Self-starting multistep methods for the numerical integration of ordinary differential equations
Self-starting multistep methods for numerical integration of ordinary differential equations
DIFFERENTIAL EQUATION↗
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Theoretical investigation of wave-vector-dependent analytical and numerical formulations of the interband impact-ionization transition rate for electrons in bulk silicon and GaAs
The electron interband impact-ionization rate for both silicon and gallium arsenide is calculated using an ensemble Monte Carlo simulation with the expressed purpose of comparing different formulations of the interband ionization transition rate. Specifically, three different treatments of the transition rate are examined: the traditional Keldysh formula, a new k-dependent analytical formulation first derived by W. Quade, E. Scholl, and M. Rudan (1993), and a more exact, numerical method of Y. Wang and K. F. Brennan (1994). Although the completely numerical formulation contains no adjustable parameters and as such provides a very reliable result, it is highly computationally intensive. Alternatively, the Keldysh formular, although inherently simple and computationally efficient, fails to include the k dependence as well as the details of the energy band structure. The k-dependent analytical formulation of Quade and co-workers overcomes the limitations of both of these models but at the expense of some new parameterization. It is found that the k-dependent analytical method of Quade and co-workers produces very similar results to those obtained with the completely numerical model for some quantities. Specifically, both models predict that the effective threshold for impact ionization in GaAs and silicon is quite soft, that the majority of ionization events originate from the second conduction band in both materials, and that the transition rate is k dependent. Therefore, it is concluded that the k-dependent analytical model can qualitatively reproduce results similar to those obtained with the numerical model yet with far greater computational efficiency. Nevertheless, there exist some important drawbacks to the k-dependent analytical model of Quade and co-workers: These are that it does not accurately reproduce the quantum yield data for bulk silicon, it requires determination of a new parameter, related physically to the overlap intergrals of the Bloch state which can only be adjusted by comparison to experiment, and fails to account for any wave-vector dependence of the overlap integrals. As such the transition rate may be overestimated at those points for which 'near vertical,' small change in k, transitions occur.
Kolnik, Jan↗
High-Order Shock-Capturing Methods for Modeling Dynamics of the Solar Atmosphere
We use one-dimensional high-order central shock capturing numerical methods to study the response of various model solar atmospheres to forcing at the solar surface. The dynamics of the atmosphere is modeled with the Euler equations in a variable-sized flux tube in the presence of gravity. We study dynamics of the atmosphere suggestive of spicule formation and coronal oscillations. These studies are performed on observationally-derived model atmospheres above the quiet sun and above sunspots. To perform these simulations, we provide a new extension of existing second- and third- order shock-capturing methods to irregular grids. We also solve the problem of numerically maintaining initial hydrostatic balance via the introduction of new variables in the model equations and a careful initialization mechanism. We find several striking results: all model atmospheres respond to a single impulsive perturbation with several strong shock waves consistent with the rebound-shock model. These shock waves lift material and the transition region well into the initial corona, and the sensitivity of this lift to the initial impulse depends non-linearly on the details of the atmosphere model. We also reproduce an observed 3-minute coronal oscillation above sunspots compared to 5-minute oscillations above the quiet sun.
Bryson, Steve↗
Reducing spurious mesh motion in Lagrangian finite volume and discontinuous Galerkin hydrodynamic methods
The Lagrangian finite volume (FV) cell-centered hydrodynamic (CCH) method and the Lagrangian discontinuous Galerkin (DG) CCH method have been demonstrated to be quite stable and capable of producing very accurate solutions on many mesh topologies. However, some challenges can arise with higher-order elements and polygonal elements that have many deformational degrees of freedom. With these types of meshes, elements can deform in unphysical ways and the mesh can tangle. In this study, we present methods for obtaining more robust Lagrangian solutions on polygonal and higher-order elements. The robustness is achieved by (1) incorporating a new iterative method that modifies the velocity reconstructions in the corners of the elements, and (2) a new multidirectional approximate Riemann solver that, when coupled with the iterative method, reduces spurious mesh motion. Lastly, the details of the numerical methods are discussed and their utility is demonstrated on a diverse suite of test problems using higher-order and polygonal elements.
