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NASA NTRS1989

Efficient estimation of diffusion during dendritic solidification

A very efficient finite difference method has been developed to estimate the solute redistribution during solidification with diffusion in the solid. This method is validated by comparing the computed results with the results of an analytical solution derived by Kobayashi (1988) for the assumptions of a constant diffusion coefficient, a constant equilibrium partition ratio, and a parabolic rate of the advancement of the solid/liquid interface. The flexibility of the method is demonstrated by applying it to the dendritic solidification of a Pb-15 wt pct Sn alloy, for which the equilibrium partition ratio and diffusion coefficient vary substantially during solidification. The fraction eutectic at the end of solidification is also obtained by estimating the fraction solid, in greater resolution, where the concentration of solute in the interdendritic liquid reaches the eutectic composition of the alloy.

Yeum, K. S.↗

NASA NTRS1996

Development of Finite Elements for Two-Dimensional Structural Analysis Using the Integrated Force Method

The Integrated Force Method has been developed in recent years for the analysis of structural mechanics problems. This method treats all independent internal forces as unknown variables that can be calculated by simultaneously imposing equations of equilibrium and compatibility conditions. In this paper a finite element library for analyzing two-dimensional problems by the Integrated Force Method is presented. Triangular- and quadrilateral-shaped elements capable of modeling arbitrary domain configurations are presented. The element equilibrium and flexibility matrices are derived by discretizing the expressions for potential and complementary energies, respectively. The displacement and stress fields within the finite elements are independently approximated. The displacement field is interpolated as it is in the standard displacement method, and the stress field is approximated by using complete polynomials of the correct order. A procedure that uses the definitions of stress components in terms of an Airy stress function is developed to derive the stress interpolation polynomials. Such derived stress fields identically satisfy the equations of equilibrium. Moreover, the resulting element matrices are insensitive to the orientation of local coordinate systems. A method is devised to calculate the number of rigid body modes, and the present elements are shown to be free of spurious zero-energy modes. A number of example problems are solved by using the present library, and the results are compared with corresponding analytical solutions and with results from the standard displacement finite element method. The Integrated Force Method not only gives results that agree well with analytical and displacement method results but also outperforms the displacement method in stress calculations.

Kaljevic, Igor↗

DOE OSTI2021

Gravitational wave propagation beyond general relativity: waveform distortions and echoes

We study the cosmological propagation of gravitational waves (GWs) beyond general relativity (GR) across homogeneous and isotropic backgrounds. We consider scenarios in which GWs interact with an additional tensor field and use a parametrized phenomenological approach that generically describes their coupled equations of motion. We analyze four distinct classes of derivative and non-derivative interactions: mass, friction, velocity, and chiral. We apply the WKB formalism to account for the cosmological evolution and obtain analytical solutions to these equations. We corroborate these results by analyzing numerically the propagation of a toy GW signal. We then proceed to use the analytical results to study the modified propagation of realistic GWs from merging compact binaries, assuming that the GW signal emitted is the same as in GR. We generically find that tensor interactions lead to copies of the originally emitted GW signal, each one with its own possibly modified dispersion relation. These copies can travel coherently and interfere with each other leading to a scrambled GW signal, or propagate decoherently and lead to echoes arriving at different times at the observer that could be misidentified as independent GW events. Depending on the type of tensor interaction, the detected GW signal may exhibit amplitude and phase distortions with respect to a GW waveform in GR, as well as birefringence effects. Furthermore, we discuss observational probes of these tensor interactions with both individual GW events, as well as population studies for both ground- and space-based detectors.

79 ASTRONOMY AND ASTROPHYSICS↗

NASA NTRS1983

Simplified Modeling of Tetrahedral Trusses

Tetrahedral-lattice structure exhibits great complexity in mathematical analysis, and complexity quickly increases with number of elements. Solutions for static and dynamic behavior of structure obtained with model consisting of analogous structure with fewer repeating elements. Advantage of continuum models is analytic solutions can be obtained, either in closed form or in relatively-tractable Taylor series.

