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Computational techniques for high-speed flows with viscous and chemical effects

Algorithms for solving the Euler and the Navier-Stokes equations in conjunction with chemical kinetic equations are presented. The convective flux is estimated from a quasi one dimensional interpolation procedure. Shock, contact, and expansion waves and thermochemical nonequilibrium phenomena are captured by the Lax-Friedrichs technique. Relaxation techniques were developed to enhance their effectiveness in dealing with spatial and temporal stiffness associated with the physical problems. Both explicit and implicit smoothers were implemented into the standard multigrid time stepping method. Unsteady and steady scalar problems are discussed. A perfect gas and equilibrium air shock tube problem is investigated. Numerical schemes and techniques are compared for the problems of shock and boundary layer interaction and three dimensional viscous, nonequilibrium flow encompassing an aerobrake. The results are comparable in accuracy against other high order non-oscillatory techniques. The multigrid methods are assessed using a Mach 8 flow over a complete planar body, a sphere, and a blunt delta wing at 20 deg incidence. Applying an implicit multigrid method on a nested grid of 128 by 64 nodes the reduction factor is 0.25. The central processing unit reduction factor is 2.2 after both the single and multigrid Runge-Kutta solutions converged to machine zero on a grid of 37 by 41 by 73 nodes.

Li, C. P.↗

Calculation methods for rotor wake formation in forward flight

The primary focus of this research centered on improving the vorticity-capturing properties of Eulerian flow solvers in convection-dominated flows. The work was motivated by the fact that excessive numerical diffusion observed in current Euler solvers on grids of feasible size is a major obstacle to accurately computing the flow about helicopter rotors, especially in the forward flight case. The work was originally supported by NASA Grant NAG2-421 for two years. The research then continued for four more years under an ONR contract. The research goal of reduction of numerical diffusion in Euler solvers remained the same, although the target application was now the prediction of internal vortical secondary flows. This was part of a larger ONR-sponsored research program at MIT aimed at reducing the noise of turbopumps. The research work culminated in the PhD thesis titled 'A Coupled Eulerian/Lagrangian Method for the Solution of Three-Dimensional Vortical Flows'.

Drela, Mark↗

Multigrid Approach to Incompressible Viscous Cavity Flows

Two-dimensional incompressible viscous driven-cavity flows are computed for Reynolds numbers on the range 100-20,000 using a loosely coupled, implicit, second-order centrally-different scheme. Mesh sequencing and three-level V-cycle multigrid error smoothing are incorporated into the symmetric Gauss-Seidel time-integration algorithm. Parametrics on the numerical parameters are performed, achieving reductions in solution times by more than 60 percent with the full multigrid approach. Details of the circulation patterns are investigated in cavities of 2-to-1, 1-to-1, and 1-to-2 depth to width ratios.

Wood, William A.↗

Analysis and Modeling of a Two-Phase Jet Pump of a Flow Boiling Test Facility for Aerospace Applications

Jet pumps are devices capable of pumping fluids to a higher pressure employing a nozzle/diffuser/mixing chamber combination. A primary fluid is usually allowed to pass through a converging-diverging nozzle where it can accelerate to supersonic speeds at the nozzle exit. The relatively high kinetic energy that the primary fluid possesses at the nozzle exit is accompanied by a low pressure region in order to satisfy Bernoulli's equation. The low pressure region downstream of the nozzle exit permits a secondary fluid to be entrained into and mixed with the primary fluid in a mixing chamber located downstream of the nozzle. Several combinations may exist in terms of the nature of the primary and secondary fluids in so far as whether they are single or two-phase fluids. Depending on this, the jet pump may be classified as gas/gas, gas/liquid, liquid/liquid, two-phase/liquid, or similar combinations. The mixing chamber serves to create a homogeneous single-phase or two-phase mixture which enters a diffuser where the high kinetic energy of the fluid is converted into pressure energy. If the fluid mixture entering the diffuser is in the supersonic flow regime, a normal shock wave usually develops inside the diffuser. If the fluid mixture is one that can easily change phase, a condensation shock would normally develop. Because of the overall rise in pressure in the diffuser as well as the additional rise in pressure across the shock layer, condensation becomes more likely. Associated with the pressure rise across the shock is a velocity reduction from the supersonic to the subsonic range. If the two-phase flow entering the diffuser is predominantly gaseous with liquid droplets suspended in it, it will transform into a predominantly liquid flow containing gaseous bubbles (bubbly flow) somewhere in the diffuser. While past researchers have been able to model the two-phase flow jet pump using the one-dimensional assumption with no shock waves and no phase change, there is no research known to the authors apart from that of Anand (1992) which accounted for condensation shocks. One of the objectives of this research effort is to develop a comprehensive model in which the effects of phase slip and inter-phase heat transfer as well as the wall friction and shock waves are accounted for. While this modeling effort is predominantly analytical in nature and is primarily intended to provide a parametric understanding of the jet pump performance under different operating scenarios, another parallel effort employing a commercial CFD code is also implemented. The latter effort is primarily intended to model an axisymmetric counterpart of the problem in question. The viability of using the CFD code to model a two-phase flow jet pump will be assessed by attempting to recreate some of the existing performance data of similar jet pumps. The code will eventually be used to generate the jet pump performance characteristics of several scenarios involving jet pump geometries as well as flow regimes in order to be able to determine an optimum design which would be suitable for a two-phase flow boiling test facility at NASA-Marshall. Because of the extensive nature of the analytical model developed, the following section will only provide very brief highlights of it, while leaving the details to a more complete report submitted to the NASA colleague. This report will also contain some of the simulation results obtained using the CFD code.