71 CLASSICAL AND QUANTUM MECHANICS, GENERAL PHYSIC↗
A Riccati Solution for the Ideal MHD Plasma Response with Applications to Real-time Stability Control
To effectuate near real-time feedback control of ideal MHD instabilities in a tokamak geometry, a rapid solution for stability analysis is a prerequisite. Toward this end, we reformulate the δW stability method with a Hamilton-Jacobi theory, elucidating analytical and numerical features of the generic tokamak ideal MHD stability problem. The plasma response matrix is demonstrated to be the solution of an ideal MHD matrix Riccati differential equation (MRDE). Since Riccati equations are prevalent in the control theory literature, such a shift in perspective brings to bear a range of numerical methods that are well-suited to the robust, fast solution of control problems. We discuss the usefulness of Riccati techniques in solving the stiff ODEs often encountered in ideal MHD stability analyses-—for example, in tokamak edge and stellarator physics. We then demonstrate the applicability of such methods to an existing 2D ideal MHD stability code—DCON—enabling its parallel operation in near real-time. Output is shown to match with high accuracy, and with wall-clock time ≪ 1s. Such speed may help enable active feedback ideal MHD stability control, especially in tokamak plasmas whose ideal MHD equilibria evolve with inductive timescale τ > 1s-—as in ITER.
Parallel↗
Partnership Center for High-Fidelity Boundary Plasma Simulation (Final Report)
Within the Partnership Center for High-Fidelity Boundary Plasma Simulation (HBPS), work at UT-Austin was aimed at improved verification, validation, and uncertainty quantification (VVUQ) for edge plasma simulations and on performing gyrokinetics simulations of pedestal instabilities and turbulence in order to expand foundational understanding of pedestal transport. Regarding VVUQ, the accomplishments can be summarized as follows. First, it was shown that the Moment Preserving Constrained Resampling technique, when applied periodically in particle-in-cell simulations in the XGC code, can dramatically improve the accuracy of the simulation at essentially equivalent computational cost. Second, a technique for estimating model correlations, which are required to solve the model selection and sample allocation problem in multifidelity UQ techniques, without sampling the highest fidelity, most computationally expensive model, was developed and demonstrated. Third, previously developed methods for estimating statistical and discretization errors were applied to numerical methods relevant to edge plasma simulations, namely in particle-in-cell-based approaches, and shown to work. Finally, benchmark studies for comparing gyrokinetic codes were developed and performed, leading to reasonable agreement between four commonly used codes. Regarding physics studies, gyrokinetic simulations to investigate microtearing modes in the DIII-D pedestal were performed using the GENE code.
70 PLASMA PHYSICS AND FUSION TECHNOLOGY↗
Lagrangian simulations of spray interaction with a surface using a stochastic multi-regime impingement model
Modeling of spray impingement upon the piston or cylinder surface is important for hydrocarbon emission and soot predictions in high-fidelity simulation of internal combustion engines with direct injections. Due to the complexity of the physical process of spray impingement, current computational models heavily rely on the empirical studies which involve large uncertainties. A computational model is presented for spray impingement on solid walls in general computational fluid dynamics (CFD) simulations. The model adopts the Marmanis-Thoroddsen theory for determining the number of secondary droplets and a stochastic method for calculating the velocities of the secondary droplets. The numerical methods and parallel implementation of the impingement model are presented in detail. The model is implemented in Grit, which is a recently developed CPU/GPU performance-portable Lagrangian particle library, for accelerating spray simulations. The model and its implementation are tested for examples of spray impingements on flat and smooth surfaces with different impingement angles.
