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All sourcesNASA NTRSDOE OSTI
NASA NTRS2015

High-Order Hyperbolic Residual-Distribution Schemes on Arbitrary Triangular Grids

In this paper, we construct high-order hyperbolic residual-distribution schemes for general advection-diffusion problems on arbitrary triangular grids. We demonstrate that the second-order accuracy of the hyperbolic schemes can be greatly improved by requiring the scheme to preserve exact quadratic solutions. We also show that the improved second-order scheme can be easily extended to third-order by further requiring the exactness for cubic solutions. We construct these schemes based on the LDA and the SUPG methodology formulated in the framework of the residual-distribution method. For both second- and third-order-schemes, we construct a fully implicit solver by the exact residual Jacobian of the second-order scheme, and demonstrate rapid convergence of 10-15 iterations to reduce the residuals by 10 orders of magnitude. We demonstrate also that these schemes can be constructed based on a separate treatment of the advective and diffusive terms, which paves the way for the construction of hyperbolic residual-distribution schemes for the compressible Navier-Stokes equations. Numerical results show that these schemes produce exceptionally accurate and smooth solution gradients on highly skewed and anisotropic triangular grids, including curved boundary problems, using linear elements. We also present Fourier analysis performed on the constructed linear system and show that an under-relaxation parameter is needed for stabilization of Gauss-Seidel relaxation.

Mazaheri, Alireza↗
DOE OSTI2020

Matrix-free subcell residual distribution for Bernstein finite elements: Monolithic limiting

This paper is focused on the aspects of limiting in residual distribution (RD) schemes for high-order finite element approximations to advection problems. Both continuous and discontinuous Galerkin methods are considered in this work. Discrete maximum principles are enforced using algebraic manipulations of element contributions to the global nonlinear system. The required modifications can be carried out without calculating the element matrices and assembling their global counterparts. The components of element vectors associated with the standard Galerkin discretization are manipulated directly using localized subcell weights to achieve optimal accuracy. Low-order nonlinear RD schemes of this kind were originally developed to calculate local extremum diminishing predictors for flux-corrected transport (FCT) algorithms. In the present paper, we incorporate limiters directly into the residual distribution procedure, which makes it applicable to stationary problems and leads to well-posed nonlinear discrete problems. To circumvent the second-order accuracy barrier, the correction factors of monolithic limiting approaches and FCT schemes are adjusted using smoothness sensors based on second derivatives. Finally, the convergence behavior of presented methods is illustrated by numerical studies for two-dimensional test problems.

97 MATHEMATICS AND COMPUTING↗
NASA NTRS2016

High-Order Residual-Distribution Schemes for Discontinuous Problems on Irregular Triangular Grids

In this paper, we develop second- and third-order non-oscillatory shock-capturing hyperbolic residual distribution schemes for irregular triangular grids, extending our second- and third-order schemes to discontinuous problems. We present extended first-order N- and Rusanov-scheme formulations for hyperbolic advection-diffusion system, and demonstrate that the hyperbolic diffusion term does not affect the solution of inviscid problems for vanishingly small viscous coefficient. We then propose second- and third-order blended hyperbolic residual-distribution schemes with the extended first-order Rusanov-scheme. We show that these proposed schemes are extremely accurate in predicting non-oscillatory solutions for discontinuous problems. We also propose a characteristics-based nonlinear wave sensor for accurately detecting shocks, compression, and expansion regions. Using this proposed sensor, we demonstrate that the developed hyperbolic blended schemes do not produce entropy-violating solutions (unphysical stocks). We then verify the design order of accuracy of these blended schemes on irregular triangular grids.

Mazaheri, Alireza↗
NASA NTRS2014

High-Order Residual-Distribution Hyperbolic Advection-Diffusion Schemes: 3rd-, 4th-, and 6th-Order

In this paper, spatially high-order Residual-Distribution (RD) schemes using the first-order hyperbolic system method are proposed for general time-dependent advection-diffusion problems. The corresponding second-order time-dependent hyperbolic advection- diffusion scheme was first introduced in [NASA/TM-2014-218175, 2014], where rapid convergences over each physical time step, with typically less than five Newton iterations, were shown. In that method, the time-dependent hyperbolic advection-diffusion system (linear and nonlinear) was discretized by the second-order upwind RD scheme in a unified manner, and the system of implicit-residual-equations was solved efficiently by Newton's method over every physical time step. In this paper, two techniques for the source term discretization are proposed; 1) reformulation of the source terms with their divergence forms, and 2) correction to the trapezoidal rule for the source term discretization. Third-, fourth, and sixth-order RD schemes are then proposed with the above techniques that, relative to the second-order RD scheme, only cost the evaluation of either the first derivative or both the first and the second derivatives of the source terms. A special fourth-order RD scheme is also proposed that is even less computationally expensive than the third-order RD schemes. The second-order Jacobian formulation was used for all the proposed high-order schemes. The numerical results are then presented for both steady and time-dependent linear and nonlinear advection-diffusion problems. It is shown that these newly developed high-order RD schemes are remarkably efficient and capable of producing the solutions and the gradients to the same order of accuracy of the proposed RD schemes with rapid convergence over each physical time step, typically less than ten Newton iterations.

