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At least 991 records · Page 55

Numerical analysis of the asymptotic two-point boundary value solution for N-body trajectories.

Previously published asymptotic solutions for lunar and interplanetary trajectories have been modified and combined to formulate a general analytical boundary value solution applicable to a broad class of trajectory problems. In addition, the earlier first-order solutions have been extended to second-order to determine if improved accuracy is possible. Comparisons between the asymptotic solution and numerical integration for several lunar and interplanetary trajectories show that the asymptotic solution is generally quite accurate. Also, since no iterations are required, a solution to the boundary value problem is obtained in a fraction of the time required for numerically integrated solutions.

Lancaster, J. E.↗

Similarity analysis of differential equations by Lie group.

Methods for transforming partial differential equations into forms more suitable for analysis and solution are investigated. The idea of Lie's infinitesimal contact transformation group is introduced to develop a systematic method which involves mostly algebraic manipulations. A thorough presentation of the application of this general method to the problem of similarity analysis in a broader sense - namely, the similarity between partial and ordinary differential equations, boundary value and initial value problems, and nonlinear and linear equations - is given with new and very general methods evolved for deriving the possible groups of transformations.

Na, T. Y.↗

CSM solutions of rotating blade dynamics using integrating matrices

The dynamic behavior of flexible rotating beams continues to receive considerable research attention as it constitutes a fundamental problem in applied mechanics. Further, beams comprise parts of many rotating structures of engineering significance. A topic of particular interest at the present time involves the development of techniques for obtaining the behavior in both space and time of a rotor acted upon by a simple airload loading. Most current work on problems of this type use solution techniques based on normal modes. It is certainly true that normal modes cannot be disregarded, as knowledge of natural blade frequencies is always important. However, the present work has considered a computational structural mechanics (CSM) approach to rotor blade dynamics problems in which the physical properties of the rotor blade provide input for a direct numerical solution of the relevant boundary-and-initial-value problem. Analysis of the dynamics of a given rotor system may require solution of the governing equations over a long time interval corresponding to many revolutions of the loaded flexible blade. For this reason, most of the common techniques in computational mechanics, which treat the space-time behavior concurrently, cannot be applied to the rotor dynamics problem without a large expenditure of computational resources. By contrast, the integrating matrix technique of computational mechanics has the ability to consistently incorporate boundary conditions and 'remove' dependence on a space variable. For problems involving both space and time, this feature of the integrating matrix approach thus can generate a 'splitting' which forms the basis of an efficient CSM method for numerical solution of rotor dynamics problems.

Lakin, William D.↗

Singular perturbations in the state regulator problem

Most of the results of singular perturbation theory have been concerned with initial value problems whereas optimal control problems are of two-point boundary value type. The portions of this theory applicable to the open loop state regulator problem are reviewed. For obtaining approximate solutions to the state regulator problem the method of matched asymptotic expansions is employed. This method has been developed in connection with certain fluid mechanics problems and is applicable to nonlinear as well as linear problems. It has been found in the past to be advantageous not to formulate this method generally but to apply it to each individual problem and this approach is adopted here. A general recipe for the method is given and its application is illustrated by using the method to obtain an approximate solution to a simple, specific state regulator problem.

Ardema, M. D.↗

Thermal instability in post-flare plasmas

The cooling of post-flare plasmas is discussed and the formation of loop prominences is explained as due to a thermal instability. A one-dimensional model was developed for active loop prominences. Only the motion and heat fluxes parallel to the existing magnetic fields are considered. The relevant size scales and time scales are such that single-fluid MHD equations are valid. The effects of gravity, the geometry of the field and conduction losses to the chromosphere are included. A computer code was constructed to solve the model equations. Basically, the system is treated as an initial value problem (with certain boundary conditions at the chromosphere-corona transition region), and a two-step time differencing scheme is used.

Antiochos, S. K.↗

Approximate solutions to the Navier-Stokes initial value problem

A Galerkin-Ritz procedure for any arbitrary system of field equations is shown to follow generically from 'two-functional' variational conditions. The initial-value problem for boundary-free incompressible Navier-Stokes flow is solved analytically in the two-parameter-function approximation.

