Hamiltonian system for the inhomogeneous plane elasticity of dodecagonal quasicrystal plates and its analytical solutions

A Hamiltonian system is derived for the plane elasticity problem of two-dimensional dodecagonal quasicrystals by introducing the simple state function. By using symplectic elasticity approach, the analytic solutions of the phonon and phason displacements are obtained further for the quasicrystal plates. In addition, the effectiveness of the approach is verified by comparison with the data of the finite integral transformation method.

Analytical solutions to the precession relaxation of magnetization with uniaxial anisotropy

Based on the Landau-Lifshitz-Gilbert(LLG)equation,the precession relaxation of magnetization is studied when the external field H is parallel to the uniaxial anisotropic field H_(k).The evolution of three-component magnetization is solved analytically under the condition of H=nH_(k)(n=3,1 and 0).It is found that with an increase of H or a decrease of the initial polar angle of magnetization,the relaxation time decreases and the angular frequency of magnetization increases.For comparison,the analytical solution for H_(k)=0 is also given.When the magnetization becomes stable,the angular frequency is proportional to the total effective field acting on the magnetization.The analytical solutions are not only conducive to the understanding of the precession relaxation of magnetization,but also can be used as a standard model to test the numerical calculation of LLG equation.

Analytical solutions fractional order partial differential equations arising in fluid dynamics

This article describes the solution procedure of the fractional Pade-Ⅱ equation and generalized Zakharov equation(GSEs)using the sine-cosine method.Pade-Ⅱ is an important nonlinear wave equation modeling unidirectional propagation of long-wave in dispersive media and GSEs are used to model the interaction between one-dimensional high,and low-frequency waves.Classes of trigonometric and hyperbolic function solutions in fractional calculus are discussed.Graphical simulations of the numerical solutions are flaunted by MATLAB.

Analytic Solutions for Hindered Rotation Quantum Problem

Normalizable analytic solutions of the quantum rotor problem with divergent potential are presented here as solution of the Schrödinger equation. These solutions, unknown to the literature, represent a mathematical advance in the description of physical phenomena described by the second derivative operator associated with a divergent interaction potential and, being analytical, guarantee the optimal interpretation of such phenomena.

Bifurcations, Analytical and Non-Analytical Traveling Wave Solutions of (2 + 1)-Dimensional Nonlinear Dispersive Boussinesq Equation

For the (2 + 1)-dimensional nonlinear dispersive Boussinesq equation, by using the bifurcation theory of planar dynamical systems to study its corresponding traveling wave system, the bifurcations and phase portraits of the regular system are obtained. Under different parametric conditions, various sufficient conditions to guarantee the existence of analytical and non-analytical solutions of the singular system are given by using singular traveling wave theory. For certain special cases, some explicit and exact parametric representations of traveling wave solutions are derived such as analytical periodic waves and non-analytical periodic cusp waves. Further, two-dimensional wave plots of analytical periodic solutions and non-analytical periodic cusp wave solutions are drawn to visualize the dynamics of the equation.

Analytical Solutions for Testing of Baroclinic and Wind-Driven Three-Dimensional Hydrodynamic Models

It is always a challenge for a model developer to verify a three-dimensional hydrodynamic model, especially for the baroclinic term over variable topography, due to a lack of observational data sets or suitable analytical solutions. In this paper, exact solutions for the periodic forcing by surface heat flux and wind stress are given by solving the linearized equations of motion neglecting the rotation, advection and horizontal diffusion terms. The temperature at the bottom is set to a prescribed periodic value and a slip condition on flow is enforced at the bottom. The geometry of the quarter annulus, which has been extensively studied for two- and three-dimensional analytical solutions of unstratified water bodies, is used with a general power law variation of the bottom slope in the radial direction and is constant in the azimuthal direction. The analytical solutions are derived in a cylindrical coordinate system, which describes the three-dimensional fluid field in a Cartesian coordinate system. The results presented in this paper should provide a foundation for studying and verifying the baroclinic term over a varied topography in a three-dimensional numerical model.

