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NASA NTRS1974

A finite element solution algorithm for the Navier-Stokes equations

A finite element solution algorithm is established for the two-dimensional Navier-Stokes equations governing the steady-state kinematics and thermodynamics of a variable viscosity, compressible multiple-species fluid. For an incompressible fluid, the motion may be transient as well. The primitive dependent variables are replaced by a vorticity-streamfunction description valid in domains spanned by rectangular, cylindrical and spherical coordinate systems. Use of derived variables provides a uniformly elliptic partial differential equation description for the Navier-Stokes system, and for which the finite element algorithm is established. Explicit non-linearity is accepted by the theory, since no psuedo-variational principles are employed, and there is no requirement for either computational mesh or solution domain closure regularity. Boundary condition constraints on the normal flux and tangential distribution of all computational variables, as well as velocity, are routinely piecewise enforceable on domain closure segments arbitrarily oriented with respect to a global reference frame.

Baker, A. J.↗

NASA NTRS1994

Renormalization Group Theory of Bolgiano Scaling in Boussinesq Turbulence

Bolgiano scaling in Boussinesq turbulence is analyzed using the Yakhot-Orszag renormalization group. For this purpose, an isotropic model is introduced. Scaling exponents are calculated by forcing the temperature equation so that the temperature variance flux is constant in the inertial range. Universal amplitudes associated with the scaling laws are computed by expanding about a logarithmic theory. Connections between this formalism and the direct interaction approximation are discussed. It is suggested that the Yakhot-Orszag theory yields a lowest order approximate solution of a regularized direct interaction approximation which can be corrected by a simple iterative procedure.

Rubinstein, Robert↗

NASA NTRS1997

Direct Numerical Simulation of Incompressible Pipe Flow Using a B-Spline Spectral Method

A numerical method based on b-spline polynomials was developed to study incompressible flows in cylindrical geometries. A b-spline method has the advantages of possessing spectral accuracy and the flexibility of standard finite element methods. Using this method it was possible to ensure regularity of the solution near the origin, i.e. smoothness and boundedness. Because b-splines have compact support, it is also possible to remove b-splines near the center to alleviate the constraint placed on the time step by an overly fine grid. Using the natural periodicity in the azimuthal direction and approximating the streamwise direction as periodic, so-called time evolving flow, greatly reduced the cost and complexity of the computations. A direct numerical simulation of pipe flow was carried out using the method described above at a Reynolds number of 5600 based on diameter and bulk velocity. General knowledge of pipe flow and the availability of experimental measurements make pipe flow the ideal test case with which to validate the numerical method. Results indicated that high flatness levels of the radial component of velocity in the near wall region are physical; regions of high radial velocity were detected and appear to be related to high speed streaks in the boundary layer. Budgets of Reynolds stress transport equations showed close similarity with those of channel flow. However contrary to channel flow, the log layer of pipe flow is not homogeneous for the present Reynolds number. A topological method based on a classification of the invariants of the velocity gradient tensor was used. Plotting iso-surfaces of the discriminant of the invariants proved to be a good method for identifying vortical eddies in the flow field.

Loulou, Patrick↗

NASA NTRS2001

Calculation of Phase Equilibria in the Y2O3-Yb2O3-ZrO2 System

Rare earth oxide stabilized zirconias find a wide range of applications. An understanding of phase equilibria is essential to all applications. In this study, the available phase boundary data and thermodynamic data is collected and assessed. Calphad-type databases are developed to completely describe the Y2O3-ZrO2, Yb2O3-ZrO2, and Y2O3-Yb2O3 systems. The oxide units are treated as components and regular and subregular solution models are used. The resultant calculated phase diagrams show good agreement with the experimental data. Then the binaries are combined to form the database for the Y2O3-Yb2O3-ZrO2 psuedo-ternary.

Jacobson, Nathan S.↗

NASA NTRS2005

Dependence of Ru2O3 Activity on Composition of Silicate Melts: Using Statistical Correlations to Infer Thermodynamic Behavior in the Melt