Rosen, G.↗

Mechanisms of interfacial reactivity in near surface and extreme environments

The local water structure surrounding ions in aqueous solutions greatly affects their chemical properties such as reaction rates, ion association, and proton and electron transport. These properties result in the behavior of ions in natural aqueous environments. For example ore transport is facilitated by chloride ion pair formation and the reaction of ions in an interface is strongly dependent on the dehydration of the ion hydration shell. We are developing the use of high-­resolution XAFS observations and 1st principles based MD-­XAFS analysis (spectra simulated using 1st principle methods with no adjustable parameters, AIMD) to interpret the solution properties of strongly interacting aqueous solutes under arbitrary pressure and temperature conditions. In the 1st principle MD-­XAFS method density functional theory (DFT) based MD simulations(Car and Parrinello, 1985) are used to generate a large ensemble of structural snap shots of the hydration region. These are then used to generate scattering intensities. I emphasize three points about this novel approach to analyzing XAFS data. 1st: As illustrated in Figure 1, the level of agreement between the calculated and observed intensities is considerably higher than has been obtained by any XAFS analysis to date (note 2nd shell region, R> 2 Å). 2nd: This result was obtained from a parameter free simulation with no fitting of the interaction potentials to any data. This supports the use of these methods for more difficult environments and more complex solutes (polyions). 3rd: New information about the shell structure (Figure 1) is now available because of this more detailed agreement. We note also that both multiple scattering and second shell features are well represented in the analysis. As far as we know this is the 1st analysis of second shell structure and multiple scattering. Excellent agreement has been obtained for most of the third row metal ions: Ca 2+ , Zn 2+ , Cu 2+ , Ni 2+ , Co 2+ , Mn 2+ , Fe 3+ , Cr 3+ . Calculations on these systems are demanding because of their open electronic shells, and high ionic charge. Principal Investigator: Professor John Weare (University of California, San Diego) The prediction of the interactions of geochemical fluids with minerals, nanoparticles, and colloids under extreme near surface conditions of temperature (T) and pressure (P) is a grand challenge research need in geosciences (U.S. DOE 2007, Basic Research Needs for Geosciences: Facilitating the 21st Energy Systems.). To evaluate the impact of these processes on energy production and management strategies it is necessary to have a high level of understanding of the interaction between complex natural fluids and mineral formations. This program emphasizes 1st principle parameter free simulations of complex chemical processes in solutions, in the mineral phase, and in the interfaces between these phases The development of new computational tools (with emphasis on oxide materials and reaction dynamics) tailored to treat wide range of conditions and time scales experienced in such geochemical applications is has been developed. Because of the sensitivity of the interaction in these systems to electronic structure and local bonding environments, and of the need to describe bond breaking/formation, our simulations are based on interactions calculated at the electronic structure level (ab-initio molecular dynamics, AIMD). The progress in the computational aspects of program may be summarized in terms of the following themes (objectives); Development of efficient parameter free dynamical simulation technology based on 1st principles force and energy calculations especially adapted for geochemical applications (e.g., mineral, interfaces and aqueous solutions) (continuing program); Calculation of the dynamics of water structure of in the surface-water interface of transition metal oxides and oxihydroxides; and Development of improved (beyond DFT+GGA) electronic structure calculations for minerals and the interface region that more accurately calculate electron correlation, spin density, and localization. The focus of the program is also on the iron oxide and oxihydroxide minerals and Fe 2+ (aq)/Fe 3+ (cr) oxidation in the mineral solution interface region. These methods included the development of model Hamilitonian methods that can be solved to near convergence for single site models (DMFT) and many-body perturbation methods (MP2, GW); Development of time decomposition methods to extend time scales of molecular dynamics (MD) simulations and support the use of high complexity electronic structure calculations (MP2, CCSD(T)) of forces for use in dynamical simulations where very high chemical accuracy is required (microsolvated reactions in absorbed surface layers); and The development of a new linear scaling finite element solver for eigenvalue problem that supports solution of quantum problems with unusual potential and boundary values. Application progress of the above new simulation technology to problems of geochemical interests includes; The prediction of metal oxide surface structure and the reduction/oxidation of Fe 3+ (cr)/Fe 2+ (aq) in metal oxide (hematite, goethite)/solution interfaces. Result: water interacts strongly with the 001 Hematite surface; interaction of water with the 100 goethite is weak; The study of ion solvation and the composition of ion hydration shells under extreme conditions (focus on Fe 3+/2+ , Al 3+ and Mg 2+ and their hydroxide speciation). Result: Ion association in water solutions can be calculated from 1st principle methods. Efficient sampling of the free energy requires more development; The continued development of new high resolution analysis of XAFS scattering of disordered systems (particularly Al, Mg) and of XANES calculations for aqueous ions. Result: EXAFS spectra can be calculated to high accuracy with DFT level dynamic simulations; The exploration of electron localization and electron transport in metal oxides (highly correlated materials). Result: Proper description of electron localization requires levels of calculation beyond DFT; and Localization of electrons in DFT type Hamiltonians was studied. Result: For very Dirac high exchange new solutions (New unphysical bifrucations) to the eigenvalue problem are found. The program was highly collaborative involving faculty and students in mathematics, physics and computer science departments as well as coworkers at the Pacific Northwest National Laboratories (PNNL). The students in this program had the opportunity to develop skills in the development of methods, the implementation of method on high performance parallel computers and the application of these methods to problem in geochemical science. Much of the software that was developed was incorporated in the NWchem software package maintained by PNNL.

