Simple Analytical Solution Research Articles

상하단이 자유롭게 수평동요하는 수중 조파판에 의해 생성된 수면파의 근사해석

To derive a simplified analytic solution which can be utilized as a fundamental solution for the wave maker design, a segment of the wave board has been idealized as a submerged line segment in a two dimensional domain of a wave flume. The lower end of the line segment could be located at arbitrary depth of the wave flume and the upper end of the board could be also submerged to any depth from the free surface. The freely oscillating motion of the wave board is assumed to be defined by determining the condition of horizontal oscillation on both ends differently. The submerged wave board oscillating in horizontal direction could be specified by selecting the amplitude, frequency and the phase lag differently on lower and upper ends of the board. The simplified two dimensional wave generated by the wave board segment has been obtained by the first order perturbation method. It is found that the general solution of the freely oscillating wave board in two dimensional domain could be decomposed into the solution of flap motion with lower end hinge and swing motion with upper end hinge. The case study of the analytic solutions has been carried out to evaluate the effect on the wave height due to the difference of oscillation frequency, phase difference and variation of stroke between for the motion of both ends. It is found that the solution of the freely oscillating wave board could be utilized for the development of high performance wavemaker especially for irregular waves.

Enhanced resilience towards ROADM-induced optical filtering using subcarrier multiplexing and optimized bit and power loading.

In flexible optical networks, lightpaths are routed from source to destination thanks to reconfigurable optical add-drop multiplexers (ROADMs) placed in every node connected by a line system. As ROADMs are based on wavelength selective switches (WSSs), which imply channel filtering, a non negligible performance reduction can be observed when lightpaths traverse many network nodes. Control plane software network controllers that base their decision on the quality of transmission of the physical layer must also take into account the impact of ROADMs cascading. The penalties caused by WSS cascading in single-carrier systems are mostly driven by uncompensated inter-symbol interference (ISI), and therefore their evaluation cannot be based on simple analytical solutions as required for real-time network control. In this paper, we consider a 200G optical system based on multi-subcarrier (MSC) modulation with hybrid modulation formats. Using a fully analytical framework, we show that it is possible to predict ROADM-induced penalties offline and accurately estimate the required settings for their transmitter-side mitigation through bit and/or power-loading. Moreover, resorting to both simulations and experiments, we also demonstrate that properly optimized MSC signals can provide an increased resilience to ROADM filtering with respect to standard single-carrier systems: applying analytically-driven bit-loading over 8×4 Gbaud and 16×2 Gbaud MSC signals after crossing 8 cascaded WSSs in a noise loading scenario, we experimentally found ∼2 dB gain in required SNR with respect to a traditional 32 Gbaud single-carrier signal.

Kinematically Redundant Hybrid Robots With Simple Singularity Conditions and Analytical Inverse Kinematic Solutions

The rotational workspace of a parallel manipulator is mainly limited by type II singularities. In this letter, we present an approach to synthesize three-legged planar $3+n$ and spatial $6+n$ degree-of-freedom kinematically redundant hybrid robots with very simple—or vanishing—type II singularity conditions. Here, $n$ denotes the number of redundancies. Moreover, the inverse kinematic problem of the proposed architectures can be solved analytically. Because of these advantages, such redundant robots are well suited for applications in physical human–robot interaction, where large rotational workspaces are typically needed.

Simple analytical solutions of one-equation modelling for steady/decaying homogeneous turbulence sensitized to small strain

This study presents a simple mathematical solution describing the turbulent kinetic energy in steady or decaying homogeneous turbulence sensitized to small strain. Small strain is found to have little effect on the anisotropy of decaying homogeneous turbulence, which is consistent with a previous experiment. The effect of small strain on dissipation in the governing equations of turbulent kinetic energy is examined using the standard k-ε model. Two temporal profiles, namely a constant profile and a linear profile, with small strain are used. The steady homogeneous turbulence is maintained by linear forcing, as previously reported. This study mainly focuses on the effects of small strain on the turbulence time scale. When the magnitude of the small strain is small, the effects on the turbulence time scale can be negligibly small, especially for a constant profile. Based on this result, the governing equations for steady or decaying homogeneous turbulence sensitized to small strain can be reduced to simple linear equations. The analytical solution of these equations, a simple exponential function, is the same for the two types of homogeneous turbulence.

