Simple Analytical Solution Research Articles

Trajectory Calculations with Lift and Drag

The trajectory of a ball in air is affected by aerodynamic drag and lift. In general, the trajectory needs to be calculated numerically since the acceleration varies with time in both the horizontal and vertical directions. If the trajectory remains approximately parabolic, then simple analytical solutions can be found, giving useful insights into the behavior of the ball as it travels through the air.

Simple analytical solutions for the , , and related queues

AbstractIn this paper we revisit some classical queueing systems such as the M $^b$ /E $_k$ /1/m and E $_k$ /M $^b$ /1/m queues, for which fast numerical and recursive methods exist to study their main performance measures. We present simple explicit results for the loss probability and queue length distribution of these queueing systems as well as for some related queues such as the M $^b$ /D/1/m queue, the D/M $^b$ /1/m queue, and fluid versions thereof. In order to establish these results we first present a simple analytical solution for the invariant measure of the M/E $_k$ /1 queue that appears to be new.

An Analytical Solution for Steel Plate Strengthened Circular Tunnels with Various Interface Slip Modes

A simplified analytical solution for steel plate strengthened circular tunnel concrete linings is presented, considering various interface slip modes. The full derivation for the analytical solution is introduced in detail, from which explicit expressions for the stress and displacement fields throughout the system are obtained. The distribution of lining deformation, internal force, and the interaction stress between tunnel lining and steel plate were investigated by considering various tunnel-steel interface scenarios, namely no slip, full slip, and elastic slip. The result demonstrates the importance of having a well-bonded interface to maximize the benefits of the steel plate strengthening approach. The maximum interface shear stress locates at the position with minimum interface normal stress. Subsequently, the influence of interface shear stiffness and steel plate thickness on the composited tunnel lining performance was investigated. It is founded that the improvement of the interface shear stiffness can enhance the overall structural stiffness of the strengthened tunnel lining, which results in increases in the interaction stresses and lining forces simultaneously. With a ratio of the steel thickness to the tunnel lining thickness less than 0.1, greater achievement in both strengthening effectiveness and cost efficiency is obtained.

Solute Transport in a Doublet‐Type Flow Configuration Through a Weakly Heterogeneous Porous Formation

AbstractSteady flow generated by an injecting and a pumping well (doublet) takes place in a porous formation where the spatially variable hydraulic conductivity K is modeled as a stationary, lognormal, random field with anisotropic two‐point autocorrelation. The latter is characterized by a vertical integral scale, that is, Iv, smaller than the horizontal one, that is, I. A solute, either passive or reactive, is injected in the medium, and we aim at computing the breakthrough curve (BTC) and its moments not only at the recovery (pumping) well, but also at any location between the two wells. The strong coupling between K and the nonuniformity of the flow renders the problem very difficult. Nevertheless, a simple (analytical) solution is obtained by adopting a few assumptions: (a) wells are replaced by lines of singularity, (b) a perturbation solution which regards the variance of the log‐conductivity Y = ln K as a perturbation parameter is employed, (c) the study is limited to strongly anisotropic heterogeneous formations (for which the anisotropy ratio λ = Iv/I is much smaller than one), and (d) the impact of pore‐scale dispersion is neglected. Central for the computation of the BTC is the statistics of the travel time of a fluid particle released at the injecting well and reaching a control plane located at any position x1 along the distance connecting the two wells. It is shown that the spatial variability of Y acts de facto like a dispersion mechanism: it enhances spreading, especially in the early arrivals. Useful closed form expressions for moments of the travel time along the central trajectory are also obtained. Finally, the theoretical framework presented in this study is applied to two transport experiments in order to compute the second‐order (temporal) moment as function of x1, and therefore to quantify dispersion occurring in the zone delimited by the two wells.

The explicit and traveling wave solutions of the nonlinear Phi-4 model and its implementations

Analysis of simple analytical wave solutions to nonlinear evolution equations is a fascinating and growing area of knowledge in highly nonlinear sciences. The Riccati equation rational expansion technique is being used in the paper to establish the exact traveling wave solutions of the Phi-4 model. We have investigated certain types of explicit solutions using this technique, such as periodic, hyperbolic, and rational solutions with some free parameters. The solitary wave solutions are derived from the traveling wave solutions when the parameters are set to particular values. In mathematical physics, it is discovered that the implemented technique is more effective for finding the analytical wave solution of nonlinear evolution models. The approach stands out for its simple and quick computation procedure.

