Coupling of boundary and finite elements for soil-structure interaction problems

The investigation of complex soil-structure interaction problems is usually carried out with numerical solution procedures such as the finite element or the boundary element method. It must be noted, however, that the choice of one or the other of these approaches is not just a matter of preferences; depending on the type of the problem under consideration, either boundary or finite elements may be more advantageous. A considerable expansion in the computational power can be obtained, on the other hand, if one resorts to hybrid schemes which retain the main advantages of the two methods and eliminate their respective disadvantages. This paper presents results obtained with a boundary element-finite element coupling procedure, and discusses its applicability to some representative soil-structure interaction problems. The structures considered are elastic systems, such as foundations, tunnels and filled trenches (modelled by finite elements), which are coupled with homogeneous elastic halfspaces (modelled by boundary elements). The examples demonstrate the importance of using a model that includes wave radiation effects. The coupling approach is formulated entirely in the time domain so that an extension of the algorithm to non-linear analyses seems to present no further difficulties.     

Stabilized finite elements for harmonic soil dynamics problems near the undrained-incompressible limit

Simple Finite Element models for soil dynamics and earthquake engineering problems in the frequency domain are a fast and valuable tool providing a first approximation before a full non-linear analysis in the time domain is performed.Quite often the problem concerns saturated soils with very small permeability and pore fluid of neglectable compressibility. In the limit, the permeability is assumed to be zero and the pore fluid incompressible. Here, engineers use standard finite element codes formulated in terms of displacements but incompressibility may result in volumetric locking of the mesh with a severe loss of accuracy.The purpose of this paper is to present a simple mixed finite element formulation in the frequency domain based on displacements and pore pressures as main variables. A suitable stabilization technique allowing for equal order interpolation of displacements and pressures has been introduced for incompressible and zero permeability limits.Of course, the range of application is limited to those problems in which the behaviour of the material can be approximated by linear models, and therefore modelling of phenomena such as liquefaction, cyclic mobility or cavitation occur is excluded.The paper shows as well an extremely simple way of coupling solid and water domains as it occurs for instance in quay walls under dynamic loading.     

Dynamic response of flexible strip-foundations by boundary and finite elements

A time domain Boundary Element-Finite method is employed to determine the dynamic response of flexible surface two-dimensional foundations under conditions of plane strain placed on an elastic soil medium and subjected either to transient external forces or to obliquely incident seismic waves. The elastic, isotropic, and homogeneous soil medium is treated by the time domain Direct Boundary Element Method, while the flexible foundation is treated by the Finite Element Method. The two methods are appropriately combined through equilibrium and compatibility considerations at the soil-foundation interface. Parametric studies examining the effect of the relative stiffness between the foundation and the soil and the spatial distribution of the dynamic disturbances on the foundation response are presented.     

Harbour resonance problems using finite elements

The problem of harbour resonance is discussed, and the probabilistic analysis of a harbour's response is carried out using a finite element technique. The results presented in this paper show the importance of considering two types of damping (radiation and friction) in the study of harbour resonance phenomena. The radiation, or energy transfer between the harbour and the open sea, is introduced using an expression similar to Sommerfeld's radiation condition, and appears as a boundary only damping term.The three analyses with or without damping and with or without radiation are compared, and it is shown that it is important to include both effects to avoid distortion of the response spectrum. The probability of exceedance for elevations is calculated for the Duncan Basin, and a way of calculating the horizontal displacements is indicated.     

Trace elements in ocean ridge basalts

A correlary of sea floor spreading is that the production rate of ocean ridge basalts exceeds that of all other volcanic rocks on the earth combined. Basalts of the ocean ridges bring with them a continuous record in space and time of the chemical characteristics of the underlying mantle. The chemical record is once removed, due to chemical fractionation during partial melting. Chemical fractionations can be evaluated by assuming that peridotite melting has proceeded to an olivine-orthopyroxene stage, in which case the ratios of a number of magmaphile elements in the extracted melt closely match the ratios in the mantle. Comparison of ocean ridge basalts and chondritic meteorites reveals systematic patterns of element fractionation, and what is probably a double depletion in some elements. The first depletion is in volatile elements and is due to high accretion temperatures of a large percentage of the earth from the solar nebula. The second depletion is in the largest, most highly charged lithophile elements (“incompatible elements”), probably because the mantle source of the basalts was melted previously, and the melt, enriched in these elements, was removed. Migration of melt relative to solid under ocean ridges and oceanic plates, element fractionation at subduction zones, and fractional melting of amphibolite in the Precambrian are possible mechanisms for depleting the mantle in incompatible elements. Ratios of transition metals in the mantle source of ocean ridge basalts are close to chondritic, and contrast to the extreme depletion of refractory siderophile elements, the reason for which remains uncertain. Variation of ocean ridge basalt chemistry along the length of the ridge has been correlated with ridge elevation. Thus chemically anomalous ridge segments up to 1000 km long appear to broadly coincide with regions of high magma production (plumes, hot spots). Basalt heterogeneity at a single location indicates mantle heterogeneity on a smaller scale. Variation of ocean ridge basalt chemistry with time has not been established, in fact, criteria for recognizing old oceanic crust in ophiolite terrains are currently under debate. The similarity of rare earth element patterns in basalt from ocean ridges, back-arc basins, some young island arcs, and some continental flood basalts illustrates the dangers of tectonic labeling by rare earth element pattern.     