58 GEOSCIENCES↗

COMOC: Three dimensional boundary region variant, programmer's manual

The three-dimensional boundary region variant of the COMOC computer program system solves the partial differential equation system governing certain three-dimensional flows of a viscous, heat conducting, multiple-species, compressible fluid including combustion. The solution is established in physical variables, using a finite element algorithm for the boundary value portion of the problem description in combination with an explicit marching technique for the initial value character. The computational lattice may be arbitrarily nonregular, and boundary condition constraints are readily applied. The theoretical foundation of the algorithm, a detailed description on the construction and operation of the program, and instructions on utilization of the many features of the code are presented.

Orzechowski, J. A.↗

Exact solutions for the static bending of Euler-Bernoulli beams using Eringen’s two-phase local/nonlocal model

Though widely used in modelling nano- and micro- structures, Eringen’s differential model shows some inconsistencies and recent study has demonstrated its differences between the integral model, which then implies the necessity of using the latter model. In this paper, an analytical study is taken to analyze static bending of nonlocal Euler-Bernoulli beams using Eringen’s two-phase local/nonlocal model. Firstly, a reduction method is proved rigorously, with which the integral equation in consideration can be reduced to a differential equation with mixed boundary value conditions. Then, the static bending problem is formulated and four types of boundary conditions with various loadings are considered. By solving the corresponding differential equations, exact solutions are obtained explicitly in all of the cases, especially for the paradoxical cantilever beam problem. Finally, asymptotic analysis of the exact solutions reveals clearly that, unlike the differential model, the integral model adopted herein has a consistent softening effect. Comparisons are also made with existing analytical and numerical results, which further shows the advantages of the analytical results obtained. Additionally, it seems that the once controversial nonlocal bar problem in the literature is well resolved by the reduction method.

71 CLASSICAL AND QUANTUM MECHANICS, GENERAL PHYSIC↗

Reduction of the Dirichlet problem to an initial value problem.

Although the derivation is concerned with solutions for plane regions with prescribed boundary values, the approach presented could by easily generalized to higher dimensions. The initial-value method is derived by a combination of invariant imbedding techniques and the Fredholm integral equation method of representation of the potential as a function of a dilayer distribution on the boundary of the region in question.

Kalaba, R.↗

Modes of vibration in a circular plate with three simple support points.

The analytical solutions for the vibrational modes of a thin circular flat plate that is simply supported at three points on the circumference are presented. The mode shapes and corresponding eigenvalues are obtained. Results show that the modes can be grouped into four different types depending on the manner by which they receive the pressure at the supported points. The problem is of the mixed boundary value type in that some portion of the boundary is free while the other portion is simply supported.

