A Review and Update of Analytical and Numerical Solutions of the Terzaghi One-Dimensional Consolidation Equation

Practical resolution of consolidation problems that we often face requires an extensive and solid knowledge of the different parameters highlighted by the Terzaghi one-dimensional consolidation theory. This theory, with its assumptions, leads to a partial differential equation of second order in space and first order in time of pore water pressure. Analytical and numerical resolutions of this equation allow determining the water pressure variation before and after the application of a charge. Numerical modeling has enabled the simulation of the whole results obtained by the two methods of resolution (pressure, degree of consolidation, time factor, among others) to have a physical analysis and a lawful observation that lead to a suitable understanding of the phenomenon of Terzaghi one-dimensional consolidation.

THE STABILITY AND CONVERGENCE OF THE FINITE ANALYTIC METHOD FOR THE NUMERICAL SOLUTION OF CONVECTIVE DIFFUSION EQUATION

In this paper we make a close study of the finite analytic method by means of the maximum principles in differential equations and give the proof of the stability and convergence of the finite analytic method.

An Analytic-Numerical Solution of Wave Transformation

An analytic-numerical solution of wave transformation in shoaling water is presented in this paper. The analytical expression for wave heights along the wave rays is derived in consideration of the combined effect of water depth shoaling, the wave refraction and the sea bottom friction. The wave rays (orthogonals) are calculated by a fourth order Runge-Kutta algorithm and the wave crest lines are computed by an iteration procedure. The numerical results are compared with analytical solution for a special case of parallel- straight contour shore and field data, and comparisons show that the proposed mathematical model and computation method are very useful and convenient for engineering application.

Analytical Solution of Helmholtz Equation in Anisotropic and Nonhomogeneous Region

The analytical solution of Helmholtz equation for magnetic vector potential in anisotropic and nonhomogeneous region is presented. The solution is built of a combination of both Bessel and power functions. There are developed two examples that proof the accuracy of the proposed analytical solution. First example is showing the electromagnetic field analysis in slot of ferromagnetic rotor of electrical induction machine. The second example approaches electromagnetic field wave in resonator of the form of rectangular cavity The analytical solution presented is treated as an exact one and is being compared with the numerical solution, e.g., given by finite element method. The analytical solution can be used as a benchmark test for numerical algorithms,

Stress analytical solution for plane problem of a double-layered thick-walled cylinder subjected to a type of non-uniform distributed pressure

The stress state around circular openings,such as boreholes,shafts,and tunnels,is usually needed to be evaluated.Solutions for stresses,strains and ultimate bearing capacities of pressurized hollow cylinder are common cases.Stress analytical method for plane problem of a double-layered thick-walled cylinder subjected to a type of non-uniform pressure on the outer surface and uniform radial pressure on the inner surface is given.The power series method of complex function is used.The stress analytical solution is obtained with the assumption that two layers of a cylinder are fully contacted.The distributions of normal and tangential contact stress along the interface,tangential stress on the inner boundary and stresses in the radial direction at θ=0°,45° and 90°,are obtained.An example indicates that,when the elastic modulus of the inner layer of a double-layered thick-walled cylinder is smaller than that of the outer layer,the tangential stress is smaller than that in the corresponding point for a traditional cylinder composed of homogeneous materials.In that way,stress concentration at the inner surface can be alleviated and the stress distribution is more uniform.This is a capable way to enhance the elastic ultimate bearing capacity of thick-walled cylinder.

Analytic elasticity solution of bi-modulus beams under combined loads

A unified stress function for bi-modulus beams is proposed based on its mechanic sense on the boundary of beams. Elasticity solutions of stress and displacement for bi-modulus beams under combined loads are derived. The example analysis shows that the maximum tensile stress using the same elastic modulus theory is underestimated if the tensile elastic modulus is larger than the compressive elastic modulus. Otherwise, the maximum compressive stress is underestimated. The maximum tensile stress using the material mechanics solution is underestimated when the tensile elastic modulus is larger than the compressive elastic modulus to a certain extent. The error of stress using the material mechanics theory decreases as the span-to-height ratio of beams increases, which is apparent when L/h ≤ 5. The error also varies with the distributed load patterns.

