Elastic solutions for stresses in a transversely isotropic half‐space subjected to three‐dimensional buried parabolic rectangular loads

In many areas of engineering practice, applied loads are not uniformly distributed but often concentrated towards the centre of a foundation. Thus, loads are more realistically depicted as distributed as linearly varying or as parabola of revolution. Solutions for stresses in a transversely isotropic half‐space caused by concave and convex parabolic loads that act on a rectangle have not been derived. This work proposes analytical solutions for stresses in a transversely isotropic half‐space, induced by three‐dimensional, buried, linearly varying/uniform/parabolic rectangular loads. Load types include an upwardly and a downwardly linearly varying load, a uniform load, a concave and a convex parabolic load, all distributed over a rectangular area. These solutions are obtained by integrating the point load solutions in a Cartesian co‐ordinate system for a transversely isotropic half‐space. The buried depth, the dimensions of the loaded area, the type and degree of material anisotropy and the loading type for transversely isotropic half‐spaces influence the proposed solutions. An illustrative example is presented to elucidate the effect of the dimensions of the loaded area, the type and degree of rock anisotropy, and the type of loading on the vertical stress in the isotropic/transversely isotropic rocks subjected to a linearly varying/uniform/parabolic rectangular load. Copyright © 2002 John Wiley & Sons, Ltd.     

三维抛物方程Douglas交替方向隐格式的稳定性和收敛性

作者研究三维变系数抛物方程 Douglas交替方向隐格式的稳定性和收敛性。采用 H1能量估计方法 ,证明格式按离散 H1范数是绝对稳定的 ,并且收敛阶为 O(Δ t2 + h2 )     

Parabolic Wave Propagation Modelling in Orthogonal Coordinate Systems

This paper presents a numerical model study of the propagation of water waves using the parabolic approximation of the mild-slope equation in the orthogonal coordinate system. Two types of coordinate systems are studied: (a) a general form of orthogonal coordinate system and (b) the conformal system, a special form of orthogonal coordinate system. Two typical examples, namely, expanded breakwaters and a circular channel, are studied to validate the model. First, the examples are studied by use of the general orthogonal coordinates. Then the same examples are computed by use of the confonnal system. The computational results show that the confonnal coordinate system generally gives better predictions than the general orthogonal system. A numerical technique for generating the conformal grid is combined with the numerical model to improve the practicability of the model. The comparison between the result from the numerical grid system and that from the analytical grid system shows that reliable computational results can be obtained by use of the numerical confonnal grid system.     

On solving Kepler's equation for nearly parabolic orbits

We deal here with the efficient starting points for Kepler's equation in the special case of nearly parabolic orbits. Our approach provides with very simple formulas that allow calculating these points on a scientific vest-pocket calculator. Moreover, srtarting with these points in the Newton's method we can calculate a root of Kepler's equation with an accuracy greater than 0.001 in 0–2 iterations. This accuracy holds for the true anomaly || 135° and |e – 1| 0.01. We explain the reason for this effect also.Dedicated to the memory of Professor G.N. Duboshin (1903–1986).     

Permeability Estimation with the Augmented Lagrangian Method for a Nonlinear Diffusion Equation

We consider numerical identification of the piecewise constant permeability function in a nonlinear parabolic equation, with the augmented Lagrangian method. By studying this problem, we aim at also gaining some insight into the potential ability of the augmented Lagrangian method to handle permeability estimation within the full two-phase porous-media flow setting. The identification is formulated as a constrained minimization problem. The parameter estimation problem is reduced to a coupled nonlinear algebraic system, which can be solved efficiently with the conjugate gradient method. The methodology is developed and numerical experiments with the proposed method are presented.     

