolution, but are slightly more sensitive to measurement error than constrained matrix methods. The iterative methods converge to the exact solution, whereas two of the matrix methods under consideration have an explicit constraint, the sensitivity of the solution to the a priori profile. Tradeoffs of these retrieval characteristics are presented.

Puliafito, E.↗

DOE OSTI2019

An a posteriori residual-based spatial discretization error estimator for S N neutron transport

The “residual source estimator”, an implicit residual-based spatial discretization error estimator for S N neutron transport methods, is introduced. The estimator uses an approximation of the residual as a fixed source for an analogous transport problem whose solution is the error estimate, using the same transport operator as the original numerical solution. The complete derivation of the residual source estimator, the approximation of the residual, and higher-order derivatives is included. The performance of the residual source estimator for 0-order DGFEM is tested against reference solutions. For problems with true solutions discontinuous in the first derivative across the singular characteristics, the residual source estimator is generally accurate, precise, and computationally affordable compared to similar estimators. It suffers in problems where solutions are discontinuous, particularly when the discontinuity inflicts significant inaccuracies on the discrete solution; however, compared to other estimators, which suffer similarly, it remains a competitive option.

42 ENGINEERING↗