Advanced numerical analyses by the Non-Smooth Contact Dynamics method of an ancient masonry bell tower

Advanced numerical analyses were carried out in order to assess the nonlinear dynamical behaviour of the bell tower of Pomposa Abbey in Codigoro, in the province of Ferrara (Italy), by means of the Non-Smooth Contact Dynamics (NSCD) method. The main purpose of the work is to investigate the capacity of the main mechanical parameter used in the analyses, namely, the friction coefficient, to have effects on the mechanical response of ancient masonry structures undergoing seismic actions. Therefore, the tower was modelled following the discrete element method (DEM) and assembling the masonry texture as rigid bodies tied by frictional joints. Thus a discontinuous approach was used to assess the dynamic properties and the vulnerability of the masonry structure, through large deformations regulated by the Signorini's law, concerning the impenetrability between the rigid bodies; and by the Coulomb's law, regarding the dry-friction model. Afterward, different values were assigned to the friction coefficient of the models, and a variety of real seismic shocks have been applied in the nonlinear analyses. Finally, it is possible to see different failure mechanisms resulting for each friction value and types of dynamic actions used, as expected.     

Evaluación sísmica de los edificios de mampostería típicos de Barcelona aplicando la metodología Risk-UE

In this work, a study about the vulnerability and seismic damage of unreinforced masonry buildings is carried out. Three models of buildings representatives of the Eixample district of Barcelona have been chosen. The seismic vulnerability is evaluated by means of the Risk-UE methodology. The seismic demand is described by elastic project spectrum, in this case, defined by the Eurocode 8. Fragility curves are obtained from a nonlinear analysis, considering the capacity spectra. Expected seismic damage is gotten with the damage probability matrices, which indicate the occurrence probability of a damage state for a seismic demand given. The analysis of the buildings has been performed by TreMuri program by means of a macroelements model, which represents of a whole masonry panel. The buildings, here, analyzed are real and detailed structural drawings and reports have been used to model them. The results shown a considerable vulnerability in this type of buildings, therefore, in spite of the seismic hazard the expected seismic risk is significant.     

Nonlinear analysis of elastic thin-walled shell structures

Theoretical principles, methodology and algorithms presented herein are to analyze and design the elastic thin-walled engineering structures and components, with emphasis on the important nonlinear behavior. The methodology of the consequent analysis of single-parametric nonlinear problems is applied to structural syntheses. The numerical algorithm for this analysis is based on the parameter continuation methods and the “control parameter subspace changing”. The effectiveness of the proposed approach is illustrated through several examples in thin-walled structures.     

The rate of convergence for the cyclic projections algorithm III: Regularity of convex sets

The cyclic projections algorithm is an important method for determining a point in the intersection of a finite number of closed convex sets in a Hilbert space. That is, for determining a solution to the “convex feasibility” problem. This is the third paper in a series on a study of the rate of convergence for the cyclic projections algorithm. In the first of these papers, we showed that the rate could be described in terms of the “angles” between the convex sets involved. In the second, we showed that these angles often had a more tractable formulation in terms of the “norm” of the product of the (nonlinear) metric projections onto related convex sets.In this paper, we show that the rate of convergence of the cyclic projections algorithm is also intimately related to the “linear regularity property” of Bauschke and Borwein, the “normal property” of Jameson (as well as Bakan, Deutsch, and Li’s generalization of Jameson’s normal property), the “strong conical hull intersection property” of Deutsch, Li, and Ward, and the rate of convergence of iterated parallel projections. Such properties have already been shown to be important in various other contexts as well.     

Sensitivity analysis for product design selection with an implicit value function

In product design selection the decision maker (DM) often does not have enough information about the end users’ needs to state the “preferences” with precision, as is required by many of the existing selection methods. We present, for the case where the DM gives estimates of the preferences, a concept for calculating a “robustness index.” The concept can be used with any iterative selection method that chooses a trial design for each iteration, and uses the DM’s preference parameters at that trial design to eliminate some design options which have lower value than the trial design. Such methods, like our previously published method, are applicable to cases where the DM’s value function is implicit. Our robustness index is a metric of the allowed variation between the actual and estimated preferences for which the set of non-eliminated trial designs (which could be singleton) will not change. The DM, through experience, can use the robustness index and other information generated in calculating the index to determine what action to take: make a final selection from the present set of non-eliminated designs; improve the precision of the preference estimates; or otherwise cope with the imprecision. We present an algorithm for finding the robustness index, and demonstrate and verify the algorithm with an engineering example and a numerical example.     