象山皇城海滩稳定性分析与防护对策

以浙江岬湾砂质海岸象山皇城海滩为例,运用抛物线模型,分析得出皇城海滩的冲淤演化动态,并通过历史岸线形态对比探讨皇城海塘对其的影响,最终提出防止海岸侵蚀的对策。研究结果表明,皇城海滩北部有一定的侵蚀,其平面形态还未达到静态平衡,处于不稳定状态;皇城海滩北段海塘修建得过于靠海,致使海滩最重要的泥沙补给源——后滨沙丘被阻断,甚至消失,同时大潮水位可直接淹没海塘堤脚,加上海塘前为近破波,滩面水流和泥沙运动剧烈,最终导致堤前滩面的冲刷。基于此,防止海岸侵蚀的最佳对策是在拆除、重建海滩北部部分海塘后再进行适当的海滩养护。通过对皇城海滩的稳定性研究,了解海塘建设的影响,可为浙江岬湾海滩资源的合理开发、海岸防护工程建设提供一定的科学依据。     

Parabolic density function in sedimentary basin modelling

For modelling sedimentary basins of large thickness from their gravity anomalies, the concept of parabolic density function which explains the variation of true density contrast of the sediments with depth in such basins is introduced inBott's (1960) procedure. The analytical expression the gravity anomaly of a two-dimensional vertical prism with parabolic density contrast needed to estimate the gravity effect of the basin in modelling procedure is derived in a closed form. Two profiles of gravity anomalies, one across San Jacinto Graben, California and the other across Tucson basin, Arizona where the density of sediments is found to vary with depth are interpreted.     

The Parabolic Three-Body Problem

In this communication we present an analytical model for the restricted three-body problem, in the case where the perturber is in a parabolic orbit with respect to the central mass. The equations of motion are derived explicitly using the so-called Global Expansion of the disturbing function, and are valid for any eccentricity of the massless body, as well as in the case where both secondary masses have crossing orbits. Integrating the equations of motion over the complete passage of the perturber through the system, we are then able to construct a first-order algebraic mapping for the change in semimajor axis, eccentricity and inclination of the perturbed body.Comparisons with numerical solutions of the exact equations show that the map yields precise results, as long as the minimum distance between both bodies is not too small. Finally, we discuss several possible applications of this model, including the evolution of asteroidal satellites due to background bodies, and simulations of passing stars on extra-solar planets.     

Stresses due to vertical subsurface loading for an inhomogeneous cross‐anisotropic half‐space

In this article, we present the solutions for the stresses induced by four different loads associated with an axially loaded pile in a continuously inhomogeneous cross‐anisotropic half‐space. The planes of cross‐anisotropy are parallel to the horizontal surface of the half‐space, and the Young's and shear moduli are assumed to vary exponentially with depth. The four loading types are: an embedded point load for an end‐bearing pile, uniform skin friction, linear variation of skin friction, and non‐linear parabolic variation of skin friction for a friction pile. The solutions for the stresses due to the pile load are expressed in terms of the Hankel integral and are obtained from the point load solutions of the same inhomogeneous cross‐anisotropic half‐space which were derived recently by the authors (Int. J. Rock Mech. Min. Sci. 2003; 40 (5):667–685). A numerical procedure is proposed to carry out the integral. For the special case of homogeneous isotropic and cross‐anisotropic half‐space, the stresses predicted by the numerical procedure agree well with the solutions of Geddes and Wang (Geotechnique 1966; 16 (3):231–255; Soils Found. 2003; 43 (5):41–52). An illustrative example is also given to investigate the effect of soil inhomogeneity, the type and degree of soil anisotropy, and the four different loading types on the vertical normal stress. The presented solutions are more realistic in simulating the actual stratum of loading problem in many areas of engineering practice. Copyright © 2004 John Wiley & Sons, Ltd.     

A pair of Langmuir cells in two laboratory tanks (II) on generation mechanism

Vertical and cross-wind profiles of mean currents were measured systematically in vertical cross-sections of two wind-wave tanks with aspect ratios of order one to study the secondary flow in the tanks. A pair of Langmuir cells turned out to be driven by a close combination of the pressure gradient along the tank and the side-wall effects. That is, part of the adverse pressure gradient produced a parabolic cross-wind profile with the smallest downwind current at the centerline and the largest current along the two sidewalls. As a result, upwelling occurred in the center zone where the return flow was strongest, probably because of the entrainment action of the wind-driven current. In order to compensate for this upwelling, downwelling occurred along the two side-walls from the flow continuity. The resulting vertical circulation formed a pair of Langmuir circulations across the span and served to maintain the parabolic profile formed by the pressure gradient. A positive feedback mechanism is thus found between the primary and secondary circulations through upwelling of the return flow in the center zone. Vertical shears of the span-averaged downwind current measured in two tanks were found to be systematically different from each other. This difference seems to depend on the magnitude of the advective Reynolds stresses in the two tanks.