Ge, Wenjun↗
A finite element formulation of Euler equations for the solution of steady transonic flows
The main objective of the considered investigation is related to the development of a relaxation scheme for the analysis of inviscid, rotational, transonic flow problems. To formulate the equations of motion for inviscid flows in a fixed coordinate system, an Eulerian type variational principle is required. The derivation of an Eulerian variational principle which is employed in the finite element formulation is discussed. The presented numerical method describes the mathematical formulation and the application of a numerical process for the direct solution of steady Euler equations. The development of the procedure as an extension of existing potential flow formulations provides the applicability of previous procedures, e.g., proper application of the artificial viscosity for supersonic elements, and the accurate modeling of the shock.
Ecer, A.↗
Numerical approach for the aerodynamic analysis if airfoils with laminar separation
A numerical method for simultaneously and efficiently coupling an external subsonic potential flow and an interior viscous flow such that the two flows match at an interfacing boundary is discussed. Both a panel method and a simple point compressible vortex model are used for the outer potential field. The interior flow solvers which were used are the Navier-Stokes and Euler codes of T. J. Coakley and the Euler code of A. Verhoff. In order to test compatibility, the panel method is coupled to the less expensive Euler codes since the coupling procedure is identical with the Navier-Stokes code. The results show significant efficiency improvements can be obtained over the uncoupled approach. Results also indicate the outer potential flow is best represented by the simple point compressible vortex model. The panel method couples smoothly to Coakley's implicit code but is numerically incompatible as coupled with the explicit Euler code. An improved Navier-Stokes code is under initial development which extends the Euler code to include the necessary viscous terms. Results are shown for all infinite length channel with one wavy periodic wall with and without laminar separation.
Halt, D. W.↗
Calculations of the unsteady, three-dimensional flow field inside a motored Wankel engine
A computer program (referred to as UF-LRC-3D) was developed for studying the unsteady, three-dimensional flow field inside the combustion chambers of motored Wankel engines as a function of engine design and operating parameters. This paper presents the details of the governing equations and the numerical method used by UF-LRC-3D. Also presented are numerical solutions generated by UF-LRC-3D showing the velocity field inside a motored Wankel engine, the mixing of nonhomogeneous fuel-air mixtures that enter through the intake port, and the mixing that takes place when a gaseous fuel is injected into the combustion chamber during compression.
Steinthorsson, Erlendur↗
An Automated Approach to Very High Order Aeroacoustic Computations in Complex Geometries
Computational aeroacoustics requires efficient, high-resolution simulation tools. And for smooth problems, this is best accomplished with very high order in space and time methods on small stencils. But the complexity of highly accurate numerical methods can inhibit their practical application, especially in irregular geometries. This complexity is reduced by using a special form of Hermite divided-difference spatial interpolation on Cartesian grids, and a Cauchy-Kowalewslci recursion procedure for time advancement. In addition, a stencil constraint tree reduces the complexity of interpolating grid points that are located near wall boundaries. These procedures are used to automatically develop and implement very high order methods (>15) for solving the linearized Euler equations that can achieve less than one grid point per wavelength resolution away from boundaries by including spatial derivatives of the primitive variables at each grid point. The accuracy of stable surface treatments is currently limited to 11th order for grid aligned boundaries and to 2nd order for irregular boundaries.
Dyson, Rodger W.↗
Searching the ASRS Database Using QUORUM Keyword Search, Phrase Search, Phrase Generation, and Phrase Discovery
To support Search Requests and Quick Responses at the Aviation Safety Reporting System (ASRS), four new QUORUM methods have been developed: keyword search, phrase search, phrase generation, and phrase discovery. These methods build upon the core QUORUM methods of text analysis, modeling, and relevance-ranking. QUORUM keyword search retrieves ASRS incident narratives that contain one or more user-specified keywords in typical or selected contexts, and ranks the narratives on their relevance to the keywords in context. QUORUM phrase search retrieves narratives that contain one or more user-specified phrases, and ranks the narratives on their relevance to the phrases. QUORUM phrase generation produces a list of phrases from the ASRS database that contain a user-specified word or phrase. QUORUM phrase discovery finds phrases that are related to topics of interest. Phrase generation and phrase discovery are particularly useful for finding query phrases for input to QUORUM phrase search. The presentation of the new QUORUM methods includes: a brief review of the underlying core QUORUM methods; an overview of the new methods; numerous, concrete examples of ASRS database searches using the new methods; discussion of related methods; and, in the appendices, detailed descriptions of the new methods.
