Shoreline change at an infinite jetty for wave time series

Future shoreline changes on a sandy beach with a structure such as a jetty or groin can be estimated when wave time series is known (i.e. sequence of wave height, period, and direction). This paper presents an extension of an existing solution (Pelnard-Considere, 1956) for the linearized partial differential equation for shoreline change at an infinite jetty where waves are time varying and when the angle of the shoreline is small with respect to the waves breaking at the shoreline. The novel solution provided in this paper allows the previous constant wave condition solution to be extended to the case where wave properties (i.e. wave direction, wave height, and wave period) are time varying. Example usage of the method presented shows that shorelines may be of different final plan form shape for time varying wave conditions even though the sediment transport along adjacent beaches is not spatially varying (i.e. spatially constant) from time step to time step. Although this difference in shape may have been known previously using numerical models, it could not be proved analytically. Reversals of wave height, period, and direction time series are shown to provide different final shoreline shapes even though the time series consists of the same waves although in different ordered time. The solution provided will allow one line numerical shoreline models to be tested using an analytic solution.     

Stability and accuracy analysis of outer loop dynamics in real‐time pseudodynamic testing of SDOF systems

Real‐time pseudodynamic (PSD) testing is an experimental technique for evaluating the dynamic behaviour of a complex structure. During the test, when the targeted command displacements are not achieved by the test structure, or a delay in the measured restoring forces from the test structure exists, the reliability of the testing method is impaired. The stability and accuracy of real‐time PSD testing in the presence of amplitude error and a time delay in the restoring force is presented. Systems consisting of an elastic single degree of freedom (SDOF) structure with load‐rate independent and dependent restoring forces are considered. Bode plots are used to assess the effects of amplitude error and a time delay on the steady‐state accuracy of the system. A method called the pseudodelay technique is used to derive the exact solution to the delay differential equation for the critical time delay that causes instability of the system. The solution is expressed in terms of the test structure parameters (mass, damping, stiffness). An error in the restoring force amplitude is shown to degrade the accuracy of a real‐time PSD test but not destabilize the system, while a time delay can lead to instability. Example calculations are performed for determining the critical time delay, and numerical simulations with both a constant delay and variable delay in the restoring force are shown to agree well with the stability limit for the system based on the critical time delay solution. The simulation models are also used to investigate the effects of a time delay in the PSD test of an inelastic SDOF system. The effect of energy dissipation in an inelastic structure increases the limit for the critical time delay, due to the energy removed from the system by the energy dissipation. Copyright © 2007 John Wiley & Sons, Ltd.     

Scale invariance and self‐similarity in kinematic wave overland flow in space and time

Fractals are famous for their self‐similar nature at different spatial scales. Similar to fractals, solutions of scale invariant processes are self‐similar at different space–time scales. This unique property of scale‐invariant processes can be utilized to translate the solution of the processes at a much larger or smaller space–time scale (domain) based on the solution calculated on the original space–time scale. This study investigates scale invariance conditions of kinematic wave overland flow process in one‐parameter Lie group of point transformations framework. Scaling (stretching) transformation is one of the one‐parameter Lie group of point transformations and it has a unique importance among the other transformations, as it leads to the scale invariance or scale dependence of a process. Scale invariance of a process yields a self‐similar solution at different space–time scales. However, the conditions for the process to be scale invariant usually dictate various relationships between the scaling coefficients of the dependent and independent variables of the process. Therefore, the scale invariance of a process does not assure a self‐similar solution at any arbitrary space and time scale. The kinematic wave overland flow process is modelled mathematically as initial‐boundary value problem. The conditions to be satisfied by the system of governing equations as well as the initial and boundary conditions of the kinematic wave overland flow process are established in order for the process to be scale invariant. Also, self‐similarity of the solution of the kinematic wave overland flow under the established invariance conditions is demonstrated by various numerical example problems. Copyright © 2011 John Wiley & Sons, Ltd.     

