Singular Value Decomposition Method for Multi-species First-order Reactive Transport with Identical Decay Rates

A new decomposition method for solution of multi-component advection- dispersion-reaction equations coupled by first-order reactions is developed. This new method overcomes some of the limitations that were implicit in previously published algorithms. The approach is based on the Singular Value Decomposition (SVD) method. We derived analytical solution formulas, which can also be applied to systems with identical reaction rates. The proposed approach is flexible for solving one-, two- or three-dimensional advection-dispersion-reaction networks. The methodology is demonstrated on the reductive biodegradation of chlorinated solvents.     

An Analytical Solution to the One-Dimensional Solute Advection-Dispersion Equation in Multi-Layer Porous Media

An analytical solution to the one-dimensional solute advection-dispersion equation in multi-layer porous media is derived using a generalized integral transform method. The solution was derived under conditions of steady-state flow and arbitrary initial and inlet boundary conditions. The results obtained by this solution agree well with the results obtained by numerically inverting Laplace transform-generated solutions previously published in the literature. The analytical solution presented in this paper provides more flexibility with regard to the inlet conditions. The numerical evaluation of eigenvalues and matrix exponentials required in this solution technique can be accurately and efficiently computed using the sign-count method and eigenvalue evaluation methods commonly available. The illustrative calculations presented herein have shown how an analytical solution can provide insight into contaminant distribution and breakthrough in transport through well defined layered column systems. We also note that the method described here is readily adaptable to two and three-dimensional transport problems.     

Theoretical analysis,numerical calculation and experimental measurement of transient elastic waves in strips subjected to dynamic loadings

In this study, the transient response of an elastic strip subjected to dynamic in-plane loadings on the surface is investigated in detail. One of the objectives of this study is to develop an effective analytical method for determining transient solutions in a strip. By applying Laplace transform, the analytical solution in the transformed domain is derived and expressed in matrix form. The solution is then decomposed into infinite wave groups in which the multiple reflected waves with the same reflection are involved. Each multi-reflected wave can be identified by a coding method and be verified by the theory of generalized ray. The inverse transform is performed by using the well-known Cagniard method. The transient solutions in time domain for stresses and displacements are expressed in a closed form and are discussed in detail by an example. The experimental results show that the early time transient responses of displacements on the surface agree very well with the numerical calculations based on the theoretical solutions.     

Wave interaction with a new type perforated breakwater

In this study examined is the wave interaction with a new modified perforated breakwater, consisting of a perforated front wall, a solid back wall and a wave absorbing chamber between them with a two-layer rock-filled core. The fluid domain is divided into three sub-domains according to the components of the breakwater. Then by means of the matched eigenfunction expansion method, an analytical solution is obtained to assess the hydrodynamic performance of the new structure. An approach based on a step approach method is introduced to solve the complex dispersion equations for water wave motions within two-layer porous media. Numerical results of the present model are compared with previous limiting cases. The effects of rock fill on the reflection coefficient and the horizontal wave force are discussed. The project supported by the Program for Changjiang Scholars and Innovative Research Teams in Universities (IRT0420). The English text was polished by Keren Wang.     

Sound transmission from stiffened double laminated composite plates

An analytical model is developed to investigate the sound transmission loss from orthogonally rib-stiffened double laminated composite plates structure under a plane sound wave excitation, in which first order shear deformation theory is presented for laminated composite plates. By using the space harmonic approach and virtual work principle, the sound transmission loss is described analytically. The validity and feasibility of the model are verified by comparing the present theoretical predictions with numerical results published previously. The influences of structure geometrical parameters on sound transmission loss are subsequently presented. Through numerical results, it can be concluded that the proposed analytical model is accurate and simple in solving the vibroacoustic behavior of an orthogonally rib-stiffened double laminated composite plates.     

Time‐domain BEM solution of convection–diffusion‐type MHD equations

The two‐dimensional convection–diffusion‐type equations are solved by using the boundary element method (BEM) based on the time‐dependent fundamental solution. The emphasis is given on the solution of magnetohydrodynamic (MHD) duct flow problems with arbitrary wall conductivity. The boundary and time integrals in the BEM formulation are computed numerically assuming constant variations of the unknowns on both the boundary elements and the time intervals. Then, the solution is advanced to the steady‐state iteratively. Thus, it is possible to use quite large time increments and stability problems are not encountered. The time‐domain BEM solution procedure is tested on some convection–diffusion problems and the MHD duct flow problem with insulated walls to establish the validity of the approach. The numerical results for these sample problems compare very well to analytical results. Then, the BEM formulation of the MHD duct flow problem with arbitrary wall conductivity is obtained for the first time in such a way that the equations are solved together with the coupled boundary conditions. The use of time‐dependent fundamental solution enables us to obtain numerical solutions for this problem for the Hartmann number values up to 300 and for several values of conductivity parameter. Copyright © 2007 John Wiley & Sons, Ltd.     