Chi, C.↗

Computing Deformations Of Rubbery Materials

Better use made of experimental data in finite-element computations. New formulation of constitutive equations of rubbery, nonlinearly elastic material enables finite-element analysis of boundary-value stress-and-strain problems involving arbitrary shapes and loads. In development of formulation, principal stretches used as arguments of strain-energy-density function.

Peng, Steven T. J.↗

Aeroelastic System Development Using Proper Orthogonal Decomposition and Volterra Theory

This research combines Volterra theory and proper orthogonal decomposition (POD) into a hybrid methodology for reduced-order modeling of aeroelastic systems. The out-come of the method is a set of linear ordinary differential equations (ODEs) describing the modal amplitudes associated with both the structural modes and the POD basis functions for the uid. For this research, the structural modes are sine waves of varying frequency, and the Volterra-POD approach is applied to the fluid dynamics equations. The structural modes are treated as forcing terms which are impulsed as part of the uid model realization. Using this approach, structural and uid operators are coupled into a single aeroelastic operator. This coupling converts a free boundary uid problem into an initial value problem, while preserving the parameter (or parameters) of interest for sensitivity analysis. The approach is applied to an elastic panel in supersonic cross ow. The hybrid Volterra-POD approach provides a low-order uid model in state-space form. The linear uid model is tightly coupled with a nonlinear panel model using an implicit integration scheme. The resulting aeroelastic model provides correct limit-cycle oscillation prediction over a wide range of panel dynamic pressure values. Time integration of the reduced-order aeroelastic model is four orders of magnitude faster than the high-order solution procedure developed for this research using traditional uid and structural solvers.

Lucia, David J.↗

Towards Understanding the Mechanism of Receptivity and Bypass Dynamics in Laminar Boundary Layers

Three problems concerning laminar-turbulent transition are addressed by solving a series of initial value problems. The first problem is the calculation of resonance within the continuous spectrum of the Blasius boundary layer. The second is calculation of the growth of Tollmien-Schlichting waves that are a direct result of disturbances that only lie outside of the boundary layer. And, the third problem is the calculation of non-parallel effects. Together, these problems represent a unified approach to the study of freestream disturbance effects that could lead to transition. Solutions to the temporal, initial-value problem with an inhomogeneous forcing term imposed upon the flow is sought. By solving a series of problems, it is shown that: A transient disturbance lying completely outside of the boundary layer can lead to the growth of an unstable Tollmien-Schlichting wave. A resonance with the continuous spectrum leads to strong amplification that may provide a mechanism for bypass transition once nonlinear effects are considered. A disturbance with a very weak unstable Tollmien-Schlichting wave can lead to a much stronger Tollmien-Schlichting wave downstream, if the original disturbance has a significant portion of its energy in the continuum modes.

Lasseigne, D. G.↗

Asymptotic theory of two-dimensional trailing-edge flows

Problems of laminar and turbulent viscous interaction near trailing edges of streamlined bodies are considered. Asymptotic expansions of the Navier-Stokes equations in the limit of large Reynolds numbers are used to describe the local solution near the trailing edge of cusped or nearly cusped airfoils at small angles of attack in compressible flow. A complicated inverse iterative procedure, involving finite-difference solutions of the triple-deck equations coupled with asymptotic solutions of the boundary values, is used to accurately solve the viscous interaction problem. Results are given for the correction to the boundary-layer solution for drag of a finite flat plate at zero angle of attack and for the viscous correction to the lift of an airfoil at incidence. A rational asymptotic theory is developed for treating turbulent interactions near trailing edges and is shown to lead to a multilayer structure of turbulent boundary layers. The flow over most of the boundary layer is described by a Lighthill model of inviscid rotational flow. The main features of the model are discussed and a sample solution for the skin friction is obtained and compared with the data of Schubauer and Klebanoff for a turbulent flow in a moderately large adverse pressure gradient.

Melnik, R. E.↗

Dynamic behavior of soap films.

Dynamic behavior of soap film stretched between two coaxial rings, assuming potential energy of this surface to be proportional to surface area

FILM↗