Benchmarking the vertically integrated ice-sheet model IMAU-ICE (version 2.0)

Abstract. Ice-dynamical processes constitute a large uncertainty in future projections of sea-level rise caused by anthropogenic climate change. Improving our understanding of these processes requires ice-sheet models that perform well at simulating both past and future ice-sheet evolution. Here, we present version 2.0 of the ice-sheet model IMAU-ICE, which uses the depth-integrated viscosity approximation (DIVA) to solve the stress balance. We evaluate its performance in a range of benchmark experiments, including simple analytical solutions and both schematic and realistic model intercomparison exercises. IMAU-ICE has adopted recent developments in the numerical treatment of englacial stress and sub-shelf melt near the grounding line, which result in good performance in experiments concerning grounding-line migration (MISMIP, MISMIP+) and buttressing (ABUMIP). This makes it a model that is robust, versatile, and user-friendly, which will provide a firm basis for (palaeo-)glaciological research in the coming years.

Programmed shape-morphing into complex target shapes using architected dielectric elastomer actuators.

Dielectric elastomer actuators (DEAs) are among the fastest and most energy-efficient, shape-morphing materials. To date, their shapes have been controlled using patterned electrodes or stiffening elements. While their actuated shapes can be analyzed for prescribed configurations of electrodes or stiffening elements (the forward problem), the design of DEAs that morph into target shapes (the inverse problem) has not been fully addressed. Here, we report a simple analytical solution for the inverse design and fabrication of programmable shape-morphing DEAs. To realize the target shape, two mechanisms are combined to locally control the actuation magnitude and direction by patterning the number of local active layers and stiff rings of varying shapes, respectively. Our combined design and fabrication strategy enables the creation of complex DEA architectures that shape-morph into simple target shapes, for instance, those with zero, positive, and negative Gaussian curvatures as well as complex shapes, such as a face.

Eclipse analysis for small-eccentricity orbits using analytical model

The orbital eclipse is an old but fundamental problem in astrodynamics, and it has been fully studied in many papers and books. However, the properties of the orbital eclipse under the critical situation and the effects of the eccentricity on the eclipse have not been clearly revealed. Applying the first-order approximation to the eccentricity, the simple analytical solution to the conical eclipse of the elliptical orbit with a small eccentricity is developed, which is similar to the circular-orbit solution in form. The proposed model is validated by numerical simulation, and the properties of the critical situation between the cases with and without orbital eclipses and the effects of the eccentricity on the eclipse are both analyzed. The results show that when the eccentricity is less than 0.01, the proposed analytical model agrees well with the numerical method in common situations. When the orbital sun angle is close to the critical value defined by the ratio of the Earth radius to the semimajor axis, the precision of the proposed analytical model would be degraded. In that situation, the penumbra arc length would be no longer a small value, and the statement in previous studies that the penumbra arc was short and neglectable would not always be right. The difference between the eclipse of the small-eccentricity orbit and that of the circular orbit increases as the eccentricity increases. Moreover, when the argument of perigee is in the same or the opposite direction with the projection direction of the Sun in the orbital plane, the variation in the eclipse length caused by the eccentricity reaches the maximum value. When the above two directions are perpendicular to each other, the variation in the eclipse arc center caused by the eccentricity reaches the maximum value.

Asymptotic analysis of multiple mode structures equipped with multiple tuned mass dampers

The design of a mitigation dissipative system composed of several tuned mass dampers (TMDs) is not a simple task. As soon as several structural modes or several additional degrees-of-freedom are considered, the analytical response of the coupled system is not tractable anymore. The generic problem involving any number of structural modes and any number of attached tuned mass dampers is considered in this paper. The appropriate re-scaling of the problem is formulated in order to highlight the governing scales in the problem. An asymptotic analysis in the frequency domain, similar to the multiple timescale spectral analysis, is developed. The resulting expression for the frequency response function of the coupled system is expressed in a lower dimensional space, which offers the possibility to derive simple and accurate analytical solutions in other cases than the well-known single-mode single-TMD case. Although the major contribution of this paper consists in the derivation of this analytical formulation, two illustrations are also provided. The first one concerns the well-known single-mode single-TMD case, which illustrates the benefits of the proposed formulation and validates the resulting approximations against established literature. The second example concerns a single-mode double-TMD, where the proposed approximation is used to discuss the accuracy with which the two damping units need to be tuned in order to provide reasonable optimality with respect to mitigation performances.