Understanding variations in activity with composition is an essential step in improving prediction of partition coefficients during magma evolution. Variations in activity with composition are complex and do not generally exhibit ideal behavior relative to a traditional melt-component set. Although deviations from component ideality can be modeled numerically by simply fitting to compositional variables (such as in a regular or subregular solution model), such models have not been particularly successful for describing variations in trace component activities. A better approach might be to try to identify components that do a better job of describing the behavior of the species in the melt. Electrochemical Measurement of Ru2O3 activities: Electrodes were inserted into silicate melt beads of various compositions (Table 1) suspended on Ptwire loops in a 1-atm gas mixing furnace. An electrical potential was imposed between the electrodes, the imposed potential increasing along a step ramp with a pulse imposed on each step (Fig. 1). Current flows between electrodes when electroactive species in the melt are oxidized or reduced at the electrodes. The resulting current was measured at the top and bottom of the voltage pulse, and the difference (the differential current) was plotted against potential. The peak of the resulting curve is related to the activity coefficient for the particular electroactive species (Ru2O3) in the melt [1, 2, 3]. A significant part of the nonideal contribution to activity is due not to intrinsic properties of the component in the melt, but to our ignorance about the state and mixing properties of the component in the melt.

Colson, R. O.↗

NASA NTRS2021

GRACE Fast Mascons from Spherical Harmonics and a Regularization Design Trade Study

Mass concentration (mascon) solutions have become a prominent medium for investigating time-variable gravity recovered by GRACE and GRACE Follow-On. While GRACE Level-2 spherical harmonic products require various post-processing techniques to eliminate correlated noise, mascon formulations employ spatial regularization strategies during the estimation step to improve signal recovery. However, mascon estimation has traditionally required large computing resources and GRACE Level-1B processing capabilities. In this study, we show that a typical mascon estimation system can be reformulated to allow for the estimation of regularized mascons from Level-2 spherical harmonics. Provided that spherical harmonic solution covariances are available, the computed mascons will be mathematically equivalent to similar mascons estimated from Level-1B observations. This method is computationally efficient, better leverages GRACE spherical harmonics than past methods, and matches the performance of typical mascon solutions without locking scientists into pre-determined regularization designs. We develop a proof-of-concept solution using ITSG-Grace2018 and compare results with traditional mascons from JPL and NASA GSFC. We then assess the effects of spherical harmonic truncation and use of regularization correlations on basin signal recovery. We find that spherical expansions to degree and order 60 provide the minimum expansion necessary to study most basins, while larger expansions help further localize signals. We also find that diagonal regularizations (i.e., regularizations that do not contain inter-mascon correlations) are adversely affected by leakage, especially across boundaries such as coastlines where signals are not highly correlated, whereas including inter-mascon correlations and regional boundaries in the regularization greatly improves signal recovery.

M. J. Croteau↗

DOE OSTI2022

A fourth-order phase-field fracture model: Formulation and numerical solution using a continuous/discontinuous Galerkin method

Modeling crack initiation and propagation in brittle materials is of great importance to be able to predict sudden loss of load-carrying capacity and prevent catastrophic failure under severe dynamic loading conditions. Second-order phase-field fracture models have gained wide adoption given their ability to capture the formation of complex fracture patterns, e.g. via crack merging and branching, and their suitability for implementation within the context of the conventional finite element method. Higher-order phase-field models have also been proposed to increase the regularity of the exact solution and thus increase the spatial convergence rate of its numerical approximation. However, they require special numerical techniques to enforce the necessary continuity of the phase field solution. In this paper, we derive a fourth-order phase-field model of fracture in two independent ways; namely, from Hamilton’s principle and from a higher-order micromechanics-based approach. The latter approach is novel, and provides a physical interpretation of the higher-order terms in the model. In addition, we propose a continuous/discontinuous Galerkin (C/DG) method for use in computing the approximate phase-field solution. This method employs Lagrange polynomial shape functions to guarantee -continuity of the solution at inter-element boundaries, and enforces the required regularity with the aid of additional variational and interior penalty terms in the weak form. Finally, the phase-field equation is coupled with the momentum balance equation to model dynamic fracture problems in hyper-elastic materials. Two benchmark problems are presented to compare the numerical behavior of the C/DG method with mixed finite element methods.