Leondis, A.↗

NASA NTRS1982

Numerical computation of exponential matrices using the Cayley-Hamilton theorem

A method for computing exponential matrices, which often arise naturally in the solution of systems of linear differential equations, is developed. An exponential matrix is generated as a linear combination of a finite number (equal to the matrix order) of matrices, the coefficients of which are scalar infinite sums. The method can be generalized to apply to any formal power series of matrices. Attention is focused upon the exponential function, and the matrix exponent is assumed tri-diagonal in form. In such cases, the terms in the coefficient infinite sums can be extracted, as recursion relations, from the characteristic polynomial of the matrix exponent. Two numerical examples are presented in some detail: (1) the three dimensional infinitesimal rotation rate matrix, which is skew symmetric, and (2) an N-dimensional tri-diagonal and symmetric finite difference matrix which arises in the numerical solution of the heat conduction partial differential equation. In the second example, the known eigenvalues and eigenvectors of the finite difference matrix permit an analytical solution for the exponential matrix, through the theory of diagonalization and similarity transformations, which is used for independent verification. The convergence properties of the scalar infinite summations are investigated for finite difference matrices of various orders up to ten, and it is found that the number of terms required for convergence increases slowly with the order of the matrix.

Walden, H.↗

NASA NTRS1984

Skin-stiffener interface stresses in composite stiffened panels

A model and solution method for determining the normal and shear stresses in the interface between the skin and the stiffener attached flange were developed. An efficient, analytical solution procedure was developed and incorporated in a sizing code for stiffened panels. The analysis procedure described provides a means to study the effects of material and geometric design parameters on the interface stresses. These stresses include the normal stress, and the shear stresses in both the longitudinal and the transverse directions. The tendency toward skin/stiffener separation may therefore be minimized by choosing appropriate values for the design variables. The most important design variables include the relative bending stiffnesses of the skin and stiffener attached flange, the bending stiffness of the stiffener web, and the flange width. The longitudinal compressive loads in the flange and skin have significant effects on the interface stresses.

Wang, J. T. S.↗

NASA NTRSDate not supplied

Convolution-Based Numerical Solutions of Transient Temperature Fields during Powder Bed Fusion Additive Manufacturing: Theory, Accuracy, and Computational Cost

Powder bed fusion (PBF) additive manufacturing has found numerous applications in the aerospace domain. However, components fabricated via PBF have a complex time-temperature history that significantly impacts subsequent mechanical performance. This study examines convolution-based numerical solutions of transient temperature fields that support simulations involving arbitrary beam shapes and paths during PBF. The convolutional approach is verified through comparisons with analytical solutions of the temperature field. The computational speed and accuracy of the method are assessed through comparisons with other explicit and implicit numerical techniques. In addition, the straightforward translation of the approach from a CPU to a GPU implementation and the resultant performance improvement are presented. The role of the technique in predicting microstructure evolution during PBF (for a greater process-structure-property-performance framework) is also demonstrated. This work supports the development of computational materials methods for understanding and controlling the time-temperature history during PBF.

powder bed fusion↗

NASA NTRS1972

N-burn optimal analytic trajectories.

Derivation of N-burn analytic solutions for propellant-optimal transfer trajectories of a vehicle in a vacuum between arbitrary boundary conditions. Variational changes in the desired boundary conditions are expressed, in general, in terms of variational changes in the control vector and in the initial state vector. All coefficient matrices are computed recursively in terms of the analytic matrices established from the subarcs of the N-burn solution. The solution is applicable to shuttle ascent (exoatmospheric), rendezvous, and deorbit problems. Consideration is also given to state-variable and control-variable inequality constraints.

