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NASA NTRS1986

Studies of electron-molecule collisions - Applications to e-H2O

Elastic differential and momentum transfer cross sections for the elastic scattering of electrons by H2O are reported for collision energies from 2 to 20 eV. These fixed-nuclei static-exchange cross sections were obtained using the Schwinger variational approach. In these studies the exchange potential is directly evaluated and not approximated by local models. The calculated differential cross sections, obtained with a basis set expansion of the scattering wave function, agree well with available experimental data at intermediate and larger angles. As used here, the results cannot adequately describe the divergent cross sections at small angles. An interesting feature of the calculated cross sections, particularly at 15 and 20 eV, is their significant backward peaking. This peaking occurs in the experimentally inaccessible region beyond a scattering angle of 120 deg. The implication of this feature for the determination of momentum transfer cross sections is described.

Brescansin, L. M.↗

NASA NTRS1986

Thermal analysis of an orthotropic engineering body

This work is an effort in determining the temperature distribution of an orthotropic engineering body. First the governing partial differential equation is derived, then, a finite element procedure using variational approach is developed for the solution of the governing equation. Finally, a computer program provides a set of satisfactory results.

Fu, K.-C.↗

NASA NTRS1986

Simulation experiments for testing the assimilation of geostationary satellite temperature retrievals into a numerical prediction model

A simulation experiment is presented which tests a procedure that assimilates temperature soundings from geostationary satellites using a method developed by Gal-Chen (1983). The simulation is used to decide whether the increased temporal resolution of geostationary satellite data can be used to improve numerical simulation of a baroclinically unstable wave, and to assess the impact of gaps in the satellite sounding fields on the model simulations. Background information on the simulation experiment is presented, including discussion of the variational approach utilized, the numerical model, updating procedures, and control run. The simulation results indicate that, for the case of a baroclinically unstable wave, considerable improvements in the short-range forecast may be realized if geostationary satellite data are inserted with a frequency near 1 hr during a 6 hr analysis cycle compared with a single insertion.

Gal-Chen, T.↗

NASA NTRS1987

Grid adaption for hypersonic flow

The methods of grid adaption are reviewed and a method is developed with the capability of adaption to several flow variables. This method is based on a variational approach and is an algebraic method which does not require the solution of partial differential equations. Also the method has been formulated in such a way that there is no need for any matrix inversion. The method is used in conjunction with the calculation of hypersonic flow over a blunt nose body. The equations of motion are the compressible Navier-Stokes equations where all viscous terms are retained. They are solved by the MacCormack time-splitting method. A movie has been produced which shows simultaneously the transient behavior of the solution and the grid adaption.

Abolhassani, Jamshid S.↗

NASA NTRS1986

Numerical solutions of three-dimensional Navier-Stokes equations for closed-bluff bodies

Methods of grid adaption are reviewed and a method is developed with the capability of adaption to several flow variables. This method is based on a variational approach and is an algebraic method which does not require the solution of partial differential equations. The method has also been formulated in such a way that there is no need for any matrix inversion. The method is used in conjunction with the calculation of hypersonic flow over a blunt nose. The equations of motion are the compressible Navier-Stokes equations where all viscous terms are retained. They are solved by the MacCormack time-splitting method and a movie has been produced which shows simultaneously the transient behavior of the solution and the grid adaption. The results are compared with the experimental and other numerical results.

Abolhassani, Jamshid S.↗

NASA NTRS1987

Thermal stresses of a wind turbine blade made of orthotropic material

This study is to investigate the thermal stress of a wind turbine blade made of wood composite material. First, the governing partial differential equation on heat conduction is stated, then, a finite element procedure using a variational approach is employed for the solution of the governing equation. Thus, the temperature field throughout the blade is determined. Next, based on the temperature field, a finite element procedure using potential energy approach is applied to determine the thermal stress field. A set of results is obtained through the use of a computer, which is considered to be satisfactory.

Fu, Kuan-Chen↗

NASA NTRS1989

Aerodynamics of engine-airframe interaction

The development of the variational approach for the solution of inviscid aerodynamic problems using solution adaptive grids is discussed. The formulation of a new, directional weighting, functional has been shown to have desirable properties. The scheme has been applied to compute the transonic flow past two-dimensional airfoils using the Euler equations of inviscid, compressible flow. Transonic flows in quasi-one-dimensional nozzles and over the two dimensional airfoils are solved on the various solution-adaptive-grids to demonstrate the applicability of the proposed directional-concentration functional and the grid adaptation process from the stand point of improving the solution accuracy and demonstrating the overall convergence.

Caughey, David A.↗

NASA NTRS1989

Significance of norms and completeness in variational based methods

By means of a simple structural problem, an important requirement often overlooked in practice on the basis functions used in Rayleigh-Ritz-Galerkin type methods is brought into focus. The problem of the static deformation of a uniformly loaded beam is solved variationally by expanding the beam displacement in a Fourier Cosine series. The potential energy functional is rendered stationary subject to the geometric boundary conditions. It is demonstrated that the variational approach does not converge to the true solution. The object is to resolve this paradox, and in so doing, indicate the practical implications of norms and completeness in an appropriate inner product space.