Physics of manipulation of spin dynamics in a driven double well made transparent

We study the coherent manipulation of quantum spin dynamics via simple analytic solutions for a spin–orbit (SO) coupled ultracold atom held in a driven double well. In the high-frequency regime, we analytically obtain the quasienergies, the Floquet states, and the non-Floquet state of this system. When the system parameters are properly selected, the non-Floquet state reduces to special quantum states of simple forms which can provide a more intuitive understanding of some interesting quantum spin dynamical phenomena, such as the intrawell spin–flipping, the interwell spin-conserving tunneling, and the interwell spin–flipping tunneling. Specially, for the interwell spin–flipping tunneling, we surprisingly find that the selective coherent destruction of tunneling (SCDT) can occur at the anti-crossing (non-collapse) points of quasienergy spectrum, which breaks the conventional understanding that the SCDT is associated with the crossing of quasienergy levels. Further, we reveal the interwell spin–flipping tunneling strongly depends on the initial spin direction and the initial position of the SO-coupled atom. These novel results can be readily tested by the current experimental setups and may be useful for the design of spintronic devices.

Introduction of modern information and communication technologies in transport flow management

The article proposes an economic mechanism for regulating the route network, the calculation of which allows you to choose a transport route, as well as calculate the number of flights based on the condition of achieving a break-even transport process for carriers of various forms of ownership. The Work deals with the solution of problems related to the choice of a route, which belong to the class of complex combinatorial optimization problems. They don't have simple analytical solutions. As the number of nodes increases, the computational complexity increases exponentially. The Work proposes an alternative method for solving routing problems based on the use of neural network models, which allow obtaining good suboptimal solutions with a significant reduction in time costs. As in the Hopfield network, a matrix of neurons is used here, but neurons interact not on the "each with each" principle, but in rows and columns.

An approximate explicit analytical solution for the frictionless swash hydrodynamics with an improved seaward boundary condition

The classical Shen and Meyer (1963) analytical solution gives much shallow and asymmetric swash flows comparing with experimental observations. In this study, after considering an improved seaward boundary condition controlled by a parameter k proposed by Guard and Baldock (2007), a simple approximate analytical solution of the dam-break induced breaking bore propagation along the frictionless swash zone was deduced. The present analytical solution, though approximate since its validity is limited to the early stage of the swash evolution, is explicit allowing immediate and proper investigation of the swash hydrodynamics under the improved seaward boundary condition. After comparing with the implicit analytical solution by Pritchard et al. (2008), it is found that the present solution can be well applied to the entire swash process for relatively small k values and the uprush stage for all k values, which slightly overestimates the water depth and flow velocity during the backwash in the case of large k values. As parameter k increases, the present solution predicts increased mass and momentum fluxes entering into the swash zone, which is found to be ascribed to two specified reasons. One is the greater water depth and larger flow velocity near the original shoreline which leads to more water into the swash zone, and the other comes from the postponed flow reversal time which extends the entire uprush process. Comparison with the shock solution proposed by Antuono (2010) reveals that the shock-induced and the present dam-break induced swashes are different in the initial incident conditions and the hydrodynamics during the swash process, ascribing to the different characteristic variable features in the calculation domain of the two solutions. • An approximate explicit analytical solution of the dam-break induced frictionless swash hydrodynamics was deduced. • Comparisons between the present solution and the analytical solution by Pritchard et al. (2008) were conducted. • Influence of the seaward boundary condition parameter on the swash hydrodynamics was quantitatively investigated. • Two reasons for the enhanced mass and momentum fluxes were specified. • Linkages and differences between the shock-induced and the present dam-break induced swash hydrodynamics were revealed.

