Possio Integral Equation of Aeroelasticity Theory

This paper presents a new abstract function space formulation of the subsonic small disturbance potential field equations of aeroelasticity and an operator theoretic treatment of the Possio integral equation in the generality of the Laplace transform variable λ. A key result is the new form of the kernel—which is shown to be analytic in the whole plane, excepting the negative real axis—using an existence and uniqueness theorem is proved valid for small ∣λ∣. The main new feature is the use of spatial Lp-Lq Fourier transforms for 1相似文献   

On the Fundamental Solution of Dynamic Poroelastic Boundary Integral Equations in the Time Domain

In this article, boundary integral equations (BIE) and Laplace transform domain fundamental solutions to the u‐p formulation of two‐dimensional dynamic poroelasticity are derived. Time domain fundamental solutions have been derived based on the Cleary reciprocal theorem. A set of numerical results is presented to highlight the salient features of the transient fundamental solutions, their components, and accuracies of proposed approximations. Several comparisons with Chen’s solution are made.     

Influence Zone of Recharging-Dewatering Actions in Unconfined Aquifer

This work deals with the recharging-dewatering problem for an unconfined aquifer. In particular, the evolution of the aquifer free surface consequent to changes in the water level in a trench near the aquifer itself has been investigated. The problem has been analyzed using a numerical approach so as not to eliminate the nonlinearities of the boundary conditions: the Laplace equation has been solved using the finite-element method coupled to an interpolation-collocation method for the solution of the free surface kinematic condition. Many configurations, which are most likely to cover cases of practical interest, have been studied, and the results have been summarized in a simple formula that permits the evaluation of the aquifer portion that is affected by recharging-dewatering actions.     

Least-Squares Finite-Element Evaluation of Flow Nets

A novel least-squares implementation of the finite-element method is presented to evaluate stream functions in the solution of field problems. The method is programmatically similar to the solution of the Laplace equation, and is based on the development of a stream field that is orthogonal to an already calculated potential field. The main advantage of the method comes from the fact that it eliminates the need of identification of boundary conditions for the stream functions. Implementation of this method requires that the Laplace equation be solved first to calculate the nodal potentials. The Laplace equation, with an identity conductivity matrix is then solved again to calculate the nodal values of the stream functions. One arbitrary boundary condition is sufficient for the second solution. Examples of cofferdam and curtained dam flow with isotropic as well as orthotropic soil conductivity are presented to demonstrate the method.     

Surface Loading of a Multilayered Viscoelastic Pavement: Semianalytical Solution

In this paper a new method is proposed to analyze the mechanical response of a linear viscoelastic pavement. The material parameters of the asphalt concrete are characterized by the relaxation modulus and creep compliance, which are further represented by the Prony series. By virtue of the Laplace transform and the correspondence principle, the solution in the Laplace domain is first derived. The interconversion between the relaxation modulus and creep compliance is then applied to treat the complicated inverse Laplace transform. The displacement, strain, and stress fields are represented concisely in terms of the convolution integral in the time domain, which is subsequently solved analytically. Therefore, responses of the viscoelastic pavement are finally expressed analytically in the time domain and numerically in space domain, called a semianalytical approach. Since both the relaxation modulus and creep compliance are used simultaneously, instead of only one parameter in the conventional methods, the present method is also called a dual-parameter method. The present formulation is verified at both the short- and long-term time limits analytically and at the other finite time numerically, as compared to the conventional numerical methods. We clearly show that the present dual-parameter and semianalytical method can predict accurately the time-dependent responses of the viscoelastic pavement, especially at the long-term time. The present formulation could also be employed to validate the widely used collocation method.     

Surface diffusion from a needle source including the effect of volume diffusion into the bulk

The problem of determining the quantity of diffusant to be found on and beneath the surface of a solid at any radial distance from a cylindrical source of constant strength on the surface has been solved using a Laplace transform technique. The movement of diffusant from the surface into the bulk through volume diffusion was taken into account. The integrals in the solution were evaluated numerically for a wide range of experimentally realizable conditions and the results given in a table. Experimental data can be analyzed with the help of this table to yield the product of the surface diffusion coefficient and the thickness of the surface layer. It is shown that, in general, these two quantities cannot be determined separately from data obtained in a conventional experiment because the amount of diffusant in the surface layer is usually only a small fraction of the total measured at any point.     

