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NASA NTRS1992

Model reduction using new optimal Routh approximant technique

An optimal Routh approximant of a single-input single-output dynamic system is a reduced-order transfer function of which the denominator is obtained by the Routh approximation method while the numerator is determined by minimizing a time-response integral-squared-error (ISE) criterion. In this paper, a new elegant approach is presented for obtaining the optimal Routh approximants for linear time-invariant continuous-time systems. The approach is based on the Routh canonical expansion, which is a finite-term orthogonal series of rational basis functions, and minimization of the ISE criterion. A procedure for combining the above approach with the bilinear transformation is also presented in order to obtain the optimal bilinear Routh approximants of linear time-invariant discrete-time systems. The proposed technique is simple in formulation and is amenable to practical implementation.

Hwang, Chyi↗

DOE OSTI2025

An adaptive, data-driven multiscale approach for dense granular flows

The accuracy of coarse-grained continuum models of dense granular flows is limited by the lack of high-fidelity closure models for granular rheology. One approach to addressing this issue, referred to as the hierarchical multiscale method, is to use a high-fidelity fine-grained model to compute the closure terms needed by the coarse-grained model. The difficulty with this approach is that the overall model can become computationally intractable due to the high computational cost of the high-fidelity model. In this work, we describe a multiscale modeling approach for dense granular flows that utilizes neural networks trained using high-fidelity discrete element method (DEM) simulations to approximate the constitutive granular rheology for a continuum incompressible flow model. Our approach leverages an ensemble of neural networks to estimate predictive uncertainty that allows us to determine whether the rheology at a given point is accurately represented by the neural network model. Additional DEM simulations are only performed when needed, minimizing the number of additional DEM simulations required when updating the rheology. This adaptive coupling significantly reduces the overall computational cost of the approach while controlling the error. In addition, the neural networks are customized to learn regularized rheological behavior to ensure well-posedness of the continuum solution. We first validate the approach using two-dimensional steady-state and decelerating inclined flows. We then demonstrate the efficiency of our approach by modeling three-dimensional sub-aerial granular column collapse for varying initial column aspect ratios, where our multiscale method compares well with the computationally expensive computational fluid dynamics (CFD)-DEM simulation.

Dense granular flows↗

DOE OSTI2021

Distributed State Estimation Over Time-Varying Graphs: Exploiting the Age-of-Information

Here, we study the problem of designing a distributed observer for an LTI system over a time-varying communication graph. The limited existing work on this topic imposes various restrictions either on the observation model or on the sequence of communication graphs. In contrast, we propose a single-time-scale distributed observer that works under mild assumptions. Specifically, our communication model only requires strong-connectivity to be preserved over non-overlapping, contiguous intervals that are even allowed to grow unbounded over time. We show that under suitable conditions that bound the growth of such intervals, joint observability is sufficient to track the state of any discrete-time LTI system exponentially fast, at any desired rate. We also develop a variant of our algorithm that is provably robust to worst-case adversarial attacks, provided the sequence of graphs is sufficiently connected over time. The key to our approach is the notion of a "freshness-index" that keeps track of the age-of-information being diffused across the network. Such indices enable nodes to reject stale estimates of the state, and, in turn, contribute to stability of the error dynamics.

42 ENGINEERING↗

DOE OSTI2023

Numerical and boundary condition effects on the prediction of detonation engine behavior using detailed numerical simulations

High-fidelity numerical simulations of an experimental rotating detonation engine with discrete fuel/air injection were conducted. A series of configurations with different feed-plenum pressures but with constant equivalence ratio were studied. Detailed chemical kinetics for the hydrogen/air system is used. A resolution study for the full rotating detonation engine (RDE) system simulation is also conducted. Two kinds of boundary conditions, a total pressure boundary and a constant mass flow rate boundary, are used to assess the effects of the inlet boundary. As mass flow rate is increased, the total pressure boundary causes more error in the axial pressure distribution while the constant mass flow rate gives a better solution for all cases ran. The simulations confirm experimental findings, and reproduce qualitative as well as some of the quantitative trends. These results demonstrate that a) fuel-air mixing is highly non-uniform within the detonation chamber, leading to variations in local equivalence ratio, b) the fuel and oxidizer injectors experience significant backflow as the detonation wave passes over, but recover at different rates which further augments the inefficiencies in mixing, and c) parasitic combustion in the mixing region makes the detonation wave weak by extending the reaction zone across the wave.