42 ENGINEERING↗

NASA NTRS2020

Development of a Daily GRACE Mascon Solution for Terrestrial Water Storage

The Gravity Recovery and Climate Experiment (GRACE) and GRACE-Follow On missions have provided a global history of terrestrial water storage changes since 2002. Traditional GRACE products resolve monthly time-variable gravity at spatial resolutions of 300-500 km, with many recent efforts focusing on regularized mass concentration (mascon) solutions to better resolve signals spatially. However, monthly resolution inhibits the applicability of GRACE to investigations of sub-monthly signals. This study presents a new daily mass change solution estimated as deviations from the NASA Goddard Space Flight Center (GSFC) converged monthly mascon product and quantifies the fundamental trade-off between temporal and spatial resolution in GRACE-only solutions. As an iteration of the GSFC product, this daily solution represents a mixing of monthly information at higher relative spatial resolution with daily information at lower relative spatial resolution, resolving each temporal scale at the best resolution achievable without requiring hydrological model dependencies in the estimation. The resolution of the daily mascons is 300-400 km at high latitudes and 600-1,000 km lower, depending on each daily orbital track and the proximity of each mascon to constraint region boundaries (e.g., coastlines). Through simulations and model comparisons, we demonstrate that daily signals over areas larger than 400,000 km(2) are recoverable, with basins larger than 800,000 km(2) exhibiting strong signal recovery relative to leakage. This analysis establishes baselines for daily signal recovery from GRACE in the context of longer time scale solutions, characterizes leakage inherent in daily GRACE information, and creates new opportunities for applying GRACE to investigations of sub-monthly signals.

M. J. Croteau↗

DOE OSTI2020

Solving gyrokinetic systems with higher-order time dependence

We discuss theoretical and numerical aspects of gyrokinetics as a Lagrangian field theory when the field perturbation is introduced into the symplectic part. A consequence is that the field equations and particle equations of motion in general depend on the time derivatives of the field. The most well-known example is when the parallel vector potential is introduced as a perturbation, where a time derivative of the field arises only in the equations of motion, so an explicit equation for the fields may still be written. We will consider the conceptually more problematic case where the time-dependent fields appear in both the field equations and equations of motion, but where the additional term in the field equations is formally small. The conceptual issues were described by Burby (J. Plasma Phys., vol. 82 (3), 2016, 905820304): these terms lead to apparent additional degrees of freedom to the problem, so that the electric field now requires an initial condition, which is not required in low-frequency (Darwin) Vlasov–Maxwell equations. Also, the small terms in the Euler–Lagrange equations are a singular perturbation, and these two issues are interlinked. For well-behaved problems the apparent additional degrees of freedom are spurious, and the physically relevant solution may be directly identified. Because we needed to assume that the system is well behaved for small perturbations when deriving gyrokinetic theory, we must continue to assume that when solving it, and the physical solutions are thus the regular ones. The spurious nature of the singular degrees of freedom may also be seen by changing coordinate systems so the varying field appears only in the Hamiltonian. We then describe how methods appropriate for singular perturbation theory may be used to solve these asymptotic equations numerically. We then describe a proof-of-principle implementation of these methods for an electrostatic strong-flow gyrokinetic system; two basic test cases are presented to illustrate code functionality.

70 PLASMA PHYSICS AND FUSION TECHNOLOGY↗

DOE OSTI2021

An Investigation of the Effects of Volume Fraction on Drag Coefficient of Non-Spherical Particles Using PR-DNS

Prediction of the drag coefficient is required in gas-particle multiphase flow modeling and simulation. Experimental data and correlations on the fixed-bed system of spherical particles with high volume fractions for various possible arrangements are available in the literature. However, the effect of volume fraction on the drag coefficient of non-spherical particles is not well studied. In solving the momentum equation, the volume fraction plays a vital role in determining the flow resistances. In this paper, we study the impact of volume fraction in the range of 0.069 to 0.65 on the drag coefficient using the computational fluid dynamics (CFD) simulation of air for Reynold number in the range of 10 to 10000 using particle resolved direct numerical solution (PR-DNS). Regular non-spherical particles such as a cube, tetrahedron, and spheroids are used in this study since their single particle’s drag coefficient data are available in the literature for comparison. For this work, the simulations are carried out in the Ansys Fluent using polyhedral mesh, which consumes significantly less computational time and power. The study showed the sphericity and volume fraction have significant impact on the bed pressure drop and average drag coefficient of the particles in the bed especially in high Reynolds number regime. The bed of the spheroid experiences the lowest drag being the most streamlined particle, and the particles with the edges result in a large drag coefficient due to flow separation at the discontinuity. The vector plots verify this behavior where large wake regions are observed behind the tetrahedron particle.

