结合边界条件的截断高阶隐式有限差分方法(英文)

有限差分方法广泛应用于求解许多科技领域所涉及的偏微分方程,高阶显式有限差分方法通常用来提高求解精度,已经提出的高阶隐式有限差分方法和截断高阶显式有限差分方法可用来进一步提高模拟精度而不增加计算量。本文首先计算了针对常规网格上的一阶导数和二阶导数、交错网格上的一阶导数的有限差分系数,发现高阶隐式有限差分系数中存在一些小的系数。频散分析结果表明:忽略这些小的差分系数能够近似维持有限差分的精度,但是显著减小了计算量。然后,引入镜像对称边界条件来提高隐式有限差分方法的精度和稳定性,采用混合吸收边界条件来减小来自模型边界所不需要的反射。最后,给出了针对均匀和非均匀介质模型的弹性波模拟例子,表明了本文方法的优点。     

A high-order time-domain transmitting boundary for cylindrical wave propagation problems in unbounded saturated poroelastic media

Based on the u–p formulation of Biot equation with an assumption of zero permeability coefficient, a high-order transmitting boundary is derived for cylindrical elastic wave propagation in infinite saturated porous media. By this transmitting boundary the total stresses on the truncated boundaries of a numerical model, such as a finite element model, are replaced by a set of spring, dashpot and mass elements, with some additionally introduced auxiliary degrees of freedom. The transmitting boundaries are incorporated into the DIANA SWANDYNE II program and an unconditionally stable implicit time integration algorithm is adopted. Despite the assumption made in the derivation of the transmitting boundary, numerical examples show that it can provide highly accurate results for cylindrical elastic wave propagation problems in infinite saturated porous medium in case the u–p formulation is applicable. Although the direct applications of the proposed transmitting boundary to general two dimensional wave problems in infinite saturated porous media are not highly accurate, acceptable accuracy can still be achieved by placing the transmitting boundary at relatively large distance from the wave source.     

考虑刚度及边界条件的短索索力求解与试验研究

建立了一个非齐次边界拉索力学模型,基于拉索线性振动理论,推导了考虑拉索抗弯刚度及边界条件的拉索频率方程;根据双曲函数的数理特性对频率方程进行了合理简化,得到了实用的频率简化方程,提出了逆过程求解索力的方法和步骤,并探讨了不同支承刚度对索力的影响。设计了室内拉索试验模型并进行了27种工况索力试验,比较了不同工况下的计算索力与实际索力,验证了本文提出的频率简化方程和逆过程求解方法的实用可靠性。     

求解混合边值振动问题的辅助函数法

提出辅助函数方法,为混合边值振动问题求解提供一种手段。以圆柱第2类边值出平面简谐振动问题为例阐述辅助函数的基本思想和定义;以半空间凹陷地形出平面散射问题为例讨论辅助函数的构造原则和正确方式;利用辅助函数法给出圆柱一般混合边值出平面简谐振动问题解答;讨论采用辅助函数法在位移和应力边界交接处级数解答的收敛情况;给出若干算例,说明辅助函数法的可行性和有效性。     

Weakly-reflective boundary conditions for two-dimensional shallow water flow problems

In this paper weakly-reflective boundary conditions are derived for the two-dimensional shallow water equations, including bottom friction and Coriolis force. The essential aspects of the derivation are given. Zeroth and first order approximations are applied to the test problem of an initially Gaussian-shaped free surface elevation. For the numerical solution a finite element program is used and various aspects of the numerical implementation are discussed. For small scale practical problems a rather simple (one parameter) formulation might be sufficient. The influence of this parameter is discussed on the weakly-reflectiveness of the boundary condition.     

Approach to constructing a system of mathematical models in solving long-term planning problems for small-scale water bodies

A general structure of mathematical models for solving the problems of water supply, water protection, and flood and freshet control are proposed for small-scale water bodies. Various aspects of these water bodies are considered, allowing them to be identified against other water bodies.     

A parallel method of solving nonlinear inverse problems in gravimetry and magnetometry

A parallel iterative method of solving nonlinear inverse logarithmic potential problems and three-dimensional problems in gravimetry and magnetometry is proposed.     

