Vibration studies
Computer program calculations of upper and lower bounds to nonzero bending frequencies of missiles
GUIDED MISSILE↗
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Sudden stretching of a four layered composite plate
An approximate theory of laminated plates is developed by assuming that the extensioral and thickness mode of vibration are coupled. The mixed boundary value crack problem of a four layered composite plate is solved. Dynamic stress intensity factors for a crack subjected to suddenly applied stress are found to vary as a function of time and depend on the material properties of the laminate. Stress intensification in the region near the crack front can be reduced by having the shear modulus of the inner layers to be larger than that of the outer layers.
Sih, G. C.↗
Investigation of Boundary Conditions for Flexible Multibody Spacecraft Dynamics
In support of both the Space Shuttle and International Space Station programs, a set of generic multibody dynamics algorithms integrated within the Trick simulation environment have addressed the variety of on-orbit manipulator simulation requirements for engineering analysis, procedures development and crew familiarization/training at the NASA Johnson Space Center (JSC). Enhancements to these dynamics algorithms are now being driven by a new set of Constellation program requirements for flexible multibody spacecraft simulation. One particular issue that has been discussed within the NASA community is the assumption of cantilever-type flexible body boundary conditions. This assumption has been commonly utilized within manipulator multibody dynamics formulations as it simplifies the computation of relative motion for articulated flexible topologies. Moreover, its use for modeling of space-based manipulators such as the Shuttle Remote Manipulator System (SRMS) and Space Station Remote Manipulator System (SSRMS) has been extensively validated against flight data. For more general flexible spacecraft applications, however, the assumption of cantilever-type boundary conditions may not be sufficient. This paper describes the boundary condition assumptions that were used in the original formulation, demonstrates that this formulation can be augmented to accommodate systems in which the assumption of cantilever boundary conditions no longer applies, and verifies the approach through comparison with an independent model previously validated against experimental hardware test data from a spacecraft flexible dynamics emulator.
Boundary-value problems↗
Satellite orbital transfer studies Final report, 29 Jun. 1965 - 30 Aug. 1966
Quasi-linearization techniques for solving boundary value satellite orbital transfer problems
ORBITAL TRANSFER↗
Space vehicle guidance-a boundary value formulation. part ii- boundary conditions with parameters
Boundary value formulation of space vehicle guidance problem
GUIDANCE↗
New Boundary Constraints for Elliptic Systems used in Grid Generation Problems
This paper discusses new boundary constraints for elliptic partial differential equations as used in grid generation problems in generalized curvilinear coordinate systems. These constraints, based on the principle of local conservation of thermal energy in the vicinity of the boundaries, are derived using the Green's Theorem. They uniquely determine the so called decay parameters in the source terms of these elliptic systems. These constraints' are designed for boundary clustered grids where large gradients in physical quantities need to be resolved adequately. It is observed that the present formulation also works satisfactorily for mild clustering. Therefore, a closure for the decay parameter specification for elliptic grid generation problems has been provided resulting in a fully automated elliptic grid generation technique. Thus, there is no need for a parametric study of these decay parameters since the new constraints fix them uniquely. It is also shown that for Neumann type boundary conditions, these boundary constraints uniquely determine the solution to the internal elliptic problem thus eliminating the non-uniqueness of the solution of an internal Neumann boundary value grid generation problem.
Kaul, Upender K.↗
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.↗
A modified sweep method for control optimization
Modified sweep second-order variational method for solving two-point boundary value trajectory optimization problem
Colunga, 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.↗
The initial value problem for the linearized equations of water waves.
Linearized system of water waves initial value problem, obtaining eigenvalue with parameter in boundary condition
LAPLACE EQUATION↗
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.↗