33 ADVANCED PROPULSION SYSTEMS↗

NASA NTRSDate not supplied

Nodal Modeling of Tank Pressurization and Draining using a Multi-Node-Ullage Approach

Pressurized expulsion tests of liquid methane were modeled by a nodal code using a multinode ullage approach. Generalized Fluid System Simulation Program (GFSSP), a finite volume based nodal code was used to model the expulsion of liquid methane from a 1.52-meter (5 ft) diameter spherical tank by pressurizing with helium and gaseous methane. The purpose of the model was to estimate the amount of pressurant required and amount of condensation and evaporation of methane during the operation. The ullage was discretized into multiple nodes and each fluid node was connected to solid nodes. Both the fluid and solid nodes grow as the tank drains. The heat and mass transfer between fluid and solid node in the tank ullage was computed. The heat and mass transfer between the bottom ullage node and the liquid methane was also computed. The model predictions were compared with data from twelve test cases. The predicted pressurant consumption for helium pressurization compares with test data with a Mean Absolute Percent Error (MAPE) of 3 %. For autogenous pressurization, where gaseous methane was used to pressurize liquid methane, the predicted pressurant consumption compares with a MAPE of 6%.

Cryogenic Propellant↗

NASA NTRSDate not supplied

Nodal Modeling of Helium Pressurization and Autogenous Pressurization and Draining using a Multi-Node-Ullage Approach

Pressurized expulsion tests of liquid methane were modeled by a nodal code using a multinode ullage approach. Generalized Fluid System Simulation Program (GFSSP), a finite volume based nodal code was used to model the expulsion of liquid methane from a 1.52-meter (5 ft) diameter spherical tank by pressurizing with helium and gaseous methane. The purpose of the model was to estimate the amount of pressurant required and amount of condensation and evaporation of methane during the operation. The ullage was discretized into multiple nodes and each fluid node was connected to solid nodes. Both the fluid and solid nodes grow as the tank drains. The heat and mass transfer between fluid and solid node in the tank ullage was computed. The heat and mass transfer between the bottom ullage node and the liquid methane was also computed. The model predictions were compared with data from twelve test cases. The predicted pressurant consumption for helium pressurization compares with test data with a Mean Absolute Percent Error (MAPE) of 3%. For autogenous pressurization, where gaseous methane was used to pressurize liquid methane, the predicted pressurant consumption compares with a MAPE of 6%.

Nodal Model↗

DOE OSTI2025

An explicit, energy-conserving particle-in-cell scheme

We present an explicit temporal discretization of particle-in-cell schemes for the non-relativistic Vlasov equation that results in exact energy conservation when combined with an appropriate spatial discretization. The scheme is inspired by a simple, second-order explicit scheme that conserves energy exactly in the Eulerian context. We show that direct translation to particle-in-cell does not result in strict conservation, but derive a simple correction based on an analytically solvable optimization problem that recovers conservation. While this optimization problem is not guaranteed to have a real solution for every particle, we provide a correction that makes imaginary values extremely rare and still admits $\mathcal{O}$(10 –12 ) fractional errors in energy for practical simulation parameters. We present the scheme in both electrostatic – where we use the Ampère formulation – and electromagnetic contexts. With an electromagnetic field solve, the field update is most naturally linearly implicit, but the more computationally intensive particle update remains fully explicit. Here, we also show how the scheme can be extended to use the fully explicit leapfrog and pseudospectral analytic time-domain (PSATD) field solvers. The scheme is tested on standard kinetic plasma problems, confirming its conservation properties.