Mahyawansi, Pratik↗

NASA NTRS2013

Advanced Signal Processing for High Temperatures Health Monitoring of Condensed Water Height in Steam Pipes

An advanced signal processing methodology is being developed to monitor the height of condensed water thru the wall of a steel pipe while operating at temperatures as high as 250deg. Using existing techniques, previous study indicated that, when the water height is low or there is disturbance in the environment, the predicted water height may not be accurate. In recent years, the use of the autocorrelation and envelope techniques in the signal processing has been demonstrated to be a very useful tool for practical applications. In this paper, various signal processing techniques including the auto correlation, Hilbert transform, and the Shannon Energy Envelope methods were studied and implemented to determine the water height in the steam pipe. The results have shown that the developed method provides a good capability for monitoring the height in the regular conditions. An alternative solution for shallow water or no water conditions based on a developed hybrid method based on Hilbert transform (HT) with a high pass filter and using the optimized windowing technique is suggested. Further development of the reported methods would provide a powerful tool for the identification of the disturbances of water height inside the pipe.

health monitoring↗

DOE OSTI2026

PDE-constrained high-order mesh optimization

Here, we present a novel framework for PDE-constrained r-adaptivity of high-order meshes. The proposed method formulates mesh movement as an optimization problem, with an objective function defined as a convex combination of a mesh quality metric and a measure of the accuracy of the PDE solution obtained via finite element discretization. The proposed formulation achieves optimized, well-defined high-order meshes by integrating mesh quality control, PDE solution accuracy, and robust gradient regularization. We adopt the Target-Matrix Optimization Paradigm to control geometric properties across the mesh, independent of the PDE of interest. To incorporate the accuracy of the PDE solution, we introduce error measures that control the finite element discretization error. The implicit dependence of these error measures on the mesh nodal positions is accurately captured by adjoint sensitivity analysis. Additionally, a convolution-based gradient regularization strategy is used to ensure stable and effective adaptation of high-order meshes. We demonstrate that the proposed framework can improve mesh quality and reduce the error by up to 10 times for the solution of Poisson and linear elasto-static problems. The approach is general with respect to the dimensionality, the order of the mesh, the types of mesh elements, and can be applied to any PDE that admits well-defined adjoint operators.

Computer science↗

NASA NTRS1986

Regularity estimates up to the boundary for elliptic systems of difference equations

Regularity estimates up to the boundary for solutions of elliptic systems of finite difference equations were proved. The regularity estimates, obtained for boundary fitted coordinate systems on domains with smooth boundary, involve discrete Sobolev norms and are proved using pseudo-difference operators to treat systems with variable coefficients. The elliptic systems of difference equations and the boundary conditions which are considered are very general in form. The regularity of a regular elliptic system of difference equations was proved equivalent to the nonexistence of eigensolutions. The regularity estimates obtained are analogous to those in the theory of elliptic systems of partial differential equations, and to the results of Gustafsson, Kreiss, and Sundstrom (1972) and others for hyperbolic difference equations.

Strikwerda, J. C.↗

NASA NTRS1988

Supersonic combusting flow - A comparison of numerical and experimental results

The flow field of an unconfined coaxial supersonic burner (SSB) was obtained numerically, using a CFD code. The explicit MacCormack algorithm, with point implicit treatment of chemistry source terms, was employed in the CFD code. Combustion of hydrogen and air was simulated by a two-step finite-rate combustion model whereas turbulence was accounted for by a Prandtl mixing length scheme. Boundary conditions and some important features of the numerical scheme were discussed. The qualitative features of the numerical solution were compared with those obtained by regular as well as schlieren photographs. General agreements of the CFD solution and the experimental photographs were observed.

Chitsomboon, T.↗

DOE OSTI2020

A Real-Time Optimization with Warm-Start of Multiperiod AC Optimal Power Flows

We present a real-time optimization strategy based on warm-start for solving a moving horizon of multi period AC optimal power flow (ACOPF) problems. In each horizon, ACOPFs are temporally interlinked via generator ramp constraints, and we assume that each horizon needs to be solved every few seconds or minutes. We introduce two approximate tracking schemes that closely follow a solution path consisting of strongly regular points. We present theoretical results bounding the tracking error by the square of the parameter changes between time periods. Experimental results for networks of sizes up to 9K buses show a fast computation time while maintaining a good solution quality, thus making our approach well suited for real-time circumstances.

moving horizon↗

DOE OSTI2026

Close-coupling approach to electron scattering with multielectron targets

Momentum-space close coupling calculations of electron scattering require removal of spurious and unphysical solutions. Here, we demonstrate here that removal of these solutions involve regulator operators that enforce Pauli exclusion selection rules in addition to removing spurious solutions. The form of a regular operator for e-H scattering has already been established, but a general extension to the multielectron case has been elusive. Here we present a general method for scattering on multielectron targets, atoms, or molecules, ensuring that the scattering solutions obey Pauli-exclusion selection rules. The regulator operator is obtained by finding the null space vectors of the 𝑁+1 electrons of the projectile and target atom scattering system. We demonstrate that this general procedure reduces to the e-H result and provide examples for He- and Li-like targets as well as guidance for implementation.