Storch, Joel A.↗

NASA NTRS1990

Derivatives of buckling loads and vibration frequencies with respect to stiffness and initial strain parameters

A uniform variational approach to sensitivity analysis of vibration frequencies and bifurcation loads of nonlinear structures is developed. Two methods of calculating the sensitivities of bifurcation buckling loads and vibration frequencies of nonlinear structures, with respect to stiffness and initial strain parameters, are presented. A direct method requires calculation of derivatives of the prebuckling state with respect to these parameters. An adjoint method bypasses the need for these derivatives by using instead the strain field associated with the second-order postbuckling state. An operator notation is used and the derivation is based on the principle of virtual work. The derivative computations are easily implemented in structural analysis programs. This is demonstrated by examples using a general purpose, finite element program and a shell-of-revolution program.

Haftka, Raphael T.↗

NASA NTRS1991

Discussion summary: Fictitious domain methods

Fictitious Domain methods are constructed in the following manner: Suppose a partial differential equation is to be solved on an open bounded set, Omega, in 2-D or 3-D. Let R be a rectangle domain containing the closure of Omega. The partial differential equation is first solved on R. Using the solution on R, the solution of the equation on Omega is then recovered by some procedure. The advantage of the fictitious domain method is that in many cases the solution of a partial differential equation on a rectangular region is easier to compute than on a nonrectangular region. Fictitious domain methods for solving elliptic PDEs on general regions are also very efficient when used on a parallel computer. The reason is that one can use the many domain decomposition methods that are available for solving the PDE on the fictitious rectangular region. The discussion on fictitious domain methods began with a talk by R. Glowinski in which he gave some examples of a variational approach to ficititious domain methods for solving the Helmholtz and Navier-Stokes equations.

Glowinski, Rowland↗

NASA NTRS1989

Topology and grid adaption for high-speed flow computations

This study investigates the effects of grid topology and grid adaptation on numerical solutions of the Navier-Stokes equations. In the first part of this study, a general procedure is presented for computation of high-speed flow over complex three-dimensional configurations. The flow field is simulated on the surface of a Butler wing in a uniform stream. Results are presented for Mach number 3.5 and a Reynolds number of 2,000,000. The O-type and H-type grids have been used for this study, and the results are compared together and with other theoretical and experimental results. The results demonstrate that while the H-type grid is suitable for the leading and trailing edges, a more accurate solution can be obtained for the middle part of the wing with an O-type grid. In the second part of this study, methods of grid adaption are reviewed and a method is developed with the capability of adapting to several variables. This method is based on a variational approach and is an algebraic method. Also, the method has been formulated in such a way that there is no need for any matrix inversion. This method is used in conjunction with the calculation of hypersonic flow over a blunt-nose body. A movie has been produced which shows simultaneously the transient behavior of the solution and the grid adaption.

Abolhassani, Jamshid S.↗

NASA NTRS1990

Configuration design sensitivity analysis of built-up structures

In the design of complex built-up structures that are made of truss, beam, membrane, shell, and solid, there are five different kinds of design variables: material property, sizing, shape, configuration, and topological variables. Previous research has shown that the improvement in performances obtained by altering the configuration of structural components can be much more significant than those obtained when the geometry is assumed to be fixed. Using the variational approach, a unified design sensitivity was developed for the first three kinds of design variables, and was further extended recently in many structural analysis problems such as nonlinear, structural dynamics, and frequency response analysis. A continuum design sensitivity analysis method is developed for the configuration design variable of built-up structures.

Choi, Kyung K.↗

NASA NTRS1999

Synergistic Effects of Physical Aging and Damage on Long-Term Behavior of Polymer Matrix Composites

The research consisted of two major parts, first modeling and simulation of the combined effects of aging and damage on polymer composites and secondly an experimental phase examining composite response at elevated temperatures, again activating both aging and damage. For the simulation, a damage model for polymeric composite laminates operating at elevated temperatures was developed. Viscoelastic behavior of the material is accounted for via the correspondence principle and a variational approach is adopted to compute the temporal stresses within the laminate. Also, the effect of physical aging on ply level stress and on overall laminate behavior is included. An important feature of the model is that damage evolution predictions for viscoelastic laminates can be made. This allows us to track the mechanical response of the laminate up to large load levels though within the confines of linear viscoelastic constitutive behavior. An experimental investigation of microcracking and physical aging effects in polymer matrix composites was also pursued. The goal of the study was to assess the impact of aging on damage accumulation, in ten-ns of microcracking, and the impact of damage on aging and viscoelastic behavior. The testing was performed both at room and elevated temperatures on [+/- 45/903](sub s) and [02/903](sub s) laminates, both containing a set of 90 deg plies centrally located to facilitate investigation of microcracking. Edge replication and X-ray-radiography were utilized to quantify damage. Sequenced creep tests were performed to characterize viscoelastic and aging parameters. Results indicate that while the aging times studied have limited ]Influence on damage evolution, elevated temperature and viscoelastic effects have a profound effect on the damage mode seen. Some results are counterintuitive, including the lower strain to failure for elevated temperature tests and the catastrophic failure mode observed for the [+/- 45/9O3](sub s), specimens. The fracture toughness for transverse cracks increases with increasing temperature for both systems: transverse cracking was completely absent prior to failure in [+/- 45/903](sub s), and was suppressed for [02/903](sub s). No significant effect of damage on aging or viscoelastic parameters was observed.

