Probabilistic Marcinkiewicz–Zygmund Inequalities on the Rotation Group

Approximation problems on the rotation group SO(3) naturally arise in various fields, like crystallography, chemistry, and biology. For example, in crystallographic texture analysis one is confronted with the problem of evaluating so-called orientation density functions (ODFs). In many situations one only has a finite number of measurements at scattered sampling nodes. In order to reconstruct ODFs over all rotations, so-called Marcinkiewicz–Zygmund inequalities on the rotation group are an important tool. These inequalities provide norm equivalences between polynomials on SO(3) and their sample values. Recently shown equivalences depend on a density parameter of the sampling set and the proven inequalities hold true for polynomials on SO(3) whose degree does not exceed an upper bound which is determined by this density parameter. In this paper, we show that we can enlarge this upper bound for the polynomial degree significantly if we are satisfied by such norm equivalences that hold with a given probability only. Moreover, we show that there are fixed sampling sets for which we get probabilistic Marcinkiewicz–Zygmund inequalities that hold for polynomials on SO(3) of all degrees.     

Seismic hazard assessment for low-seismicity areas — Case study: Northern Germany

The problem of assessing seismic hazard in low-seismicity areas becomes obvious in many practical applications. A typical low-seismicity area, which experienced damaging earthquakes in historical times, is the North German Plain, for which a case study is presented. It is shown how seismic hazard assessments are influenced by different interpretations of historical key earthquakes, changes in b-value as well as variations of the upper bound magnitude assumed for the seismic source regions. The latter strongly influences the hazard results in the case of very low b-values for long return periods.     

基于刚体平动运动单元的上限有限元研究

刚性块体极限分析上限法常应用于岩土工程稳定性研究,然而应用时需假定刚性块体破坏模式并递推繁琐的几何关系。为此,提出一种适应性更广的基于非线性规划模型的刚体平动运动单元上限有限元法,并解决了其优化模型初始值的确定问题。通过引入有限单元思想,将计算区域离散成刚体单元,同时以单元速度和节点坐标作为决策变量,由上限定理建立非线性规划模型获得上限解。利用编制的上限有限元程序进行边坡和浅埋隧道稳定性算例验证,表明运动单元上限有限元法能调整速度间断线至较优方位,所得破坏模式特征鲜明,上限解精度高,可广泛应用于边坡、隧道等稳定性分析研究。     

A homogenization method for estimating the bearing capacity of soils reinforced by columns

The ultimate bearing capacity problem of a strip foundation resting on a soil reinforced by a group of regularly spaced columns is investigated in the situation when both the native soil and reinforcing material are purely cohesive. Making use of the yield design homogenization approach, it is shown that such a problem may be dealt with as a plane strain yield design problem, provided that the reinforced soil macroscopic strength condition has been previously determined. Lower and upper bound estimates for such a macroscopic criterion are obtained, thus giving evidence of the reinforced soil strong anisotropy. Performing the upper bound kinematic approach on the homogenized bearing capacity problem, by using the classical Prandtl's failure mechanism, makes it then possible to derive analytical upper bound estimates for the reinforced foundation bearing capacity, as a function of the reinforced soil parameters (volume fraction and cohesion ratio), as well as of the relative extension of the reinforced area. It is shown in particular that such an estimate is closer to the exact value of the ultimate bearing capacity, than that derived from a direct analysis which implicitly assumes that the reinforced soil is an isotropic material. Copyright © 2005 John Wiley & Sons, Ltd.     

Calculation of equilibrium distributions of chemical species in aqueous solutions by means of monotone sequences

A numerical method for the calculation of equilibrium distributions of chemical species in aqueous solutions of electrolytes is presented. This method is constructed by transforming the problem of determining the set of unknown concentrations satisfying the mass balance and mass action equations into the equivalent problem of finding the limits of certain well-behaved mathematical sequences in a multivariable direct iteration scheme. The total (analytical) concentrations are taken as the starting estimates, and the recursive equations are constructed from the starting equations. It is shown that the sequences so constructed are both monotonic and bounded, hence convergent in an orderly fashion by a mathematical axiom. Each unknown free ion concentration (that is, each limit) is approached simultaneously from above and from below, being effectively sandwiched in an ever tightening manner. Strict error bounds therefore are easily constructed. The method has been found to be exceedingly efficient in practice, with the error bound (maximum fractional error in taking the upper estimate as the final answer) decreasing in an approximately exponential fashion with respect to iteration number, commonly 0.5–1.5 orders of magnitude per iteration. Correction for nonideality is presented, and the possibility of this giving rise to more than one solution is discussed in connection with three examples of natural waters of low-, medium-, and high-ionic strengths.     