A method for constructing high-order differential boundary conditions for solving external boundary value problems in geoelectromagnetism

A general approach to the construction of differential boundary conditions for vector fields satisfying the Helmholtz equation is proposed on the basis of the field expansion in multipole series and the application of annihilating operators to them. The resulting differential constraints can be used as boundary conditions in solving external boundary value problems. Examples of their application to the solution of forward geoelectric problems in three-dimensionally inhomogeneous media are examined. Their use at a finite distance from the source of an anomaly is shown to yield more accurate results than those obtained under the assumption that the anomalous field at this distance vanishes. Another effect of their application is a substantial decrease in the dimensions of the modeling domain and therefore in the time required to solve the forward problem. The “safe” distance for using the Dirichlet-type boundary conditions is estimated.     

Coupling method of finite and infinite elements for strip foundation wave problems

Starting from two-dimensional wave equations and making use of Galerkin weighted residual approximations, discretized formulations for wave problems of a visco-elastic foundation have been derived. With considerations of geometrical and mechanical characteristics of a semi-infinite domain, a frequency-dependent compatible infinite element is also presented. Finally, by coupling the infinite elements with ordinary finite elements the system is used for simulation of propagating waves in a semi-infinite foundation. This model is not only suitable for simulations of complicated variations of geometrical conditions, but also for describing the unbounded behaviour of arbitrary multiple layers. Examples given indicate the model has excellent computational accuracy and feasibility for analysing the effects on foundation response due to the existence of faults or any other soft layers.     

Surface semi-geostrophic dynamics in the ocean

The surface quasi-geostrophic approximation is re-written in an oceanic context using the two-dimensional semi-geostrophic approximation. The new formulation allows to take into account the presence of out-of-balance flow features at scales comparable to or smaller than the Rossby radius of deformation and for small bulk Richardson numbers. Analytical solutions show that, while the surface quasi-geostrophic approximation tends to underestimate the buoyancy anomaly, the inclusion of finite Rossby number allows for larger values of the buoyancy anomaly at depth. The projection of the surface semi-geostrophic solution on the first baroclinic modes is calculated. The result of the projection is a functional form that decreases with the values of the Rossby number and toward smaller scales. Solutions for constant and exponential profile for the background potential vorticity are compared. Results of the comparison show that, in agreement with the results found for balanced flows, even for large Rossby number the exponential profile for the background potential vorticity retains smaller values for the buoyancy anomaly at depth than the solution found using a constant potential vorticity profile.     

The surface boundary condition in nonhydrostatic ocean models

With increasing resolution in numerical ocean models, nonhydrostatic pressure effects have to be accounted for. In sigma-coordinate mode split ocean models, this pressure may be regarded as a pressure correction. An elliptic equation must be solved for the nonhydrostatic pressure, and the gradients are used to correct the provisional hydrostatic velocity components in each time step. The focus in the present work is on the surface boundary condition for the elliptic equation. In the literature, both Dirichlet and Neumann boundary conditions are suggested and applied. To investigate the sensitivity of the numerical results to the choice of boundary condition, three numerical experiments are performed. The first and second experiments are studies of the propagation and steepening of nonlinear internal waves. The first study is on tank scale and the second experiment is on ocean scale. In the tank-scale experiment, the density and the flow fields are very robust to the choice of boundary condition. In the ocean-scale experiment, the waves produced with a Dirichlet boundary condition become more damped than the waves produced with a Neumann boundary condition. The third study involves a surface buoyant jet. It is shown that well-known characteristics of the plume front are reproduced with a Neumann boundary condition, but the rotating turbulent core of this front is lost with a Dirichlet condition. It is accordingly argued that the appropriate surface boundary condition in mode split nonhydrostatic ocean models is the Neumann condition.     