A 2.5-D dynamic model for a saturated porous medium. Part II: Boundary element method

The 3-D boundary integral equation is derived in terms of the reciprocal work theorem and used along with the 2.5-D Green’s function developed in Part I [Lu, J.F., Jeng, D.S., Williams, S., submitted for publication. A 2.5-D dynamic model for a saturated porous medium: Part I. Green’s function. Int. J. Solids Struct.] to develop the 2.5-D boundary integral equation for a saturated porous medium. The 2.5-D boundary integral equations for the wave scattering problem and the moving load problem are established. The Cauchy type singularity of the 2.5-D boundary integral equation is eliminated through introduction of an auxiliary problem and the treatment of the weakly singular kernel is also addressed. Discretisation of the 2.5-D boundary integral equation is achieved using boundary iso-parametric elements. The discrete wavenumber domain solution is obtained via the 2.5-D boundary element method, and the space domain solution is recovered using the inverse Fourier transform. To validate the new methodology, numerical results of this paper are compared with those obtained using an analytical approach; also, some numerical results and corresponding analysis are presented.     

A σ‐coordinate three‐dimensional numerical model for surface wave propagation

A three‐dimensional numerical model based on the full Navier–Stokes equations (NSE) in σ‐coordinate is developed in this study. The σ‐coordinate transformation is first introduced to map the irregular physical domain with the wavy free surface and uneven bottom to the regular computational domain with the shape of a rectangular prism. Using the chain rule of partial differentiation, a new set of governing equations is derived in the σ‐coordinate from the original NSE defined in the Cartesian coordinate. The operator splitting method (Li and Yu, Int. J. Num. Meth. Fluids 1996; 23 : 485–501), which splits the solution procedure into the advection, diffusion, and propagation steps, is used to solve the modified NSE. The model is first tested for mass and energy conservation as well as mesh convergence by using an example of water sloshing in a confined tank. Excellent agreements between numerical results and analytical solutions are obtained. The model is then used to simulate two‐ and three‐dimensional solitary waves propagating in constant depth. Very good agreements between numerical results and analytical solutions are obtained for both free surface displacements and velocities. Finally, a more realistic case of periodic wave train passing through a submerged breakwater is simulated. Comparisons between numerical results and experimental data are promising. The model is proven to be an accurate tool for consequent studies of wave‐structure interaction. Copyright © 2002 John Wiley & Sons, Ltd.     

寒区非圆形隧道冻胀力的解析解

利用复变函数相应理论引入一种求解考虑衬砌结构的季节性冻土地区非圆形隧道冻胀力和冻胀变形的方法. 方法克服了隧道断面形状为非圆形状并且同时考虑非圆形隧道支护、冻胀圈、未冻围岩时, 经典的复变函数理论不能直接应用求解非圆形隧道应力和位移的问题. 方法通过将经典复变函数理论与连续性条件结合, 导出了非圆形隧道衬砌-冻胀圈-未冻围岩系统在正交曲线坐标系$\zeta $平面内的解析式, 然后通过保角变换求得直角坐标系$Z$平面上考虑衬砌支护的寒区非圆形隧道冻胀力和冻胀变形. 将推导的解析式应用于鹧鸪山隧道洞口段研究中, 得到鹧鸪山隧道洞口段冻胀应力和冻胀位移解析解, 并将解析解与数值解进行对比,验证解析解的正确性. 由结果可知: 围岩冻胀力对衬砌影响明显, 在拱顶、拱脚、拱底处因冻胀造成的环向附加应力显著增大; 在拱脚及两侧拱肩处会受到较大的法向附加应力; 由于衬砌几何形状的原因, 造成了衬砌冻胀变形的不均匀, 进而造成冻胀力分布不均.应用复变函数理论将冻胀圈考虑为马蹄形、直墙拱形等非圆形状较以往将冻胀圈考虑为单一圆环的冻胀力研究更符合实际, 较以往研究更能反映实际工况. 研究结果为季节性冻土地区非圆形隧道冻胀力的弹塑性分析奠定了基础.      