Revisiting the analytical solutions for ultimate bearing capacity of pile embedded in rocks

This paper investigates the validity and shortcomings of the existing analytical solution for the ultimate bearing capacity of a pile embedded in a rock mass using the modified HoekeBrown failure criterion.Although this criterion is considered a reference value for empirical and numerical calculations,some limitations of its basic simplifications have not been clarified yet.This research compares the analytical results obtained from the novel discontinuity layout optimization(DLO)method and the numerical solutions from the finite difference method(FDM).The limitations of the analytical solution are considered by comparing different DLO failure modes,thus allowing for the first time a critical evaluation of its scope and conditioning for implementation.Errors of up to 40%in the bearing capacity and unrealistic failure modes are the main issues in the analytical solution.The main aspects of the DLO method are also analyzed with an emphasis on the linearization of the rock failure criterion and the accuracy resulting from the discretization size.The analysis demonstrates DLO as a very efficient and accurate tool to address the pile tip bearing capacity,presenting considerable advantages over other methods.

Analytical wave solutions of an electronically and biologically important model via two efficient schemes

We search for analytical wave solutions of an electronically and biologically important model named as the Fitzhugh–Nagumo model with truncated M-fractional derivative, in which the expafunction and extended sinh-Gordon equation expansion(ESh GEE) schemes are utilized. The solutions obtained include dark, bright, dark-bright, periodic and other kinds of solitons. These analytical wave solutions are gained and verified with the use of Mathematica software. These solutions do not exist in literature. Some of the solutions are demonstrated by 2D, 3D and contour graphs. This model is mostly used in circuit theory, transmission of nerve impulses, and population genetics. Finally, both the schemes are more applicable, reliable and significant to deal with the fractional nonlinear partial differential equations.

Bi-Objective Optimization: A Pareto Method with Analytical Solutions

Multiple objectives to be optimized simultaneously are prevalent in real-life problems. This paper develops a new Pareto Method for bi-objective optimization which yields analytical solutions. The Pareto optimal front is obtained in closed-form, enabling the derivation of various solutions in a convenient and efficient way. The advantage of analytical solution is the possibility of deriving accurate, exact and well-understood solutions, which is especially useful for policy analysis. An extension of the method to include multiple objectives is provided with the objectives being classified into two types. Such an extension expands the applicability of the developed techniques.

Exact solutions for magnetohydrodynamic nanofluids flow and heat transfer over a permeable axisymmetric radially stretching/shrinking sheet

We report on the magnetohydrodynamic impact on the axisymmetric flow of Al_(2)O_(3)/Cu nanoparticles suspended in H_(2)O past a stretched/shrinked sheet.With the use of partial differential equations and the corresponding thermophysical characteristics of nanoparticles,the physical flow process is illustrated.The resultant nonlinear system of partial differential equations is converted into a system of ordinary differential equations using the suitable similarity transformations.The transformed differential equations are solved analytically.Impacts of the magnetic parameter,solid volume fraction and stretching/shrinking parameter on momentum and temperature distribution have been analyzed and interpreted graphically.The skin friction and Nusselt number were also evaluated.In addition,existence of dual solution was deduced for the shrinking sheet and unique solution for the stretching one.Further,Al_(2)O_(3)/H_(2)O nanofluid flow has better thermal conductivity on comparing with Cu/H_(2)O nanofluid.Furthermore,it was found that the first solutions of the stream are stable and physically realizable,whereas those of the second ones are unstable.

Analytical solution for the effective elastic properties of rocks with the tilted penny-shaped cracks in the transversely isotropic background

Seismic prediction of cracks is of great significance in many disciplines,for which the rock physics model is indispensable.However,up to now,multitudinous analytical models focus primarily on the cracked rock with the isotropic background,while the explicit model for the cracked rock with the anisotropic background is rarely investigated in spite of such case being often encountered in the earth.Hence,we first studied dependences of the crack opening displacement tensors on the crack dip angle in the coordinate systems formed by symmetry planes of the crack and the background anisotropy,respectively,by forty groups of numerical experiments.Based on the conclusion from the experiments,the analytical solution was derived for the effective elastic properties of the rock with the inclined penny-shaped cracks in the transversely isotropic background.Further,we comprehensively analyzed,according to the developed model,effects of the crack dip angle,background anisotropy,filling fluid and crack density on the effective elastic properties of the cracked rock.The analysis results indicate that the dip angle and background anisotropy can significantly either enhance or weaken the anisotropy degrees of the P-and SH-wave velocities,whereas they have relatively small effects on the SV-wave velocity anisotropy.Moreover,the filling fluid can increase the stiffness coefficients related to the compressional modulus by reducing crack compliance parameters,while its effects on shear coefficients depend on the crack dip angle.The increasing crack density reduces velocities of the dry rock,and decreasing rates of the velocities are affected by the crack dip angle.By comparing with exact numerical results and experimental data,it was demonstrated that the proposed model can achieve high-precision estimations of stiffness coefficients.Moreover,the assumption of the weakly anisotropic background results in the consistency between the proposed model and Hudson's published theory for the orthorhombic rock.