Analytic solution of phreatic surface in the slope of reservoir bank

In most cases, the slope stability of reservoir bank is analyzed on the premise that the location of phreatic surface is obtained. But many designers generalize a line as the phreatie surface through their experience to analyze the stability, which is unsafe in the project. To find a solution of the phreatic surface which is convenient to put into use and in accordance with the practice, the article, based on Boussinesq equation, infers analytic solutions suitable to the water level at different ratios and achieves an analytic solution equation through fitting curves. The correctness of the equation is also proved by the experiments of sand and sand-clay models and the inaccuracy of empirical generalization is analyzed quantitatively. The calculation results show that the inaccuracy through the method of experiential generalizing is so large that the designers should be awake to it.

Numerical Solution for Fractional-Order Differential Systems with Time-Domain and Frequency-Domain Methods

For a general nonlinear fractional-order differential equation, the numerical solution is a good way to approximate the trajectory of such systems. In this paper, a novel algorithm for numerical solution of fractional-order differential equations based on the definition of Grunwald-Letnikov is presented. The results of numerical solution by using the novel method and the frequency-domain method are compared, and the limitations of frequency-domain method are discussed.

AN ANALYTICAL SOLUTION TO LARGE DEFLECTION EQUATIONS OF SIMPLY－SUPPORTED RECTANGULAR HYPERBOLOIDAL SHALLOW SHELLS OF ORTHOTROPIC COMPOSITES

Based on the product rule of the Fourier series and some relevant results inreferences[1, 2]. a method on solving the large deflection equations of plates and shells by means of the Fourier series is proposed in the present paper,Applying this method .we derive a type solution to the Navier’s solution of the nonlinear differential equations of the rectangular hyperboloidal shallow shells of the orthotropic compositessimply supported .This solution is suitable for plates and shells with large deflection orsmall deflection whether it is isotropic or orthotropic.Their data processing results are correlative with those found in the classical examples and from the experiments.

Rigorous Axially Symmetric Consolidation Solution of Vacuum Combined Linear Loading Surcharge Preloading

Vacuum combined surcharge preloading is an effective method which has been used in ground treatment. This study presents the analytical solution of vertical drains with vacuum combined surcharge preloading. The surcharge was considered to be time-dependent preloading due to its reflection of the real situation.Moreover,the free strain,instead of equal strain hypothesis of the surface,was adopted for the mathematical rigor. Both vertical and horizontal drainages,along with the well resistance were considered.Besides,the time-dependent surcharge preloading was resolved with the impulse theorem which has been presented in the study,and the solutions for pore water pressure and degree of consolidation were derived by applying the approach of separation of variables.Furthermore,the solution is compared with previous solutions and numerical solutions,and the results verify the correctness of the proposed model.

Numerical simulation of flooding induced uplift for abandoned coal mines:simulation schemes and parameter sensitivity

Numerical simulation approaches have been widely applied to study mining induced subsidence,and they are potential methods to study the flooding induced uplift for abandoned mines.This paper gives an overview about different numerical approaches to simulate uplift induced by flooding abandoned underground mines,including three different hydraulic conditions,considering both unconfined and confined water conditions.Four basic simulation schemes using 1-dimensional rock column models verified by analytical solutions demonstrate these procedures.The results reveal that flooding induced uplift is mainly related to the pore pressure in the mine goaf.The parameter study documents that height and stiffness of the mine goaf have the strongest influence on maximum surface uplift.

Numerical Simulation of Diffusion Type Traffic Flow Model Using Second-Order Lax-Wendroff Scheme Based on Exponential Velocity Density Function

In order to control traffic congestion, many mathematical models have been used for several decades. In this paper, we study diffusion-type traffic flow model based on exponential velocity density relation, which provides a non-linear second-order parabolic partial differential equation. The analytical solution of the diffusion-type traffic flow model is very complicated to approximate the initial density of the Cauchy problem as a function of x from given data and it may cause a huge error. For the complexity of the analytical solution, the numerical solution is performed by implementing an explicit upwind, explicitly centered, and second-order Lax-Wendroff scheme for the numerical solution. From the comparison of relative error among these three schemes, it is observed that Lax-Wendroff scheme gives less error than the explicit upwind and explicit centered difference scheme. The numerical, analytical analysis and comparative result discussion bring out the fact that the Lax-Wendroff scheme with exponential velocity-density relation of diffusion type traffic flow model is suitable for the congested area and shows a better fit in traffic-congested regions.

