A lumped mass Chebyshev spectral element method and its application to structural dynamic problems

A diagonal or lumped mass matrix is of great value for time-domain analysis of structural dynamic and wave propagation problems, as the computational efforts can be greatly reduced in the process of mass matrix inversion. In this study, the nodal quadrature method is employed to construct a lumped mass matrix for the Chebyshev spectral element method (CSEM). A Gauss-Lobatto type quadrature, based on Gauss-Lobatto-Chebyshev points with a weighting function of unity, is thus derived. With the aid of this quadrature, the CSEM can take advantage of explicit time-marching schemes and provide an efficient new tool for solving structural dynamic problems. Several types of lumped mass Chebyshev spectral elements are designed, including rod, beam and plate elements. The performance of the developed method is examined via some numerical examples of natural vibration and elastic wave propagation, accompanied by their comparison to that of traditional consistent-mass CSEM or the classical finite element method (FEM). Numerical results indicate that the proposed method displays comparable accuracy as its consistent-mass counterpart, and is more accurate than classical FEM. For the simulation of elastic wave propagation in structures induced by high-frequency loading, this method achieves satisfactory performance in accuracy and efficiency.

     

Lumped hysteretic model for subsurface stormflow developed using downward approach

The terms ‘downward’ and ‘upward’ (synonymous with ‘top‐down’ and ‘bottom‐up’ respectively) are sometimes used when describing methods for developing hydrological models. A downward approach is used here to develop a lumped catchment‐scale model for subsurface stormflow at the 0·94 km² Slapton Wood catchment. During the development, as few assumptions as possible are made about the behaviour of subsurface stormflow at the catchment scale, and no assumptions are made about its behaviour at smaller scales. (In an upward approach, in contrast, the modelling would be based on assumptions about, and data for, the behaviour at smaller scales, such as the hillslope, plot, and point scales.) The model has a single store with a relatively simple relationship between discharge and storage, based on equations describing hysteretic patterns seen in a graph of discharge against storage. Double‐peaked hydrographs have been observed at the catchment outlet. Rainfall on the channel and infiltration‐excess and saturation‐excess runoff give a rapid response, and shallow subsurface stormflow gives a delayed response. Hydrographs are successfully simulated for the large delayed responses observed in 1971–1980 and 1989–1991, then a lumped model for the rapid response is coupled to the lumped hysteretic model and some double‐peaked hydrographs simulated. A physical interpretation is developed for the lumped hysteretic model, making use of information on patterns of perched saturation observed in 1982 on a hillslope at the Slapton Wood catchment. Downward and upward approaches are complementary, and the most robust way to develop and improve lumped catchment models is to iterate between downward and upward steps. Possible next steps are described. Copyright © 2006 John Wiley & Sons, Ltd.     

POTENTIAL FIELD OF A STATIONARY ELECTRIC CURRENT USING FREDHOLM'S INTEGRAL EQUATIONS OF THE SECOND KIND*

An integral equation method is described for solving the potential problem of a stationary electric current in a medium that is linear, isotropic and piecewise homogeneous in terms of electrical conductivity. The integral equations are Fredholm's equations of the ‘second kind’ developed for the potential of the electric field. In this method the discontinuity-surfaces of electrical conductivity are divided into ‘sub-areas’ that are so small that the value of their potential can be regarded as constant. The equations are applied to 3-D galvanic modeling. In the numerical examples the convergence is examined. The results are also compared with solutions derived with other integral equations. Examples are given of anomalies of apparent resistivity and mise-a-la-masse methods, assuming finite conductivity contrast. We show that the numerical solutions converge more rapidly than compared to solutions published earlier for the electric field. This results from the fact that the potential (as a function of the location coordinate) behaves more regularly than the electric field. The equations are applicable to all cases where conductivity contrast is finite.     

The finite element method: Is weighted volume integration essential?