An improved analytical solution for solving the shield tunnel uplift problem

The problem of shield tunnel uplift is a common issue in tunnel construction.Due to the decrease in shear stiffness at the joints between the rings,uplift is typically observed as bending and dislocation deformation at these joints.Existing modeling methods typically rely on the Euler-Bernoulli beam theory,only considering the bending effect while disregarding shear deformation.Furthermore,the constraints on the shield tail are often neglected in existing models.In this study,an improved theoretical model of tunnel floating is proposed.The constraint effect of the shield machine shell on the tunnel structure is considered using the structural forms of two finite long beams and one semi-infinite long beam.Furthermore,the Timoshenko beam theory is adopted,providing a more accurate description of tunnel deformation,including both the bending effect and shear deformation,than existing models.Meanwhile,the buoyancy force and stratum resistance are calculated in a nonlinear manner.A reliable method for calculating the shear stiffness correction factor is proposed to better determination of the calculation parameters.The proposed theoretical model is validated through five cases using sitemonitored data.Its applicability and effectiveness are demonstrated.Furthermore,the influences of soil type,buried depth,and buoyancy force on the three key indicators of tunnel floating(i.e.the maximum uplift magnitude,the ring position with the fastest uplift race,and the ring position with the maximum uplift magnitude)are analyzed.The results indicate that the proposed model can provide a better understanding of the floating characteristics of the tunnel structure during construction.

Analytical and Experimental Research on Wave Scattering by an Open-Type Rectangular Breakwater with Horizontal Perforated Plates

This study proposes a novel open-type rectangular breakwater combined with horizontal perforated plates on both sides to enhance the sheltering effect of the rectangular box-type breakwaters against longer waves.The hydrodynamic characteristics of this breakwater are analyzed through analytical potential solutions and experimental tests.The quadratic pressure drop conditions are exerted on the horizontal perforated plates to facilitate assessing the effect of wave height on the dissipated wave energy of breakwater through the analytical solution.The hydrodynamic quantities of the breakwater,including the reflection,transmission,and energyloss coefficients,together with vertical and horizontal wave forces,are calculated using the velocity potential decomposition method as well as an iterative algorithm.Furthermore,the reflection and transmission coefficients of the breakwater are measured by conducting experimental tests at various wave periods,wave heights,and both porosities and widths of the horizontal perforated plates.The analytical predicted results demonstrate good agreement with the iterative boundary element method solution and measured data.The influences of variable incident waves and structure parameters on the hydrodynamic characteristics of the breakwater are investigated through further calculations based on analytical solutions.Results indicate that horizontal perforated plates placed on the water surface for both sides of the rectangular breakwater can enhance the wave dissipation ability of the breakwater while effectively decreasing the transmission and reflection coefficients.

Analytical modeling of piezoelectric meta-beams with unidirectional circuit for broadband vibration attenuation

Broadband vibration attenuation is a challenging task in engineering since it is difficult to achieve low-frequency and broadband vibration control simultaneously.To solve this problem,this paper designs a piezoelectric meta-beam with unidirectional electric circuits,exhibiting promising broadband attenuation capabilities.An analytical model in a closed form for achieving the solution of unidirectional vibration transmission of the designed meta-beam is developed based on the state-space transfer function method.The method can analyze the forward and backward vibration transmission of the piezoelectric meta-beam in a unified manner,providing reliable dynamics solutions of the beam.The analytical results indicate that the meta-beam effectively reduces the unidirectional vibration across a broad low-frequency range,which is also verified by the solutions obtained from finite element analyses.The designed meta-beam and the proposed analytical method facilitate a comprehensive investigation into the distinctive unidirectional transmission behavior and superb broadband vibration attenuation performance.

