Nonlinear Analytical Solution Research Articles

Nonlinear Analytical Solution for Cable Truss

In this technical note the nonlinear closed-form static solution of the suspended biconvex and biconcave cable trusses with unmovable, movable, or elastic yielding supports subjected to vertical distributed load applied over the entire span is presented. Irvine's linearized forms of the deflection and the cable equations are modified because the effects of the nonlinear truss behavior needed to be incorporated in them. The concrete form of the system of two nonlinear cubic cable equations is derived and presented. From a solution of a nonlinear vertical equilibrium equation for a loaded cable truss, the additional vertical deflection is determined. The transformation analytical model serves to determine the response, i.e., horizontal components of cable forces and deflection of the geometrically nonlinear truss, due to the applied loading, considering effects of elastic deformations, temperature changes, and elastic supports. The deflection of asymmetric prestressed cable trusses has been compared with Irvine's linear solution as well as the nonlinear finite element model results.

Temporal Adaptive Three-Grid Correction Method for Transient Nonlinear Nodal Calculations

As an efficiency enhancement numerical scheme of transient nonlinear nodal calculations, a three-grid correction scheme (3GCS) using a modified W cycle based on three grid structures of three-dimensional (3-D) four-node-per-assembly (4N/A), 3-D 1N/A, and two-dimensional (2-D) 1N/A is developed. Its computational efficiency is compared with a single-grid biconjugate gradient stabilized (BICGSTAB) iteration scheme in popular use in terms of 3-D 4N/A nonlinear analytical nodal method solutions to Nuclear Energy Agency Committee on Reactor Physics pressurized water reactor rod ejection benchmark problems. It is shown that in computational efficiency, the 3GCS excels the BICGSTAB iteration method using preconditioners such as Jacobi, incomplete lower and upper (ILU), and 3-D block incomplete lower and upper (BILU3D) preconditioners. It is also shown that coarse-grid residual equations based on the 3-D 1N/A grid structure can predict temporal truncation errors as accurately as the 3-D 4N/A fine-grid residual equation but with considerably less overhead computing time for variable time-step size control calculations by a step doubling method. In addition, incorporation of preconditioners into the 3GCS is shown to enhance further efficiency of the nonpreconditioned 3GCS. From these results, it is concluded that the temporal adaptive 3GCS employing coarse-grid residual equations for temporal step-size control as well as the preconditioner like the BILU3D can provide a very efficient iterative solution scheme for transient nonlinear nodal calculations.

The Momentum Imbalance Paradox Revisited

Abstract The classical problem of a point source situated along a southern boundary emptying buoyant water into a (β plane) ocean is revisited. Pichevin and Nof (PN) have shown that, in contrast to the view prevailing at the time, such an inviscid outflow does not simply turn to the right. Rather, it bifurcates into two branches: a steady branch that does turn to the right (eastward) and an unsteady branch that periodically sheds eddies to the left (westward). This partition is because a simple turn to the right of the entire outflow leaves the outflow’s long-shore momentum flux unbalanced, creating a paradox. In contrast, the branching allows the westward-drifting eddies (westward branch) to balance the momentum flux of the steady current (eastward branch). Although the analytical PN solution is useful and informative, it is cumbersome and difficult to apply to actual outflows. Here, a considerably simpler nonlinear analytical solution is presented. Using the idea that the eddies grow slowly relative to their rotation rate, it is shown that an intense (i.e., large Rossby number) and large relative vorticity outflow dumps most of its mass flux (Q) into the eddies (66%). (The remaining 33% goes into the eastward long-shore current.) By contrast, a weak outflow (i.e., an outflow with weak anticyclonic vorticity −αf, where α is analogous to the Rossby number and is much smaller than unity and f is the Coriolis parameter) dumps most of its water into the downstream current [(1–2α)Q]. Unexpectedly, this partition of mass turns out to be the same as the one taking place on an f plane. (Note that this is not at all the case for southward outflow nor is it the case for either eastward or westward outflow, where β alters the balance drastically.) Although the above partition of mass is independent of β, the size of the eddies generated by the above process is a function of β. It is given by [768g′Q/βπf 2α(2 − α)(1 + 2α)]1/5, where g′ is the reduced gravity. This gives a reasonable estimate for the Loop Current eddies’ size and generation frequency. Numerical simulations are in agreement with the above nonlinear solution, though the agreement is not necessarily any better than that of PN.