General Analytical Shakedown Analysis of a Structure Subjected to Combined Loading with Constant and Cyclic Loads

The shakedown behavior of structures subjected to a combined loading of constant and cyclic loads has been well researched.For some specified problems,shakedown limit loads have been obtained.However,the general effect of combined loading on the structural shakedown has not yet been presented.The general analytic solution of the elastic shakedown limit load is thus derived for a structure subjected to combined loading.Polizzotto's extended static shakedown theorem for combined loading is applied.The stress field in equilibrium with the external constant load required in Polizzotto's extended theorem is constructed by subtracting the reference elastic stress field of the peak cyclic load from the elastic-plastic stress field of the combined constant load and peak cyclic load.The shakedown condition of the stress field is then derived according to the extended theorem.Through the analytical analysis of the shakedown condition,the structural shakedown behavior under combined loading is investigated.A general solution of the shakedown limit load is then derived,and the effects of the combined loading on the shakedown behavior are proposed.The obtained general analytical result is applied to a hollow tension specimen under constant tension and alternating torsion and a plate with a central hole under constant and cyclic tension.The results are consistent with the solutions reported in the literature.

Analytical Research of Temperature Distribution and Thermal Stresses within Circular Micro-hotplates

Micro-hotplate （MHP） technology is one key part in the manufacturing of gas sensors. The pursuit of analytical solutions for the temperature distribution and also thermal stresses within the MHP is of intrinsic scientific interest. In this study, analytical solutions for the temperature field, and both radial and tangential stresses and van Mises stress for circular MHP were obtained. Two geometries were considered： one had a circular heater at the center and the other had a circular heater at the center and an annular heater within the membrane part. Internal heat generation was incorporated in the energy equation for the MHP and different values of convection heat transfer coefficient were used at the upper and lower surfaces of the MHP. It has been shown that the MHP with two heaters can provide more uniform temperature field compared with the MHP with one heater. The main objective of this work is to provide an exact analytical solution for thermal stresses within the circular micro-hcater with a simple geometry as a benchmark, from mathematical point of view, against which the accuracy of new numerical schemes can be checked. To make sure that the analytical procedure is correct, the analytical results are checked against numerical solutions derived from finite element simulation. Since the analytical models for the temperature field and especially for the thermal stresses of MHP ace seldom investigated in the literature, the obtained results are believed to facilitate the design and performance evaluation of MHPs as well.

On the Construction of Analytic-Numerical Approximations for a Class of Coupled Differential Models in Engineering

In this paper, a method to construct an analytic-numerical solution for homogeneous parabolic coupled systems with homogeneous boundary conditions of the type ut = Auxx, A1u(o,t) + B1ux(o,t) = 0, A2u(1,t) + B2ux(1,t) = 0, ot>0, u (x,0) = f(x), where A is a positive stable matrix and A1, B1, B1, B2,? ?are arbitrary matrices for which the block matrix is non-singular, is proposed.

Jeffery-Hamel flow of non-Newtonian fluid with nonlinear viscosity and wall friction

A Jeffery-Hamel （J-H） flow model of the non-Newtonian fluid type inside a convergent wedge （inclined walls） with a wall friction is derived by a nonlinear ordinary differential equation with appropriate boundary conditions based on similarity relationships. Unlike the usual power law model, this paper develops nonlinear viscosity based only on a tangential coordinate function due to the radial geometry shape. Two kinds of solutions are developed, i.e., analytical and semi-analytical （numerical） solutions with suitable assumptions. As a result of the parametric examination, it has been found that the Newtonian normalized velocity gradually decreases with the tangential direction progress. Also, an increase in the friction coefficient leads to a decrease in the normalized Newtonian velocity profile values. However, an increase in the Reynolds number causes an increase in the normalized velocity function values. Additionally, for the small values of wedge semi-angle, the present solutions are in good agreement with the previous results in the literature.