An approximate analytical model for unconventional reservoir considering variable matrix blocks and simultaneous matrix depletion

In regard to unconventional oil reservoirs,the transient dual-porosity and triple-porosity models have been adopted to describe the fluid flow in the complex fracture network.It has been proven to cause inaccurate production evaluations because of the absence of matrix-macrofracture communication.In addition,most of the existing models are solved analytically based on Laplace transform and numerical inversion.Hence,an approximate analytical solution is derived directly in real-time space considering variable matrix blocks and simultaneous matrix depletion.To simplify the derivation,the simultaneous matrix depletion is divided into two parts:one part feeding the macrofractures and the other part feeding the microfractures.Then,a series of partial differential equations(PDEs)describing the transient flow and boundary conditions are constructed and solved analytically by integration.Finally,a relationship between oil rate and production time in real-time space is obtained.The new model is verified against classical analytical models.When the microfracture system and matrix-macrofracture communication is neglected,the result of the new model agrees with those obtained with the dual-porosity and triple-porosity model,respectively.Certainly,the new model also has an excellent agreement with the numerical model.The model is then applied to two actual tight oil wells completed in western Canada sedimentary basin.After identifying the flow regime,the solution suitably matches the field production data,and the model parameters are determined.Through these output parameters,we can accurately forecast the production and even estimate the petrophysical properties.

Analytical solutions of transient heat conduction in multilayered slabs and application to thermal analysis of landfills

The study of transient heat conduction in multilayered slabs is widely used in various engineering fields. In this paper, the transient heat conduction in multilayered slabs with general boundary conditions and arbitrary heat generations is analysed. The boundary conditions are general and include various combinations of Dirichlet, Neumann or Robin boundary conditions at either surface. Moreover, arbitrary heat generations in the slabs are taken into account. The solutions are derived by basic methods, including the superposition method, separation variable method and orthogonal expansion method. The simplified double-layered analytical solution is validated by a numerical method and applied to predicting the temporal and spatial distribution of the temperature inside a landfill. It indicates the ability of the proposed analytical solutions for solving the wide range of applied transient heat conduction problems.

Time-domain analytic solutions of two-wire transmission line excited by a plane-wave field

This paper reports that an analytic method is used to calculate the load responses of the two-wire transmission line excited by a plane-wave directly in the time domain. By the frequency-domain Baum Liu-Tesehe （BLT） equation, the time-domain analytic solutions are obtained and expressed in an infinite geometric series. Moreover, it is shown that there exist only finite nonzero terms in the infinite geometric series if the time variate is at a finite interval. In other word, the time-domain analytic solutions are expanded in a finite geometric series indeed if the time variate is at a finite interval. The computed results are subsequently compared with transient responses obtained by using the frequency-domain BLT equation via a fast Fourier transform, and the agreement is excellent.

ANALYTICAL SOLUTIONS TO STRESS CONCENTRATION PROBLEM IN PLATES CONTAINING RECTANGULAR HOLE UNDER BIAXIAL TENSIONS

The stress concentration problem in structures with a circular or elliptic hole can be investigated by analytical methods. For the problem with a rectangular hole, only approximate results are derived. This paper deduces the analytical solutions to the stress concentration problem in plates with a rectangular hole under biaxial tensions. By using the U-transformation technique and the finite element method, the analytical displacement solutions of the finite element equations are derived in the series form. Therefore, the stress concentration can then be discussed easily and conveniently. For plate problem the bilinear rectangular element with four nodes is taken as an example to demonstrate the applicability of the proposed method. The stress concentration factors for various ratios of height to width of the hole are obtained.

Analytical solutions of simply supported magnetoelectroelastic circular plate under uniform loads

In this paper, the axisymmetric general solutions of transversely isotropic magnetoelectroelastic media are expressed with four harmonic displacement functions at first. Then, based on the solutions, the analytical three-dimensional solutions are provided for a simply supported magnetoelectroelastic circular plate subjected to uniform loads. Finally, the example of circular plate is presented.

Analytical solutions for elastic binary nanotubes of arbitrary chirality

Analytical solutions for the elastic properties of a variety of binary nanotubes with arbitrary chirality are obtained through the study of systematic molecular mechanics. This molecular mechanics model is first extended to chiral binary nanotubes by introducing an additional out-of-plane inversion term into the so-called stick-spiral model, which results from the polar bonds and the buckling of binary graphitic crystals. The closed-form expressions for the longitudinal and circumferential Young's modulus and Poisson's ratio of chiral binary nanotubes are derived as functions of the tube diameter. The obtained inversion force constants are negative for all types of binary nanotubes, and the predicted tube stiffness is lower than that by the former stick-spiral model without consideration of the inversion term, reflecting the softening effect of the buckling on the elastic properties of binary nanotubes. The obtained properties are shown to be comparable to available density functional theory calculated results and to be chirality and size sensitive. The developed model and explicit solutions provide a systematic understanding of the mechanical performance of binary nanotubes consisting of III-V and II-VI group elements.