Energy conservation↗

DOE OSTI2020

Scalable line and plane relaxation in a parallel structured multigrid solver

The efficient solution of sparse, linear systems that arise through the discretization of partial differential equations remains a key challenge for a range of high performance scientific simulations. One approach for reducing data movement and improving performance is by exposing and exploiting structure in a problem through the use of robust structured multilevel solvers. By choosing coarsening that preserves the structure of the problem, these methods maintain efficient structured computation and communication throughout the multigrid hierarchy. However, when coarsening is not permitted to be dependent on the operator, anisotropy must be addressed by the smoother — producing error compatible for coarse-grid correction with structured coarsening. Here, the components required in a scalable parallel structured solver are described with a focus on memory and communication efficiency of robust smoothers. While the implementation of communication and memory reduction techniques in smoothers integrated in a complete 3D solver present a significant engineering challenge, a novel approach is proposed that addresses these challenges systematically through a change to the solver’s execution model. Enabled by user-level threading paired with a set of data and communication abstractions, this approach permits seamless aggregation of communication in plane smoothers — directly reusing code for a 2D distributed multilevel cycle. Results show an effective reduction in communication costs for coarse-grid problems, and result in a speedup of 8.7x in smoothing routines shown in Fig. 12 using this approach. This produces a significant improvement to strong scalability while maintaining favorable weak scaling behavior. Finally, a parallel scaling study using a series of refined meshes is included that demonstrates the effectiveness of this approach in an application of interest.

97 MATHEMATICS AND COMPUTING↗

DOE OSTI2024

Autonomous stabilization with programmable stabilized state

Reservoir engineering is a powerful technique to autonomously stabilize a quantum state. Traditional schemes involving multi-body states typically function for discrete entangled states. In this work, we enhance the stabilization capability to a continuous manifold of states with programmable stabilized state selection using multiple continuous tuning parameters. We experimentally achieve 84.6% and 82.5% stabilization fidelity for the odd and even-parity Bell states as two special points in the manifold. We also perform fast dissipative switching between these opposite parity states within 1.8 μs and 0.9 μs by sequentially applying different stabilization drives. Our result is a precursor for new reservoir engineering-based error correction schemes.

71 CLASSICAL AND QUANTUM MECHANICS, GENERAL PHYSIC↗

NASA NTRS1994

An Adaptively-Refined, Cartesian, Cell-Based Scheme for the Euler and Navier-Stokes Equations

A Cartesian, cell-based scheme for solving the Euler and Navier-Stokes equations in two dimensions is developed and tested. Grids about geometrically complicated bodies are generated automatically, by recursive subdivision of a single Cartesian cell encompassing the entire flow domain. Where the resulting cells intersect bodies, polygonal 'cut' cells are created. The geometry of the cut cells is computed using polygon-clipping algorithms. The grid is stored in a binary-tree data structure which provides a natural means of obtaining cell-to-cell connectivity and of carrying out solution-adaptive refinement. The Euler and Navier-Stokes equations are solved on the resulting grids using a finite-volume formulation. The convective terms are upwinded, with a limited linear reconstruction of the primitive variables used to provide input states to an approximate Riemann solver for computing the fluxes between neighboring cells. A multi-stage time-stepping scheme is used to reach a steady-state solution. Validation of the Euler solver with benchmark numerical and exact solutions is presented. An assessment of the accuracy of the approach is made by uniform and adaptive grid refinements for a steady, transonic, exact solution to the Euler equations. The error of the approach is directly compared to a structured solver formulation. A non smooth flow is also assessed for grid convergence, comparing uniform and adaptively refined results. Several formulations of the viscous terms are assessed analytically, both for accuracy and positivity. The two best formulations are used to compute adaptively refined solutions of the Navier-Stokes equations. These solutions are compared to each other, to experimental results and/or theory for a series of low and moderate Reynolds numbers flow fields. The most suitable viscous discretization is demonstrated for geometrically-complicated internal flows. For flows at high Reynolds numbers, both an altered grid-generation procedure and a different formulation of the viscous terms are shown to be necessary. A hybrid Cartesian/body-fitted grid generation approach is demonstrated. In addition, a grid-generation procedure based on body-aligned cell cutting coupled with a viscous stensil-construction procedure based on quadratic programming is presented.

Coirier, William John↗

NASA NTRSDate not supplied

High-Lift Common Research Model: RANS, HRLES and WMLES Perspectives for CLmax Prediction Using LAVA