The Effects of Photon Bubble Instability in Radiation‐dominated Accretion Disks

We examine the effects of photon bubble instability in radiation-dominated accretion disks such as those found around black holes in active galactic nuclei and X-ray binary star systems. Two- and three-dimensional numerical radiation MHD calculations of small patches of disk are used. Modes with wavelengths shorter than the gas pressure scale height grow faster than the orbital frequency in the disk surface layers. The fastest growth rate observed is 5 times the orbital frequency and occurs on nearly vertical magnetic fields. The spectrum of linear modes is in good agreement with a WKB analysis that indicates still faster growth at unresolved scales, with a maximum growth rate proportional to the gravitational acceleration and inversely proportional to the gas sound speed. Disturbances reaching nonlinear amplitudes steepen into trains of shocks similar to a one-dimensional periodic nonlinear analytic solution. Variations in propagation speed result in merging of adjacent fronts, and over time the shock spacing and amplitude increase. Growth is limited by the strength of the magnetic field. The shock train structure is disrupted when the ram pressure of the disturbances exceeds the magnetic pressure. The maximum horizontal density variations are comparable to the ratio of magnetic to gas pressure and in our calculations exceed 100. Under the conditions considered, radiation diffuses through the inhomogeneneous flow 5 times faster than through the initial hydrostatic equilibrium, and the net cooling rate is several times greater than in a similar calculation without magnetic fields that shows the effects of convection. These results indicate that photon bubbles may be important in cooling radiation-dominated accretion disks. The Shaviv type I global instability grows faster than the orbital frequency in calculations of the disk surface layers with lower boundaries of fixed temperature, but is weak or absent in calculations spanning the disk thickness.

Solving Relative Two-Point Boundary Value Problems: Spacecraft Formulation Flight Transfers Application

The dynamics of reconfiguration of a formation of spacecraft is studied. The problem is posed as a two-point boundary value problem where the initial and final positions and times are known, and the necessary velocities for the reconfiguration transfers must be found. An algorithm is presented that solves this problem for reconfiguration about a nominal trajectory in arbitrary gravity fields, providing a nonlinear analytical solution to this problem in the spatial domain. The technique being developed is based on a novel application of Hamiltonian dynamics, generating functions, canonical transformations, and the Hamilton‐Jacobi equation. A specific example of reconfiguration of spacecraft about an Earth‐sun libration point is studied that shows that the optimal transfer time depends strongly

Linear regression with spatial constraint to generate parametric images of ligand-receptor dynamic PET studies with a simplified reference tissue model

For the quantitative analysis of ligand-receptor dynamic positron emission tomography (PET) studies, it is often desirable to apply reference tissue methods that eliminate the need for arterial blood sampling. A common technique is to apply a simplified reference tissue model (SRTM). Applications of this method are generally based on an analytical solution of the SRTM equation with parameters estimated by nonlinear regression. In this study, we derive, based on the same assumptions used to derive the SRTM, a new set of operational equations of integral form with parameters directly estimated by conventional weighted linear regression (WLR). In addition, a linear regression with spatial constraint (LRSC) algorithm is developed for parametric imaging to reduce the effects of high noise levels in pixel time activity curves that are typical of PET dynamic data. For comparison, conventional weighted nonlinear regression with the Marquardt algorithm (WNLRM) and nonlinear ridge regression with spatial constraint (NLRRSC) were also implemented using the nonlinear analytical solution of the SRTM equation. In contrast to the other three methods, LRSC reduces the percent root mean square error of the estimated parameters, especially at higher noise levels. For estimation of binding potential (BP), WLR and LRSC show similar variance even at high noise levels, but LRSC yields a smaller bias. Results from human studies demonstrate that LRSC produces high-quality parametric images. The variance of R 1 and k 2 images generated by WLR, WNLRM, and NLRRSC can be decreased 30%–60% by using LRSC. The quality of the BP images generated by WLR and LRSC is visually comparable, and the variance of BP images generated by WNLRM can be reduced 10%–40% by WLR or LRSC. The BP estimates obtained using WLR are 3%–5% lower than those estimated by LRSC. We conclude that the new linear equations yield a reliable, computationally efficient, and robust LRSC algorithm to generate parametric images of ligand-receptor dynamic PET studies.

Analytical Solution for Nonlinear Schrödinger Vortex Reconnection

Analysis of the nonlinear Schrodinger vortex reconnection is given in terms of coordinate-time power series. The lowest order terms in these series correspond to a solution of the linear Schrodinger equation and provide several interesting properties of the reconnection process, in particular the nonsingular character of reconnections, the anti-parallel configuration of vortex filaments and a square-root law of approach just before/after reconnections. The complete infinite power series represents a fully nonlinear analytic solution in a finite volume which includes the reconnection point, and is valid for finite time provided the initial condition is an analytic function. These series solutions are free from the periodicity artifacts and discretization error of the direct computational approaches and they are easy to analyze using a computer algebra program.