Extraction of the anomaly magnetic field of the earth from stratospheric balloon magnetic surveys at altitudes of 20–40 km

The solution to the problem of extraction of the anomaly Earth’s magnetic field (EMF) from stratospheric balloon magnetic surveys with the help of global analytical models of the normal EMF is proposed. In the problem solution, errors for the analytical models of the normal EMF and its secular variation at a set moment of time are assessed; the found error is introduced as a correction to the extracted anomaly EMF. The error of the model is determined in the places where significant magnetic anomalies are absent. In this case, the error of the model corresponds to deviations of the normal EMF components, synthesized by coefficients of analytical models, and to deviations of the EMF secular variations from the measured values at quite a low value of the variable EMF or one being taken into account. These places are determined when carrying out additional measurements in vertical gradients of the EMF with the use of scalar magnetometers at the gauge length of 6 km. It has been shown that the found places can be considered as nonanomaly, if the difference of values of the anomaly EMF at the gauge length of 6 km does not exceed 1.5 nT within the profile’s portion of about 100 km in length. An experiment in nature has revealed that errors for the IGRF-2005 and IGRF-2010 models, corrected for secular variation of the EMF, can reach 200 and 140 nT, respectively, within the limits of the territory where the Kama-Emba magnetic anomaly is located; these errors are determined by the considered causes. Comparison of aerostatic profiles of magnetic anomalies with data on the anomaly EMF, derived from the maps, has shown that the realizations derived from the maps contain overestimated negative values of the anomaly EMF, because they reflect processes in the near-surface layer of the Earth’s crust. This fact causes the situation when attempts to recalculate the anomaly EMF into the upper half-space by the near-surface data still have not been successful. Only realizations derived at the altitudes comparable to the thickness of the Earth’s crust can give an adequate model of the anomaly EMF in the circumterrestrial space and enable us to recalculate magnetic anomalies reliably into any altitude levels.     

Estimating an upper bound to particle size in sediment populations

A sediment-size measure based on an estimate of an upper-size bound is suggested as a useful alternative to the largest sample observation. Two estimation procedures are described which are compatible with the usual sediment sampling techniques that yield very large samples of unknown and variable size. One method requires only the first two order statistics from a single sample, while the other uses the maxima from a number of samples and estimates the upper bound as the location parameter of a Weibull distribution. In the latter case it is shown that the effect of random sample size can be overcome provided the expected sample size is sufficiently large.     

Upper bound finite element analysis of slope stability using a nonlinear failure criterion

In this study, upper bound finite element (FE) limit analysis is applied to stability problems of slopes using a nonlinear criterion. After formulating the upper bound analysis as the dual form of a second-order cone programming (SOCP) problem, the stress field and corresponding shear strength parameters can be determined iteratively. Thus, the nonlinear failure criterion is represented by the shear strength parameters associated with stress so that the analysis of slope stability using a nonlinear failure criterion can be transformed into the traditional upper bound method with a linear Mohr–Coulomb failure criterion. Comparison with published solutions illustrates the accuracy and feasibility of the proposed method for a simple homogeneous slope stability problem. The proposed approach is also applied to a seismic stability problem for a rockfill dam to study the influence of different failure criterions on the upper bound solutions. The results show that the seismic stability coefficients obtained using two different nonlinear failure criteria are similar but that the convergence differs significantly.     

Upper bound limit analysis using finite elements and linear programming

This paper describes a technique for computing rigorous upper bounds on limit loads under conditions of plane strain. The method assumes a perfectly plastic soil model, which is either purely cohesive or cohesive-frictional, and employs finite elements in conjunction with the upper bound theorem of classical plasticity theory. The computational procedure uses three-noded triangular elements with the unknown velocities as the nodal variables. An additional set of unknowns, the plastic multiplier rates, is associated with each element. Kinematically admissible velocity discontinuities are permitted along specified planes within the grid. The finite element formulation of the upper bound theorem leads to a classical linear programming problem where the objective function, which is to be minimized, corresponds to the dissipated power and is expressed in terms of the velocities and plastic multiplier rates. The unknowns are subject to a set of linear constraints arising from the imposition of the flow rulé and velocity boundary conditions. It is shown that the upper bound optimization problem may be solved efficiently by applying an active set algorithm to the dual linear programming problem. Since the computed velocity field satisfies all the conditions of the upper bound theorem, the corresponding limit load is a strict upper bound on the true limit load. Other advantages include the ability to deal with complicated loading, complex geometry and a variety of boundary conditions. Several examples are given to illustrate the effectiveness of the procedure.     