Sherif, S. A.↗

Higher Order Time Integration Schemes for the Unsteady Navier-Stokes Equations on Unstructured Meshes

The rapid increase in available computational power over the last decade has enabled higher resolution flow simulations and more widespread use of unstructured grid methods for complex geometries. While much of this effort has been focused on steady-state calculations in the aerodynamics community, the need to accurately predict off-design conditions, which may involve substantial amounts of flow separation, points to the need to efficiently simulate unsteady flow fields. Accurate unsteady flow simulations can easily require several orders of magnitude more computational effort than a corresponding steady-state simulation. For this reason, techniques for improving the efficiency of unsteady flow simulations are required in order to make such calculations feasible in the foreseeable future. The purpose of this work is to investigate possible reductions in computer time due to the choice of an efficient time-integration scheme from a series of schemes differing in the order of time-accuracy, and by the use of more efficient techniques to solve the nonlinear equations which arise while using implicit time-integration schemes. This investigation is carried out in the context of a two-dimensional unstructured mesh laminar Navier-Stokes solver.

Jothiprasad, Giridhar↗

Flutter Stability of the Efficient Low Noise Fan Calculated

The TURBO-AE aeroelastic code has been used to verify the flutter stability of the Efficient Low Noise Fan (ELNF), which is also referred to as the trailing-edge blowing fan. The ELNF is a unique technology demonstrator being designed and fabricated at the NASA Glenn Research Center for testing in Glenn's 9-by-15-Foot Low-Speed Wind Tunnel. In the ELNF, air can be blown out of slots near the trailing edges of the fan blades to fill in the wakes downstream of the rotating blades. This filling of the wakes leads to a reduction of the rotor-stator interaction (tone) noise that results from the interaction of wakes with the downstream stators. The ELNF will demonstrate a 1.6-EPNdB1 reduction in tone noise through wake filling, without increasing the broadband noise. Furthermore, the reduced blade row interaction will decrease the possibility of forced response and enable closer spacing of blade rows, thus reducing engine length and weight. During the design of the ELNF, the rotor blades were checked for flutter stability using the detailed aeroelastic analysis capability of the three-dimensional Navier-Stokes TURBOAE code. The aeroelastic calculations were preceded by steady calculations in which the blades were not allowed to vibrate. For each rotational speed, as the back-pressure was increased, the mass flow rate decreased, and the operating point moved along the constant speed characteristic (speed-line) from choke to stall as shown on the fan map. The TURBO-AE aeroelastic analyses were performed separately for the first two vibration modes (bending and torsion) and covered the complete range of interblade phase angles or nodal diameters at which flutter can occur. The results indicated that the ELNF blades would not encounter flutter at takeoff conditions. The calculations were then repeated for a part-speed condition (70-percent rotational speed), and the results again showed no flutter in the operating region. On the fan map (shown), the predicted flutter point at part speed condition was located beyond the stall line, which means that the ELNF will not encounter flutter since it will never operate beyond the stall line. All the calculations done so far have been for the nonblowing case, and selected calculations will be repeated with air blowing from the trailing edge of the fan.