A note on Bödewadt–Hartmann layers

This paper addresses the problem of axisymetric rotating flows bounded by a fixed horizontal plate and subject to a permanent, uniform, vertical magnetic field (the so-called Bödewadt–Hartmann problem). The aim is to find out which one of the Coriolis or the Lorentz force dominates the dynamics (and hence the boundary layer thickness) when their ratio, represented by the Elsasser number A, varies. After a short review of existing linear solutions of the semi infinite Ekman–Hartmann problem, weakly non-linear analytical solutions as well as fully non-linear numerical solutions are given. The case of a rotating vortex in a finite depth fluid layer is then studied, first when the flow is steady under a forced rotation and second for spin-down from some initial state. The angular velocity in the first case and decay time in the second are obtained analytically as a function of A using the weakly non linear results of the semi-infinite Bödewadt–Hartmann problem.

Exact Solution for Nonlinear Thermal Diffusion and Its Use for Verification

An analytical solution is provided to the nonlinear diffusion equation, with the thermal conductivity given as a linear function of temperature. The derivation of the solution, and implications of it, are presented. The boundary and initial conditions associated with the solution provide applicability to specific cases. The solution is useful for verifying numerical (computer) solutions to thermal diffusion with temperature-dependent thermal conductivity. The (nonlinear) analytical solution is compared to a numerical solution from a finite element code to verify the accuracy of the code and to establish the order of convergence for the spatial discretization error

Stability of Elastomeric Isolation Bearings: Experimental Study

Elastomeric isolation bearings are required to be stable at high shear strains, which occur during strong earthquakes. Hence, rigorous determination of the critical axial load during design is important. Currently, the critical load is determined using the small displacement Haringx theory and modified to account for large shear strains by an approximate correction factor. The objective of this study is to experimentally determine the effect of horizontal displacement or shear strain on critical load and to study the validity of the approximate correction factor. Experiments were conducted on a series of elastomeric bearings with low shape factors. Test procedure and test results are presented in detail. It is shown that the critical load decreases with increasing horizontal displacement or shear strain. It is also shown that substantial critical load capacity exists at a horizontal displacement equal to the width of the bearing and is not zero, as predicted by the correction factor. It is further shown that the approximate formula is not conservative at smaller displacements and overly conservative at larger displacements. The critical loads obtained from experiments are compared with results from finite element analyses and nonlinear analytical solutions; the comparisons indicate that the effect of large horizontal displacements on the critical load can be reliably predicted.

The Ballooning of Outflows

Abstract It has been recently shown that when an inviscid outflow empties into the ocean, a steady alongshore current (in the Kelvin wave sense) cannot be established. This is due to the impossibility of balancing the alongshore momentum flux. To offset this momentum-flux deficit the outflow balloons near its source, forming an anticyclonic bulge. Using 1½-layer analytical and numerical models, the authors show that, on an f plane, the Coriolis force associated with the offshore movement of the growing bulge (which pushes itself away from the wall) compensates for the momentum flux of the longshore current downstream. With the aid of the slowly varying approximation, an inviscid nonlinear analytical solution is constructed. Numerical simulations with the Bleck and Boudra model are also performed. It is found that an outflow with an intense anticyclonic vorticity (i.e., a zero potential vorticity outflow with a relative vorticity of −f) produces a steep gyre that balloons (i.e., its radius increases with t...

A nonlinear analytical and numerical study for an open system with coherent off-resonance fields

An exact nonlinear analytical solution of an open V-type inversionless lasing system with two incoherent pump fields, off-resonant driving and probe fields is given. Through qualitative and quantitative comparison of the gain, dispersion and population differences of the linear and nonlinear cases, some conclusions concerning the Rabi frequencies of the driving and the probe fields are drawn and discussed in detail.