Simplified analytical solution for the optimal design of Tuned Mass Damper Inerter for base isolated structures

Abstract In this paper the use of the Tuned Mass Damper Inerter (TMDI) to control the response of base isolated structures under stochastic horizontal base acceleration is examined. Notably, the TMDI, recently introduced as a generalization of the classical Tuned Mass Damper, allows to achieve enhanced performance compared to the other passive vibration control devices. Thus, it represents an ideal alternative for reducing displacements of base isolated structures. To this aim, firstly a straightforward numerical approach is developed for the optimal design of this device considering a white noise base excitation. Further, a simplified analytical solution for the optimal design of TMDI parameters for base isolated structures is proposed minimizing the displacement variance of the corresponding undamped base isolated system. A thorough numerical analysis is performed and related results, in terms of optimal parameters and control performance, are compared with pertinent data obtained by a more computationally demanding iterative optimization procedure on the original damped system, considering both white noise and coloured noise stationary base excitation. Analytical and numerical results are found in good agreement, especially in terms of control performance, thus establishing the reliability and efficiency of the proposed approach. Finally, numerical analyses on a five-story benchmark base isolated structure controlled with an optimally designed TMDI are performed considering real recorded ground motions as base excitation. It is concluded that the TMDI, properly optimized with the proposed procedure, can effectively reduce the response of base isolated structures even under strong earthquakes.

An Investigation of Nanomechanical Properties of Materials using Nanoindentation and Artificial Neural Network

Mechanical properties of materials can be derived from the force-displacement relationship through instrumented indentation tests. Complications arise when establishing the full elastic-plastic stress-strain relationship as the accuracy depends on how the material’s and indenter’s parameters are incorporated. For instance, the effect of the material work-hardening phenomenon such as the pile-up and sink-in effect cannot be accounted for with simplified analytical indentation solutions. Due to this limitation, this paper proposes a new inverse analysis approach based on dimensional functions analysis and artificial neural networks (ANNs). A database of the dimensional functions relating stress and strain parameters of materials has been developed. The database covers a wide range of engineering materials that have the yield strength-to-modulus ratio (σy/E) between 0.001 to 0.5, the work-hardening power (n) between 0–0.5, Poisson’s ratio (v) between 0.15–0.45, and the indentation angle (θ) between 65–80 degrees. The proposed algorithm enables determining the nanomechanical stress-strain parameters using the indentation force-displacement relationship, and is applicable to any materials that the properties are within the database range. The obtained results are validated with the conventional test results of steel and aluminum samples. To further demonstrate the application of the proposed algorithm, the nanomechanical stress-strain parameters of ordinary Portland cement phases were determined.

Simple Analytical Solution of Čerenkov-Type Terahertz Wave Generation via Difference Frequency Generation in Dielectric Waveguides

We newly analyze the Cerenkov-type terahertz wave generation via the optical difference frequency generation in nonlinear dielectric waveguides based on the coupled-mode theory. The nonlinear coupled equations are derived, and the simple analytical solution for the terahertz wave is obtained in a closed form. The validity of our solution is also checked by the nonlinear beam propagation method. The analytical solution shows that for planar waveguides, the coupling coefficient has pole singularities at the propagation constants of the leaky modes at the terahertz frequency of the waveguides, and thus, the terahertz wave generation can be significantly enhanced as the phase-matching point approaches to the propagation constant of the lowest order leaky mode, whose imaginary part is very small compared to the real part of that.

Stress heterogeneities and failure mechanisms induced by temperature and pore-pressure increase in volcanic regions