Mazaheri, Alireza R.↗
NASA NTRS2000

Residual Distribution Schemes for Conservation Laws Via Adaptive Quadrature

This paper considers a family of nonconservative numerical discretizations for conservation laws which retains the correct weak solution behavior in the limit of mesh refinement whenever sufficient order numerical quadrature is used. Our analysis of 2-D discretizations in nonconservative form follows the 1-D analysis of Hou and Le Floch. For a specific family of nonconservative discretizations, it is shown under mild assumptions that the error arising from non-conservation is strictly smaller than the discretization error in the scheme. In the limit of mesh refinement under the same assumptions, solutions are shown to satisfy an entropy inequality. Using results from this analysis, a variant of the "N" (Narrow) residual distribution scheme of van der Weide and Deconinck is developed for first-order systems of conservation laws. The modified form of the N-scheme supplants the usual exact single-state mean-value linearization of flux divergence, typically used for the Euler equations of gasdynamics, by an equivalent integral form on simplex interiors. This integral form is then numerically approximated using an adaptive quadrature procedure. This renders the scheme nonconservative in the sense described earlier so that correct weak solutions are still obtained in the limit of mesh refinement. Consequently, we then show that the modified form of the N-scheme can be easily applied to general (non-simplicial) element shapes and general systems of first-order conservation laws equipped with an entropy inequality where exact mean-value linearization of the flux divergence is not readily obtained, e.g. magnetohydrodynamics, the Euler equations with certain forms of chemistry, etc. Numerical examples of subsonic, transonic and supersonic flows containing discontinuities together with multi-level mesh refinement are provided to verify the analysis.

Barth, Timothy↗
DOE OSTI2023

Residuals-based distributionally robust optimization with covariate information

We consider data-driven approaches that integrate a machine learning prediction model within distributionally robust optimization (DRO) given limited joint observations of uncertain parameters and covariates. Our framework is flexible in the sense that it can accommodate a variety of regression setups and DRO ambiguity sets. We investigate asymptotic and finite sample properties of solutions obtained using Wasserstein, sample robust optimization, and phi-divergence-based ambiguity sets within our DRO formulations, and explore cross-validation approaches for sizing these ambiguity sets. Through numerical experiments, we validate our theoretical results, study the effectiveness of our approaches for sizing ambiguity sets, and illustrate the benefits of our DRO formulations in the limited data regime even when the prediction model is misspecified.

97 MATHEMATICS AND COMPUTING↗
DOE OSTI2025

Residual stress distribution in an additively manufactured complex structure by neutron diffraction measurement

Residual stress in an aerodynamically shaped Ni-based superalloy airfoil fabricated by laser powder bed fusion was measured by neutron diffraction. The experiment was conducted by considering the complex shape, implementing computer aided experiment planning, and automatic alignment at each rapid measurement. The 3-dimensional (3D) residual stress distribution in the airfoil is presented in this work, which lacks symmetry due to the complex geometry of the airfoil. In conclusion, the results provide theoretical thermal processing models a complete residual stress dataset of simulation validation on 3D shape complex structure.

Residual stress↗
DOE OSTI2021

AFRL Additive Manufacturing Modeling Series: Challenge 1, Characterization of Residual Strain Distribution in Additively-Manufactured Metal Parts Using Energy-Dispersive Diffraction

A machine component fabricated by additive manufacturing (AM) of metal powders often possesses steep residual stress gradients with significant magnitudes due to large temperature gradients that transpire within a localized area during fabrication. These processing-induced residual stresses can cause distortion of the part, and if they are of significant magnitude, they could induce cracking of the component, degrade printability, and/or diminish subsequent mechanical performance. The ability to predict these residual stresses imparted by AM is an important step in permeating AM technology for advanced manufacturing. Calibration and validation of AM process models used for prediction are, therefore, a critical step in understanding the origin and mitigating the challenges associated with residual stresses inherent to the AM process. In the present work, the residual strain distributions in components with simple geometries fabricated by a laser powder bed fusion (LBPF) process were characterized non-destructively using energy-dispersive X-ray diffraction in support of the US Air Force Research Laboratory Additive Manufacturing Modeling Challenge Series Groeber et al. (JOM 70:441-448, 2018). The measurement setup and approach are described in detail so that the data can be used as a benchmark to calibrate and validate models for the prediction of macro-scale residual stresses due to the LPBF process.