Islands as eddy splitters

Previous theoretical work has shown that, in an unbounded domain, anticyclones are prohibited from splitting on their own due to limitations imposed by the conservation of angular momentum, By explicitly considering the role of angular momentum exchange between eddies and boundaries (neglected by previous theories), splitting criteria for an anticyclonic lens colliding with a long and thin island are established analytically. The inviscid analytical model consists of an isolated patch of fluid in a reduced gravity regime. Nonlinear analytical solutions are constructed by connecting the initial and final states using conserved quantities (integrated angular momentum, vorticity and mass) and the familiar slowly varying approximation. For the conceptual case of a lens pierced by a thin moving wall, the result is that, in order for a zero potential vorticity lens (with a radius R 1 ) to split into two equal offspring, the wall length must be at least 1.19R 1 . Even for infinitesimal splitting, which arises from weak collisions (where the wall merely brushes the lens), the wall must be @ (R 1 ). This is because the parent lens can split into two offspring only when the wall allows sufficient spreading of the lens, which increases its relative angular momentum, and thereby enables the lens to form two distinct offspring. Numerical experiments employing Lagrangian floats reveal that the splitting is accomplished by a jet that leaks fluid along the wall, forming a second lens. The fluid initially found along the rim of the parent lens occupies both the core and the rim of the second lens; the fluid found at an intermediate radius in the second lens is derived from fluid situated at an intermediate radius in the parent lens. In general, a very good agreement between the numerics and the analytical theory is found. The numerical simulations demonstrate that the integrated angular momentum is a far stronger constraint than energy conservation. Using the numerics, we extend the moving wall theory to the splitting of finite vorticity lenses and lenses on a β-plane. We find that the basic requirement of mass redistribution by a wall is relevant in all the regimes that we examined, and, therefore, is likely to also be relevant to collisions of eddies with actual islands. This supports our application of the theory to Meddy splitting by seamounts, where we find that the seamounts can provide the necessary torque for the recently observed Meddy splitting and destruction.

Equivalent initial conditions for compatibility between analytical and computational solutions of convection in porous media

The derivation of a set of compatibility conditions for the equivalence between a weak non-linear analytical solution and any computational or numerical solution is presented. Both direct and inverse transformations are derived and shown to apply well for arbitrary initial conditions, provided that a validity condition of the asymptotic expansion associated with the weak non-linear solution is not violated. The results presented by using these compatibility conditions for a comparison between computational and analytical transitional values of a scaled Rayleigh number, that represents the point of transition from steady-to-chaotic solutions, show very good agreement within the validity domain of the asymptotic expansion.

Thermal convection in a layer bounded by uniform heat flux: Application of a strongly nonlinear analytical solution

A steady analytical solution for two-dimensional thermal convection in a layer with boundaries of uniform heat flux is presented. The solution applies to the lateral flow between the updrafts and downdrafts where the buoyancy varies linearly with horizontal distance. The solution in the lateral flow also predicts the mass flux and buoyancy fluctuation in the updraft, but not the width. Unlike the popular weakly nonlinear theory, the solution here is valid for all supercritical Rayleigh numbers. The steady analytical solution is confirmed to within 1% of two-dimensional numerical solutions at up to 1024 times critical, where the weakly nonlinear theory is erroneous by more than a factor of 1000. The analytical solution also predicts well the magnitude and evolution of the buoyancy fluctuations seen in three-dimensional simulations, including those from a turbulent LES. The analytical solution shows the buoyancy amplitude increases with wavelength, and is maximized for a (effective) Rayleigh number of about twice critical. The observed cell-broadening can be explained by the analytical result of smaller growth rates, but larger equilibrium amplitudes, for the longer wavelengths.

Intense nonlinear migrations of the Pacific warm pool

An upper bound for the nonlinear eastward propagation rate of the Pacific warm pool is derived analytically using an inertial two-and-a-half-layer model on a β plane. The model is based on the familiar idea that, in most years, the eastward migration tendency is arrested by the drag imposed on the ocean by the westward trade winds. During El Nino years, however, when the wind partially (or completely) relaxes, the pool is freed to move toward the east. The upper bound that we focus on corresponds to a rapid migration associated with a complete relaxation of the westward winds. Nonlinear analytical solutions to the above state are constructed by integrating the (inviscid) horizontal momentum equations over a control volume in a coordinate system moving steadily toward the east. A balance between the eastward flow-force (i.e., the momentum flux resulting from the eastward density gradient) and the opposing westward form-drag (exerted by the westward flowing intermediate fluid diving under the pool) is examined. It involves integrated pressure forces, integrated inertia and the integrated Coriolis forces. In the limit of a control volume with an infinitesimal north–south extent, no recirculation (i.e., no lateral exchange of mass between the fraction of the pool occupying the immediate vicinity of the equator and regions immediately to the north and south), and no cross-equatorial flows, the governing equations reduce to the equations that govern the nonrotating (i.e., β≡0) intrusion of warm water into a resting two-layer system. This essentially means that the Coriolis force does not have any zonal component along the equator. For such conditions, the nonlinear eastward speed is found to be [2g(Δρ 1/ρ)H 1] 1/2 [1−(H 2/H 1)] 3/2 , where Δρ 1 is the density difference between the pool and the intermediate water underneath (i.e., the so-called intermediate layer), H 1 the undisturbed thickness of the intermediate layer ahead of the pool, and H 2 is the intermediate layer thickness under the pool. Typical values for the Pacific give a bounding propagation rate of 50–60 cm s −1, which is in good agreement with the observed migration rate during both the 1982–83 El Nino and the 1997 El Nino, the only ones in history that are known to result from an almost complete relaxation of the winds.

