A SEMI-ANALYTICAL AND SEMI-NUMERICAL METHOD FOR SOLVING 2-D SOUND-STRUCTURE INTERACTION PROBLEMS

Based on the transfer matrix method and the virtual source simulation technique, this paper proposes a novel semi-analytical and semi-numerical method for solving 2-D sound- structure interaction problems under a harmonic excitation.Within any integration segment, as long as its length is small enough,along the circumferential curvilinear coordinate,the non- homogeneous matrix differential equation of an elastic ring of complex geometrical shape can be rewritten in terms of the homogeneous one by the method of extended homogeneous capacity proposed in this paper.For the exterior fluid domain,the multi-circular virtual source simulation technique is adopted.The source density distributed on each virtual circular curve may be ex- panded as the Fourier's series.Combining with the inverse fast Fourier transformation,a higher accuracy and efficiency method for solving 2-D exterior Helmholtz's problems is presented in this paper.In the aspect of solution to the coupling equations,the state vectors of elastic ring induced by the given harmonic excitation and generalized forces of coefficients of the Fourier series can be obtained respectively by using a high precision integration scheme combined with the method of extended homogeneous capacity put forward in this paper.According to the superposition princi- ple and compatibility conditions at the interface between the elastic ring and fluid,the algebraic equation of system can be directly constructed by using the least square approximation.Examples of acoustic radiation from two typical fluid-loaded elastic rings under a harmonic concentrated force are presented.Numerical results show that the method proposed is more efficient than the mixed FE-BE method in common use.

General semi-analytical solutions to one-dimensional consolidation for unsaturated soils

This paper presents general semi-analytical solutions to Fredlund and Hasan＇s one-dimensional （1D） consolidation equations for unsaturated soils subject to different initial conditions, homogeneous boundaries and time-dependent loadings. Two variables are introduced to transform the two-coupled governing equations of pore-water and poreair pressures into an equivalent set of partial differential equations （PDEs）, which are solved with the Laplace transform method. The pore-water and pore-air pressures and settlement are obtained in the Laplace transform domMn. The Crump＇s method is used to perform inverse Laplace transform to obtain the solutions in the time domain. The present solutions are more general in practical applications and show good agreement with the previous solutions in the literature.

Semi-analytical solutions to one-dimensional consolidation for unsaturated soils with semi-permeable drainage boundary

The semi-analytical solutions to Fredlund and Hasan＇s one-dimensional （1D） consolidation for unsaturated soils with a semi-permeable drainage boundary are pre- seated. Two variables are introduced to transform the two coupled governing equations of pore-water and pore-air pressures into an equivalent set of partial differential equations （PDFs）, which are easily solved by the Laplace transform method. Then, the pore-water pressure, pore-air pressure, and soil settlement are obtained in the Laplace domain. The Crump method is adopted to perform the inverse Laplace transform in order to obtain the semi-analytical solutions in the time domain. It is shown that the proposed solutions are more applicable to various types of boundary conditions and agree well with the existing solutions from the literature. Several numerical examples are provided to investigate the consolidation behavior of an unsaturated single-layer soil with single, double, mixed, and semi-permeable drainage boundaries. The changes in the pore-air and pore-water pres- sures and the soil settlement with the time factor at different values of the semi-permeable drainage boundary parameters are illustrated. In addition, parametric studies are con- ducted on the pore-air and pore-water pressures at different ratios （the air permeability coefficient to the water permeability coefficient） and depths.

Hamiltonian parametric element and semi-analytical solution for smart laminated plates

Based on the Hellinger-Reissner （H-R） mixed variational principle for piezoelectric material, a unified 4-node Hamiltonian isoparametric element of anisotropy piezoelectric material is established. A new semi-analytical solution for the natural vibration of smart laminated plates and the transient response of the laminated cantilever with piezoelectric patch is presented. The major steps of mathematical model are as follows： the piezoelectric layer and host layer of laminated plate are considered as unattached three-dimensional bodies and discretized by the Hamiltonian isoparametric elements. The control equation of whole structure is derived by considering the compatibility of generalized displacements and generalized stresses on the interface between layers. There is no restriction for the side-face geometrical boundaries, the thickness and the number of layers of plate by the use of the present isoparametric element. Present method has wide application area.

