An endpoint sufficiency condition for multiple stationary solutions
Multiple solutions to problem of bolza in calculus of variation
Brusch, R. G.↗
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On the Relationship of Multiple Solutions in Tropospheric Photochemistry to Observable Phenomena
Multiple steady-state solutions are a fairly robust feature of simplified models of tropospheric photochemistry and have been reported for a range of different modeling assumptions. Multiple solutions occur through a bifurcation as changes in a control parameter induce a transition from low to high NO(x) conditions. The usual control parameters are the sources of NO and of CO, CH4 or non-methane hydrocarbons. Typically, with increasing NO source, bifurcations occur at NO(x). concentrations that are higher than would be expected of even heavily polluted conditions. However, there are other ways of inducing a low NO(x) - high NO(x) transition. In this paper the primary control parameter is solar zenith angle. This is varied throughout the year by computing noontime steady states on successive days. Background NO(x), varied by assuming different NO(x) sources values, is used as a secondary control parameter. It is found that bifurcations can occur from high to low NO(x) conditions, for reasonable background NO(x) levels, as the model progresses through spring and then from low to high NO,, during the progression through fall. A time dependent version of this model has been run for the same parameter values as in the steady state runs. This shows rapid spring and fall transitions between high and low NO(x) states. H2O2, for example, rises from sub-ppb levels to about 2 ppb over a five day period in spring and declines quickly, but less precipitously, in fall. This study supports the suggestions that the rapid change in peroxide concentrations between summer and winter conditions may be understood as a manifestation of different underlying steady-state behavior.
Stewart, Richard W.↗
The nature of multiple solutions for surface wind speed over the oceans from scatterometer measurements
The satellite SEASAT-A will carry a radar scatterometer in order to measure microwave backscatter from the sea surface. From pairs of radar measurements at angles separated by 90 deg in azimuth the surface wind speed and direction may be inferred, though not uniquely. The character of the solutions for wind speed and direction is displayed, as well as the nature of the ambiguities of these solutions. An economical procedure for handling such data is described, plus a criterion for the need for conventional (surface) data in order to resolve the ambiguities of solutions.
Price, J. C.↗
The nature of multiple solutions for surface wind speed over the oceans from scatterometer measurements
The satellite SEASAT-A will carry a radar scatterometer in order to measure microwave backscatter from the sea surface. From pairs of radar measurements at angles separated by 90 deg in azimuth the surface wind speed and direction may be inferred, though not uniquely. In this paper the character of the solutions for wind speed and direction is displayed, as well as the nature of the ambiguities of these solutions. An economical procedure for handling such data is described, plus a criterion for the need for conventional (surface) data in order to resolve the ambiguities of solutions.
Price, J. C.↗
Multiple solutions in a rotating annulus flow model
A series of numerical experiments is conducted for rotating annulus flow using Miller et al.'s (1992) Geophysical Flow Simulation (GFS) model; a mixture of 25-percent upwind-differencing and 75-percent center-differencing is employed to approximate the temperature advective terms. Attention is given to the wavenumber selection time and wavenumber regimes, the sensitivity in the wavenumber transition regions, and hysteresis and irregular wavenumber selections.
Lu, Huei-Iin↗
Finding Every Root of a Broad Class of Real, Continuous Functions in a Given Interval
One of the most pervasive needs within the Deep Space Network (DSN) Metric Prediction Generator (MPG) view period event generation is that of finding solutions to given occurrence conditions. While the general form of an equation expresses equivalence between its left-hand and right-hand expressions, the traditional treatment of the subject subtracts the two sides, leaving an expression of the form Integral of(x) = 0. Values of the independent variable x satisfying this condition are roots, or solutions. Generally speaking, there may be no solutions, a unique solution, multiple solutions, or a continuum of solutions to a given equation. In particular, all view period events are modeled as zero crossings of various metrics; for example, the time at which the elevation of a spacecraft reaches its maximum value, as viewed from a Deep Space Station (DSS), is found by locating that point at which the derivative of the elevation function becomes zero. Moreover, each event type may have several occurrences within a given time interval of interest. For example, a spacecraft in a low Moon orbit will experience several possible occultations per day, each of which must be located in time. The MPG is charged with finding all specified event occurrences that take place within a given time interval (or pass ), without any special clues from operators as to when they may occur, for the entire spectrum of missions undertaken by the DSN. For each event type, the event metric function is a known form that can be computed for any instant within the interval. A method has been created for a mathematical root finder to be capable of finding all roots of an arbitrary continuous function, within a given interval, to be subject to very lenient, parameterized assumptions. One assumption is that adjacent roots are separated at least by a given amount, xGuard. Any point whose function value is less than ef in magnitude is considered to be a root, and the function values at distances xGuard away from a root are larger than ef, unless there is another root located in this vicinity. A root is considered found if, during iteration, two root candidates differ by less than a pre-specified ex, and the optimum cubic polynomial matching the function at the end and at two interval points (that is within a relative error fraction L at its midpoint) is reliable in indicating whether the function has extrema within the interval. The robustness of this method depends solely on choosing these four parameters that control the search. The roots of discontinuous functions were also found, but at degraded performance.
Tausworthe, Robert C.↗
Multiple transonic solutions and a new class of shock transitions in solar and stellar winds
The steady isothermal solar wind equations are shown to admit, under certain circumstances, mutliple transonic solutions when, for example, momentum deposition gives rise to multiplee critical points in the flow. These multiple solutions consist of a continuous solution and solutions which involve shock transitions between critical solutions. The ambiguity arising from the multiplicity of the solutions can be resolved by following the time evolution of a wind profile with one critical point. Results of the numerical integration of the time-dependent equations with momentum addition show that each of these multiple solutions is physically accessible and depends on the rate of change of momentum deposition. These results suggest that standing shocks are likely to be present in the inner solar wind flow.