A marching in space and time (MAST) solver of the shallow water equations. Part I: The 1D model

A new approach is presented for the numerical solution of the complete 1D Saint-Venant equations. At each time step, the governing system of partial differential equations (PDEs) is split, using a fractional time step methodology, into a convective prediction system and a diffusive correction system. Convective prediction system is further split into a convective prediction and a convective correction system, according to a specified approximated potential. If a scalar exact potential of the flow field exists, correction vanishes and the solution of the convective correction system is the same solution of the prediction system. Both convective prediction and correction systems are shown to have at each x − t point a single characteristic line, and a corresponding eigenvalue equal to the local velocity. A marching in space and time (MAST) technique is used for the solution of the two systems. MAST solves a system of two ordinary differential equations (ODEs) in each computational cell, using for the time discretization a self-adjusting fraction of the original time step. The computational cells are ordered and solved according to the decreasing value of the potential in the convective prediction step and to the increasing value of the same potential in the convective correction step. The diffusive correction system is solved using an implicit scheme, that leads to the solution of a large linear system, with the same order of the cell number, but sparse, symmetric and well conditioned. The numerical model shows unconditional stability with regard of the Courant–Friedrichs–Levi (CFL) number, requires no special treatment of the source terms and a computational effort almost proportional to the cell number. Several tests have been carried out and results of the proposed scheme are in good agreement with analytical solutions, as well as with experimental data.     

Wellbore flow‐rate solution for a constant‐head test in two‐zone finite confined aquifers

The solution describing the wellbore flow rate in a constant‐head test integrated with an optimization approach is commonly used to analyze observed wellbore flow‐rate data for estimating the hydrogeological parameters of low‐permeability aquifers. To our knowledge, the wellbore flow‐rate solution for the constant‐head test in a two‐zone finite‐extent confined aquifer has never been reported so far in the literature. This article is first to develop a mathematical model for describing the head distribution in the two‐zone aquifer. The Laplace domain solutions for the head distributions and wellbore flow rate in a two‐zone finite confined aquifer are derived using the Laplace transform, and their corresponding time domain solutions are then obtained using the Bromwich integral method and residue theorem. These new solutions are expressed in terms of an infinite series with Bessel functions and not straightforward to calculate numerically. A large‐time solution for the wellbore flow rate is therefore developed by employing the relationship of small Laplace variable versus large time variable and L'Hospital's rule. The result shows that the large‐time solution is identical to the steady‐state solution obtained after applying the Tauberian theorem into the Laplace domain solution. This large‐time solution can reduce to the Thiem equation in the case of no skin. Finally, the newly developed solution is used to investigate the effects of outer boundary distance and conductivity ratio on the wellbore flow rate. Copyright © 2011 John Wiley & Sons, Ltd.     

Calculating Groundwater Response Times for Flow in Heterogeneous Porous Media

Predicting the amount of time required for a transient groundwater response to take place is a practical question that is of interest in many situations. This time scale is often called the response time. In the groundwater hydrology literature, there are two main methods used to calculate the response time: (1) both the transient and steady‐state groundwater flow equations are solved, and the response time is taken to be amount of time required for the transient solution to approach the steady solution within some tolerance; and (2) simple scaling arguments are adopted. Certain limitations restrict both of these approaches. In this study, we outline a third method, based on the theory of mean action time. We derive the governing boundary value problem for both the mean and variance of action time for confined flow in two‐dimensional heterogeneous porous media. Importantly, we show that these boundary value problems can be solved using widely available software. Applying these methods to a test case reveals the advantages of the theory of mean action time relative to standard methods.     

Dynamic interaction numerical models in the time domain based on the high performance scaled boundary finite element method

Consideration of structure-foundation-soil dynamic interaction is a basic requirement in the evaluation of the seismic safety of nuclear power facilities. An efficient and accurate dynamic interaction numerical model in the time domain has become an important topic of current research. In this study, the scaled boundary finite element method （SBFEM） is improved for use as an effective numerical approach with good application prospects. This method has several advantages, including dimensionality reduction, accuracy of the radial analytical solution, and unlike other boundary element methods, it does not require a fundamental solution. This study focuses on establishing a high performance scaled boundary finite element interaction analysis model in the time domain based on the acceleration unit-impulse response matrix, in which several new solution techniques, such as a dimensionless method to solve the interaction force, are applied to improve the numerical stability of the actual soil parameters and reduce the amount of calculation. Finally, the feasibility of the time domain methods are illustrated by the response of the nuclear power structure and the accuracy of the algorithms are dynamically verified by comparison with the refinement of a large-scale viscoelastic soil model.     