McGreevy, Michael W.↗
Automated Approach to Very High-Order Aeroacoustic Computations
Computational aeroacoustics requires efficient, high-resolution simulation tools. For smooth problems, this is best accomplished with very high-order in space and time methods on small stencils. However, the complexity of highly accurate numerical methods can inhibit their practical application, especially in irregular geometries. This complexity is reduced by using a special form of Hermite divided-difference spatial interpolation on Cartesian grids, and a Cauchy-Kowalewski recursion procedure for time advancement. In addition, a stencil constraint tree reduces the complexity of interpolating grid points that am located near wall boundaries. These procedures are used to develop automatically and to implement very high-order methods (> 15) for solving the linearized Euler equations that can achieve less than one grid point per wavelength resolution away from boundaries by including spatial derivatives of the primitive variables at each grid point. The accuracy of stable surface treatments is currently limited to 11th order for grid aligned boundaries and to 2nd order for irregular boundaries.
Dyson, Rodger W.↗
Machine Vision for High Precision Volume Measurement Applied to Levitated Containerless Materials Processing
By combining the best practices in optical dilatometry with new numerical methods, a high-speed and high precision technique has been developed to measure volume of levitated, containerlessly processed samples with sub- pixel resolution. Containerless processing provides the ability to study highly reactive materials without the possibility of contamination affecting thermo-physical properties. Levitation is a common technique used to isolate a sample as it is being processed. Noncontact optical measurement of thermo-ophysical properties is very important as traditional measuring methods cannot be used. Modern, digitally recorded images require advanced numerical routines to recover the sub-pixel locations of sample edges and, in turn produce high precision measurements.
Bradshaw, R. C.↗
Simulations of Sea-Ice Dynamics Using the Material-Point Method
In recent years, the availability of large volumes of recorded ice motion derived from high-resolution SAR data has provided an amazingly detailed look at the deformation of the ice cover. The deformation is dominated by the appearance of linear kinematic features that have been associated with the presence of leads. These remarkable data put us in a position to begin detailed evaluation of current coupled mechanical and thermodynamic models of sea ice. This presentation will describe the material point method (MPM) for solving these model equations. MPM is a numerical method for continuum mechanics that combines the best aspects of Lagrangian and Eulerian discretizations. The material points provide a Lagrangian description of the ice that models convection naturally. Thus, properties such as ice thickness and compactness are computed in a Lagrangian frame and do not suffer from errors associated with Eulerian advection schemes, such as artificial diffusion, dispersion, or oscillations near discontinuities. This desirable property is illustrated by solving transport of ice in uniform, rotational and convergent velocity fields. Moreover, the ice geometry is represented by unconnected material points rather than a grid. This representation facilitates modeling the large deformations observed in the Arctic, as well as localized deformation along leads, and admits a sharp representation of the ice edge. MPM also easily allows the use of any ice constitutive model. The versatility of MPM is demonstrated by using two constitutive models for simulations of wind-driven ice. The first model is a standard viscous-plastic model with two thickness categories. The MPM solution to the viscous-plastic model agrees with previously published results using finite elements. The second model is a new elastic-decohesive model that explicitly represents leads. The model includes a mechanism to initiate leads, and to predict their orientation and width. The elastic-decohesion model can provide similar overall deformation as the viscous-plastic model; however, explicit regions of opening and shear are predicted. Furthermore, the efficiency of MPM with the elastic-decohesive model is competitive with the current best methods for sea ice dynamics. Simulations will also be presented for an area of the Beaufort Sea, where predictions can be validated against satellite observations of the Arctic.