Bakhle, Milind A.↗

CFD Analysis of Nozzle Jet Plume Effects on Sonic Boom Signature

An axisymmetric full Navier-Stokes computational fluid dynamics study is conducted to examine nozzle exhaust jet plume effects on the sonic boom signature of a supersonic aircraft. A simplified axisymmetric nozzle geometry, representative of the nozzle on the NASA Dryden NF-15B Lift and Nozzle Change Effects on Tail Shock research airplane, is considered. The computational fluid dynamics code is validated using available wind-tunnel sonic boom experimental data. The effects of grid size, spatial order of accuracy, grid type, and flow viscosity on the accuracy of the predicted sonic boom pressure signature are quantified. Grid lines parallel to the Mach wave direction are found to give the best results. Second-order accurate upwind methods are required as a minimum for accurate sonic boom simulations. The highly underexpanded nozzle flow is found to provide significantly more reduction in the tail shock strength in the sonic boom N-wave pressure signature than perfectly expanded and overexpanded nozzle flows. A tail shock train in the sonic boom signature is observed for the highly underexpanded nozzle flow. Axisymmetric computational fluid dynamics simulations show the flow physics inside the F-15 nozzle to be nonisentropic and complex. Although the one-dimensional isentropic nozzle plume results look reasonable, they fail to capture the sonic boom shock train in the highly underexpanded nozzle flow.

Bui, Trong T.↗

Real-time three-dimensional echocardiographic study of left ventricular function after infarct exclusion surgery for ischemic cardiomyopathy

BACKGROUND: Infarct exclusion (IE) surgery, a technique of left ventricular (LV) reconstruction for dyskinetic or akinetic LV segments in patients with ischemic cardiomyopathy, requires accurate volume quantification to determine the impact of surgery due to complicated geometric changes. METHODS AND RESULTS: Thirty patients who underwent IE (mean age 61+/-8 years, 73% men) had epicardial real-time 3-dimensional echocardiographic (RT3DE) studies performed before and after IE. RT3DE follow-up was performed transthoracically 42+/-67 days after surgery in 22 patients. Repeated measures ANOVA was used to compare the values before and after IE surgery and at follow-up. Significant decreases in LV end-diastolic (EDVI) and end-systolic (ESVI) volume indices were apparent immediately after IE and in follow-up (EDVI 99+/-40, 67+/-26, and 71+/-31 mL/m(2), respectively; ESVI 72+/-37, 40+/-21, and 42+/-22 mL/m(2), respectively; P:<0.05). LV ejection fraction increased significantly and remained higher (0.29+/-0.11, 0.43+/-0.13, and 0.42+/-0.09, respectively, P:<0.05). Forward stroke volume in 16 patients with preoperative mitral regurgitation significantly improved after IE and in follow-up (22+/-12, 53+/-24, and 58+/-21 mL, respectively, P:<0.005). New York Heart Association functional class at an average 285+/-144 days of clinical follow-up significantly improved from 3.0+/-0.8 to 1.8+/-0.8 (P:<0.0001). Smaller end-diastolic and end-systolic volumes measured with RT3DE immediately after IE were closely related to improvement in New York Heart Association functional class at clinical follow-up (Spearman's rho=0.58 and 0.60, respectively). CONCLUSIONS: RT3DE can be used to quantitatively assess changes in LV volume and function after complicated LV reconstruction. Decreased LV volume and increased ejection fraction imply a reduction in LV wall stress after IE surgery and are predictive of symptomatic improvement.