Stability and accuracy of two-dimensional kinematic wave overland flow modeling

A two-dimensional finite element based overland flow model was developed and used to study the accuracy and stability of three numerical schemes and watershed parameter aggregation error. The conventional consistent finite element scheme results in oscillations for certain time step ranges. The lumped and the upwind finite element schemes are tested as alternatives to the consistent scheme. The upwind scheme did not improve on the stability or the accuracy of the solution, while the lumped scheme provided stable and accurate solutions for time steps twice the size of time steps needed for the consistent scheme. A new accuracy based dynamic time step estimate for the two-dimensional overland flow kinematic wave solution is developed for the lumped scheme. The newly developed dynamic time step estimates are functions of the mesh size, and time of concentration of the watershed hydrograph. Due to lack of analytical solutions, the time step was developed by comparing numerical solutions of various levels of discretization to a reference solution using a very fine mesh and a very small time step. The time step criteria were tested on a different set of problems and proved to be adequate for accurate and stable solutions. A sensitivity analysis for the watershed slope, Manning’s roughness coefficient and excess rainfall rate was conducted in order to test the effect of parameter aggregation on the stability and accuracy of the solution. The results of this analysis show that aggregation of the slope data resulted in the highest error. The roughness coefficient had a smaller effect on the solution while the rainfall intensity did not show any significant effect on the flow rate solution for the range of rainfall intensity used. This work pioneers the challenge of providing guidelines for accurate and stable numerical solutions of the two-dimensional kinematic wave equations for overland flow.     

Estimating Groundwater Mounding in Sloping Aquifers for Managed Aquifer Recharge

Design of managed aquifer recharge (MAR) for augmentation of groundwater resources often lacks detailed data, and simple diagnostic tools for evaluation of the water table in a broad range of parameters are needed. In many large‐scale MAR projects, the effect of a regional aquifer base dip cannot be ignored due to the scale of recharge sources (e.g., wadis, streams, reservoirs). However, Hantush's (1967) solution for a horizontal aquifer base is commonly used. To address sloping aquifers, a new closed‐form analytical solution for water table mound accounts for the geometry and orientation of recharge sources at the land surface with respect to the aquifer base dip. The solution, based on the Dupiuit‐Forchheimer approximation, Green's function method, and coordinate transformations is convenient for computing. This solution reveals important MAR traits in variance with Hantush's solution: mounding is limited in time and space; elevation of the mound is strongly affected by the dip angle; and the peak of the mound moves over time. These findings have important practical implications for assessment of various MAR scenarios, including waterlogging potential and determining proper rates of recharge. Computations are illustrated for several characteristic MAR settings.     

Time evolution of the first‐order linear acoustic/elastic wave equation using Lie product formula and Taylor expansion

We propose a new numerical solution to the first‐order linear acoustic/elastic wave equation. This numerical solution is based on the analytic solution of the linear acoustic/elastic wave equation and uses the Lie product formula, where the time evolution operator of the analytic solution is written as a product of exponential matrices where each exponential matrix term is then approximated by Taylor series expansion. Initially, we check the proposed approach numerically and then demonstrate that it is more accurate to apply a Taylor expansion for the exponential function identity rather than the exponential function itself. The numerical solution formulated employs a recursive procedure and also incorporates the split perfectly matched layer boundary condition. Thus, our scheme can be used to extrapolate wavefields in a stable manner with even larger time‐steps than traditional finite‐difference schemes. This new numerical solution is examined through the comparison of the solution of full acoustic wave equation using the Chebyshev expansion approach for the matrix exponential term. Moreover, to demonstrate the efficiency and applicability of our proposed solution, seismic modelling results of three geological models are presented and the processing time for each model is compared with the computing time taking by the Chebyshev expansion method. We also present the result of seismic modelling using the scheme based in Lie product formula and Taylor series expansion for the first‐order linear elastic wave equation in vertical transversely isotropic and tilted transversely isotropic media as well. Finally, a post‐stack migration results are also shown using the proposed method.     

On the use and error of approximation in the Domenico (1987) solution

A mathematical solution for solute transport in a three-dimensional porous medium with a patch source under steady-state, uniform ground water flow conditions was developed by Domenico (1987). The solution derivation strategy used an approximate approach to solve the boundary value problem, resulting in a nonexact solution. Variations of the Domenico (1987) solution are incorporated into the software programs BIOSCREEN and BIOCHLOR, which are frequently used to evaluate subsurface contaminant transport problems. This article mathematically elucidates the error in the approximation and presents simulations that compare different versions of the Domenico (1987) solution to an exact analytical solution to demonstrate the potential error inherent in the approximate expressions. Results suggest that the accuracy of the approximate solutions is highly variable and dependent on the selection of input parameters. For solute transport in a medium-grained sand aquifer, the Domenico (1987) solution underpredicts solute concentrations along the centerline of the plume by as much as 80% depending on the case of interest. Increasing the dispersivity, time, or dimensionality of the system leads to increased error. Because more accurate exact analytical solutions exist, we suggest that the Domenico (1987) solution, and its predecessor and successor approximate solutions, need not be employed as the basis for screening tools at contaminated sites.     