Sulsky, D.↗
Effect of Delayed Energy Release on Power Normalization in Reactor Depletion Calculations
Reactor depletion calculations typically assume that the thermal power of the reactor is directly proportional to the instantaneous fission rate within the core. However, approximately 7% of the sensible energy released per fission comes from the radioactive decay of fission products, which occurs alter the fission event. Thus, the assumed direct relationship between reactor power and fission rate implies that all fission products have reached secular equilibrium within the core. In situations where fission product concentrations have not reached equilibrium (e.g., shortly after reactor startup or a change in reactor power during operation), the delayed energy release due to fission product decay creates a nonlinear relationship between reactor power and fission rate. This non-linear relationship, in turn, affects the neutron flux levels and fuel consumption rate required to maintain the desired thermal power output for the reactor. For example, during initial startup of a commercial light water reactor the apparent energy release per fission is ∼7{sup %} lower than the typical value of ∼200 MeV/fission due to the potential energy stored in radioactive fission products. As a consequence, the flux level (and, therefore, uranium consumption rate) in the reactor must be correspondingly higher to preserve the desired reactor power. As the reactor operates and fission product inventories increase, the decay heat produced by the reactor will rise, producing a corresponding increase in the apparent energy release per fission. The characteristic time-dependent behavior of energy release due to the decay of fission products (i.e., decay heat) following reactor startup, shutdown, or changes in reactor power is well-known and widely-used in reactor safety analyses. However, to our knowledge, the effects of time-dependent decay heating rates on reactor depletion calculations have not been thoroughly investigated. In this paper, we present a simple methodology for explicitly accounting for delayed energy release due to the decay of fission products in reactor depletion calculations. In the proposed method, the balance equations for fission product decay energy are incorporated into the depletion coefficient matrix, using a strategy similar to that described by Isotalo in 2016. The modified system of differential equations can be solved via a matrix exponential approximation (or other suitable numerical method) to yield solutions for both nuclide densities and decay energy emission densities as a function of time. Numerical results from the method are presented for simplified (one-group point (0-D) and 1-D) representations of a commercial pressurized water reactor (PWR), in order to estimate the effect of delayed energy release on fuel consumption and flux level immediately following reactor startup.
21 SPECIFIC NUCLEAR REACTORS AND ASSOCIATED PLANTS↗
Predicting Spacecraft Trajectories by the WeavEncke Method
A combination of methods is proposed of predicting spacecraft trajectories that possibly include multiple maneuvers and/or perturbing accelerations, with greater speed, accuracy, and repeatability than were heretofore achievable. The combination is denoted the WeavEncke method because it is based on unpublished studies by Jonathan Weaver of the orbit-prediction formulation of the noted astronomer Johann Franz Encke. Weaver evaluated a number of alternatives that arise within that formulation, arriving at an orbit-predicting algorithm optimized for complex trajectory operations. In the WeavEncke method, Encke's method of prediction of perturbed orbits is enhanced by application of modern numerical methods. Among these methods are efficient Kepler s-equation time-of-flight solutions and self-starting numerical integration with time as the independent variable. Self-starting numerical integration satisfies the requirements for accuracy, reproducibility, and efficiency (and, hence, speed). Self-starting numerical integration also supports fully analytic regulation of integration step sizes, thereby further increasing speed while maintaining accuracy.
Weaver, Jonathan K.↗
The Space-Time CE/SE Method for Solving Maxwell's Equations in Time-Domain
An innovative finite-volume-type numerical method named as the space-time conservation element and solution element (CE/SE) method is applied to solve time-dependent Maxwell's equations in this paper. Test problems of electromagnetics scattering and antenna radiation are solved for validations. Numerical results are presented and compared with the analytical solutions, showing very good agreements.
Wang, X. Y.↗