Isometric method: Efficient tool for solving non-linear inverse problems

A novel algorithm called Isometric Method (IM) for solving smooth real-valued non-linear inverse problems has been developed. Model and data spaces are represented by using m + 1 corresponding vectors at a time (m is the dimension of model space). Relations among vectors in the data space are set up and then transferred into the model space thus generating a new model. If the problem is truly linear, this new model is the exact solution of the inverse problem. If the problem is non-linear, the whole procedure has to be repeated iteratively. The basic underlying idea of IM is to postulate the distance in the model space in such a way that the model and data spaces are isometric, i.e. distances in both spaces have the same measure. As all model-data vector pairs are used many times in successive iterations, the number of the forward problem computations is minimized. There is no necessity to deal with derivatives. The requirement for the computer memory is low. IM is suitable especially for solving smooth medium non-linear problems when forward modelling is time-consuming and minimizing the number of function evaluations is topical. Applications of IM on synthetic and real geophysical problems are also presented. malek@irsm.cas.cz     

Online numerical simulation: A hybrid simulation method for incomplete boundary conditions

Hybrid simulation is a powerful and cost‐effective simulation technique to evaluate structural dynamic performance. However, it is sometimes rather difficult to guarantee all the boundaries on the physical substructures, especially when the boundary conditions are very complex, due to limited laboratory resources. Lacking of boundary conditions is bound to change the stress state of the structure and eventually result in an inaccurate evaluation of structural performance. A model updating‐based online numerical simulation method is proposed in this paper to tackle the problem of incomplete boundary conditions. In the proposed method, 2 sets of finite element models with the same constitutive model are set up for the overall analysis of the whole structure and the constitutive model parameter estimation of the physical substructure, respectively. The boundary conditions are naturally satisfied because the response is calculated from the overall structural model, and the accuracy is improved as the material constitutive parameters are updated. The effectiveness of the proposed method is validated via numerical simulations and actual hybrid tests on a RC frame structure, and the results show that the negative effect of incomplete boundary conditions is almost eliminated and the accuracy of hybrid simulation is very much improved.     

Open boundary value problems in ocean dynamics by finite elements

This paper presents recent results of application of the finite element models to wave overtopping and wave run-up problems in ocean dynamics. Open boundaries are prescribed as natural boundary condition obtained from the continuity equation of the Galerkin finite element formulation. The numerical results are, in general, reasonably good agreements with the histrical field data.     

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...     

The neural network approximation method for solving multidimensional nonlinear inverse problems of geophysics

The iterative approximation neural network method for solving conditionally well-posed nonlinear inverse problems of geophysics is presented. The method is based on the neural network approximation of the inverse operator. The inverse problem is solved in the class of grid (block) models of the medium on a regularized parameterization grid. The construction principle of this grid relies on using the calculated values of the continuity modulus of the inverse operator and its modifications determining the degree of ambiguity of the solutions. The method provides approximate solutions of inverse problems with the maximal degree of detail given the specified degree of ambiguity with the total number of the sought parameters ~n × 10³ of the medium. The a priori and a posteriori estimates of the degree of ambiguity of the approximated solutions are calculated. The work of the method is illustrated by the example of the three-dimensional (3D) inversion of the synthesized 2D areal geoelectrical (audio magnetotelluric sounding, AMTS) data corresponding to the schematic model of a kimberlite pipe.     

高阶谱混合方法地震子波估计及处理

凭借高阶累积量对噪声的不敏感性,依据地震记录高阶谱中包含的子波相位、幅值信息,提出了基于双谱和三谱混合的子波恢复方法,并对理论合成记录进行子波估计及处理模拟试验,其结果论证了该方法的可行性.同时,用本文介绍的方法对野外采集的地震资料进行处理后,其分辨率明显提高,说明了高阶谱技术在资料处理方面有相当大的潜力.     