Time-domain solution for transient dynamic response of a large-diameter thin-walled pipe pile

The propagation of stress waves in a large-diameter pipe pile for low strain dynamic testing cannot be explained properly by traditional 1D wave theories. A new computational model is established to obtain a wave equation that can describe the dynamic response of a large-diameter thin-walled pipe pile to a transient point load during a low strain integrity test. An analytical solution in the time domain is deduced using the separation of variables and variation of constant methods. The validity of this new solution is verifi ed by an existing analytical solution under free boundary conditions. The results of this time domain solution are also compared with the results of a frequency domain solution and fi eld test data. The comparisons indicate that the new solution agrees well with the results of previous solutions. Parametric studies using the new solution with reference to a case study are also carried out. The results show that the mode number affects the accuracy of the dynamic response. A mode number greater than 10 is required to enable the calculated dynamic responses to be independent of the mode number. The dynamic response is also greatly affected by soil properties. The larger the side resistance, the smaller the displacement response and the smaller the refl ected velocity wave crest. The displacement increases as the stress waves propagate along the pile when the pile shaft is free. The incident waves of displacement and velocity responses of the pile are not the same among different points in the circumferential direction on the pile top. However, the arrival time and peak value of the pile tip refl ected waves are almost the same among different points on the pile top.     

A marching in space and time (MAST) solver of the shallow water equations. Part II: The 2D model

A novel methodology for the solution of the 2D shallow water equations is proposed. The algorithm is based on a fractional step decomposition of the original system in (1) a convective prediction, (2) a convective correction, and (3) a diffusive correction step. The convective components are solved using a Marching in Space and Time (MAST) procedure, that solves a sequence of small ODEs systems, one for each computational cell, ordered according to the cell value of a scalar approximated potential. The scalar potential is sought after computing first the minimum of a functional via the solution of a large linear system and then refining locally the optimum search. Model results are compared with the experimental data of two laboratory tests and with the results of other simulations carried out for the same tests by different authors. A comparison with the analytical solution of the oblique jump test has been also considered. Numerical results of the proposed scheme are in good agreement with measured data, as well as with analytical and higher order approximation methods results. The growth of the CPU time versus the cell number is investigated successively refining the elements of an initially coarse mesh. The CPU specific time, per element and per time step, is found out to be almost constant and no evidence of Courant–Friedrichs–Levi (CFL) number limitation has been detected in all the numerical experiments.     

Response of a non-linear system with restoring forces governed by fractional derivatives—Time domain simulation and statistical linearization solution

In this paper, the random response of a non-linear system comprising frequency dependent restoring force terms is examined. These terms are accurately modeled in seismic isolation and in many other applications using fractional derivatives. In this context, an efficient numerical approach for determining the time domain response of the system to an arbitrary excitation is first proposed. This approach is based on the Grunwald–Letnikov representation of a fractional derivative and on the well-known Newmark numerical integration scheme for structural dynamic problems. Next, it is shown that for the case of a stochastic excitation, in addition to the time domain solutions, a frequency domain solution can be readily determined by the method of statistical linearization. The reliability of this solution is established in a Monte Carlo simulation context using the herein adopted time domain solution scheme. Furthermore, several related parameter studies are reported.     