74 ATOMIC AND MOLECULAR PHYSICS↗

NASA NTRS2017

Imaging a Free-Geometry Magma Reservoir Using FEM-Based Linear Inverse Modeling of InSAR Data: Application to Rabaul Caldera, Papua New Guinea

We test an innovative inversion scheme using Green’s functions from an array of pressure sources embedded in finite-element method (FEM) models to image, without assuming an a-priori geometry, the composite and complex shape of a volcano deformation source. We invert interferometric synthetic aperture radar (InSAR) data to estimate the pressurization and shape of the magma reservoir of Rabaul caldera, Papua New Guinea. The results image the extended shallow magmatic system responsible for a broad and long-term subsidence of the caldera between 2007 February and 2010 December. Elastic FEM solutions are integrated into the regularized linear inversion of InSAR data of volcano surface displacements in order to obtain a 3-D image of the source of deformation. The Green’s function matrix is constructed from a library of forward line-of-sight displacement solutions for a grid of cubic elementary deformation sources. Each source is sequentially generated by removing the corresponding cubic elements from a common meshed domain and simulating the injection of a fluid mass flux into the cavity, which results in a pressurization and volumetric change of the fluidfilled cavity. The use of a single mesh for the generation of all FEM models avoids the computationally expensive process of non-linear inversion and remeshing a variable geometry domain. Without assuming an a-priori source geometry other than the configuration of the 3-D grid that generates the library of Green’s functions, the geodetic data dictate the geometry of the magma reservoir as a 3-D distribution of pressure (or flux of magma) within the source array. The inversion of InSAR data of Rabaul caldera shows a distribution of interconnected sources forming an amorphous, shallow magmatic system elongated under two opposite sides of the caldera. The marginal areas at the sides of the imaged magmatic system are the possible feeding reservoirs of the ongoing Tavurvur volcano eruption of andesitic products on the east side and of the past Vulcan volcano eruptions of more evolved materials on the west side. The interconnection and spatial distributions of sources correspond to the petrography of the volcanic products described in the literature and to the dynamics of the single and twin eruptions that characterize the caldera. The ability to image the complex geometry of deformation sources in both space and time can improve our ability to monitor active volcanoes, widen our understanding of the dynamics of active volcanic systems and improve the predictions of eruptions.

Inverse theory↗

NASA NTRS2017

Imaging the Complex Geometry of a Magma Reservoir Using Fem-Based Linear Inverse Modeling of InSAR Data: Application to Rabaul Caldera, Papua New Guinea

We test an innovative inversion scheme using Green’s functions from an array of pressure sources embedded in finite-element method (FEM) models to image, without assuming an a-priori geometry, the composite and complex shape of a volcano deformation source. We invert interferometric synthetic aperture radar (InSAR) data to estimate the pressurization and shape of the magma reservoir of Rabaul caldera, Papua New Guinea. The results image the extended shallow magmatic system responsible for a broad and long-term subsidence of the caldera between 2007 February and 2010 December. Elastic FEM solutions are integrated into the regularized linear inversion of InSAR data of volcano surface displacements in order to obtain a 3-D image of the source of deformation. The Green’s function matrix is constructed from a library of forward line-of-sight displacement solutions for a grid of cubic elementary deformation sources. Each source is sequentially generated by removing the corresponding cubic elements from a common meshed domain and simulating the injection of a fluid mass flux into the cavity, which results in a pressurization and volumetric change of the fluid-filled cavity. The use of a single mesh for the generation of all FEM models avoids the computationally expensive process of non-linear inversion and remeshing a variable geometry domain. Without assuming an a-priori source geometry other than the configuration of the 3-D grid that generates the library of Green’s functions, the geodetic data dictate the geometry of the magma reservoir as a 3-D distribution of pressure (or flux of magma) within the source array. The inversion of InSAR data of Rabaul caldera shows a distribution of interconnected sources forming an amorphous, shallow magmatic system elongated under two opposite sides of the caldera. The marginal areas at the sides of the imaged magmatic system are the possible feeding reservoirs of the ongoing Tavurvur volcano eruption of andesitic products on the east side and of the past Vulcan volcano eruptions of more evolved materials on the west side. The interconnection and spatial distributions of sources correspond to the petrography of the volcanic products described in the literature and to the dynamics of the single and twin eruptions that characterize the caldera. The ability to image the complex geometry of deformation sources in both space and time can improve our ability to monitor active volcanoes, widen our understanding of the dynamics of active volcanic systems and improve the predictions of eruptions.

Inverse theory↗