In developing finite element equations for steady state and transient diffusion-type processes, weighted volume integration is generally assumed to be an intrinsic requirement. It is shown that such finite element equations can be developed directly and with ease on the basis of the elementary notion of a surface integral. Although weighted volume integration is mathematically correct, the algebraic equations stemming from it are no more informative than those derived directly on the basis of a surface integral. An interesting upshot is that the derivation based on surface integration does not require knowledge of a partial differential equation but yet is logically rigorous. It is commonly stated that weighted volume integration of the differential equation helps one carry out analyses of errors, convergence and existence, and therefore, weighted volume integration is preferable. It is suggested that because the direct derivation is logically consistent, numerical solutions emanating from it must be testable for accuracy and internal consistency in ways that the style of which may differ from the classical procedures of error- and convergence-analysis. In addition to simplifying the teaching of the finite element method, the thoughts presented in this paper may lead to establishing the finite element method independently in its own right, rather than it being a surrogate of the differential equation. The purpose of this paper is not to espouse any one particular way of formulating the finite element equations. Rather, it is one of introspection. The desire is to critically examine our traditional way of doing things and inquire whether alternate approaches may reveal to us new and interesting insights.     

Further perspectives on the evolution of bed material waves in alluvial rivers

In one‐dimensional mathematical models of ?uvial ?ow, sediment transport and morphological evolution, the governing equations based on mass and momentum conservation laws constitute a hyperbolic system. Succinctly, the hyperbolic nature excludes dispersion or diffusion operators, which is well known in the context of differential equations. There is no doubt that the so‐called ‘dispersion’ argument for bed material wave evolution is questionable, as we have explicitly asserted. Surprisingly, in a recent communication, the authors of the ‘dispersion’ argument suggest that dispersion is not precluded in hyperbolic systems. We provide herein further perspectives to help explain that the dispersion argument is neither appropriate nor necessary for interpreting bed material wave evolution. Also the continuity equations involved are addressed to prompt wider understanding of their signi?cance. In particular, the continuity equation of the water–sediment mixture proposed by the authors of the ‘dispersion’ argument is proved to be incorrect, and inevitably their reasoning based on it is problematic. Copyright © 2005 John Wiley & Sons, Ltd.     

Rockfall rebound: comparison of detailed field experiments and alternative modelling approaches

The accuracy of rockfall trajectory simulations mainly rests on the calculation of the rebound of fragments following their impact on the slope. This paper is dedicated to the comparative analysis of two rebound modelling approaches currently used in rockfall simulation using field experiments of single rebounds. The two approaches consist in either modelling the rock as a single material point (lumped mass approach) or in explicitly accounting for the fragment shape (rigid body approach). A lumped mass model accounting for the coupling between translational and rotational velocities and introducing a slope perturbation angle was used. A rigid body approach modelling the rocks as rigid locally deformable (in the vicinity of the contact surface) assemblies of spheres was chosen. The comparative analysis of the rebound models shows that both of them are efficient with only a few parameters. The main limitation of each approach are the calibration of the value of the slope perturbation (‘roughness’) angle, for the lumped mass approach, and the estimation of the rock length and height from field geological and historical analyses, for the rigid body approach. Finally, both rebound models require being improved in a pragmatic manner to better predict the rotational velocities distribution. Copyright © 2012 John Wiley & Sons, Ltd.     

基于全局弱式无单元法直流电阻率正演模拟

全局弱式无单元法是在有限单元法基础上发展起来的一种数值模拟方法,它采用局部支持域内的节点信息来构造形函数实现局部精确逼近,摆脱了单元,仅依赖于节点信息,具有预处理简单、模拟精度高、灵活性强的特点,适用于复杂地电条件下直流电阻率正演模拟.本文采用RPIM构造直流电阻率全局弱式无单元法形函数,利用RPIM形函数推导了直流电阻率全局弱式无单元法方程.然后,编制了直流电阻率全局弱式无单元法正演模拟Fortran程序,利用该程序对典型的地电模型进行了正演模拟,并将正演结果与基于线性插值的FEM正演结果及解析解进行对比,结果表明采用RPIM形函数的全局弱式无单元法用于直流电阻率正演模拟的正确性及有效性,且在同等条件下,全局弱式无单元法模拟精度高于矩形剖分的FEM,更有利于指导电法勘探的数据解译;利用该程序对复杂地电模型进行了正演模拟,结果表明全局弱式无单元法对复杂地电模型模拟效果良好,适应性强,灵活性高,可任意加密节点提高模拟精度.     