Brinson, L. Cate↗

NASA NTRS2000

Guidance Concept for a Mars Ascent Vehicle First Stage

This paper presents a guidance concept for use on the first stage of a Mars Ascent Vehicle (MAV). The guidance is based on a calculus of variations approach similar to that used for the final phase of the Apollo Earth return guidance. A three degree-of-freedom (3DOF) Monte Carlo simulation is used to evaluate performance and robustness of the algorithm.

Queen, Eric M.↗

NASA NTRS2001

Ab-Initio Description and Prediction of Properties of Carbon-Based and Other Non-Metallic Materials

We have resolved the long-standing problem consisting of 30%-50% theoretical underestimates of the band gaps of non-metallic materials. We describe the Bagayoko, Zhao, and Williams (BZW) method that rigorously circumvents the basis-set and variational effect presumed to be a cause of these underestimates. We present ab-initio, computational results that are in agreement with experiment for diamond (C), silicon (Si), silicon carbides (3C-SiC and 4H-SiC), and other semiconductors (GaN, BaTiO3, AlN, ZnSe, ZnO). We illustrate the predictive capability of the BZW method in the case of the newly discovered cubic phase of silicon nitride (c-Si3N4) and of selected carbon nanotabes [(10,0), and (8,4)]. Our conclusion underscores the inescapable need for the BZW method in ab-initio calculations that employ a basis set in a variational approach. Current nanoscale trends amplify this need. We estimate that the potential impact of applications of the BZW method in advancing our understanding of nonmetallic materials, in informing experiment, and particularly in guiding device design and fabrication is simply priceless.

Bagayoko, D.↗

NASA NTRS2004

Angle-of-Attack-Modulated Terminal Point Control for Neptune Aerocapture

An aerocapture guidance algorithm based on a calculus of variations approach is developed, using angle of attack as the primary control variable. Bank angle is used as a secondary control to alleviate angle of attack extremes and to control inclination. The guidance equations are derived in detail. The controller has very small onboard computational requirements and is robust to atmospheric and aerodynamic dispersions. The algorithm is applied to aerocapture at Neptune. Three versions of the controller are considered with varying angle of attack authority. The three versions of the controller are evaluated using Monte Carlo simulations with expected dispersions.

Queen, Eric M.↗

NASA NTRS2004

Assimilative modeling of low latitude ionosphere

In this paper we present an observation system simulation experiment for modeling low-latitude ionosphere using a 3-dimensional (3-D) global assimilative ionospheric model (GAIM). The experiment is conducted to test the effectiveness of GAIM with a 4-D variational approach (4DVAR) in estimation of the ExB drift and thermospheric wind in the magnetic meridional planes simultaneously for all longitude or local time sectors. The operational Global Positioning System (GPS) satellites and the ground-based global GPS receiver network of the International GPS Service are used in the experiment as the data assimilation source. 'The optimization of the ionospheric state (electron density) modeling is performed through a nonlinear least-squares minimization process that adjusts the dynamical forces to reduce the difference between the modeled and observed slant total electron content in the entire modeled region. The present experiment for multiple force estimations reinforces our previous assessment made through single driver estimations conducted for the ExB drift only.

data assimilation↗

NASA NTRS2008

Application of an Ensemble Smoother to Precipitation Assimilation

Assimilation of precipitation in a global modeling system poses a special challenge in that the observation operators for precipitation processes are highly nonlinear. In the variational approach, substantial development work and model simplifications are required to include precipitation-related physical processes in the tangent linear model and its adjoint. An ensemble based data assimilation algorithm "Maximum Likelihood Ensemble Smoother (MLES)" has been developed to explore the ensemble representation of the precipitation observation operator with nonlinear convection and large-scale moist physics. An ensemble assimilation system based on the NASA GEOS-5 GCM has been constructed to assimilate satellite precipitation data within the MLES framework. The configuration of the smoother takes the time dimension into account for the relationship between state variables and observable rainfall. The full nonlinear forward model ensembles are used to represent components involving the observation operator and its transpose. Several assimilation experiments using satellite precipitation observations have been carried out to investigate the effectiveness of the ensemble representation of the nonlinear observation operator and the data impact of assimilating rain retrievals from the TMI and SSM/I sensors. Preliminary results show that this ensemble assimilation approach is capable of extracting information from nonlinear observations to improve the analysis and forecast if ensemble size is adequate, and a suitable localization scheme is applied. In addition to a dynamically consistent precipitation analysis, the assimilation system produces a statistical estimate of the analysis uncertainty.

Zhang, Sara↗