Consolidation of partially saturated ground improved by impervious column inclusion:Governing equations and semi-analytical solutions

This study focuses on the consolidation behavior and mathematical interpretation of partially-saturated ground improved by impervious column inclusion.The constitutive relations for soil skeleton,pore air and pore water for partially saturated soils are proposed in the context of partially-saturated ground improved by impervious column inclusion.Settlement equation and dissipation equations of excess pore air/water pressures for a partially saturated improved ground are then derived.The semi-analytical solutions for ground settlement and pore pressure dissipation are then obtained through the Laplace transform and validated by the existing solutions for two special cases in the literature and the numerical results obtained from the finite difference method.A series of parametric studies is finally conducted to investigate the influence of some key factors on consolidation of partially saturated ground improved by impervious column inclusion.Based on the parametric study,it can be found that a higher value of the area replacement ratio or modulus of the pile results in a longer dissipation time of excess pore air pressure(PAP),a shorter dissipation time of excess pore water pressure(PWP),and a lower normalized settlement.

Semi-analytical solution to one-dimensional consolidation in unsaturated soils

In this paper, a series of semi-analytical solutions to one-dimensional consolidation in unsaturated soils are obtained. The air governing equation by Fredlund for unsaturated soils consolidation is simplified. By applying the Laplace transform and the Cayley-Hamilton theorem to the simplified governing equations of water and air, Darcy＇s law, and Fick＇s law, the transfer function between the state vectors at top and at any depth is then constructed. Finally, by the boundary conditions, the excess pore-water pressure, the excess pore-air pressure, and the soil settlement are obtained under several kinds of boundary conditions with the large-area uniform instantaneous loading. By the Crump method, the inverse Laplace transform is performed, and the semi-analytical solutions to the excess pore-water pressure, the excess pore-air pressure, and the soils settlement are obtained in the time domain. In the case of one surface which is permeable to air and water, comparisons between the semi-analytical solutions and the analytical solutions indicate that the semi-analytical solutions are correct. In the case of one surface which is permeable to air but impermeable to water, comparisons between the semi-analytical solutions and the results of the finite difference method are made, indicating that the semi-analytical solution is also correct.

A Semi-Analytical Potential Solution for Wave Resonance in Gap Between Floating Box and Vertical Wall

Based on potential flow theory, a dissipative semi-analytical solution is developed for the wave resonance in the narrow gap between a fixed floating box and a vertical wall by using velocity potential decompositions and matched eigenfunction expansions. The energy dissipation near the box is modelled in the potential flow solution by introducing a quadratic pressure loss condition on the gap entrance. Such a treatment is inspired by the classical local head loss formula for the sudden change of cross section in channel flow, where the energy dissipation is assumed to be proportional to the square of local velocity for high Reynolds number flows. The dimensionless energy loss coefficient is calibrated based on experimental data. And it is found to be insensitive to the incident wave height and wave frequency. With the calibrated energy loss coefficient, the resonant wave height in gap and the reflection coefficient are calculated by the present dissipative semi-analytical solution. The predictions are in good agreement with experimental data. Case studies suggest that the maximum relative energy dissipation occurs near the resonant frequency, which leads to the minimum reflection coefficient. The horizontal wave forces on the box and the vertical wall attain also maximum values near the resonant frequency, while the vertical wave force on the box decreases abruptly there to a small value.

A semi-analytical solution for electric double layers near an elliptical cylinder

A theoretical analysis on the electric double layer formed near the surface of an infinite cylinder with an elliptical cross section and a prescribed electric potential in an ionic conductor was performed using the linearized Gouy–Chapman theory. A semi-analytical solution in terms of the Mathieu functions was obtained. The distributions of the electric potential, cations, anions, and electric field were calculated. The effects of various physical and geometric parameters were examined. The fields vary rapidly near the elliptical boundary and are nearly uniform at far field. Electric field concentrations were found at the ends of the semi-major and semi-minor axes of the ellipse. These concentrations are sensitive to the physical and geometric parameters.

A semi-analytical solution for frost heave prediction of clay soil

Frost heave is one of the main freezing problems for construction in permafrost regions.The Konrad-Morgenstern segregation potential（SP） model is being used in practice for frost heave using numerical techniques.However,the heat release from in-situ and migrated water in the freezing zone could result in some numerical instability,so the simulation of frost fringe is not ideal.In this study,a semi-analytical solution is developed for frost heave prediction of clay soil.The prediction results to the two tests with different freezing mode with clay soil agree well with the tested behavior,which indicates the feasibility of the solution.

Semi-analytic Solution of Steady Temperature Fields During Thin Plate Welding

Usually, it is very difficult to find out an analytical solution to thermal conduction problems during high temperature welding. Therefore, as an important numerical approach, the method of lines （MOLs） is introduced to solve the temperature field characterized by high gradients. The basic idea of the method is to semi-discretize the governing equation of the problem into a system of ordinary differential equations （ODEs） defined on discrete lines by means of the finite difference method, by which the thermal boundary condition with high gradients are directly embodied in formulation. Thus the temperature field can be obtained by solving the ODEs. As a numerical example, the variation of an axisymmetrical temperature field along the plate thickness can be obtained.