Dynamic transient behaviour of two- and three-dimensional structures including plasticity,large deformation effects and fluid interaction

The finite element method is employed in the prediction of the dynamic transient response of two- and three-dimensional solids exhibiting geometric (large deformations) and material (elasto-plastic) non-linearities. Explicit time marching schemes are adopted for integration of the dynamic equilibrium equation and a diagonal ‘lumped’ mass matrix is employed with a special procedure applicable to parabolic isoparametric elements. A variety of problems are presented including a solid/fluid interaction situation, and the method is shown to be able to solve economically many problems of dynamic or catastrophic nature which can occur in such structures as nuclear reactors, containment vessels, etc.     

Does Poisson’s downward continuation give physically meaningful results?

The downward continuation (DWC) of the gravity anomalies from the Earth’s surface to the geoid is still probably the most problematic step in the precise geoid determination. It is this step that motivates the quasi-geoid users to opt for Molodenskij’s rather than Stokes’s theory. The reason for this is that the DWC is perceived as suffering from two major flaws: first, a physically meaningful DWC technique requires the knowledge of the irregular topographical density; second, the Poisson DWC, which is the only physically meaningful technique we know, presents itself mathematically in the form of Fredholm integral equation of the 1st kind. As Fredholm integral equations are often numerically ill-conditioned, this makes some people believe that the DWC problem is physically ill-posed. According to a revered French mathematician Hadamard, the DWC problem is physically well-posed and as such gives always a finite and unique solution. The necessity of knowing the topographical density is, of course, a real problem but one that is being solved with an ever increasing accuracy; so sooner or later it will allow us to determine the geoid with the centimetre accuracy.     

A semi-analytical artificial boundary for time-dependent elastic wave propagation in two-dimensional homogeneous half space

This paper presents a time-dependent semi-analytical artificial boundary for numerically simulating elastic wave propagation problems in a two-dimensional homogeneous half space. A polygonal boundary is considered in the half space to truncate the semi-infinite domain, with an appropriate boundary condition imposed. Using the concept of the scaled boundary finite element method, the wave equation of the truncated semi-infinite domain is represented by the partial differential equation of non-constant coefficients. The resulting partial differential equation has only one spatial coordinate variable and time variable. Through introducing a few auxiliary functions at the truncated boundary, the resulting partial differential equations are further transformed into linear time-dependent equations. This allows an artificial boundary to be derived from the time-dependent equations. The proposed artificial boundary is local in time, global at the truncated boundary and semi-analytical in the finite element sense. Compared with the scaled boundary finite element method, the main advantage in using the proposed artificial boundary is that the requirement for solving a matrix form of Lyapunov equation to obtain the unit-impulse response matrix is avoided, so that computer efforts are significantly reduced. The related numerical results from some typical examples have demonstrated that the proposed artificial boundary is of high accuracy in dealing with time-dependent elastic wave propagation in two-dimensional homogeneous semi-infinite domains.     

Curved beam finite elements for coupled bending and torsional vibration

Curved beam finite elements are presented for out of plane coupled bending and torsional vibration. The element formulation is based upon the exact differential equations of an infinitesimal element in static equilibrium. The effects of shear deformation and rotary inertia are allowed for in the analysis. The element stiffness and mass matrices can be easily restricted to those of a ‘thin’ beam without the secondary effects. Frequencies obtained using either formulation are shown to converge onto exact values using ‘thick’ or ‘thin’ beam theories.     

A stationary principle for non conservative systems

A stationary principle is described to yield governing integral formulations for dissipative systems. Variation is applied on selective terms of energy or momentum functionals resulting with force or mass balance equations respectively. Applying the principle for a motion of a viscous fluid yields the Navier-Stokes equations as an approximation of the functional (i.e. equating to zero part of the integrand). When a Darcy's flow regime in a porous media is considered, implementing a space averaging method on the resultant integral derived by the principle, Forchheimer's law for energy accumulation and solute transport equation for momentum assembling are yielded in differential form approximation of a more extended functional formulation.     