Enhancement of Smith-Purcell radiation from cylindrical gratings by quasi-bound states in the continuum.

Smith-Purcell radiation (SPR) is an important means of generating terahertz waves, and the enhancement of SPR is an attractive topic nowadays. Inspired by the phenomenon of special SPR, where the enhancement is achieved by using a high-duty-cycle grating, we describe a new, to the best of our knowledge, but more effective approach to this challenging problem. By deriving a simple analytical solution for the SPR from an annular electron beam passing through a cylindrical metallic grating, we show that the inverse structure, a low-duty-cycle grating can exhibit rather high SPR efficiencies in the presence of quasi-bound states in the continuum (quasi-BICs). The analytical prediction is supported by particle-in-cell simulations, which show that the quasi-BICs can enhance the superradiant SPR generated by a train of electron bunches by orders of magnitude. These results present an interesting mechanism for enhancing the SPR from metallic gratings, and may find applications in terahertz free-electron lasers.

Analysis of thermal dilution experiments with distributed temperature sensing for fractured rock characterization

Thermal dilution experiments with Fiber-Optic Distributed Temperature Sensing (FO-DTS) were conducted at the In-situ Stimulation and Circulation (ISC) rock laboratory, at the Grimsel Test Site (GTS) in Switzerland. The experiment consists in replacing the total volume of a borehole with a warmer water and monitoring the rate of temperature increase and decrease during the heating and cooling periods, respectively. The changes in temperature monitored in depth and time under various hydraulic conditions and in different boreholes, are used to investigate the information provided by thermal dilution experiments in terms of groundwater flow and thermal properties in low-permeable fractured crystalline rock. The data analysis, and the use of analytical and numerical solutions for reproducing this data in the context of pure diffusion and advection–diffusion scenarios, lead to the following improvements and conclusions. (i) The formation thermal conductivity is estimated along the borehole by inverting the data collected under ambient conditions with a simple analytical solution. The estimated values are consistent with laboratory estimates. The method presents the advantage of requiring much shorter experiments than existing methods based on standard active-line-source (ALS) experiments, i.e., several hours versus the traditional 1–2 days. (ii) Hydraulically active fractures connecting boreholes are detected from experiments conducted under cross-borehole forced hydraulic conditions. (iii) The formation thermal conductivity and fracture flow velocity have a distinguished impact on the temperature anomalies for some ranges of these property values, implying that both properties can be estimated from well-parametrized experiments.

Two-component self-gravitating isothermal slab models

We revisit the problem of the isothermal slab (in standard Cartesian coordinates, density distributions and mean gravitational potential are considered to be independent of x and y and to be a function of z, symmetric with respect to the {{z}} = {{0}} plane) in the context of the general issues related to the role of weak collisionality in inhomogeneous self-gravitating stellar systems. We thus consider the two-component case, that is a system of heavy and light stars with assigned mass ratio ({varvec{mu }}) and assigned global relative abundance ({{alpha }}; the ratio of the total mass of the heavy and light stars). The system is imagined to start from an initial condition in which the two species are well mixed and have identical spatial and velocity distributions and to evolve into a final configuration in which collisions have generated equipartition and mass segregation. Initial and final distribution functions are assumed to be Maxwellian. Application of mass and energy conservation allows us to derive the properties of the final state from the assumed initial conditions. In general, the derivation of these properties requires a simple numerical integration of the Poisson equation. Curiously, the case in which the heavy stars are exactly twice as massive as the light stars ({{mu }} = {{2}}) turns out to admit a relatively simple analytic solution. Although the general framework of this investigation is relatively straightforward, some non-trivial issues related to energy conservation and the possible use of a virial constraint are noted and clarified. The formulation and the results of this paper prepare the way to future studies in which the evolution induced by weak collisionality will be followed either by considering the action of standard collision operators or by means of dedicated numerical simulations.