A unified assessment of three turbulence treatments: Reynolds Averaged Navier-Stokes(RANS), Hybrid RANS/LES (HRLES) and Equilibrium Wall-Modelled Large Eddy Simulation(WMLES) is presented for the High-Lift Common Research Model (CRM-HL). For the free-air configuration, steady-state RANS simulations show very accurate drag polar predictions in the low-𝛼linear regime. However, strong grid sensitivity is reported near the maximum lift-state(𝐶𝐿max), with finer-grids showing larger errors and predicting erroneous flow topologies on the wing. Our RANS simulations show that several corrections for the Spalart-Allmaras (SA)turbulence model widely used in the community lead to more erroneous results compared to the baseline closure, without exception. Both scale-resolving methods (HRLES and WMLES)address these drawbacks and predict an outboard separation pattern on the main element that is in good agreement with the oil flow photographs taken from the QinetiQ wind tunnel experiments, when LES-appropriate grids and numerical discretizations are used. While RANS simulations with the baseline SA closure do not show any wing-root separation post𝐶𝐿max, both HRLES and WMLES show onset of corner flow separation with varying degrees of progression, along with a weak pitch break in the wing-contribution of the overall pitching moment. This post-𝐶𝐿max pitch break seen in the free-air simulations is weaker than the break observed in experiments, with a weaker break reported in WMLES for each iteration of grid-refinement. In-tunnel simulations using both SA-baseline RANS and WMLES show a much stronger post-𝐶𝐿max break with the WMLES predictions showing excellent agreement with the experiment in terms of both the flow-topology observed and the pressure-coefficients at various spanwise stations. Sensitivity to the tunnel wall boundary layer is characterized via comparisons between viscous and inviscid treatments for the tunnel walls. WMLES predictions show moderate sensitivity at the predicted inboard flow-state at 𝐶𝐿max along with the progression towards a post-𝐶𝐿max stall; however, this stalled state at 𝛼≈20◦(inside the tunnel) obtained with both tunnel wall treatments appears to be largely identical.

TTT↗

DOE OSTI2021

Discrete Green’s functions and spectral graph theory for computationally efficient thermal modeling

Here, this work concerns solutions of the heat equation with the spectral graph method, for which the temperature is defined at discrete points in the domain and the spatial relationship among the points is described by a graph. The heat equation on the graph is solved using matrix techniques involving the eigenvectors and eigenvalues of the Laplacian matrix. The spectral graph approach precludes the computationally intensive meshing and numerous time-integration steps of the finite element method. In the present work, the spectral graph method is extended to include heat loss at the boundaries with a generalized boundary condition, and physics-based edge weights are introduced which simplify the calibration process. From this approach a discrete Green’s function is defined which allows for solutions under a variety of heating conditions including: space-varying initial conditions; time-and-space varying internal heating; and, time-and-space-varying heating at boundaries of type 1 (Dirichlet), type 2 (Neumann) and type 3 (Robin). Results are provided for benchmark heat transfer problems in one spatial dimension and in three spatial dimensions, and verification is provided by comparison with exact analytical solutions and finite difference solutions. The spectral graph method converges within 0.4% error of the analytical solution. The practical utility of the approach is demonstrated by thermal simulation of a multilayer additive manufacturing process. The spectral graph results are compared to experimentally-obtained temperature data for two metal parts, with error less than 5% of the experimental measurements, with computation time less than one minute on a desktop computer.

36 MATERIALS SCIENCE↗

NASA NTRS1992

Discrete distributed strain sensing of intelligent structures

Techniques are developed for the design of discrete highly distributed sensor systems for use in intelligent structures. First the functional requirements for such a system are presented. Discrete spatially averaging strain sensors are then identified as satisfying the functional requirements. A variety of spatial weightings for spatially averaging sensors are examined, and their wave number characteristics are determined. Preferable spatial weightings are identified. Several numerical integration rules used to integrate such sensors in order to determine the global deflection of the structure are discussed. A numerical simulation is conducted using point and rectangular sensors mounted on a cantilevered beam under static loading. Gage factor and sensor position uncertainties are incorporated to assess the absolute error and standard deviation of the error in the estimated tip displacement found by numerically integrating the sensor outputs. An experiment is carried out using a statically loaded cantilevered beam with five point sensors. It is found that in most cases the actual experimental error is within one standard deviation of the absolute error as found in the numerical simulation.