Forced Vertical Vibration of Rigid Circular Disc on a Transversely Isotropic Half-Space

Vertical vibration of a rigid circular disc attached to the surface of a transversely isotropic half-space is considered in such a way that the axis of material symmetry is normal to the surface of the half-space and parallel to the vibration direction. By using Hankel integral transforms, the mixed boundary-value problem is transformed to a pair of integral equations termed dual integral equations in the literature, which generally can be reduced to a Fredholm integral equation of the second kind. With the aid of complex variable or contour integration the governing integral equation is numerically solved in the general dynamic case. The reduced static case of the dual integral equations is solved analytically and the vertical displacement, the contact pressure, and the static impedance/compliance function are explicitly solved. The dynamic contact pressure under the disc and the impedance function are numerically evaluated, and it is shown that the singularity that exists at the edge of the disc is the same as the one obtained for the static case. In addition, the impedance functions evaluated here are identical to the solution given by Luco and Mita for the isotropic domain. To show the effect of different material anisotropy, the numerical evaluations are given for some different transversely isotropic materials and compared.     

Consolidation of a Finite Transversely Isotropic Soil Layer on a Rough Impervious Base

A semianalytical solution to axisymmetric consolidation of a transversely isotropic soil layer resting on a rough impervious base and subjected to a uniform circular pressure at the ground surface is presented. The analysis uses Biot’s fully coupled consolidation theory for a transversely isotropic soil. The general solutions for the governing consolidation equations are derived by applying the Hankel and Laplace transform techniques. These general solutions are then used to solve the corresponding boundary value problem for the consolidation of a transversely isotropic soil layer. Once solutions in the transformed domain have been found, the actual solutions in the physical domain for displacements and stress components of the solid matrix, pore-water pressure and fluid discharge can finally be obtained by direct numerical inversions of the integral transforms. The accuracy of the present numerical solutions is confirmed by comparison with an existing exact solution for an isotropic and saturated soil that is a special case of the more general problem addressed. Further, some numerical results are presented to show the influence of the nature of material anisotropy, the surface drainage condition, and the layer thickness on the consolidation settlement and the pore pressure dissipation.     

Diffusion fields associated with size and shape coarsening of oblate spheroids

The diffusion field or solute concentration distributed around an oblate spheroidal particle simulating a disc-shaped precipitate has been solved for varying particle aspect ratios and varying concentrations along the precipitate surface because of the curvature effect. With oblate spheroidal coordinates, the principal curvatures of the oblate spheroidal surface are derived as functions of the angular variable, and the Laplace field equation is separated into two Legendre equations on the angular variable and on the radial variable. The analytical solution to the Laplace equation, fitting the present boundary conditions, is secured as the sum of a Legendre function and a Legendre series composed of Legendre functions of the second kind with imaginary arguments. The Legendre function gives the concentration distribution with an ignored curvature effect, whereas the series shows the contribution from the curvature effect. Numerical results of normalized concentrations are presented as functions of the radial and angular variables for selected aspect ratios. The concentration distributions around both oblate and prolate spheroidal particles are shown to reduce to the concentration distributed around a spherical particle when the aspect ratio of the spheroids approaches unity.     

Analytical Solution of the Possio Integral Equation for the Sonic Case

We present a closed form analytical solution of the Possio integral equation of aeroelasticity for the sonic case M = 1, both time-domain and Laplace domain, including a numerical example for the flutter speed, where the flutter mode differs from that for M = 0. A noteworthy feature is that the Kutta condition is not satisfied and is not needed for uniqueness of solution.     