Integral equations in MHD: theory and application

The induction equation of magnetohydrodynamics (MHD) is mathematically equivalent to a system of integral equations for the magnetic field in the bulk of the fluid and for the electric potential at its boundary. We summarize the recent developments concerning the numerical implementation of this scheme and its applications to various forward and inverse problems in dynamo theory and applied MHD.     

A general approach to advective–dispersive transport with multirate mass transfer

A numerical solution that is significantly more general than other semi-analytical solutions is presented for governing equations describing advective–dispersive transport with multirate mass transfer between mobile and immobile domains. The new solution approach is general in the sense that it does not impose any restrictive assumption on the spatial or temporal variability of advective and dispersive processes in the mobile domain. A single integro-differential equation (IDE) is developed for the concentration in the mobile domain by separating the concentration in the immobile domain from the set of two partial differential equations. The solution to the IDE requires the evaluation of a temporal integral of the concentration in the mobile domain, which is a function of the Laplace transform of the distribution of the mass transfer rate coefficient. The Laplace transform is not limited to flow fields with known constant velocities. The solutions for one- and two-dimensional examples obtained using the new approach agree with those obtained by existing semi-analytical and numerical approaches.     

Mechanistic interpretation of alpine glacierized environments: Part 1. Model formulation and related dynamical properties

Understanding the long-term seasonal dynamics of alpine glacierized basins is essential to evaluating their relation to climatic forcing. We focus on process knowledge by following a minimalist approach, and propose a spatially lumped nonlinear differential model (MIAGE) to describe the link between the volume V of water that is stored on the basin and the river runoff Q at the seasonal scale. We formulate the model structure by mathematically describing the link relating precipitation P, temperature T, river runoff Q and stored volumes V. Beside reproducing some typical features of the catchment hydrology of glacierized basins, MIAGE offers an explanation of their seasonal hydroclimatic behaviour and of the origin of their dissipative properties from a dynamical system perspective. By studying the model nonlinear properties, characteristics, and performances, we show that climatic change has both direct and feedback effects on such basins. Eventually a synchronization of the runoffs with either the precipitation trend or the temperature trend may occur depending on the storage conditions. This model is subsequently used in a companion paper in order to investigate the potential impact of climatic change scenarios on basins of the Italian and Swiss Alps [Mechanistic interpretation of alpine glacierized environments: Part 2. Hydrologic interpretation and model parameters identification on case study, this issue].     

大型储液罐摩擦摆基底隔震控制分析

针对弹性钢制圆柱储液罐,基于Haroun-Housner模型,将连续流体质量等效为3种集中质量,分别为:对流质量、脉冲质量和刚性质量,与这些集中质量连接的相应刚度取值依赖于储罐壁和流体质量.在水平地震激励下,在储罐底部加摩擦单摆支座,给出了简化的液体 - 储罐-隔震支座的力学分析模型,建立了摩擦摆支座基底隔震体系的振动控制方程,并利用Newmark逐步积分法对控制方程进行了数值求解,研究了摩擦摆支座基底隔震的储液罐地震反应,验证了FPB隔震的有效性.     

Non-Fickian mass transport in fractured porous media

The paper provides an introduction to fundamental concepts of mathematical modeling of mass transport in fractured porous heterogeneous rocks. Keeping aside many important factors that can affect mass transport in subsurface, our main concern is the multi-scale character of the rock formation, which is constituted by porous domains dissected by the network of fractures. Taking into account the well-documented fact that porous rocks can be considered as a fractal medium and assuming that sizes of pores vary significantly (i.e. have different characteristic scales), the fractional-order differential equations that model the anomalous diffusive mass transport in such type of domains are derived and justified analytically. Analytical solutions of some particular problems of anomalous diffusion in the fractal media of various geometries are obtained. Extending this approach to more complex situation when diffusion is accompanied by advection, solute transport in a fractured porous medium is modeled by the advection-dispersion equation with fractional time derivative. In the case of confined fractured porous aquifer, accounting for anomalous non-Fickian diffusion in the surrounding rock mass, the adopted approach leads to introduction of an additional fractional time derivative in the equation for solute transport. The closed-form solutions for concentrations in the aquifer and surrounding rocks are obtained for the arbitrary time-dependent source of contamination located in the inlet of the aquifer. Based on these solutions, different regimes of contamination of the aquifers with different physical properties can be readily modeled and analyzed.     