The influence of strip load on the seismic design of cantilever sheet pile walls: a simplified analytical solution

The influence of strip load on the seismic design of cantilever sheet pile walls: a simplified analytical solution

Implications of Underground Nuclear Explosion Cavity Evolution for Radioxenon Isotopic Composition

Isotopic ratios of radioxenons sampled in the atmosphere or subsurface can be used to verify the occurrence of an underground nuclear explosion (UNE). Differences in the half-lives of radioactive xenon precursors and their decay-chain networks produce different time-dependent concentration profiles of xenon isotopes allowing isotopic ratios to be used for tracking UNE histories including estimating the time of detonation. In this study, we explore the potential effects of post-detonation cavity processes: precipitation of iodine precursors, gas seepage, and prompt venting on radioxenon isotopic evolution which influences UNE histories. Simplified analytical models and closed-form solutions yielding a potentially idealized radioactive decay/ingrowth chain in a closed and well-mixed system typically have limited application by not including the partitioning of the radionuclide inventory between a gas phase and rock melt created by the detonation and by ignoring gas transport from the cavity to host rock or ground surface. In reality, either subsurface transport or prompt release that is principally responsible for gas signatures violates the closed-system (or batch-mode) assumption. A closed-form solution representing time-dependent source-term activities is extended by considering the cavity partitioning process, slow seepage, and/or prompt release of gases from the cavity and applied to realistic systems.

Calibration of a Rigid-Trilinear Cohesive Material Law to Describe the Matrix-Fiber Bond Behavior in FRCM Composites

Fiber-reinforced cementitious matrix (FRCM) composites have been increasingly adopted as externally bonded reinforcement (EBR) of existing concrete and masonry members. Being debonding at the matrix-fiber interface one of the most frequent failure mechanisms of externally bonded FRCM, the matrix-fiber bond behavior represents a fundamental aspect for the effectiveness of the external reinforcement. A cohesive material law (CML) that describes the interface where debonding occurs can be used to model the bond behavior observed. In this paper, a rigid-trilinear CML is used to solve the differential equation that governs the bond problem at the matrix-fiber interface of an FRCM composite. The CML adopted has peculiar characteristics that entail for a finite length of the bond stress transfer zone (BSTZ). Furthermore, it allows for a simple and accurate analytical solution of the bond problem. The analytical solution obtained is compared with the results of an experimental campaign comprising single-lap direct shear tests of a polyparaphenylene benzobisoxazole (PBO) FRCM composite specifically designed for masonry substrates. Different calibrations of the rigid-trilinear CML are proposed, also considering the matrix-fiber free end slip.

Cyclicity in Earth sciences, quo vadis? Essay on cycle concepts in geological thinking and their historical influence on stratigraphic practices

Abstract. The archetype of a cycle has played an essential role in explaining observations of nature over thousands of years. At present, this perception significantly influences the worldview of modern societies, including several areas of science. In the Earth sciences, the concept of cyclicity offers simple analytical solutions in the face of complex events and their respective products, in both time and space. Current stratigraphic research integrates several methods to identify repetitive patterns in the stratigraphic record and to interpret oscillatory geological processes. This essay proposes a historical review of the cyclic conceptions from the earliest phases in the Earth sciences to their subsequent evolution into current stratigraphic principles and practices, contributing to identifying opportunities in integrating methodologies and developing future research mainly associated with quantitative approaches.

Simplified Analytical Solutions for Deep Tunnels Subjected to vertically incident shear wave with arbitrary vibration direction

A circular tunnel in a homogeneous infinite ground subjected to a vertically incident shear wave with an arbitrary vibration direction in the horizontal plane is considered in this study. A simplified analytical solution with explicit formulations for the deformation and stress of the tunnel lining is presented here. The superposition of two solutions under SH and SV waves are derived first. In the analytical solutions, the tunnel lining is taken as a thick-walled cylinder, which enables more precise forecasts and is much closer to practical engineering than the thin shell model adopted in other previous solutions. Moreover, the analytical solutions consider the slippage effect at the ground-lining interface by introducing a spring-type flexibility coefficient into the interaction force-displacement relationship. The results of the analytical solutions are verified by a comparison with the results of the dynamic numerical solutions via a 3-D ground-lining interaction model under earthquakes; the results show good agreement. By using the analytical solutions, the effects of the vibration direction, ground condition, lining thickness and flexibility coefficient on the seismic response of the tunnel are investigated. Additionally, the appropriate range of seismic wavelengths for the analytical solutions is discussed. The proposed solutions can be applied to the seismic analysis and design of tunnel structures.