基于有限元方法的陆海大地水准面衔接

大陆上用重力数据和GPS水准数据确定(似)大地水准面,海洋上用卫星测高数据确定(似)大地水准面.由于沿海地区和近岸海域往往缺少完好的重力数据,近岸海域卫星测高数据质量相对较差,两类大地水准面在陆海相接区域精度偏低且存在拼合差.纯几何方法拟合陆海局部区域大地水准面,不能顾及大地水准面的物理特性,拟合结果不稳定.顾及到大地水准面的物理特性,依据其在局部所应满足的数学物理方程,拟合陆海局部区域大地水准面问题,转化为Laplace第一边值问题.讨论了有限元法衔接陆海局部区域大地水准面的数学思想,给出了相应的数学模型.     

ABAQUS中动力问题边界条件的选取

数值模拟是研究半无限域中结构地震反应的重要手段,而边界条件的选取则直接影响到数值模拟结果的可靠性.本文应用有限元软件ABAQUS,分别采用固定边界条件、黏性边界条件和远置边界条件对简单弹性半无限介质的波源问题和散射问题进行了数值模拟分析,结果显示:使用固定边界时,波动不能穿过边界面进入半无限介质,这与波动在半无限介质中...     

Transient elastodynamic analysis of three-dimensional problems by boundary element method

An advanced implementation of the direct boundary element method applicable to transient problems involving three-dimensional solids of arbitrary shape and connectivity is presented. The work first focuses on the formulation of the method, followed by a discussion of the fundamental singular solutions. Subsequently, a family of isoparametric boundary elements is introduced, along with the necessary numerical integration techniques as well as the solution algorithm. Numerical examples are presented, which demonstrate the unconditional stability and high accuracy of this dynamic analysis technique.     

Uniform half-plane elastodynamic problems by an approximate boundary element method

In this paper the applicability of an approximate Boundary Element Method to uniform half-plane elastodynamic problems is investigated. This method employs the concept of images to construct approximate fundamental solutions for the half-plane and does not require any half-plane surface discretization. The method is formulated in the frequency domain for the case of harmonic disturbances or the Laplace transform domain for the case of transient disturbances. In the latter case a numerical inversion of the transformed solution is necessary to obtain the time domain response. The proposed method can be used as an alternative to boundary element methods that either utilize the infinite plane fundamental solution and thus require a half-plane surface discretization, or employ the exact half-plane fundamental solution, which even though leads to no surface discretization, is of a very lengthy and complicated form. Two characteristics numerical examples are used to illustrate the proposed method and study its advantages and disadvantages.     

无限域地基有限元分析的简化粘弹性边界

本文提出一种能用于无限域地基动、静力有限元分析的简单人工边界——粘弹边界，采用一层边界单元来实现。该边界弹簧部分采用弹性半空间Mindlin解计算，边界单元采用具有粘性阻尼的粘弹材料，通过确定合理的弹性模量、泊松比和粘性阻尼系数，不仅可以使边界单元能够模拟无限域远场的弹性恢复力，同时可以模拟粘性边界。通过算例分析，验证了该边界的有效性，它可用于土-结构动力相互作用分析。     

Mixed boundary value problem in an elastic quarter-space

Summary The stress and displacement throughout a homogeneous, isotropic, elastic quarterspace have been evaluated when the stress is prescribed on one of the plane boundaries and the displacement on the other. The variations of the displacement and stress on the boundaries with time and distance from the origin have been shown graphically.     

Axisymmetric soil–structure interaction by global–local finite elements

A version of the global–local finite element method is presented for studying dynamic steady-state soil–structure interaction wherein the soil medium extends to infinity. Herein, only axisymmetric behaviour is considered. In this approach, conventional finite elements are used to model the structure and some portion of the surrounding soil medium considered to be homogeneous and isotropic. A complete set of outgoing waves in the form of spherical harmonics for the entire space is used to represent the behaviour in the half-space beyond the finite element mesh and these are termed the global functions. Full traction and displacement continuity is enforced at the finite element mesh interface with the outer region. On the free surface of the half-space in the outer field, traction-free surface conditions are enforced by demanding that a sequence of integrals of the weighted-average tractions must vanish. Numerical examples are presented for the response of different shaped foundations, resting on the free surface or at various submerged levels, due to a normal seismic plane compressional wave. Plots of differential scattering cross-sections show the angular distribution of the energy (its directional nature) of the scattered field.