基于VMD和SVM的煤厚预测方法研究

煤厚变化对煤炭安全高效开采具有重要的影响。针对三维地震数据中含有噪声时,易导致煤厚预测结果具有较大误差的问题,提出一种利用变模态分解(VMD)和支持向量机(SVM)方法结合进行煤厚预测的方法。首先,构建煤厚楔形模型并对其进行地震正演模拟,当煤层厚度较薄时,振幅属性和频带宽度属性与煤厚之间具有较好的正相关性,而瞬时频率属性与煤厚具有较好的负相关性;对正演地震记录增加噪声,结果表明噪声对利用地震属性进行煤厚预测具有较大影响。利用VMD进行去噪之后,基于SVM进行煤厚预测,实际地震资料的煤厚预测结果与已有钻孔揭露的煤层信息较为吻合,预测煤厚最小绝对误差仅为0.02 m,最大绝对误差0.52 m,验证了方法的可行性和有效性。研究成果可为低信噪比区的煤厚反演提供参考。      

On Improving the Solution by Using Lower Bound Finite Elements Limit Analysis and Linear Programming

This study presents an improved technique for obtaining the collapse loads for any geotechnical stability problem with the application of the lower bound finite elements limit analysis in combination with linear programming. In the proposed method, a lower order polygon is initially used throughout the problem domain to model the Mohr–Coulomb yield function; the order of the polygon refers to its total number of sides. After obtaining the initial solution, the problem domain is then discretized into a number of zones in which a different order polygon is used to model the yield surface. It is noted that a higher order polygon needs to be chosen only in a region, not everywhere, where the stress state approaches towards the yield. The total memory usage as well as the computational time needed to solve the problem with the proposed technique becomes significantly smaller. In order to check the validity of the method, the bearing capacity factor N_γ was computed for smooth as well as rough strip footing and the obtained computational results were found to be quite encouraging.     

不排水双层粘性土地基极限承载力的数值分析

采用弹塑性有限元分析了条形基础作用下不排水条件的双层粘性土地基极限承载力性状。采用修正的地基承载力系数表征,并将不同的几何与土层参数条件下的数值解与上下限解和经典的经验解进行比较。表明弹塑性位移有限元法可以很好地求解地基的极限承载力问题,其求解得到的修正地基承载力系数与基于下限原理的有限元解很接近,而上限解高估了地基的极限荷载值,传统的经验解在某些条件下却偏小。     

基于塑性极限分析上限法理论的土质边坡可靠度分析

将极限分析的上限定理、有限元离散思想、随机规划理论和蒙特卡洛方法这四者结合起来,提出了一种土质边坡可靠度分析的上限数值方法。首先采用三节点有限单元离散土质边坡,然后将土体的抗剪参数设为随机变量,根据上限定理构建同时满足三角形单元的塑性流动约束条件、单元公共边的塑性流动约束条件和单元速度边界条件的机动许可速度场,并根据内功功率等于外功功率条件建立目标函数,构建土质边坡可靠度分析的上限法随机规划模型。采用蒙特卡洛方法求解上限法随机规划模型,同时提出了一种基于上限法速度场的边坡失效风险系数估算方法,该方法特别适用于具有多种失效模式的边坡风险分析。对2个经典算例进行了深入分析,验证了方法的正确性。     

Analysis of input data and seismic hazard assessment for the low seismicity area ‘Belgium,The Netherlands and NW Germany’

From the earthquake catalogue of the region of interest 2–8°E, 49–52°N, only events with intensity > 3 are considered. The fore/aftershocks and the induced seismicity are removed. The completeness and other properties of the remaining data set are investigated. The seismic data, as well as the geological, tectonic, and other geophysical data of the area are used for the determination of the seismotectonic zones. For each zone, the cumulative intensity-frequency relation (taking into account the completeness of the catalogue), the attenuation depending on the direction, the upper bound of intensity and the average depth are calculated. When the seismic activity is changing within a zone, this zone is divided into subzones with an almost constant activity and a b value equal to the value of the whole zone. When necessary for each subzone, a different attenuation coefficient can be used, to take into account more regional effects. Then the seismic hazard is assessed with the modified McGuire program. Annual probability versus intensity for several sites and hazard intensity and probability maps are calculated. Finally, an error discussion of the whole procedure for one test site is given.     

Upper bound limit analysis of cohesive soils using mesh-free method

A mesh-free based limit analysis approach is proposed, to determine the upper bound solutions for the collapse loads associated with cohesive soils, under plane strain conditions. In the presented technique, the geometry of problem is just simulated by nodes and there is no need of mesh in the traditional sense. The process of finding an upper bound solution consists of combining limit analysis theory and a mesh-free numerical technique as a discretisation tool. To satisfy the required conditions for the admissibility of the discretised velocity field at the entire problem domain, a strain rate smoothing technique has been adopted. The outcome of proposed combination is a nonlinear optimisation problem which is solved by a direct iterative technique. The solution found by an iterative algorithm is an upper bound for limit load of the stability problem. The efficiency of the proposed method is demonstrated by solving different example problems in the soil mechanics engineering field, at the end of the paper.     