We study the strain and stress fields produced by temperature and pore pressure increases within and outside a Thermo-Poro-Elastic (TPE) inclusion (the source region), embedded within a medium (the matrix) in isothermal drained conditions. This model is suitable to describe a crustal region in a volcanic environment pervaded by hot pressurized fluids released by an underlying magma chamber. After introducing the pertinent constitutive relations, a formal solution for the displacement field is provided in terms of the Green's function for an elastic medium with drained isothermal elastic moduli, employing a generalization of Eshelby (1961) procedure. If an unbounded medium is considered, a displacement potential can be introduced, obeying the Laplace equation within the source region and the Poisson equation within the matrix. If a spherically symmetric source region is considered, simple analytical solutions are obtained for the displacement, the strain and the stress fields, showing that thrust faulting mechanisms are promoted within the source region while normal faulting mechanisms prevail in the embedding matrix. Employing reasonable numerical values for the thermo-poro-elastic parameters, suitable to describe highly porous sedimentary rock, strain and stress variations are found to be significant even for moderate changes of temperature and pore pressure. Variations of the Coulomb failure function are high in the TPE region and are strongly dependent on the friction coefficient and pore pressure. Application of these results to the 1982-84 and 2011-13 unrest episodes at Campi Flegrei caldera (Italy) suggests that an oblique dike intrusion across a previously unfaulted TPE region took place with a mixed tensile-thrust dislocation mechanism in both events, as previously inferred from accurate inversion of geodetic data.

Karst conduit contribution to spring discharge and aquifer cross-sectional area

Groundwater flow discharging to springs from carbonate aquifers is governed by the interaction of slow matrix flow and fast fracture/conduit flow, which creates highly complex flow and transport conditions. An important unknown is the relative contribution of matrix and conduit flow to the total discharge. This study experimentally investigated groundwater fluxes in the Floridan aquifer within the springshed of Silver Springs, FL, one of the largest freshwater springs in the world with mean discharge of approximately 20 m3/s. Using in situ passive flux meters (PFMs, n = 48 at 16 wells) and a new karstic borehole dilution (KBHD, n = 21 at 7 wells) technique we measured groundwater fluxes in rock matrix and non-matrix (conduit and fracture) zones of 0.06 ± 0.003 m/day and 3.05 ± 1.8 m/day (mean ± standard error). These data, combined with previously conducted tracer tests (n = 12 at 3 sites), were coupled with simple analytical and numerical solutions to identify the proportion of the aquifer that contributes most significantly to water flow to the spring with three different modeling scenarios: single domain, dual domain including matrix and non-matrix zones, and triple domain including matrix, fracture, and conduit zones. The analytical and numerical models coupled with the in situ measured fluxes for the dual and triple domain scenarios showed good agreement with measured head profiles (Nash-Sutcliffe E > 0.90), when compared to the homogeneous porous domain scenario (E = −1.84). Conduit and fracture zones were estimated to represent between 2% and 22% of the aquifer cross-sectional area (at radial distance of 3 km from the spring outlet), yet these zones contributed between 75 and 96% of the total groundwater flow. The results of this study offer field-measured hydrogeologic data that can be used for active resource management in springsheds, and the simple modeling approach presented here may be applicable to other springsheds with fairly simple geometry to estimate the relative contributions of fast and slow water flow and solute transport pathways to the spring outlet.

Simple Chaotic Systems with Specific Analytical Solutions

In this paper, a new structure of chaotic systems is proposed. There are many examples of differential equations with analytic solutions. Chaotic systems cannot be studied with the classical methods. However, in this paper we show that a system that has a simple analytical solution can also have a strange attractor. The main goal of this paper is to show examples of chaotic systems with a simple analytical solution that is unstable so that the chaotic orbit does not track it. We believe the structures presented here are new. Two categories of chaotic systems are described, and their dynamical properties are investigated. The proposed systems have analytic solutions that exist far from the equilibrium. Of course, all strange attractors are dense in unstable periodic orbits, but mostly the equations that describe these orbits are unknown and difficult to calculate. The analytical solutions provide examples where the orbits can be calculated despite their instability.

Modeling Study of Temperature and Fracture-Propagation Effects on the Fracture-Surface Dissolution Patterns and Fractured-Well Productivity in Acid Fracturing