Non-NASA Center↗

THE LAST MINUTES OF OXYGEN SHELL BURNING IN A MASSIVE STAR

We present the first 4 π– three-dimensional (3D) simulation of the last minutes of oxygen shell burning in an 18 M {sub ⊙} supernova progenitor up to the onset of core collapse. A moving inner boundary is used to accurately model the contraction of the silicon and iron core according to a one-dimensional stellar evolution model with a self-consistent treatment of core deleptonization and nuclear quasi-equilibrium. The simulation covers the full solid angle to allow the emergence of large-scale convective modes. Due to core contraction and the concomitant acceleration of nuclear burning, the convective Mach number increases to ∼0.1 at collapse, and an ℓ = 2 mode emerges shortly before the end of the simulation. Aside from a growth of the oxygen shell from 0.51 M {sub ⊙} to 0.56 M {sub ⊙} due to entrainment from the carbon shell, the convective flow is reasonably well described by mixing-length theory, and the dominant scales are compatible with estimates from linear stability analysis. We deduce that artificial changes in the physics, such as accelerated core contraction, can have precarious consequences for the state of convection at collapse. We argue that scaling laws for the convective velocities and eddy sizes furnish good estimates for the state of shell convection at collapse and develop a simple analytic theory for the impact of convective seed perturbations on shock revival in the ensuing supernova. We predict a reduction of the critical luminosity for explosion by 12% – 24% due to seed asphericities for our 3D progenitor model relative to the case without large seed perturbations.

79 ASTRONOMY AND ASTROPHYSICS↗

Bayesian learning of orthogonal embeddings for multi-fidelity Gaussian Processes

Uncertainty propagation in complex engineering systems often poses significant computational challenges related to modeling and quantifying probability distributions of model outputs, as those emerge as the result of various sources of uncertainty that are inherent in the system under investigation. Gaussian Processes regression (GPs) is a robust meta-modeling technique that allows for fast model prediction and exploration of response surfaces. Multi-fidelity variations of GPs further leverage information from cheap and low fidelity model simulations in order to improve their predictive performance on the high fidelity model. In order to cope with the high volume of data required to train GPs in high dimensional design spaces, a common practice is to introduce latent design variables that are typically projections of the original input space to a lower dimensional subspace, and therefore substitute the problem of learning the initial high dimensional mapping, with that of training a GP on a low dimensional space. Here in this paper, we present a Bayesian approach to identify optimal transformations that map the input points to low dimensional latent variables. The \projection" mapping consists of an orthonormal matrix that is considered a priori unknown and needs to be inferred jointly with the GP parameters, conditioned on the available training data. The proposed Bayesian inference scheme relies on a two-step iterative algorithm that samples from the marginal posteriors of the GP parameters and the projection matrix respectively, both using Markov Chain Monte Carlo (MCMC) sampling. In order to take into account the orthogonality constraints imposed on the orthonormal projection matrix, a Geodesic Monte Carlo sampling algorithm is employed, that is suitable for exploiting probability measures on manifolds. We extend the proposed framework to multi-fidelity models using GPs including the scenarios of training multiple outputs together. We validate our framework on three synthetic problems with a known lower-dimensional subspace. The benefits of our proposed framework, are illustrated on the computationally challenging aerodynamic optimization of a last-stage blade for an industrial gas turbine, where we study the effect of an 85-dimensional shape parameterization of a three-dimensional airfoil on two output quantities of interest, specifically on the aerodynamic efficiency and the degree of reaction

42 ENGINEERING↗

A Pore‐Scale Investigation of Mineral Precipitation Driven Diffusivity Change at the Column‐Scale

Abstract Mineral precipitation affects the pore structure and thus transport properties of porous media. In this study, we investigated the pore‐scale dynamics of precipitation in diffusion controlled systems and the resulting impacts on the effective diffusivity, using a micro‐continuum reactive transport model. Forty two‐dimensional pore structures representing both idealized and realistic geometries were simulated with consideration of different precipitation scenarios and rates. A homogeneous nucleation scenario reproduced patterns observed in previous experimental study showing mixing‐induced precipitation, and a surface growth scenario captured the pattern for mineral precipitation on a substrate with the same or similar mineral structures. In all cases, local precipitation resulted in the reduction in the average porosity of the domain ( Φ ) until the cessation of diffusive transport and the termination of precipitation. The minimum porosity reached was referred to as the critical porosity ( Φ c ). The effective diffusivity ( D eff ) decreased with Φ and dropped sharply to effectively zero, that is, the critical effective diffusivity ( D eff c ), as Φ c was reached. These pore‐scale dynamics can be captured by a revised relationship that explicitly considers the critical porosity and the corresponding effective diffusivity, and the pre‐exponential coefficient and the exponent of the relationship varied with initial pore structure and the precipitation kinetics. Overall, the homogeneous nucleation scenario results in systematically larger Φ c and coefficients that give rise to a sharper decrease in diffusivity as Φ c is approached, compared to the surface growth scenario. The revised relationship was also implemented at continuum scale and used to examine column scale diffusivity change and reactions.