Dynamic Axial Load Transfer from Elastic Bar to Poroelastic Medium

The time-harmonic response of a cylindrical elastic bar (pile) partially embedded in a homogeneous poroelastic medium and subjected to a vertical load is considered. The bar is modeled using 1D elastic theory valid for long bars in the low-frequency range, and the porous medium using Biot's 3D elastodynamic theory. The bar is bonded to the surrounding medium along the contact surface. The problem is formulated by decomposing the bar∕porous medium system into a fictitious bar and an extended porous medium. A Fredholm's integral equation of the second kind governs the distribution of axial force in the fictitious bar. The integral equation involves kernels that are displacement and strain influence functions of a poroelastic half-space subjected to a buried, uniform vertical patch load. The governing integral equation is solved by applying numerical quadrature. The solutions for axial displacement and axial force of the bar, and the pore pressure are also derived. Selected numerical results for vertical impedance, axial force, and pore pressure profiles are presented to portray the influence of bar stiffness and length∕radius ratio, frequency of excitation, and poroelastic properties.     

Consolidation of a Double-Layered Compressible Foundation Partially Penetrated by Deep Mixed Columns

Deep mixed columns often penetrate partially into the soft soil as floating columns due to the depth of the end-bearing layer. Partially penetrated soft soil by columns and the underlying compressible soft soil create a double-layered compressible foundation. So far, no reasonable solution is available to estimate the consolidation of such a double-layered foundation. This paper proposes an analytical solution for consolidation of a double-layered compressible foundation partially penetrated by deep mixed columns considering one-side or two-side vertical drainage The Laplace transform method was used to solve the consolidation equation for the double-layered system while Stehfest’s algorithm was used to solve the inverse Laplace transform for time-dependent loading. A consolidation algorithm was used to calculate the time-settlement relationship of an embankment constructed upon the double-layered foundation partially penetrated by deep mixed columns. The calculated settlements were compared well with field measurements.     

RANS∕Laplace Calculations of Nonlinear Waves Induced by Surface-Piercing Bodies

An interactive zonal numerical method has been developed for the prediction of free surface flows around surface-piercing bodies, including both viscous and nonlinear wave effects. In this study, a Laplace solver for potential flow body-wave problems is used in conjunction with a Reynolds-averaged Navier-Stokes (RANS) method for accurate resolution of viscous, nonlinear free surface flows around a vertical strut and a series 60 ship hull. The Laplace equation for potential flow is solved in the far field to provide the nonlinear waves generated by the body. The RANS method is used in the near field to resolve the turbulent boundary layers, wakes, and nonlinear waves around the body. Both the kinematic and dynamic boundary conditions are satisfied on the exact free surface to ensure accurate resolution of the divergent and transverse waves. The viscous-inviscid interaction between the potential flow and viscous flow regions is captured through a direct matching of the velocity and pressure fields in an overlapping RANS and potential flow computational region. The numerical results demonstrate the capability of an interactive RANS∕Laplace coupling method for accurate and efficient resolution of the body boundary layer, the viscous wake, and the nonlinear waves induced by surface-piercing bodies.     

广义二阶流体的脉冲泊肃叶流动分析

讨论了广义二阶流体的脉冲泊肃叶流动,引入黎曼-刘维尔分数阶微分建立本构方程,结合不可压缩流体时间分数阶动量方程,得到控制方程.利用傅里叶正弦变换和分数阶拉普拉斯变换,获得流体速度解析解.利用Stehfest算法对结果进行数值模拟,通过图像讨论了分数阶参数以及延迟时间对流动的影响.结果表明速度过冲现象主要取决于动量方程的时间分数阶参数.      

Analysis of Point-Source and Boundary-Source Solutions of One-Dimensional Groundwater Transport Equation

The solute transport equation is commonly used to describe the migration and fate of solutes in a groundwater flow system. Depending on the problem nature, the source of the solute may be represented as a point source term in the equation or specified as the first-type or third-type boundary condition. The solutions derived under the condition that the solute introduced into the flow system is from the boundary is herein considered as the boundary-source solutions. The solution obtained when solving the transport equation with a point-source term is considered as the point-source solution. The Laplace transform technique is employed to derive the formulas for those solutions expressed in terms of the normalized mass release rate. The underlying nature of different source release modes and the differences among those boundary-source solutions and the constant point-source solution can be easily and clearly differentiated based on the derived formulas for one-dimensional transport. The methodology could, however, be easily extended to two- and three-dimensional problems.     