Summary The acid–fracturing operations in carbonate formations are modeled to evaluate possible improvements in well productivity. Models are developed mainly to estimate the acid–penetration length and fracture–surface etched–width profiles. Variable combinations of these two parameters produce significant differences in fracture productivity. To estimate these parameters better, a reliable fracture–propagation model should be coupled with an acid reaction/transport model. Simulating weak acids or dolomite–formation reactivity requires the inclusion of a heat–transfer model. The model provided in this study couples these factors as fractures propagate to obtain the fracture–conductivity distribution along the length. The fracture–propagation model was created to update the domain of the acid model continuously. In the process, a transient acid convection and diffusion equation is solved and the fracture's etched–width profile is calculated. An iterative procedure is then implemented in a temperature–dependent kinetic model that stops when both the temperature and the acid solutions converge. When the injection stops, the acid etching and fluid temperatures are updated as the fracture closes. As the final etching profile is drawn, conductivity is calculated using a correlation that considers formation heterogeneity. The conductivity distribution along the fracture surface can be used to predict the fractured–well productivity for given reservoir properties. Coupling of the fracture propagation shows a significant difference in the acid–model solutions, as compared to those assuming a constant–fracture geometry. For an extremely high Péclet number that represents a very retarded acid system, a constant drop in the etched–width value until reaching zero at the fracture tip is theoretically obtainable. For lower Péclet numbers, the etching profile sharply declines toward the fracture end. This is in contrast to the noncoupled approach, from which a uniform etching profile is obtained at moderate–to–high Péclet numbers. In this research, it was observed that the simulation of acid injection in a noncoupled, constant–fracture geometry always overestimates the acid–penetration distance. The etched–width distribution and acid–penetration length are temperature sensitive, especially in dolomite formations. Temperature coupling shows that the maximum etching in dolomite formations occurs away from the fracture entrance, as acid reactivity increases. It also shows that the cooling effects of the first–stage pad fluid on improving the acid–penetration distance are limited. This work illustrates that the impact of fracture propagation, heat transfer, and simulation of fracture after shut–in on productivity prediction is substantial. Simulating acid–fracturing operations, assuming a constant final fracture geometry and average single temperature, is time efficient but results in inaccurate solutions. This research quantifies the effects of integrating fracture–propagation and heat–transfer models on the acid–etching pattern. It also emphasizes the significance of simulating acid fracture during the closure, especially for dolomite formations or weak–acid cases, from which a better estimation of the fracture's productivity can be expected. A simplified analytical solution for multiple–fluid simulation is also introduced in this work.

Optimization of Thermal Protection Panels Subjected to Intense Heating and Mechanical Loading

In this work we solve a higly-nonlinear structural optimization problem for the sandwich panel with external thermal protection layer that can be used in the spacecraft systems. Objective function of the problem is the mass per unit area of the panel. Constraints are formulated based on the simplified analytical solutions of structural mechanics and heat transfer problems, which are suitable for the preliminary design considerations. The set of design variables includes the geometric parameters of the panel and additional microstructural parameter—porosity of the heat protection material. Direct random search and simulated annealing method are applied to solve considered problem. Change of limit states and optimal configurations of the panel are studied for different levels of the mechanical loading.

An Analytical Study for Nonlinear Free and Forced Vibration of Electrostatically Actuated MEMS Resonators

This work aims to construct accurate and simple lower-order analytical approximation solutions for the free and forced vibration of electrostatically actuated micro-electro-mechanical system (MEMS) resonators, in which geometrical and material nonlinearities are induced by the mid-plane stretching, dynamic pull-in characteristics, electrostatic forces and other intrinsic properties. Due to the complexity of nonlinear MEMS systems, the quest of exact closed-form solutions for these problems is hardly obtained for system design and analysis, in particular for harmonically forced nonlinear systems. To examine the simplicity and effectiveness of the present analytical solutions, two illustrative cases are taken into consideration. First, the free vibration of a doubly clamped microbeam suspended on an electrode due to a suddenly applied DC voltage is considered. Based on the Euler–Bernoulli beam theory and the von Karman type nonlinear kinematics, the dynamic motion of the microbeam is further discretized by the Galerkin method to an autonomous system with general nonlinearity, which can be solved analytically by using the Newton harmonic balance method. In addition to large-amplitude free vibration, the primary resonance response of a doubly clamped microbeam driven by two symmetric electrodes is also investigated, in which the microbeam is actuated by a bias DC voltage and a harmonic AC voltage. Following the same decomposition approach, the governing equation of a harmonically forced beam model can be transformed to a nonautonomous system with odd nonlinearity only. Then, lower-order analytical approximation solutions are derived to analyze the steady-state resonance response of such a problem under a combination of various DC and AC voltage effects. Finally, the analytical approximation results of both cases are validated, and they are in good agreement with those obtained by the standard Runge–Kutta method.