Closed-form solution of mid-potential between two parallel charged plates with more extensive application

Efficient calculation of the electrostatic interactions including repulsive force between charged molecules in a biomolecule system or charged particles in a colloidal system is necessary for the molecular scale or particle scale mechanical analyses of these systems. The electrostatic repulsive force depends on the mid-plane potential between two charged particles. Previous analytical solutions of the mid-plane potential, including those based on simplified assumptions and modern mathematic methods, are reviewed. It is shown that none of these solutions applies to wide ranges of interparticle distance from 0 to 10 and surface potential from 1 to 10. Three previous analytical solutions are chosen to develop a semi-analytical solution which is proven to have more extensive applications. Furthermore, an empirical closed-form expression of mid-plane potential is proposed based on plenty of numerical solutions. This empirical solution has extensive applications, as well as high computational efficiency.

HAMILTONIAN SYMPLECTIC SEMI-ANALYTICAL METHOD AND ITS APPLICATION IN INHOMOGENEOUS ELECTROMAGNETIC WAVEGUIDES

Dual vectors are applied in Hamilton system of applied mechanics. Electric and magnetic field vectors are the dual vectors in electromagnetic field. The Hamilton system method is introduced into the analysis of electromagnetism waveguide with inhomogeneous materials. The transverse electric and magnetic fields are regarded as the dual. The basic equations are solved in Hamilton system and symplectic geometry. With the Hamilton variational principle, the symplectic semi_analytical equations are derived and preserve their symplectic structures. The given numerical example demonstrates the solution of LSE (Longitudinal Section Electric) mode in a dielectric waveguide. (

Pumping-induced Well Hydraulics and Groundwater Budget in a Leaky Aquifer System with Vertical Heterogeneity in Aquitard Hydraulic Properties

In groundwater hydrology,aquitard heterogeneity is often less considered compared to aquifers,despite its significant impact on groundwater hydraulics and groundwater resources evaluation.A semi-analytical solution is derived for pumping-induced well hydraulics and groundwater budget with consideration of vertical heterogeneity in aquitard hydraulic conductivity(K)and specific storage(S_(s)).The proposed new solution is innovative in its partitioning of the aquitard into multiple homogeneous sub-layers to enable consideration of various forms of vertically heterogeneous K or S_(s).Two scenarios of analytical investigations are explored:one is the presence of aquitard interlayers with distinct K or S_(s) values,a common field-scale occurrence;another is an exponentially depth-decaying aquitard S_(s),a regional-scale phenomenon supported by statistical analysis.Analytical investigations reveal that a low-K interlayer can significantly increase aquifer drawdown and enhance aquifer/aquitard depletion;a high-S_(s) interlayer can noticeably reduce aquifer drawdown and increase aquitard depletion.Locations of low-K or high-S_(s) interlayers also significantly impact well hydraulics and groundwater budget.In the context of an exponentially depth-decaying aquitard S_(s),a larger decay exponent can enhance aquifer drawdown.When using current models with a vertically homogeneous aquitard,half the sum of the geometric and harmonic means of exponentially depth-decaying aquitard S_(s) should be used to calculate aquitard depletion and unconfined aquifer leakage.

Dynamic characteristics of multi-span spinning beams with elastic constraints under an axial compressive force

A theoretical model for the multi-span spinning beams with elastic constraints under an axial compressive force is proposed.The displacement and bending angle functions are represented through an improved Fourier series,which ensures the continuity of the derivative at the boundary and enhances the convergence.The exact characteristic equations of the multi-span spinning beams with elastic constraints under an axial compressive force are derived by the Lagrange equation.The efficiency and accuracy of the present method are validated in comparison with the finite element method(FEM)and other methods.The effects of the boundary spring stiffness,the number of spans,the spinning velocity,and the axial compressive force on the dynamic characteristics of the multi-span spinning beams are studied.The results show that the present method can freely simulate any boundary constraints without modifying the solution process.The elastic range of linear springs is larger than that of torsion springs,and it is not affected by the number of spans.With an increase in the axial compressive force,the attenuation rate of the natural frequency of a spinning beam with a large number of spans becomes larger,while the attenuation rate with an elastic boundary is lower than that under a classic simply supported boundary.

Analytical solution and semi-analytical solution for anisotropic functionally graded beam subject to arbitrary loading

Analytical and semi-analytical solutions are presented for anisotropic functionally graded beams subject to an arbitrary load,which can be expanded in terms of sinusoidal series.For plane stress problems,the stress function is assumed to consist of two parts,one being a product of a trigonometric function of the longitudinal coordinate(x) and an undetermined function of the thickness coordinate(y),and the other a linear polynomial of x with unknown coefficients depending on y.The governing equations satisfied by these y-dependent functions are derived.The expressions for stresses,resultant forces and displacements are then deduced,with integral constants determinable from the boundary conditions.While the analytical solution is derived for the beam with material coefficients varying exponentially or in a power law along the thickness,the semi-analytical solution is sought by making use of the sub-layer approximation for the beam with an arbitrary variation of material parameters along the thickness.The present analysis is applicable to beams with various boundary conditions at the two ends.Three numerical examples are presented for validation of the theory and illustration of the effects of certain parameters.