Habbal, S. R.↗
Multiple CA Solutions: Evaluating Data Quality and Creating Single Solutions for CA Operations
Briefing describes why having multiple solutions for an event is not advantageous as suggested by some entities.
conjunction assessment↗
Exact and approximate solutions for multiple scattering by cloudy and hazy planetary atmospheres.
Multiple light scattering solutions accuracy by diffraction peak omission from cloud and haze analytic phase functions compared for optically thick and thin planetary atmospheres
Hansen, J. E.↗
Calculation of asymmetric vortex separation on cones and tangent ogives based on a discrete vortex model
The boundary value problem for vortex separation at zero sideslip on cones and tangent ogives is set up by means of a discrete vortex model. The nonlinear algebraic equations for the boundary value problem admit multiple, physically feasible solutions, including the symmetric and asymmetric vortex solutions. Multiple solutions are proposed as an alternative explanation of the existence of asymmetric vortex separation at zero sideslip.
Chin, Suei↗
A block-based algorithm for the solution of compressible flows in rotor-stator combinations
A block-based solution algorithm is developed for the solution of compressible flows in rotor-stator combinations. The method allows concurrent solution of multiple solution blocks in parallel machines. It also allows a time averaged interaction at the stator-rotor interfaces. Numerical results are presented to illustrate the performance of the algorithm. The effect of the interaction between the stator and rotor is evaluated.
Akay, H. U.↗
Multiple transonic solutions with a new class of shock transitions in steady isothermal solar and stellar winds
It is shown that a new class of shock transitions arises in the transonic solutions of the steady isothermal solar wind equations when momentum deposition and/or nonradial flow tube divergence give rise to multiple critical points in the flow. These shock transitions between critical solutions occur for a certain range of the parameters which characterize the momentum deposition function. The isothermal wind equations allow multiple transonic solutions in the presence of such shock transitions, yielding a continuous solution passing through an inner critical point and solutions involving a shock transition between critical solutions. It is determined that these multiple transonic solutions have the same flow speed at the base but different supersonic flow speeds at infinity. It is found that the nonradial flow tube divergence and momentum addition are equivalent, which gives rise to multiple critical points and hence to multiple transonic solutions with shock transitions. In addition, the physical relevance of these properties are examined for astrophysical systems such as the inner solar wind, flows in extragalactic jets, and accretion discs.
Habbal, S. R.↗
Multiple CA Solutions: Technical Issues and Potential Remedies
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conjunction assessment↗
On the multiplicity of solutions of the inverse problem for a vibrating beam
The paper considers a specific inverse problem for a vibrating beam with three nonsympathetic spectra. More specifically, the spectral data consist of the natural frequencies of vibration of the beam in the following three configurations: (1) clamped-clamped, (2) clamped-supported, and (3) clamped-free. The basic equations for an inhomogeneous discrete beam are derived, where this simple mechanical system provides valuable insight into the inverse problem. All the necessary ingredients for a consideration of the inverse problem are given. Attention is given to an analysis of the particular inverse problem for which the spectral data are associated with the natural frequencies of vibrations in the above configurations. It is shown that with such data, the inverse problem has a 2exp(N-1)-fold multiplicity of solutions.
Barcilon, V.↗
Stress Intensity Factor Solutions for Multiple Edge Cracks in Ceramic Matrix Composites
NASA Lewis Research Center conducted a study to determine the stress intensity factor solutions for periodic arrays of bridged cracks for various crack spacings and crack lengths. Initially, the stress intensity factor of an array of unbridged multiple edge cracks was determined under constant global displacement as well as at a point load along the crack wake. These solutions are expected to contribute toward the development of a damage-based life-prediction methodology for CMC engine components.
Ghosn, Louis↗
Multiplicity of Solutions in Aerodynamics
One of the most striking phenomena which accompany the flow of fluids such as air and water about bodies, is that of changes in character. This phenomena is still very little understood. We will first discuss the nature of these changes and then show that pure theory leads to a multiplicity of characters of flow, among which we will endeavor to indicate those bearing some analogy to experimental results.
AERODYNAMICS↗
Contributions to HiLiftPW-3 Using Structured, Overset Grid Methods
The High-Lift Common Research Model (HL-CRM) and the JAXA Standard Model (JSM) were analyzed computationally using both the OVERFLOW and LAVA codes for the third AIAA High-Lift Prediction Workshop. Geometry descriptions and the test cases simulated are described. With the HL-CRM, the effects of surface smoothness during grid projection and the effect of partially sealing a flap gap were studied. Grid refinement studies were performed at two angles of attack using both codes. For the JSM, simulations were performed with and without the nacelle/pylon. Without the nacelle/pylon, evidence of multiple solutions was observed when a quadratic constitutive relation is used in the turbulence modeling; however, using time-accurate simulation seemed to alleviate this issue. With the nacelle/pylon, no evidence of multiple solutions was observed. Laminar-turbulent transition modeling was applied to both JSM configuration, and had an overall favorable impact on the lift predictions.
HiLiftPW-3↗
Modal interaction in postbuckled plates. Theory
Plates can have more than one buckled solution for a fixed set of boundary conditions. The theory for the identification and the computation of multiple solutions in buckled plates is examined. The theory predicts modal interaction (which is also called change in buckle pattern or secondary buckling) in experiments on certain plates with multiple theoretical solutions. A set of coordinate functions is defined for Galerkin's method so that the von Karman plate equations are reduced to a coupled set of cubic equations in generalized coordinates that are uncoupled in the linear terms. An iterative procedure for solving modal interaction problems is suggested based on this cubic form.
Thurston, Gaylen A.↗