弹性介质中几何和运动非线性问题的类孤波解

将弹性介质 的几何和运动非线性方程简化成具有电磁场中的Ｂｏｒｎ－Ｉｎｆｅｌｄ方程的形式，并证明了该方程的类孤波解的存在．     

New numerical method for volterra integral equation of the second kind in piezoelastic dynamic problems

The elastodynamic problems of piezoelectric hollow cylinders and spheres under radial deformation can be transformed into a second kind Volterra integral equation about a function with respect to time, which greatly simplifies the solving procedure for such elastodynamic problems. Meanwhile, it becomes very important to find a way to solve the second kind Volterra integral equation effectively and quickly. By using an interpolation function to approximate the unknown function, two new recursive formulae were derived, based on which numerical solution can be obtained step by step. The present method can provide accurate numerical results efficiently. It is also very stable for long time calculating.     

Numerical simulation of ground water flow via a new approach to the local radial point interpolation meshless method

A novel approach to local radial point interpolation meshless (LRPIM) method is introduced to investigate the influence of leakage on tidal response in a coastal leaky confined aquifer system, based on a local weighted residual method with the Heaviside step function as the weighting function over a local sub-domain. The present approach is a truly meshless method based only on a number of randomly located nodes. In this approach, neither global background integration mesh nor domain integration is needed. Radial basis functions (RBFs) interpolation is employed in shape function and its derivatives construction for evaluating the local weak form integrals. Due to satisfaction of kronecker delta property in RBF interpolation, no special treatment is needed to impose the essential boundary conditions. In order to obtain the optimum parameters, shape parameters of multiquadrics (MQ)-RBF are tuned and studied. The leakage has a significant impact on the tidal behaviour of the confined aquifer. The numerical results of this research indicate that both tidal amplitude of groundwater head in the aquifer and the distance over which the aquifer can be disturbed by the tide are considerably reduced by leakage. The novelty of the approach is the use of a local Heaviside weight function in the LRPIM which does not need local domain integration and only integrations on the boundary of the local domains are needed. Therefore, in this research a new local Heaviside weight function has been proposed. Numerical results are presented and compared with the results of analytical solution. It is observed that the obtained results agreed very well with the results of analytical solution. The numerical results show that the use of a local Heaviside weight function in the LRPIM is highly accurate, fast and robust. It is also noticed that this novel meshless approach using MQ radial basis is very stable.     

Theoretical statistical solution and numerical simulation of heterogeneous brittle materials

The analytical stress-strain relation with heterogeneous parameters is derived for theheterogeneous brittle materials under a uniaxial extensional load, in which the distributions of theelastic modulus and the failure strength are assumed to be statistically independent. This theoreticalsolution gives an approximate estimate of the equivalent stress-strain relations for 3-D heterogeneousmaterials. In one-dimensional cases it may provide comparatively accurate results. The theoreticalsolution can help us to explain how the heterogeneity influences the mechanical behaviors, Further, anumerical approach is developed to model the non-linear behavior of three-dimensional heterogeneousbrittle materials. The lattice approach and statistical techniques are applied to simulate the initialheterogeneity of heterogeneous materials. The load increment in each loading stage is adaptivelydetermined so that the better approximation of the failure process can be realized. When the maximumtensile principal strain exceeds the failure strain, the elements are considered to be broken, which canbe carried out by replacing its Young‘s modulus with a very small value. A 3-D heterogeneous brittlematerial specimen is simulated during a full failure process. The numerical results are in good agreementwith the analytical solutions and experimental data.     

Random wave propagation in a viscoelastic layered half space

An extremely efficient and accurate solution method is presented for the propagation of stationary random waves in a viscoelastic, transversely isotropic and stratified half space. The efficiency and accuracy are obtained by using the pseudo excitation method (PEM) with the precise integration method (PIM). The solid is multi-layered and located above a semi-infinite space. The excitation sources form a random field which is stationary in the time domain. PEM is used to transform the random wave equation into deterministic equations. In the frequency-wavenumber domain, these equations are ordinary differential equations which can be solved precisely by using PIM. The power spectral densities (PSDs) and the variances of the ground responses can then be computed. The paper presents the full theory and gives results for instructive examples. The comparison between the analytical solutions and the numerical results confirms that the algorithm presented in this paper has exceptionally high precision. In addition, the numerical results presented show that: surface waves are very important for the wave propagation problem discussed; the ground displacement PSDs and variances are significant over bigger regions in the spatial domain when surface waves exist; and as the depth of the source increases the ground displacement PSDs decrease and the regions over which they have significant effect become progressively more restricted to low frequencies while becoming more widely distributed in the spatial domain.     