58 GEOSCIENCES↗

Unraveling the mysterious failure of Cu/SAPO-34 selective catalytic reduction catalysts

Commercial Cu/SAPO-34 selective catalytic reduction (SCR) catalysts have experienced unexpected and quite perplexing failure. Understanding the causes at an atomic level is vital for the synthesis of more robust Cu/SAPO-34 catalysts. Here we show, via application of model catalysts with homogeneously dispersed isolated Cu ions, that Cu transformations resulting from low-temperature hydrothermal aging and ambient temperature storage can be semi-quantitatively probed with 2-dimensional pulsed electron paramagnetic resonance. Coupled with kinetics, additional material characterizations and DFT simulations, we propose the following catalyst deactivation steps: (1) detachment of Cu(II) ions from cationic positions in the form of Cu(OH)2; (2) irreversible hydrolysis of the SAPO-34 framework forming terminal Al species; and (3) interaction between Cu(OH)2 and terminal Al species forming SCR inactive, Cu-aluminate like species. Especially significant is that these reactions are greatly facilitated by condensed water molecules under wet ambient conditions, causing low temperature failure of the commercial Cu/SAPO-34 catalysts.

37 INORGANIC, ORGANIC, PHYSICAL, AND ANALYTICAL CH↗

Helicon plasma thruster discharge model

By considering particle, momentum, and energy balance equations, we develop a semi-empirical quasi one-dimensional analytical discharge model of radio-frequency and helicon plasma thrusters. The model, which includes both the upstream plasma source region as well as the downstream diverging magnetic nozzle region, is compared with experimental measurements and confirms current performance levels. Analysis of the discharge model identifies plasma power losses on the radial and back wall of the thruster as the major performance reduction factors. These losses serve as sinks for the input power which do not contribute to the thrust, and which reduce the maximum plasma density and hence propellant utilization. With significant radial plasma losses eliminated, the discharge model (with argon) predicts specific impulses in excess of 3000 s, propellant utilizations above 90%, and thruster efficiencies of about 30%.

70 PLASMA PHYSICS AND FUSION TECHNOLOGY↗

Xyloglucan Deficiency Disrupts Microtubule Stability and Cellulose Biosynthesis in Arabidopsis, Altering Cell Growth and Morphogenesis

Here, xyloglucan constitutes most of the hemicellulose in eudicot primary cell walls and functions in cell wall structure and mechanics. Although Arabidopsis ( Arabidopsis thaliana ) xxt1 xxt2 mutants lacking detectable xyloglucan are viable, they display growth defects that are suggestive of alterations in wall integrity. To probe the mechanisms underlying these defects, we analyzed cellulose arrangement, microtubule patterning and dynamics, microtubule- and wall-integrity-related gene expression, and cellulose biosynthesis in xxt1 xxt2 plants. We found that cellulose is highly aligned in xxt1 xxt2 cell walls, that its three-dimensional distribution is altered, and that microtubule patterning and stability are aberrant in etiolated xxt1 xxt2 hypocotyls. We also found that the expression levels of microtubule-associated genes, such as MAP70-5 and CLASP , and receptor genes, such as HERK1 and WAK1 , were changed in xxt1 xxt2 plants and that cellulose synthase motility is reduced in xxt1 xxt2 cells, corresponding with a reduction in cellulose content. Our results indicate that loss of xyloglucan affects both the stability of the microtubule cytoskeleton and the production and patterning of cellulose in primary cell walls. These findings establish, to our knowledge, new links between wall integrity, cytoskeletal dynamics, and wall synthesis in the regulation of plant morphogenesis.

59 BASIC BIOLOGICAL SCIENCES↗

The Dependence of H-mode Energy Confinement and Transport on Collisionality in NSTX