Rapid population transfer of a two-level system by a polychromatic driving field

We propose a simple exact analytical solution for a model consisting of a two-level system and a polychromatic driving field. It helps us to realize a rapid complete population transfer from the ground state to the excited state, and the system can be stable at the excited state for an extremely long time. A combination of the mechanism and the Rydberg atoms successfully prepares the Bell state and multipartite W state, and the experimental feasibility is discussed via the current experimental parameters. Finally, the simple exact analytical solution is generalized into a three-level system, which leads to a significant enhancement of the robustness against dissipation.

A novel energy pile: The thermo-syphon helical pile

This study focuses on assessing the heat transfer rate and structural stability of a novel self-operating energy pile based on the principle of a thermo-syphon. Specifically, this new energy pile, referred to as a “thermo-syphon helical pile” (THP), is formed by pressurizing a hollow helical pile with carbon dioxide (CO2) to form a heat pipe, where spontaneous liquid-vapor phase change and natural convection inside the pile will facilitate self-operating heat transfer from the pile tip to the pile head. Based on the theories of heat transfer and fluid dynamics, a simplified analytical solution was developed to calculate the heat transfer rates within THPs with different geometries, which can be further converted into equivalent thermal conductivities. The results indicate that heat transfer within THPs is a function of the boundary temperature applied to the pile head, CO2 pressure, working fluid properties, and THP geometry. The results also revealed that the equivalent thermal conductivity of the THP is 1000 times higher than that of most metals due to the latent heat transfer of working fluid. An analysis of the structural stability of a THP under pressure indicates that bifurcation is not a problem if the ratio of the diameter to thickness of a THP is less than 90. While this analytical feasibility study demonstrates that THPs are a promising alternative for energy piles for both new and retrofitted buildings, future studies on soil-pile thermal and mechanical interaction under operational thermal gradients are needed to evaluate the range of heat transfer rates from the subsurface to a building when using a THP.

Comparative Study of Methods for Delineating the Wellhead Protection Area in an Unconfined Coastal Aquifer

Various delineation methods, ranging from simple analytical solutions to complex numerical models, have been applied for wellhead protection area (WHPA) delineation. Numerical modeling is usually regarded as the most reliable method, but the uncertainty of input parameters has always been an obstacle. This study aims at examining the results from different WHPA delineation methods and addressing the delineation uncertainty of numerical modeling due to the uncertainty from input parameters. A comparison and uncertainty analysis were performed at two pumping sites—a single well and a wellfield consisting of eight wells in an unconfined coastal aquifer in Israel. By appointing numerical modeling as the reference method, a comparison between different methods showed that a semi-analytical method best fits the reference WHPA, and that analytical solutions produced overestimated WHPAs in unconfined aquifers as regional groundwater flow characteristics were neglected. The results from single well and wellfield indicated that interferences between wells are important for WHPA delineation, and thus, that only semi-analytical and numerical modelling are recommended for WHPA delineation at wellfields. Stochastic modeling was employed to analyze the uncertainty of numerical method, and the probabilistic distribution of WHPAs, rather a deterministic protection area, was generated with considering the uncertain input hydrogeological parameters.

C2 Continuous Blending of Time-Dependent Parametric Surfaces

Surface blending is widely applied in mechanical engineering. Creating a smooth transition surface of C2 continuity between time-dependent parametric surfaces that change their positions and shapes with time is an important and unsolved topic in surface blending. In order to address this issue, this paper develops a new approach to unify both time-dependent and time-independent surface blending with C2 continuity. It proposes a new surface blending mathematical model consisting of a vector-valued sixth-order partial differential equation and blending boundary constraints and investigates a simple and efficient approximate analytical solution of the mathematical model. A number of examples are presented to demonstrate the effectiveness and applications. The proposed approach has the advantages of (1) unifying time-independent and time-dependent surface blending, (2) always maintaining C2 continuity at trimlines when parametric surfaces change their positions and shapes with time, (3) providing effective shape control handles to achieve the expected shapes of blending surfaces but still exactly satisfy the given blending boundary constraints, and (4) quickly generating C2 continuous blending surfaces from the approximate analytical solution with easiness, good accuracy, and high efficiency.