Non-linear closed-form computational model of cable trusses

In this paper the non-linear closed-form static computational model of the pre-stressed suspended biconvex and biconcave cable trusses with unmovable, movable, or elastic yielding supports subjected to vertical distributed load applied over the entire span and over a part (over the half) of the span is presented. The paper is an extension of the previously published work of authors [S. Kmet, Z. Kokorudova, Non-linear analytical solution for cable trusses, Journal of Engineering Mechanics ASCE 132 (1) (2006) 119-123]. Irvine's linearized forms of the deflection and the cable equations are modified because the effects of the non-linear truss behaviour needed to be incorporated in them. The concrete forms of the system of two non-linear cubic cable equations due to the load type are derived and presented. From a solution of a non-linear vertical equilibrium equation for a loaded cable truss, the additional vertical deflection is determined. The computational analytical model serves to determine the response, i.e. horizontal components of cable forces and deflection of the geometrically non-linear biconvex or biconcave cable truss to the applied loading, considering effects of elastic deformations, temperature changes and elastic supports. The application of the derived non-linear analytical model is illustrated by numerical examples. Resulting responses of the symmetric and asymmetric cable trusses with various geometries (shallow and deep profiles) obtained by the present non-linear closed-form solution are compared with those obtained by Irvine's linear solution and those by the non-linear finite element method. The conditions for the use of the linear and non-linear approach are briefly specified.     

An analytical solution and analysis of characters for viscoelastic fluid flow in annular pipe

I.Introducti0nLuiCiqunandHuangJunqiI'I(l989),ZhuWeihuiandLuiCiquri1'l(l992)sequentiallystudiedtheaxialflowofsecondorderandMaxwellfluidsandanalyzedtheflowcharactersofthesefluids.Thispaperwillstudyunsteadyrotat0ryflowofsecondorderandMaxwellfluidsinannularpi…     

VALIDATION OF NUMERICAL SHALLOW WATER MODELS FOR STRATIFIED SEICHES

A new analytical solution is presented for the case of a stratified seiche. This solution, especially its energetics, is useful for the validation of numerical shallow water models under stratified conditions. The utility of the analytical solution for validation is shown by using it to validate a simple finite difference numerical model. A comparison of the energetics of the numerical and analytical solutions reveals that the model results converge rapidly to the analytical solution with increasing resolution, such that a grid size of 30×30 would appear adequate for validation. In addition to properly resolving the spatial features, good temporal resolution is also necessary for validation, i.e use of a Courant number (Cr) less than one. For example, owing to the numerical dispersion of the present model, using Cr=5/4 rather than Cr =1/4 for the 50×50 grid resulted in 3·6 times larger RMS errors of model versus analytical barotropic available potential energy. This new analytical solution should be applied to a test suite of such validation tools before using such numerical models to simulate the more realistic geophysical flows encountered in lakes, bays, harbours and semi-enclosed seas under stratified conditions. © 1997 by John Wiley & Sons, Ltd.     

Infiltration of Salt Solute in Homogeneous and Saturated Porous Media – An Analytical Solution Evaluated by Numerical Simulations

In many cases various land disposal activities (e.g. infiltration, injection wells) constitute an important potential source of groundwater contamination. Using a 2D physical model, the behaviour of the infiltration of a salt solute, locally injected in a homogeneous and saturated porous medium, has been analysed. Under various experimental conditions (density effects, injection flow rate) the salt solute penetrates the porous media and leads to a steady-state regime inside the mixing zone. By using experimental observations, the basic equations describing the flow and transport phenomena can be simplified and an analytical solution obtained. Its validity is subject to numerical verification. The numerical model, based on the development of the mass balance equation expressed by its conservative form, uses a combination of the mixed hybrid finite element (MHFE) and discontinuous finite element (DFE) methods. The efficiency of this numerical model was previously verified on standard benchmarks, for example Elder's problem and Henry's problem. In the first step, the qualitative good agreement between the experimental and numerical results enabled us to use the numerical model in order to verify some hypotheses resulting from visual observations. Thus, the numerical results reveal the existence of a steady-state regime inside the mixing zones. Nevertheless, both its vertical and longitudinal extensions are less than those observed in the physical model. In the second step, the numerical results enable to establish the validity domain as well as the accuracy of the proposed analytical solution.