Boundary integral equation method for the diffraction of elastic waves using simplified green's functions

An alternate formulation of the ‘substructure deletion method’ suggested by Dasgupta in 1979¹ has been successfully implemented. The idea is to utilize simple Green's functions developed for a surface problem to replace the more complicated Green's functions required for embedded problems while still being able to generate an accurate solution. Since the exterior medium is usually represented by a continuum model, the interior medium in the present approach will also be represented by a continuum model rather than a finite element model as suggested originally, thereby eliminating the incompatibility between the solutions of the interior and exterior media. Detailed studies of the method's accuracy and limitations were performed using two-dimensional examples in wave scattering of canyons and alluvial valleys, problems which are more suitable for this method than the embedded foundation problem. The results obtained indicate that the alternate formulation gives accurate results only when the vertical dimension of the scattering object is not too large; if the aspect ratio (vertical over lateral) exceeds a certain limit, the results will not approach the known results given by boundary integral equation solutions or indirect boundary integral equations no matter what the refinement of the model may be. The greatest advantage of the present method is that the task of calculating Green's functions is reduced significantly; computational time using this new formulation is approximately five times less than for conventional boundary integral equation methods.     

Dependence of lumped mass transfer coefficient on scale and reactions kinetics for biologically enhanced NAPL dissolution

Residual dense non-aqueous liquids (NAPLs) in aquifers constitute a great challenge for groundwater cleanup. Active engineered treatment of regions that contain residual NAPLs is often required to shorten the long-term impact of NAPLs on groundwater quality. Enhanced residual NAPL cleanup can be achieved by promoting biodegradation of NAPL components in the aqueous phase, thereby increasing contaminant fluxes from the NAPL phase. Reaction-enhanced NAPL dissolution is often mathematically simulated under the assumption that lumped mass transfer coefficients, used to describe the dissolution behavior of the NAPL phase, are independent of the reactions. However, this assumption is not warranted because reactions occurring near the water–NAPL interface can reduce characteristic mass transfer lengths, which tend to enhance mass transfer over the no-reaction case.In this study, we mathematically investigated the connections between lumped mass transfer coefficients and reaction kinetics over an idealized residual NAPL domain. Since mass transfer is frequently a scale-dependent process, we also examined the influence of system extent on mass transfer coefficients. For our idealized domain with an assumed first-order decay reaction, the results show that lumped mass transfer coefficients depend on reaction kinetics and system scale. The mass transfer coefficient derived from the non-reactive case cannot properly represent the mass transfer process under the reactive conditions. When the advection time scale is long in comparison to the transverse dispersion time scale in the system, a fast reaction can increase significantly the lumped mass transfer coefficient. The mass transfer coefficient used for simulation was also found to be affected by the nature of the numerical scheme used.     

A parcel formulation of Hamiltonian layer models

Starting from the three-dimensional hydrostatic primitive equations, we derive Hamiltonian N-layer models with isentropic tropospheric and isentropic or isothermal stratospheric layers. Our construction employs a new parcel Hamiltonian formulation which describes the fluid as a continuum of Hamiltonian ordinary differential equations bound together by integral transport laws. In particular, we show that this parcel Hamiltonian structure is compatible with the stacking of layers under isentropic or isothermal constraints. The appeal of the parcel formulation is the simplification of various calculations, in particular the derivation of the continuum Poisson bracket and the proof of the Jacobi identity. A comparison and connection is made between the Hamiltonian dynamics of fluid parcels and the Hamiltonian system of partial differential equations. The parcel formulation can be seen as a precursor and tool for the study of Hamiltonian numerical schemes.