Granite strength and toughness from small notched three-point-bend specimens of geometry dissimilarity

Maximum fracture loads (Pmax) of small notched granite specimens under three-point-bending (3-p-b) conditions can be easily measured with any notch/size ratio. In this study, we report a simple closed-form solution of a non-Linear Elastic Fracture Mechanics (non-LEFM) model emphasizing the influence of average grain size G on quasi-brittle fracture of granite. This simple analytical solution containing the grain size G can be conveniently used to determine granite tensile strength ft and fracture toughness KIC from Pmax measurements of small notched 3-p-b specimens of geometry dissimilarity. The span/width (S/W) ratios of small 3-p-b specimens can vary, e.g. 2.5 or 4. The notch/width (a0/W) can also vary, e.g. the notch a0 can be as short as the average grain size (G), or close to width W. However, specimens with α-ratio (=a0/W) around 0.2 are recommended to minimize the boundary influence/effect from both the front and back specimen boundaries as proven by the Boundary Effect Model (BEM). Blue granite with the average grain size around 2 (mm) was selected to test the new method. Total 64 granite samples from four different groups (different 3-p-b sample designs) were tested, with W = 27, 40 and 70 (mm), S/W = 2.5, 4.0 and a0 = 4, 6 and 8 (mm). The tensile strength ft and fracture toughness KIC estimated from every group is fairly close to the values determined from the entire population of 64 tests. Therefore, tests from any specimen group of given geometry and size are sufficient. Estimations for G = 1.5 and 2.5 (mm) were also provided and compared with those for G = 2 (mm) to show the grain size influence. Advantages and disadvantages of BEM and well-known SEL (size effect law) are also discussed using the granite results.

Radioactive Heating and Late Time Kilonova Light Curves

Abstract Compact object mergers can produce a thermal electromagnetic counterpart (a “kilonova”) powered by the decay of freshly synthesized radioactive isotopes. The luminosity of kilonova light curves depends on the efficiency with which beta-decay electrons are thermalized in the ejecta. Here we derive a simple analytic solution for thermalization by calculating how accumulate electrons lose energy adiabatically and via plasma interactions. The thermalization efficiency is well described by where the timescale is a function of the ejecta mass and velocity and the exponent depends on the electron energies and the thermalization cross-sections. For a statistical distribution of r-process isotopes with radioactive power and n = 1, the late time kilonova luminosity asymptotes to and depends super-linearly on the ejecta mass, . If a kilonova is instead powered by a single dominate isotope, we show that the late time luminosity can deviate substantially from the underlying exponential decay and the heating from the accumulation of trapped electrons eventually exceeds the instantaneous radioactivity. Applied to the kilonova associated with the gravitational wave source GW170817, these results imply that a possible steepening of the light curve at ≳7 days is unrelated to thermalization effects and instead could mark the onset of translucency in a high opacity component of ejecta. The analytic results should be convenient for estimating the properties of observed kilonovae and assessing the potential late time detectability of future events.

Analytical and Numerical Modeling of Solute Intrusion, Recovery, and Rebound in Fractured Bedrock.

Contaminated groundwater in fractured bedrock can expose ecosystems to undesired levels of risk for extended periods due to prolonged back-diffusion from rock matrix to permeable fractures. Therefore, it is key to characterize the diffusive mass loading (intrusion) of contaminants into the rock matrix for successful management of contaminated bedrock sites. Even the most detailed site characterization techniques often fail to delineate contamination in rock matrix. This study presents a set of analytical solutions to estimate diffusive mass intrusion into matrix blocks, it is recovered by pumping and concentration rebound when pumping ceases. The analytical models were validated by comparing the results with (1) numerical model results using the same model parameters and (2) observed chloride mass recovery, rebound concentration, and concentration in pumped groundwater at a highly fractured bedrock site in Alberta, Canada. It is also demonstrated that the analytical solutions can be used to estimate the total mass stored in the fractured bedrock prior to any remediation thereby providing insights into site contamination history. The predictive results of the analytical models clearly show that successful remediation by pumping depends largely on diffusive intrusion period. The results of initial mass from the analytical model was used to successfully calibrate a three-dimensional discrete fracture network numerical model further highlighting the utility of the simple analytical solutions in supplementing the more detailed site numerical modeling. Overall, the study shows the utility of simple analytical methods to support long-term management of a contaminated fractured bedrock site including site investigations and complex numerical modeling.