Linear Integral Analysis of Bar Rough Rolling by Strain Rate Vector

 A new linear integral method for bar hot rolling on roughing train was obtained. Above all, for inner deformation power, equivalent strain rate about Kobayashi’s three-dimensional velocity field was expressed by two-dimensional strain rate vector then inverted it into inner product and integrated term by term. During those processes, boundary equations and mean value theorem were introduced; but solution by definite integral was applied to integral procedure for friction and shear power. Then minimized the total upper bound power, and sequentially, analytical expressions of roll torque and separating force, and stress state factor were obtained. The calculated results by the expressions were compared with those of experimental values. The results show that this new linear integral method is available for analysis of bar rough rolling and the calculated results by the method are higher than those of experimental ones. However, the maximum error percentage between them is less than 10%.     

小型钢铁联合生产线

一个由箱式烧经机－氧气高炉－直流电弧炉－水平连铸组成的钢铁联合生产线可提供一个适合于小型钢铁厂的合理生产模式。此生产模式解决了用高炉生铁供电炉炼钢的难题以及机械化烧结作业和轧钢开坯的设备选型问题。     

Fluid Flow and Heat Transfer Modeling of AC Arc in Ferrosilicon Submerged Arc Furnace

 arc has been developed and used to predict heat transfer from the arc to the molten bath in ferrosilicon AC submerged-arc furnace. In this model the time-dependent conservation equations for mass, momentum and energy in the specified domain of plasma zone have been solved numerically coupled with the Maxwell and Laplace equations for magnetic filed and electric potential respectively. A control volume based finite difference method was used to solve the governing equations in cylindrical coordinates. The reliability of the developed model was tested by comparison with the data available in the literature. The present model showed a better consistency with the data given in the literature because of solving the Maxwell and Laplace equations simultaneously for calculation of current density. Parametric studies were carried out to evaluate the effect of electrical current and arc length on flow field and temperature distribution within the arc. According to computed results, a lower power input lead to the higher arc efficiency.     

Slopping resulting from gas injection in a peircesmith converter: The period of the standing wave

The periods of the transverse standing waves occurring in the Peirce-Smith converter geometry as a function of bath depth were solved using linear wave theory by formulating an integral equation of the second kind. The eigenvalues of the integral equation gave the wave periods which were calculated to four-figure accuracy. The eigenvalues were approximated using Chebyshev polynomials for the first four wave modes. They include the complete range of eigenvalues for the first two symmetric standing waves, which has not been reported in the literature. The asymptotic eigenvalues for the transverse standing waves are given. Experimental results from a water model agreed with the theoretical predictions over a wide range of bath depths. The first symmetric standing wave was enhanced by the presence of the longitudinal standing wave, which showed that the three-dimensional geometry of the model was important in the modeling of the Peirce-Smith converter. Data from a Peirce-Smith copper converter were in reasonable agreement with the theory. An analysis of the standing wave frequency showed that the presence of standing waves could be minimized by a change in the geometry.     

Analytical solutions for steady and non-steady heat conduction with temperature dependent material values

A transformation well-known for a long time is used to transform the differential equation for heat conduction for temperature dependent material values into a differential equation written in a more simplified form. It is investigated how the initial and boundary conditions are transformed and in which cases the problem with variable material values can be solved by applying the solutions known for constant material values. For constant thermal diffusivity, this reduction is possible for boundary conditions of the first and second kind without limitation, for conditions of the third kind only in some cases. The results of different methods of calculation are compared in an example calculation. The series solution with transformation lies generally much closer to the numerically calculated temperature than the solution without transformation. The computing times for the series solution with transformation are reduced by a factor of 100. If the first minutes of the heating period are disregarded, the first term constitutes an excellent approximation. The computing time for the finite-difference method with transformation is 20% lower than without transformation.