Anderson, Mark S.↗

NASA NTRS2011

Reconfigurable Microwave Phase Delay Element for Frequency Reference and Phase-Shifter Applications

A technique was developed to provide a reconfigurable high-precision micro - wave electrical phase delay for resonators and phase shifters. The invention employs multiple branches of transmission lines with open-ended or ground-ended terminations as configurable bits or digits. This technique minimizes the errors due to limited precision of switching devices. In addition, the proposed linear analytical approach significantly produces a much simpler design than that of other prior inventions at the time of this reporting. Microwave components such as filters, phase delay elements, or resonators require a method that can accurately adjust their frequency responses. Most tuning techniques offer very wide frequency tuning range; however, it is often difficult and expensive to tune their response in a very narrow operating frequency, especially when the tuning element reaches its minimum discrete step due to fabrication tolerances. The problem becomes worse as the operating frequency is in mm-wave frequency range (>26 GHz). The electrical tuning sensitivity of a microwave line is dependent on the position of the tuning element with respect to the reference termination. By placing this tuning element away from this reference with the main transmission line connecting the two elements together the sensitivity of the tuning element can change significantly. This concept can be used in the system that requires multiple tuning sensitivities. In this case, multiple tuning branches are superimposed in the main transmission line. The proposed invention allows the transmission-line electrical length to be accurately programmed using switching elements that have limited accuracy. The invention consists of multiple branches of transmission lines connected to discrete switching devices with open-ended terminations. They are used as discrete tuning elements. These elements are connected to the main microwave transmission line and are separated by a well-defined electrical degree spacing. Each branch is programmed to have different electrical degree sensitivity, such as a combination of discrete steps in each branch, which results in a reflective line with a unique effective phase response. To reduce the number of switching devices, it is desirable to program the devices in binary configuration where each branch represents one bit in the base-2 number system. This invention allows the transmission line electrical length to be tuned precisely with customizable sensitivity based on the known sensitivity of the base tuning circuit. The tuning resolution is dependent on the distance among tuning branches.

Hsieh, Wen-Ting↗

DOE OSTI2023

Phonon-informed Neural Thermal Scattering (NeTS) Optimization for Crystalline Graphite and Beryllium Metal

Fast neutrons born from fission lose energy through scattering interactions in the process of slowing-down. As neutrons thermalize to the order of $k$ $b$ $T$ (where $k$ $b$ is the Boltzmann constant, and $T$ is the temperature of the medium), their de Broglie wavelength and energy approaches the order of inter-atomic spacing and quantized lattice vibrations, i.e., phonons. At thermal energies, the thermal scattering law (TSL), i.e., $S$($α, β$), captures crystal binding contributions to the total reaction rate, or cross section. This dimensionless material property describes the energy ($β$) and momentum ($α$) exchanges available in a medium. Currently, $S$($α, β$) is evaluated in the Full Law Analysis Scattering System Hub (FLASSH) code for discrete inputs and stored as ENDF/B File 7 for 0-phonon elastic (MT 2) and n-phonon inelastic (MT 4) processes. Further processing recasts $S$($α, β$) into cumulative distribution functions for sampling post-collision scattering kinematics. In practice, interpolation schemes are employed to access data between tabulated values. An improvement to this juncture of the nuclear data pipeline is supplying cross sections on-the-fly (OTF), as has been developed for the un-resolved resonance region to minimize non-physical interpolation errors. This capability may improve simulation accuracy for accident and transient analyses, where rapidly varying changes in temperature and pressure are difficult to predict beforehand. To do so, deep artificial neural networks (ANNs) can be employed which collapse non-linear, complex data into a lightweight dictionary of neural weights and biases. This has been successfully demonstrated for the hydrogen in light water $S$($α, β$) dataset in the form of a Neural Thermal Scattering (NeTS) module. In this work, the NeTS framework is extended to consider the impact of material-dependent dynamical features on optimal neural pre-processing and architecture design decisions, such as number of neurons per hidden layer, residual skip connections and neural depth. New NeTS modules for crystalline graphite and beryllium metal illuminate a novel correlation between dynamical nonlinearity and optimal neural parametrization when deploying $S$($α, β$) on-the-fly.

73 NUCLEAR PHYSICS AND RADIATION PHYSICS↗

NASA NTRS1992

Discrete time learning control in nonlinear systems

In this paper digital learning control methods are developed primarily for use in single-input, single-output nonlinear dynamic systems. Conditions for convergence of the basic form of learning control based on integral control concepts are given, and shown to be satisfied by a large class of nonlinear problems. It is shown that it is not the gross nonlinearities of the differential equations that matter in the convergence, but rather the much smaller nonlinearities that can manifest themselves during the short time interval of one sample time. New algorithms are developed that eliminate restrictions on the size of the learning gain, and on knowledge of the appropriate sign of the learning gain, for convergence to zero error in tracking a feasible desired output trajectory. It is shown that one of the new algorithms can give guaranteed convergence in the presence of actuator saturation constraints, and indicate when the requested trajectory is beyond the actuator capabilities.