Mathematical Study of Medicine Propagation in Biological Tissue and Some of Its Applications

The paper deals with the problem of the distribution of the medicine (enzyme) in the damaged biological tissue where the reaction enzyme—substrat takes place. The biological problem is reduced to a singular degenerate initial-boundary value problem for two coupled ordinary differential equations. Analytical solution of the singular degenerated IBV-problem was obtained by power series. The solution demonstrates the real situation and found suitable to depict the degeneration of singular system, caused by low concentration of the enzyme.

Dynamic simulation of hydrodynamic model of drum level wave action and sloshing

In order to build the model of the drum level wave action and sloshing, based on the method of modularization modeling, the hydrodynamic model of drum level wave action and sloshing was developed, and dynamic simulation researches were carried out based on the model. The results indicate that both drum level and drum length have functional relations with period of drum level wave action and sloshing. When the drum level decreases or drum length increases, the period of drum level wave action and sloshing increases, density of liquid and number of sub-module division have little influence on the period of drum level wave action and sloshing. The model was validated by the analytical solution theory of liquid’s wave action and sloshing in cuboid container, and the 3D graphics of drum level wave action and sloshing was also obtained. The model can dynamically reflect the rules of wave action and sloshing of water in the container exactly.

索-悬链线拱联合结构的受力机理与力学特征

早期拱桥已运营多年,缺乏有效的养护,且设计荷载等级偏低;随着交通量的增加及车辆超载,其在活载作用下的力学性能已不能满足现代交通需求。为改善此类拱桥的受力性能,提出一种索-悬链线拱联合结构,在拱肋两侧对称设置索,使索和拱肋形成新的受力体系,从而改变悬链线拱结构的传力路径,进而降低拱结构的内力。在索-拱联合结构图式下,基于弹性中心法对其力法方程进行简化,采用近似曲线积分方法推导了竖向移动荷载作用下拱肋的内力解析解,并借助ANSYS有限元软件验证解析解的准确性;分析了在车道荷载作用下索约束位置、拱轴系数、矢跨比和轴向刚度比等设计参数对拱结构的影响,揭示了索-拱联合结构的受力机理和内力变化规律。研究表明,内力解析解与有限元结果的相对误差在1%以内;索的设置改变了拱结构弯矩影响线量值的正、负区间分布,而且有效地降低了拱结构弯矩影响线的峰值,从而使拱结构的整体弯矩得到较大幅度的降低,弯矩分布更均匀;沿拱脚至拱顶,拱结构弯矩的降低幅度和轴力的增长幅度均逐渐减小;轴向刚度比从0.02增加至0.10,拱脚负弯矩的减小幅度呈非线性增大,最大降幅达63.7%;索力的增长幅度与其数值大小成反比,当索设置在距拱顶0.3L(L为拱跨经)时,索力可增至轴向刚度比为0.02时的1.9倍。矢跨比对拱结构内力带来的影响不容忽视,矢跨比越大拱结构的内力变化幅度越大;拱轴系数对拱结构内力的影响可以忽略。

碎石桩-排水板组合型复合地基固结解析解

碎石桩-排水板组合型复合地基以成本低、固结速率快及加固效果好等优势被广泛应用于工程实际。虽然目前竖井地基或碎石桩复合地基固结理论已较为完善,但同时包含以上两种排水体的复合地基固结理论还鲜见报道。此外,以往固结模型中土体内孔隙水在径向仅发生单向渗流,这与实际情况存在差别。基于此,以碎石桩-排水板组合型复合地基为研究对象,考虑外荷载作用下土中孔隙水分别向碎石桩和排水板发生径向渗流,在复合地基顶面排水、底面不透水条件下建立碎石桩-排水板复合地基固结模型。利用桩体、桩周土体和竖井间流量相等的连续条件,获得该模型的解析解答。在一定条件下,该解答可退化为碎石桩复合地基固结解或竖井地基固结解,表明该解答的普适性及正确性。将该解答应用于工程实例及室内模型试验的沉降计算,计算结果与实测值吻合较好。开展大量计算研究此类复合地基不同工况下的固结性状,结果表明:在碎石桩桩间打设排水板,能有效提高复合地基固结速率,复合地基固结速率随排水板数量增加而加快;排水板渗透系数对复合地基固结速率及沉降速率的影响较大;与传统复合地基技术相比,碎石桩桩间增设排水板的工法具有更好的经济价值。