Comparison of approaches to spatial estimation in a bivariate context

The problem of estimating a regionalized variable in the presence of other secondary variables is encountered in spatial investigations. Given a context in which the secondary variable is known everywhere (or can be estimated with great precision), different estimation methods are compared: regression, regression with residual simple kriging, kriging, simple kriging with a mean obtained by regression, kriging with an external drift, and cokriging. The study focuses on 19 pairs of regionalized variables from five different datasets representing different domains (geochemical, environmental, geotechnical). The methods are compared by cross-validation using the mean absolute error as criterion. For correlations between the principal and secondary variable under 0.4, similar results are obtained using kriging and cokriging, and these methods are superior slightly to the other approaches in terms of minimizing estimation error. For correlations greater than 0.4, cokriging generally performs better than other methods, with a reduction in mean absolute errors that can reach 46% when there is a high degree of correlation between the variables. Kriging with an external drift or kriging the residuals of a regression (SKR) are almost as precise as cokriging.     

Upper bound limit analysis using linear finite elements and non‐linear programming

A new method for computing rigorous upper bounds on the limit loads for one‐, two‐ and three‐dimensional continua is described. The formulation is based on linear finite elements, permits kinematically admissible velocity discontinuities at all interelement boundaries, and furnishes a kinematically admissible velocity field by solving a non‐linear programming problem. In the latter, the objective function corresponds to the dissipated power (which is minimized) and the unknowns are subject to linear equality constraints as well as linear and non‐linear inequality constraints. Provided the yield surface is convex, the optimization problem generated by the upper bound method is also convex and can be solved efficiently by applying a two‐stage, quasi‐Newton scheme to the corresponding Kuhn–Tucker optimality conditions. A key advantage of this strategy is that its iteration count is largely independent of the mesh size. Since the formulation permits non‐linear constraints on the unknowns, no linearization of the yield surface is necessary and the modelling of three‐dimensional geometries presents no special difficulties. The utility of the proposed upper bound method is illustrated by applying it to a number of two‐ and three‐dimensional boundary value problems. For a variety of two‐dimensional cases, the new scheme is up to two orders of magnitude faster than an equivalent linear programming scheme which uses yield surface linearization. Copyright © 2001 John Wiley & Sons, Ltd.     

用无定向磁力仪测量磁化率张量与剩余磁化强度矢量

岩矿石的磁化率和剩余磁性是岩石物理性质的基本参数,它们包含着丰富的地质信息。但是常常不合理地把磁化率作为标量来测量。本文提出在无定向磁力仪的“As”位置上作多个方位的测量,为减小误差,再用最小二乘法计算出其最优概值—磁化率张量与剩余磁化强度矢量。实验结果良好。     

Certified solutions for hydraulic structures using the node‐based smoothed point interpolation method (NS‐PIM)

A meshfree node‐based smoothed point interpolation method (NS‐PIM), which has been recently developed for solid mechanics problems, is applied to obtain certified solutions with bounds for hydraulic structure designs. In this approach, shape functions for displacements are constructed using the point interpolation method (PIM), and the shape functions possess the Kronecker delta property and permit the straightforward enforcement of essential boundary conditions. The generalized smoothed Galerkin weak form is then applied to construct discretized system equations using the node‐based smoothed strains. As a very novel and important property, the approach can obtain the upper bound solution in energy norm for hydraulic structures. A 2D gravity dam problem and a 3D arch dam problem are solved, respectively, using the NS‐PIM and the simulation results of NS‐PIM are found to be the upper bounds. Together with standard fully compatible FEM results as a lower bound, we have successfully determined the solution bounds to certify the accuracy of numerical solutions. This confirms that the NS‐PIM is very useful for producing certified solutions for the analysis of huge hydraulic structures. Copyright © 2009 John Wiley & Sons, Ltd.     

一种二阶锥线性化方法在上限有限元中的应用研究

对于平面应变条件下岩土稳定性分析,基于线性规划的上限有限元需对常用的摩尔-库仑屈服准则形成的二阶锥约束进行线性化,直接地处理方法是以外接多边形替代锥体投影形成的圆域。为了提高线性化精度往往需直接增加外接多边形边数,从而造成线性规划模型中决策变量包含大量的塑性乘子变量,使计算难度大为增加甚至变得不可行。为此,引入Ben-Tal和Nemirovsky提出的一种二阶锥线性化方法,并将其嵌入到自编的上限有限元程序。经算例分析发现,该法与外接多边形线性化方法所获计算结果相互印证,且其通过适量的增加决策变量和等式约束数目,能保证摩尔-库仑屈服准则线性化精度,同时形成的线性规划规模更小,可望应用于基于线性规划模型的上限有限元中。