Strange encounters of eddies with walls

The impingement of a westward drifting lens-like anticyclonic ring on a meridional boundary is examined analytically and numerically using an (inviscid) layer-and-a-half model. A nonlinear analytical solution is constructed with the aid of a slowly varying perturbation expansion and numerical simulations are performed with the Bleck and Boudra model. It is found that, in contrast to conventional wisdom which implies that an anticyclonic ring encountering a wall moves poleward due to the image effect, the ring stays roughly in a e xed latitude. As it slowly moves into the wall due to b , it gradually leaks e uid toward the equator until it loses all of its mass. As expected, different eddies lose their mass in different rates. The (nondimensional) central thickness of an intense zero potential vorticity ring decays as 1/[1 1 b (2g8Hi) 1/2 t/18f ] 2 where Hi is the initial central thickness, t is time, g8 the ‘ ‘ reduced gravity,’ ’f the Coriolis parameter, andb is the familiar variation of the Coriolis parameter with latitude. Due to the gradually decreasing depth, the ring’ s westward drift into the wall also gradually slows down; i.e., the ring advances toward the wall and loses mass like a gradually peeling onion. The ‘ ‘ peeling’ ’ rate is 2 b Rd /9 [where Rd is the instantaneous Rossby radius based on the instantaneous maximum thickness] implying that the eddy migrates toward the wall at a rate that is one-third of the free open ocean rate. Numerical simulations are in excellent agreement with the above solution.As expected, they show that frictional forces decrease the intensity of the leakage and, hence, slow down the draining process. Unfortunately, no analytical solution can be found for weak or moderately strong rings because, for such lenses, there is no known analytical solution even for the nonmigrating state (i.e., rings on an f-plane). Numerical simulations show, however, that such weak or moderately strong rings display (qualitatively) similar characteristics to those of intense zero potential vorticity rings. It is argued that rings such as warm-core Gulf Stream rings, Loop Current rings,Agulhas rings, as well as other eddies (e.g., Meddies) may all experience similar encounters. As the walls that they encounter are subsurface (e.g., the continental rise, the Walvis Ridge and the mid-Atlantic Ridge), the leakages are not expected to be easily detected.

Assessment of models for analysis of singly-curved sandwich panels

Overall behaviour of a simply-supported singly-curved shallow sandwich panel under lateral loading is considered. Three types of linear models — general shell, shallow shell and curved plate theories — are employed to describe a static overall behaviour of such a panel. The shallow panel with a constant curvature loaded by a uniform pressure is used as a trial case to assess the accuracy of the models. A nonlinear model for a panel of arbitrary shape under an arbitrary loading is developed on the basis of the curved plate theory. A closed-form nonlinear analytic solution for the panel of constant curvature under a uniform lateral pressure is obtained, and its accuracy is estimated. An experimental investigation of a sandwich beam under a uniform lateral loading is carried out, and the data obtained are compared with the theoretical calculations.

Nonlinear drift chains associated with sheared flow of magnetized plasma with magnetic shear

Equations describing nonlinear drift waves in a strongly inhomogeneous plasma, with a sheared flow profile perpendicular to the sheared magnetic field, are derived. In such configurations, the diamagnetic drift velocity is a spatially dependent function, and in the linear domain, the magnetic shear stabilisation criteria are substantially changed. In the nonlinear regime, possible analytical and numerical vortex chain solutions, periodic along the plasma flow and localised in the perpendicular direction, are found.

Non-linear analytical solutions for laterally loaded sandwich plates

This work focuses on the response of orthotropic sandwich composite plates with large deflections due to high lateral loads. The results have special application to the design of ship structures. A geometrical non-linear theory is outlined, on the basis of the classical sandwich plate theory expanded by the higher-order terms in the strain displacement relations, including shear deformation. By use of the principle of minimum potential energy, two different analytical solutions are derived for the simply supported and the clamped cases. The solutions are presented as simple design formulae. The results of the analytical calculations are discussed and compared with numerical non-linear finite difference calculations and large-deflection experiments of equivalent plates. The presented methods lead to good results for plate response and provide the engineer with an alternative method for the design of sandwich plates subjected to high lateral loading.