36 MATERIALS SCIENCE↗
DOE OSTI2026

Mapping the Residual Stress Distribution in Polycrystalline Quartz Tiger's Eye Using Raman Spectroscopy

Stress distribution maps in polycrystalline materials are needed to reveal stress pathways caused by short‐and long‐range interactions of crystallites. Stress induced changes to wavenumbers in Raman spectroscopy is a well‐known phenomenon that occurs when a crystalline material undergoes elastic strain. Using spatially resolved Raman spectroscopy, we have developed a technique to experimentally measure grain‐scale residual stresses across a sample of polycrystal quartz Tiger's Eye. The results of this technique are validated using multiple criteria, including evaluation of the residual stress map itself and the accuracy of the measurements taken. The residual map of Tiger's Eye shows striking similarity to observed characteristics of other polycrystalline residual stress distributions, revealing heterogenous areas of high and low magnitude stresses. The stress magnitudes are consistent with residual stress magnitudes previously measured from polycrystalline quartz. Characterization of Tiger's Eye quartz also revealed the geometric nature of the iron oxide inclusions, not previously observed in Tiger's Eye quartz. With this technique, we present a residual stress map of Tiger's Eye quartz that shows with accuracy a heterogeneous stress state with inter‐ and intra‐granular detail.

36 MATERIALS SCIENCE↗
NASA NTRS1991

Distributed Minimal Residual (DMR) method for acceleration of iterative algorithms

A new method for enhancing the convergence rate of iterative algorithms for the numerical integration of systems of partial differential equations was developed. It is termed the Distributed Minimal Residual (DMR) method and it is based on general Krylov subspace methods. The DMR method differs from the Krylov subspace methods by the fact that the iterative acceleration factors are different from equation to equation in the system. At the same time, the DMR method can be viewed as an incomplete Newton iteration method. The DMR method was applied to Euler equations of gas dynamics and incompressible Navier-Stokes equations. All numerical test cases were obtained using either explicit four stage Runge-Kutta or Euler implicit time integration. The formulation for the DMR method is general in nature and can be applied to explicit and implicit iterative algorithms for arbitrary systems of partial differential equations.

Lee, Seungsoo↗
NASA NTRS1990

Numerical experiments in geomagnetic modeling

Numerical tests were made, using least squares fitting of a spherical harmonic model, to a selection of Magsat data to determine the practical limits of this technique with modern computers. The resulting (M102189) model, whose coefficients were adjusted up to n = 50, was compared with M07AV6, a previous model which used least squares (on vector data) for coefficients up to n = 29, and Gauss-Legendre quadrature (on Z residuals) to adjust the coefficients up to n = 63. For the new least squares adjustment to n = 50 a condition number of 115 was obtained for the solution matrix, with a resulting precision of 11 significant figures. The M102189 model shows a lower and more Gaussian residual distribution than did M07AV6, though the Gaussian envelope fits to the residual distributions, even for the scalar field, gives "standard deviations' never lower than 6 nT, a factor of three higher than the estimated Magsat observational errors. Ionospheric currents are noted to have a significant effect on the coefficients of the internal potential functions.

Cain, Joseph C.↗
DOE OSTI2020

Deformation-enhanced hierarchical multiscale structure heterogeneity in a Pd-Si bulk metallic glass

Here, the multiscale structures in a Pd 82 Si 18 binary bulk metallic glass before and after deformation were studied using electron microscopies, high-energy synchrotron X-ray diffraction, and small-angle scattering techniques. The experimental results revealed an enhancement of hierarchical structure heterogeneities on multiple length scales after deformation. Hierarchical multiple shear bands of high number density were observed after bending, introducing complex but periodically distributed residual strain. Pair distribution function analysis revealed that the connectivity of the short-range clusters on the medium-range scale determines the packing density difference between the tension side and the compression side in the sample after bending. In-situ synchrotron X-ray diffraction study also revealed a transformation of connection modes among short-range clusters under uniaxial tension and compression, which is consistent with those of triaxial tension/compression parts upon bending in Pd 82 Si 18 glassy alloys. The nanoscale heterogeneities for metallic glasses after deformation observed by small-angle scattering and transmission electron microscopy may be attributed to the nanoscale amorphous phase separation and interacting multiple shear bands enhanced by plastic deformation. Our findings suggested that the enhancement of hierarchical heterogeneous structure on multiple length scales may explain the excellent plasticity of Pd-Si glassy alloys, deepening the understanding of structure-property relation during plastic deformation in metallic glasses.