Jezewski, D. J.↗

NASA NTRS1974

Automatic numerical generation of body-fitted curvilinear coordinate system for field containing any number of arbitrary two-dimensional bodies

A method for automatic numerical generation of a general curvilinear coordinate system with coordinate lines coincident with all boundaries of a general multi-connected region containing any number of arbitrarily shaped bodies is presented. With this procedure the numerical solution of a partial differential system may be done on a fixed rectangular field with a square mesh with no interpolation required regardless of the shape of the physical boundaries, regardless of the spacing of the curvilinear coordinate lines in the physical field, and regardless of the movement of the coordinate system. Numerical solutions for the lifting and nonlifting potential flow about Joukowski and Karman-Trefftz airfoils using this coordinate system generation show excellent comparison with the analytic solutions. The application to fields with multiple bodies is illustrated by a potential flow solution for multiple airfoils.

Thompson, J. F.↗

NASA NTRS1985

A leaky magnetohydrodynamic waveguide model for the acceleration of high-speed solar wind streams in coronal holes

It is now reasonably well established that there is a correlation between high-speed solar wind streams and coronal holes. It has been concluded that a significant addition of momentum and/or energy in the region of supersonic flow is needed to explain the observed particle flux and flow speed observed in the high-speed streams. The most likely source of this additional momentum appears to be magnetohydrodynamic (MHD) waves propagating up from the solar surface. The present investigation is concerned with the propagation of MHD waves in a structure of finite transverse size, taking into account the consequences for the acceleration of high-speed solar wind streams. A waveguide solution for a model coronal hole is described, giving attention to a geometric or ray analysis of the slab waveguide, a wave mode analysis, an analytic solution of the dispersion relation for high-frequency waves, and the calculation of the time-averaged wave force.

Davila, J. M.↗

NASA NTRS2000

Aerodynamic Lift and Moment Calculations Using a Closed-Form Solution of the Possio Equation

In this paper, we present closed-form formulas for the lift and moment coefficients of a lifting surface in two dimensional, unsteady, compressible, subsonic flow utilizing a newly developed explicit analytical solution of the Possio equation. Numerical calculations are consistent with previous numerical tables based on series expansions or ad hoc numerical schemes. More importantly, these formulas lend themselves readily to flutter analysis, compared with the tedious table-look-up schemes currently in use.

Lin, Jensen↗

DOE OSTI2021

Development, Implementation, and Verification of Partially-Ionized Collisional Multifluid Plasma Models in Drekar

We consider the development of multifluid models for partially ionized multispecies plasmas. The models are composed of a standard set of five-moment fluid equations for each species plus a description of electromagnetics. The most general model considered utilizes a full set of fluid equations for each charge state of each atomic species, plus a set of fluid equations for electrons. The fluid equations are coupled through source terms describing electromagnetic coupling, ionization, recombination, charge exchange, and elastic scattering collisions in the low-density coronal limit. The form of each of these source terms is described in detail, and references for required rate coefficients are identified for a diverse range of atomic species. Initial efforts have been made to extend these models to incorporate some higher-density collisional effects, including ionization potential depression and three- body recombination. Some reductions of the general multifluid model are considered. First, a reduced multifluid model is derived which averages over all of the charge states (including neutrals) of each atomic species in the general multifluid model. The resulting model maintains full consistency with the general multifluid model from which it is derived by leveraging a quasi-steady-state collisional ionization equilibrium assumption to recover the ionization fractions required to make use of the general collision models. Further reductions are briefly considered to derive certain components of a single-fluid magnetohydrodynamics (MHD) model. In this case, a generalized Ohm's law is obtained, and the standard MHD resistivity is expressed in terms of the collisional models used in the general multifluid model. A number of numerical considerations required to obtain robust implementations of these multifluid models are discussed. First, an algebraic flux correction (AFC) stabilization approach for a continuous Galerkin finite element discretization of the multifluid system is described in which the characteristic speeds used in the stabilization of the fluid systems are synchronized across all species in the model. It is demonstrated that this synchronization is crucial in order to obtain a robust discretization of the multifluid system. Additionally, several different formulations are considered for describing the electromagnetics portion of the multifluid system using nodal continuous Galerkin finite element discretizations. The formulations considered include a parabolic divergence cleaning method and an implicit projection method for the traditional curl formulation of Maxwell's equations, a purely- hyperbolic potential-based formulation of Maxwell's equations, and a mixed hyperbolic-elliptic potential-based formulation of Maxwell's equations. Some advantages and disadvantages of each formulation are explored to compare solution robustness and the ease of use of each formulation. Numerical results are presented to demonstrate the accuracy and robustness of various components of our implementation. Analytic solutions for a spatially homogeneous damped plasma oscillation are derived in order to verify the implementation of the source terms for electromagnetic coupling and elastic collisions between fluid species. Ionization balance as a function of electron temperature is evaluated for several atomic species of interest by comparing to steady-state calculations using various sets of ionization and recombination rate coefficients. Several test problems in one and two spatial dimensions are used to demonstrate the accuracy and robustness of the discretization and stabilization approach for the fluid components of the multifluid system. This includes standard test problems for electrostatic and electromagnetic shock tubes in the two-fluid and ideal shock-MHD limits, a cylindrical diocotron instability, and the GEM challenge magnetic reconnection problem. A one-dimensional simplified prototype of an argon gas puff configuration as deployed on Sandia's Z-machine is used as a demonstration to exercise the full range of capabilities associated with the general multifluid model.