Longman, Richard W.↗

NASA NTRS1997

Efficient Gradient-Based Shape Optimization Methodology Using Inviscid/Viscous CFD

The formerly developed preconditioned-biconjugate-gradient (PBCG) solvers for the analysis and the sensitivity equations had resulted in very large error reductions per iteration; quadratic convergence was achieved whenever the solution entered the domain of attraction to the root. Its memory requirement was also lower as compared to a direct inversion solver. However, this memory requirement was high enough to preclude the realistic, high grid-density design of a practical 3D geometry. This limitation served as the impetus to the first-year activity (March 9, 1995 to March 8, 1996). Therefore, the major activity for this period was the development of the low-memory methodology for the discrete-sensitivity-based shape optimization. This was accomplished by solving all the resulting sets of equations using an alternating-direction-implicit (ADI) approach. The results indicated that shape optimization problems which required large numbers of grid points could be resolved with a gradient-based approach. Therefore, to better utilize the computational resources, it was recommended that a number of coarse grid cases, using the PBCG method, should initially be conducted to better define the optimization problem and the design space, and obtain an improved initial shape. Subsequently, a fine grid shape optimization, which necessitates using the ADI method, should be conducted to accurately obtain the final optimized shape. The other activity during this period was the interaction with the members of the Aerodynamic and Aeroacoustic Methods Branch of Langley Research Center during one stage of their investigation to develop an adjoint-variable sensitivity method using the viscous flow equations. This method had algorithmic similarities to the variational sensitivity methods and the control-theory approach. However, unlike the prior studies, it was considered for the three-dimensional, viscous flow equations. The major accomplishment in the second period of this project (March 9, 1996 to March 8, 1997) was the extension of the shape optimization methodology for the Thin-Layer Navier-Stokes equations. Both the Euler-based and the TLNS-based analyses compared with the analyses obtained using the CFL3D code. The sensitivities, again from both levels of the flow equations, also compared very well with the finite-differenced sensitivities. A fairly large set of shape optimization cases were conducted to study a number of issues previously not well understood. The testbed for these cases was the shaping of an arrow wing in Mach 2.4 flow. All the final shapes, obtained either from a coarse-grid-based or a fine-grid-based optimization, using either a Euler-based or a TLNS-based analysis, were all re-analyzed using a fine-grid, TLNS solution for their function evaluations. This allowed for a more fair comparison of their relative merits. From the aerodynamic performance standpoint, the fine-grid TLNS-based optimization produced the best shape, and the fine-grid Euler-based optimization produced the lowest cruise efficiency.

Baysal, Oktay↗

NASA NTRS2010

Parallel Processing of Broad-Band PPM Signals

A parallel-processing algorithm and a hardware architecture to implement the algorithm have been devised for timeslot synchronization in the reception of pulse-position-modulated (PPM) optical or radio signals. As in the cases of some prior algorithms and architectures for parallel, discrete-time, digital processing of signals other than PPM, an incoming broadband signal is divided into multiple parallel narrower-band signals by means of sub-sampling and filtering. The number of parallel streams is chosen so that the frequency content of the narrower-band signals is low enough to enable processing by relatively-low speed complementary metal oxide semiconductor (CMOS) electronic circuitry. The algorithm and architecture are intended to satisfy requirements for time-varying time-slot synchronization and post-detection filtering, with correction of timing errors independent of estimation of timing errors. They are also intended to afford flexibility for dynamic reconfiguration and upgrading. The architecture is implemented in a reconfigurable CMOS processor in the form of a field-programmable gate array. The algorithm and its hardware implementation incorporate three separate time-varying filter banks for three distinct functions: correction of sub-sample timing errors, post-detection filtering, and post-detection estimation of timing errors. The design of the filter bank for correction of timing errors, the method of estimating timing errors, and the design of a feedback-loop filter are governed by a host of parameters, the most critical one, with regard to processing very broadband signals with CMOS hardware, being the number of parallel streams (equivalently, the rate-reduction parameter).

Gray, Andrew↗