一种确定震源中心的方法:逆时成像技术(一)——原理与数值实验

研究地震断层的精细结构需要对地震活动精确定位,然而,盖戈类标准定位方法已经不能胜任.现今计算机技术的发展使我们能够直接面对地震定位这个非线性问题,所以,我们提出一种称为逆时成像技术的确定地震震源中心的非线性方法.首先,从位移表示定理出发,阐述了逆时成像技术的原理;然后,通过多组数值实验,论证了这种技术的可行性.由于直接采用直达波信号构建包络信号,进而采用互相关技术测量观测到时,因此,观测到时的准确性和客观性得到了提升;由于采用波形聚束方法直接建立观测到时和震源位置的非线性关系,因此,绕开了盖戈类方法的线性化过程,从而杜绝了非线性问题线性化过程造成的误差;由于采用波形聚束方法而不是经典的最小二乘法求解,所以,克服了最小二乘解对于少数或者个别"出格数据(outlier)"敏感的缺点;由于采用非均匀网格搜索的方法确定非线性系统的解,所以,可以利用解集的特征半径描述解的分辨率,进而利用观测到时的标准差和分辨率来描述解的不确定性,避免了以观测误差为正态分布的假设为前提的统计方法,克服了这类方法常常给出脱离实际意义的结果的不足.然而,由于采用网格搜索方法求解非线性方程,所以,与盖戈类方法相比,计算效率相对较低.例如:这里的每次定位过程在普通的个人计算机上需要大约30 s.不过,用时间换取精度也是惯常的选择.     

Dynamic initialization for barotropic flow crossing a barrier

Summary A method to obtain balanced initial state in the presence of mountains is presented. Using a prescribed initial state the primitive equations are iterated forward and backward about the initial time, using an Euler-backward time integration scheme. During this iteration the mountains are built-up steadily.The method is tested against a steady-state analytical solution of the non-linear equations of the homogeneous atmosphere. The results of this test demonstrate that the method is capable of generating the theoretical steady-state numerical solution with a good degree of accuracy.     

Travel time to a well pumping an unconfined aquifer without recharge

A solution is given for the travel time to a well pumping an ideal, horizontal unconfined aquifer, under steady-state conditions, when recharge from infiltration is negligible. Three forms of the solution are provided: a closed-form solution, an integral to be calculated in a worksheet, and a simple equation. The three forms of the solution give travel times nearly identical to those obtained using a finite-element code for saturated and unsaturated flow and particle tracking.     

注聚合物井井下温度分布数值模拟研究

本文提出了一种计算注聚合物井井下温度分布的方法,该方法视井筒内的聚合物溶液为非牛顿幂律流体,考虑注入流体在井筒中同一截面上的速度变化,根据聚合物溶液在多孔介质中的流变性,依据广义达西定律求取渗流速度,基于能量守恒方程建立柱坐标系下注聚合物井井筒内流体、注入层及围岩的二维温度场模型.通过合理的边界条件,将三部分模型耦合起来,采用交替方向半隐式有限差分法求解建立的井下温度场模型.考查了注入量、注入时间、幂律指数和稠度系数、注入液温度等因素对井下温度场分布的影响,结果表明,当注入的聚合物溶液温度低于注入层的原始温度时,随注入量和注入时间的增大以及粘度的减小,注入层的温度降低;注入聚合物溶液温度与注入层原始温度差越大,注入层处的温度剖面异常越明显.本文数值计算结果可用于指导注聚合物井的井温测井应用.     

考虑关断时间的回线源激发TEM三维时域有限差分正演

从麦克斯韦旋度方程出发可以直接导出瞬变电磁场扩散方程,然而扩散方程不含电场对时间的一阶导数,不能构成显式的时域有限差分方程,借鉴du Fort-Frankel有限差分离散方法引入虚拟位移电流项构建显式时域有限差分方程.对Wang和Hohmann的经典时域算法进行了两点改进:第一,通过将矩形回线源电流密度加入麦克斯韦方程组的安培环路定理方程,实现回线源瞬变电磁激发源加入;第二,在计算中考虑关断时间.第一点改进使时域有限差分方程考虑了一次场的计算,并且源的计算不再依赖均匀半空间模型响应作为初始条件,使算法能够适应表层电阻率不均匀时的三维复杂模型.由于实际观测中不可能出现阶跃电流的关断形式,第二点改进可以方便设置发射电流下降沿.采用改进的三维时域有限差分正演算法对均匀半空间模型、四类三层模型、均匀半空间中含有低阻块体模型进行了计算并分别与解析解、线性数字滤波解、积分方程解和Wang的三维时域有限差分解进行了对比验证.以H模型为例,采用建立的三维时域有限差分正演算法计算了不同关断时间的斜阶跃脉冲回线源瞬变电磁中心点感应电动势衰减曲线.以实际地质资料为基础,构建包含两层采空区的三维复杂模型,以1 μs的极短关断时间进行了复杂模型定回线源瞬变电磁响应计算,并计算了该复杂模型的视电阻率曲线.