70 PLASMA PHYSICS AND FUSION TECHNOLOGY↗

NASA NTRS1979

Computations of unsteady transonic flow governed by the conservative full potential equation using an alternating direction implicit algorithm

A development was the time linearization of the density function. This linearization reduces the solution process from solving just a single equation. Two sample cases were computed. First, a one dimensional traveling shock wave was computed and compared with the analytic solution. Second, a two dimensional case was calculated for a flow field which resulted from a thickening and subsequently, thinning airfoil. The resulting flow field, which included a traveling shock wave, was compared to the flow field obtained from the low frequency, small disturbance, transonic equation.

Goorjian, P. M.↗

NASA NTRS1990

Finite element two-dimensional panel flutter at high supersonic speeds and elevated temperature

A consistent finite element formulation for flutter of two-dimensional panels including effects of temperature variation along panel length is presented. The von Karman nonlinear strain-displacement relation is used to account for the large-deflections, and the quasi-steady first-order piston theory is employed for aerodynamic loading in the finite-element formulation. The panel flutter under combined thermal/aerodynamic loading can be separated into solving two problems in sequence. They are: (1) postbuckling deflection, stresses and buckling stability boundary, and (2) critical dynamic pressure and linear flutter boundary. The solution procedures in solving these problems are presented in detail. Finite element results of two-dimensional panels with uniform temperature change are compared with classic analytical solutions. Influence of nonuniform temperature distributions on critical buckling temperature, flutter deflection and stresses, and flutter boundary is also given.

Xue, David Y.↗

NASA NTRS1999

Time of Arrival Retrievals on an Oblate Spheroidal Earth

The current National Lightning Detection Network (NLDN) retrieval algorithm is based on a chi sq minimization. This numerical approach finds the lightning ground strike location on an ellipsoidal Earth that optimally agrees with multiple station time of arrival (TOA) measurements and, if available, magnetic bearing data. In addition, an analytic solution for determining lightning ground strike locations on a spherical Earth using only TOA data has been recently introduced. In the current work, a quasi-analytic approach is suggested for determining lightning ground strike locations on an oblate spheroidal Earth by perturbing the spherical model results proposed by Koshak. Latitude, longitude, time, and the associated perturbed quantities are related through terms in a Taylor series. The correction terms may be considered collectively as a vector which may be calculated by an overconstrained inversion. Expressions for the derivatives contained in the Taylor series are given in terms of elliptic integrals.