An investigation on multilayer shape memory polymers under finite bending through nonlinear thermo-visco-hyperelasticity

This study presents a semi-analytical solution to describe the behavior of shape memory polymers(SMPs) based on the nonlinear thermo-visco-hyperelasticity which originates from the concepts of internal state variables and rational thermodynamics.This method is developed for the finite bending of multilayers in a dual-shape memory effect(SME) cycle.The layer number and layering order are investigated for two different SMPs and a hyperelastic material.In addition to the semi-analytical solution,the finite element simulation is performed to verify the predicted results,where the outcomes demonstrate the excellent accuracy of the proposed solution for predicting the behavior of the multilayer SMPs.Since this method has a much lower computational cost than the finite element method(FEM),it can be used as an effective tool to analyze the SMP behavior under different conditions,including different materials,different geometries,different layer numbers,and different layer arrangements.

Two-dimensional plane strain consolidation of unsaturated soils considering the depth-dependent stress

In practical engineering,the total vertical stress in the soil layer is not constant due to stress diffusion,and varies with time and depth.Therefore,the purpose of this paper is to investigate the effect of stress diffusion on the two-dimensional(2D)plane strain consolidation properties of unsaturated soils when the stress varies with time and depth.A series of semi-analytical solutions in terms of excess pore air and water pressures and settlement for 2D plane strain consolidation of unsaturated soils can be derived with the joint use of Laplace transform and Fourier sine series expansion.Then,the inverse Laplace transform of the semi-analytical solution is given in the time domain using a self-programmed code based on Crump’s method.The reliability of the obtained solutions is proved by the degeneration.Finally,the 2D plots of excess pore pressures and the curves of settlement varying with time,considering different physical parameters of unsaturated soil stratum and depth-dependent stress,are depicted and analyzed to study the 2D plane strain consolidation properties of unsaturated soils subjected to the depthdependent stress.

MODIFIED H-R MIXED VARIATIONAL PRINCIPLE FOR MAGNETOELECTROELASTIC BODIES AND STATE-VECTOR EQUATION

Based upon the Hellinger-Reissner (H-R) mixed variational principle for three-dimensional elastic bodies, the modified H-R mixed variational theorem for magnetoelectroelastic bodies was established. The state-vector equation of magnetoelectroelastic plates was derived from the proposed theorem by performing the variational operations. To lay a theoretical basis of the semi-analytical solution applied with the magnetoelectroelastic plates, the state-vector equation for the discrete element in plane was proposed through the use of the proposed principle. Finally, it is pointed out that the modified H-R mixed variational principle for pure elastic, single piezoelectric or single piezomagnetic bodies are the special cases of the present variational theorem.

Linear, Cubic and Quintic Coordinate-Dependent Forces and Kinematic Characteristics of a Spring-Mass System

By combining a pair of linear springs we devise a nonlinear vibrator. For a one dimensional scenario the nonlinear force is composed of a polynomial of odd powers of position-dependent variable greater than or equal three. For a chosen initial condition without compromising the generality of the problem we analyze the problem considering only the leading cubic term. We solve the equation of motion analytically leading to The Jacobi Elliptic Function. To avoid the complexity of the latter, we propose a practical, intuitive-based and easy to use alternative semi-analytic method producing the same result. We demonstrate that our method is intuitive and practical vs. the plug-in Jacobi function. According to the proposed procedure, higher order terms such as quintic and beyond easily may be included in the analysis. We also extend the application of our method considering a system of a three-linear spring. Mathematica [1] is being used throughout the investigation and proven to be an indispensable computational tool.

Vibration and Acoustic Radiation from Submerged Spherical Double-Shell

Based on the motion differential equations of vibration and acoustic coupling system for a thin elastic spherical double-shell with several elastic plates attached to the shells, in which Dirac-δ functions are employed to introduce the forces and moments applied by the attachments, and by means of expanding field quantities as the Legendre series, a semi-analytic solution is derived for the solution to the vibration and acoustic radiation from a submerged spherical double-shell. This solution has a satisfying computational effectiveness and precision for arbitrary frequency range excitation. It is concluded that the internal plates attached to shells can change significantly the mechanical and acoustical characteristics of shells, and make the coupling system have a very rich resonance frequency spectrum. Moreover, the present method can be used to study the acoustic radiation mechanism of the type of structure.