A simplified method of constructing fragility curves for highway bridges

Fragility curves are found to be useful tools for predicting the extent of probable damage. They show the probability of highway structure damage as a function of strong motion parameters, and they allow the estimation of a level of damage probability for a known ground motion index. In this study, an analytical approach was adopted to develop the fragility curves for highway bridges based on numerical simulation. Four typical RC bridge piers and two RC bridge structures were considered, of which one was a non‐isolated system and the other was an isolated system, and they were designed according to the seismic design code in Japan. From a total of 250 strong motion records, selected from Japan, the United States, and Taiwan, non‐linear time history analyses were performed, and the damage indices for the bridge structures were obtained. Using the damage indices and ground motion parameters, fragility curves for the four bridge piers and the two bridge structures were constructed assuming a lognormal distribution. It was found that there was a significant effect on the fragility curves due to the variation of structural parameters. The relationship between the fragility curve parameters and the over‐strength ratio of the structures was also obtained by performing a linear regression analysis. It was observed that the fragility curve parameters showed a strong correlation with the over‐strength ratio of the structures. Based on the observed correlation between the fragility curve parameters and the over‐strength ratio of the structures, a simplified method was developed to construct the fragility curves for highway bridges using 30 non‐isolated bridge models. The simplified method may be a very useful tool to construct the fragility curves for non‐isolated highway bridges in Japan, which fall within the same group and have similar characteristics. Copyright © 2003 John Wiley & Sons, Ltd.     

Eulerian-Lagrangian method for solving transport in aquifers

A Eulerian-Langrangian scheme is used to reformulate the equation of solute transport with ground water in saturated soils. The governing equation is decomposed into advection along characteristic path lines and propagation of the residue at a fixed grid.The method was employed to simulate transport of a conservative pollutant in a hypothetical aquifer, subject to the equivalence of real conditions. Implementation was based on data involving parameters of a heterogeneous aquifer, heavy flux stresses of densed pumpage/recharge wells, precipitation and seasonally changing flow regimes. Simulation, with coarse grid and high Peclet numbers yielded minute mass balance errors.     

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.     

耦合PML边界条件的三维大地电磁二次场有限差分法

本文将大地电磁场分解为一次场和二次场,应用交错网格有限差分法模拟计算大地电磁二次场,并引入各向异性最佳匹配层（PML）吸收边界条件作为二次场边界条件,实现了耦合PML吸收边界条件的三维大地电磁二次场有限差分正演模拟.为了确保正演的稳定性和效率,QMR求解器和磁感应矢量散度校正技术被用于PML吸收边界条件下系数矩阵的快速求解.三维模型正演响应表明,基于二次场的三维大地电磁有限差分算法具有较高的计算精度和可靠性.通过计算分析不同PML吸收因子条件的大地电磁正演结果,显示在适当的吸收因子下,PML吸收边界条件可较大幅度的减小外边界距离,从而有效的压缩模型求解空间,最终提高三维大地电磁正演模拟的效率.

     

The method of linear integral representations for solving 3D structural inverse problems of potential theory

The algorithm for the solution of a three-dimensional (3D) structural inverse problem in potential theory is described. The algorithm is based on the method of linear integral representations and on the method of extending compacts. The proposed approach is verified on model examples for an anomalous gravity field.     

Tidal boundary conditions in SEAWAT

SEAWAT, a U.S. Geological Survey groundwater flow and transport code, is increasingly used to model the effects of tidal motion on coastal aquifers. Different options are available to simulate tidal boundaries but no guidelines exist nor have comparisons been made to identify the most effective approach. We test seven methods to simulate a sloping beach and a tidal flat. The ocean is represented in one of the three ways: directly using a high hydraulic conductivity (high-K) zone and indirect simulation via specified head boundaries using either the General Head Boundary (GHB) or the new Periodic Boundary Condition (PBC) package. All beach models simulate similar water fluxes across the upland boundary and across the sediment-water interface although the ratio of intertidal to subtidal flow is different at low tide. Simulating a seepage face results in larger intertidal fluxes and influences near-shore heads and salinity. Major differences in flow occur in the tidal flat simulations. Because SEAWAT does not simulate unsaturated flow the water table only rises via flow through the saturated zone. This results in delayed propagation of the rising tidal signal inland. Inundation of the tidal flat is delayed as is flow into the aquifer across the flat. This is severe in the high-K and PBC models but mild in the GHB models. Results indicate that any of the tidal boundary options are fine if the ocean-aquifer interface is steep. However, as the slope of that interface decreases, the high-K and PBC approaches perform poorly and the GHB boundary is preferable.