Solakiewicz, Richard J.↗

NASA NTRS2016

Introducing a Degree of Freedom in the Family of Generalized Logarithmic Spirals

The versatility of the family of generalized logarithmic spirals is improved by introducing a degree of freedom in the solution. The low-thrust acceleration profile now includes a control term that affects both the magnitude and the direction of the thrust. Exact and fully analytic solutions to the trajectory, the velocity, the time of flight, etc. are made available. Two integrals of motion are preserved. The first one is a generalization of the equation of the energy and depends on the values of the control parameter. The second one relates to the equation of the angular momentum. The problem of finding spiral transfers between two arbitrary state vectors reduces to solving one algebraic equation with one unknown. The degree of freedom allows fixing the time of flight of the transfer. If the time of flight is fixed, then there are two equations with two unknowns. No other iterative procedures are required. Coast arcs can be introduced in the solution naturally. An explicit expression for the maximum acceleration reached along the transfer is provided. Thanks to the symmetry properties of the solution a simple algorithm for generating periodic orbits is presented. An arbitrary number of intermediate nodes can be introduced to improve the flexibility of the solution when facing optimization problems. An example of a low-thrust gravity-assist Earth-Mars-Ceres trajectory shows that the solution is comparable to that obtained with other preliminary design techniques.

Roa, Javier↗

DOE OSTI2020

Spray Mechanism of Contained-Electrospray Ionization

Analytical characteristics of contained electrospray ionization (ESI) are summarized in terms of its potential to modify the analyte solution during the stages of droplet formation to provide opportunities to generate native versus denatured biomolecular gas-phase ions, without the need for bulk-phase analyte modifications. The real-time modification of the charged microdroplets occurs in a cavity that is included in the outlet of the contained-ESI ion source. Close examination of the inside of the cavity using a high-speed camera revealed the formation of discrete droplets as well as thin liquid films in the droplets wake. When operated at 20 psi N2 pressure, the droplets were observed to move at an average speed of 8 mm/s providing ~1 s mixing time in a 10 mm cavity length. Evidence is provided for the presence of highly reactive charged droplets based on myoglobin charge state distribution, apo-myoglobin contents, and ion mobility drift time profiles under different spray conditions. Mechanistic insights for the capture of vapor-phase reagents and droplet dynamics as influenced by different operational modes are also described.

37 INORGANIC, ORGANIC, PHYSICAL, AND ANALYTICAL CH↗

DOE OSTI2021

SAS4A/SASSYS-1 Lead Verification Test Suite

SAS4A/SASSYS-1 is a simulation tool used to perform deterministic analyses of anticipated events as well as design basis and beyond design basis accidents for advanced liquid-metal-cooled nuclear reactors. With its origin as SAS1A in the late 1960s, the SAS series of codes has been under continuous use and development for over forty-five years and represents a critical investment in safety analysis capabilities for the U.S. Department of Energy. Although SAS4A/SASSYS-1 was developed to support the analysis of any liquid-metal-cooled nuclear reactor, it has primarily been utilized to design and analyze Sodium Fast Reactors (SFRs). As a result, most of the verification basis for SAS4A/SASSYS-1 utilizes sodium as a coolant and geometry descriptions that are typical of a SFR facility. In this work, new verification problems are defined to extend the applicability of the SAS4A/SASSYS-1 verification basis into Lead Fast Reactor (LFR) design space. A review of the existing SAS4A/SASSYS-1 SFR verification test suite is performed to determine the number of test cases that are SFR specific. SFR specific test cases are replicated using lead as a coolant and system layouts that are representative of LFRs. This report provides definitions and reference solutions for seven new test cases that scope lead-specific features and capabilities of the software. For a verification test to be considered acceptable, comparisons of analytical solutions to SAS4A/SASSYS-1 predictions must produce negligible or justifiable errors. All LFR specific test cases are found to be acceptable.

22 GENERAL STUDIES OF NUCLEAR REACTORS↗