36 MATERIALS SCIENCE↗
NASA NTRS1993

Hypervelocity impact study: The effect of impact angle on crater morphology

The Space Power Institute (SPI) of Auburn University has conducted preliminary tests on the effects of impact angle on crater morphology for hypervelocity impacts. Copper target plates were set at angles of 30 deg and 60 deg from the particle flight path. For the 30 deg impact, the craters looked almost identical to earlier normal incidence impacts. The only difference found was in the apparent distribution of particle residue within the crater, and further research is needed to verify this. The 60 deg impacts showed marked differences in crater symmetry, crater lip shape, and particle residue distribution. Further research on angle effects is planned, because the particle velocities for these shots were relatively slow (7 km/s or less).

Crawford, Gary↗
DOE OSTI2025

Investigation of residual stress distribution in wire-arc directed energy deposited refractory molybdenum alloy utilizing numerical thermo-mechanical analysis and neutron diffraction method

Directed energy deposition (DED), a metal additive manufacturing (AM) technique, offers higher deposition rates and energy efficiency, making it suitable for fabricating components from refractory molybdenum alloys, such as molybdenum-titanium-zirconium (TZM). However, large thermal gradients and non-equilibrium thermal cycles in DED could generate high residual stress in the component, potentially deteriorating quality and performance. Thus, this study aims to investigate residual stress generation and its distribution in wire-arc DED of TZM thin-wall, utilizing thermo-mechanical analysis and high-fidelity neutron diffraction (ND) method. Two interpass temperatures (50°C and 200°C) have been considered to investigate their impact on residual stress formation. During experiments, in-situ thermal data has been recorded using thermocouples, which have been utilized for calibrating the thermal model. Thermocouple data shows a good agreement with the simulation results, having a difference of less than 10 %. Post-deposition part deformation has been observed, which is measured using a coordinate measuring machine, showing maximum values of 0.93 mm and 0.78 mm for interpass temperatures of 50°C and 200°C, respectively. Numerical predictions of distortion deviated by less than 15% from the experimental results. ND measurement and simulation results indicate that residual stress magnitude and evolution vary across the TZM deposits, revealing microstructural anisotropy in both conditions. Notably, lower interpass temperatures resulted in higher residual stresses, confirmed by experimental and simulation data. Further, this study demonstrated that an integrated experimental and thermo-mechanical analysis can potentially reveal the temperature history, part deformation, and residual stress formation in wire-arc DED TZM alloy.

36 MATERIALS SCIENCE↗
DOE OSTI2022

Simultaneous optimization of hatching orientations and lattice density distribution for residual warpage reduction in laser powder bed fusion considering layerwise residual stress stacking

The reduction in the residual warpage generated via fabrication is an emerging issue in metals’ laser powder bed fusion additive manufacturing (AM). One solution from a design for additive manufacturing (DfAM) perspective is to arrange a hatching strategy that utilizes the asymmetry of the residual stress generated from a molten pool. Another approach is to arrange the geometry of the parts, such as forming an inner lattice structure to arrange the effective stiffness. In this study, the simultaneous optimization of the laser hatching orientation and lattice density distribution was conducted to minimize the residual warpage in lattice infill structures and confirm their synergetic effect. The mutual effect of each layer was highlighted when considering the mechanical background of the optimal results because an imbalance in the residual stress caused the warpage of the AM parts between the layers. The proposed methodology was validated via experimental verification using a quasi-2D plate, 3D bracket, and 3D connecting-rod design problems. The residual warpage in the quasi-2D study, measured as the edge’s vertical deformation, was reduced from 23.4% to 39.4% compared with the benchmark design. In the 3D studies, the warpage reductions ranged from 13.1% to 20.7%.

36 MATERIALS SCIENCE↗
NASA NTRS1990

Charge exchange of low-energy ions in thin carbon foils

In order to calibrate a time-of-flight mass spectrometer which is to be flown in the solar wind, the charge exchange properties of low-energy ions in thin carbon foils have been investigated. Incident ions of He, C, N, O, Ne, and Ar with energies in the range 0.5-2 keV/nucleon have been used to measure charge-state distribution, residual energy, and angular distribution after transmission through thin (1-6 microgram/sq cm) carbon foils. Within such foils, an equilibrium between ionization and recombination of the projectile is rapidly established, and, consequently, the charge state of the emerging particle depends essentially on its residual velocity. A comparison of the charge exchange properties of Ne-22 with Ne-20 demonstrates that indeed the velocity (and not the energy) of the emerging particle determines its final charge. A comparison of properties of different elements provides an indication of an electron shell effect. Predictions for the energy loss of ions within the carbon foils made with the TRIM code are in good agreement with the experimental results presented in this paper.

Buergi, Alfred↗