Understanding the dependence of confi nement on collisionality in tokamaks is important for the design of next-step devices, which will operate at collisionalities at least one order of magnitude lower than in present generation. A wide range of collisionality has been obtained in the National Spherical Torus Experiment (NSTX) by employing two different wall conditioning techniques, one with boronization and between-shot helium glow discharge conditioning (HeGDC+B), and one using lithium evaporation (Li EVAP). Previous studies of HeGDC+B plasmas indicated a strong and favorable dependence of normalized con nement on collisionality. Discharges with lithium conditioning discussed in the present study gen- erally achieved lower collisionality, extending the accessible range of collisionality by almost an order of unity. While the confinement dependences on dimensional, engineering variables of the HeGDC+B and Li EVAP datasets differed, collisionality was found to unify the trends, with the lower collisionality lithium conditioned discharges extending the trend of increasing normalized confi nement time with decreasing collisionality when other dimension less variables were held as fi xed as possible. This increase of confi nement with decreasing collisionality was driven by a large reduction in electron transport in the outer region of the plasma. This result is consistent with gyrokinetic calculations that show microtearing and Electron Temperature Gradient modes to be more stable for the lower collisionality discharges. Ion transport, near neoclassical at high collisionality, became more anomalous at lower collisionality, possibly due to the growth of hybrid TEM/KBM modes in the outer regions of the plasma.

70 PLASMA PHYSICS AND FUSION TECHNOLOGY↗

A decentralized linear quadratic control design method for flexible structures

A decentralized suboptimal linear quadratic control design procedure which combines substructural synthesis, model reduction, decentralized control design, subcontroller synthesis, and controller reduction is proposed for the design of reduced-order controllers for flexible structures. The procedure starts with a definition of the continuum structure to be controlled. An evaluation model of finite dimension is obtained by the finite element method. Then, the finite element model is decomposed into several substructures by using a natural decomposition called substructuring decomposition. Each substructure, at this point, still has too large a dimension and must be reduced to a size that is Riccati-solvable. Model reduction of each substructure can be performed by using any existing model reduction method, e.g., modal truncation, balanced reduction, Krylov model reduction, or mixed-mode method. Then, based on the reduced substructure model, a subcontroller is designed by an LQ optimal control method for each substructure independently. After all subcontrollers are designed, a controller synthesis method called substructural controller synthesis is employed to synthesize all subcontrollers into a global controller. The assembling scheme used is the same as that employed for the structure matrices. Finally, a controller reduction scheme, called the equivalent impulse response energy controller (EIREC) reduction algorithm, is used to reduce the global controller to a reasonable size for implementation. The EIREC reduced controller preserves the impulse response energy of the full-order controller and has the property of matching low-frequency moments and low-frequency power moments. An advantage of the substructural controller synthesis method is that it relieves the computational burden associated with dimensionality. Besides that, the SCS design scheme is also a highly adaptable controller synthesis method for structures with varying configuration, or varying mass and stiffness properties.

Su, Tzu-Jeng↗

Oxide Two‐Dimensional Electron Gas with High Mobility at Room‐Temperature

Abstract The prospect of 2‐dimensional electron gases (2DEGs) possessing high mobility at room temperature in wide‐bandgap perovskite stannates is enticing for oxide electronics, particularly to realize transparent and high‐electron mobility transistors. Nonetheless only a small number of studies to date report 2DEGs in BaSnO 3 ‐based heterostructures. Here, 2DEG formation at the LaScO 3 /BaSnO 3 (LSO/BSO) interface with a room‐temperature mobility of 60 cm 2 V −1 s −1 at a carrier concentration of 1.7 × 10 13 cm –2 is reported. This is an order of magnitude higher mobility at room temperature than achieved in SrTiO 3 ‐based 2DEGs. This is achieved by combining a thick BSO buffer layer with an ex situ high‐temperature treatment, which not only reduces the dislocation density but also produces a SnO 2 ‐terminated atomically flat surface, followed by the growth of an overlying BSO/LSO interface. Using weak beam dark‐field transmission electron microscopy imaging and in‐line electron holography technique, a reduction of the threading dislocation density is revealed, and direct evidence for the spatial confinement of a 2DEG at the BSO/LSO interface is provided. This work opens a new pathway to explore the exciting physics of stannate‐based 2DEGs at application‐relevant temperatures for oxide nanoelectronics.

2-dimensional electron gas↗

A modal reduction method for use with nonlinear simulations of flexible multibody spacecraft

This paper presents a modal truncation method that can be used to prepare the structural flexibility data required as input to flexible, multibody spacecraft simulation programs. The method allows one to truncate the structural models of each flexible body in the system such that the assembled system is free of undesired high frequency system modes. This truncation method enables reasonable simulation costs to be incurred while maintaining dynamic simulation fidelity in the frequency range of interest. A simple one-dimensional example is presented which provides a physical interpretation of the method's workings. In addition, results are presented from the application of the method to a complex spacecraft.

Macala, G. A.↗

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