Nonlinear sloshing and passage through resonance in a shallow water tank

This paper is concerned with the effect of slowly changing the length of a tank on the nonlinear standing waves (free vibrations) and resonant forced oscillations of shallow water in the tank. The analysis begins with the Boussinesq equations. These are reduced to a nonlinear differential-difference equation for the slow variation of a Riemann invariant on one end. Then a multiple scale expansion yields a KdV equation with slowly changing coefficients for the standing wave problem, which is reduced to a KdV equation with a variable dispersion coefficient. The effect of changing the tank length on the number of solitons in the tank is investigated through numerical solutions of the variable coefficient KdV equation. A KdV equation which is “periodically” forced and slowly detuned results for the passage through resonance problem. Then the amplitude-frequency curves for the fundamental resonance and the first overtone are given numerically, as well as solutions corresponding to multiple equilibria. The evolution between multiple equilibria is also examined.Received: December 16, 2003     

Nonlinear sloshing and passage through resonance in a shallow water tank

This paper is concerned with the effect of slowly changing the length of a tank on the nonlinear standing waves (free vibrations) and resonant forced oscillations of shallow water in the tank. The analysis begins with the Boussinesq equations. These are reduced to a nonlinear differential-difference equation for the slow variation of a Riemann invariant on one end. Then a multiple scale expansion yields a KdV equation with slowly changing coefficients for the standing wave problem, which is reduced to a KdV equation with a variable dispersion coefficient. The effect of changing the tank length on the number of solitons in the tank is investigated through numerical solutions of the variable coefficient KdV equation. A KdV equation which is “periodically” forced and slowly detuned results for the passage through resonance problem. Then the amplitude-frequency curves for the fundamental resonance and the first overtone are given numerically, as well as solutions corresponding to multiple equilibria. The evolution between multiple equilibria is also examined.     

Design of induction motors using a mixed-variable approach

In this paper we are concerned with the problem of optimally designing three-phase induction motors. This problem can be formulated as a mixed variable programming problem. Two different solution strategies have been used to solve this problem. The first one consists in solving the continuous nonlinear optimization problem obtained by suitably relaxing the discrete variables. On the opposite, the second strategy tries to manage directly the discrete variables by alternating a continuous search phase and a discrete search phase. The comparison between the numerical results obtained with the above two strategies clearly shows the fruitfulness of taking directly into account the presence of both continuous and discrete variables.This work was supported by CNR/MIUR Research Program “Metodi e sistemi di supporto alle decisioni”, Rome, Italy.     

Analytical solution for Van der Pol–Duffing oscillators

In this paper, the problem of single-well, double-well and double-hump Van der Pol–Duffing oscillator is studied. Governing equation is solved analytically using a new kind of analytic technique for nonlinear problems namely the “Homotopy Analysis Method” (HAM), for the first time. Present solution gives an expression which can be used in wide range of time for all domain of response. Comparisons of the obtained solutions with numerical results show that this method is effective and convenient for solving this problem. This method is a capable tool for solving this kind of nonlinear problems.     

Semi-parametric multivariate modelling when the marginals are the same

A model is developed for multivariate distributions which have nearly the same marginals, up to shift and scale. This model, based on “interpolation” of characteristic functions, gives a new notion of “correlation”. It allows straightforward nonparametric estimation of the common marginal distribution, which avoids the “curse of dimensionality” present when nonparametically estimating the full multivariate distribution. The method is illustrated with environmental monitoring network data, where multivariate modelling with common marginals is often appropriate.     

The optimum shape of a bending beam

The problem of minimizing the static deflection of an elastic beam of variable cross-section and fixed volume in the case of free supported and rigidly clamped ends is considered. In the first case it is proved that the solutions obtained earlier, based on the necessary conditions for an extremum, satisfy the sufficient conditions. In the case of clamped ends, which is of the most interest from the point of view of applications, it is proved that the optimum solutions must necessarily have points inside the solution range in which the distribution of the beam thicknesses degenerates to zero (“internal hinges”). A qualitative, analytical and numerical analysis of this phenomenon is given. In particular, in the case of clamped ends for a class of point loads, analytical solutions for which the beam splits into two cantilevers are obtained.     

A comparison of procedures for estimating the parent probability distribution from a given set of fractiles

On one hand, eliciting subjective probabilities (fractiles) is a well-established procedure. On the other hand, knowledge of a subjective variable's central moments or distribution function is often assumed. However, the problem of converting elicited fractiles into the required moments or distribution function has been largely ignored. We show that this conversion problem is far from trivial, and that the most commonly used conversion procedures often produce huge errors. Alternative procedures are proposed; the “Tocher's curve” and “linear function of fractiles” methods are shown to be much more accurate than the commonly used procedures.     

Experimental studies of variable selection strategies based on constraint weights

Variable ordering heuristics that sample information before or during search in order to inform subsequent decisions have shown better performance and greater robustness than standard heuristics. One such strategy, the “weighted degree heuristic,” is based on weighting constraints according to their involvement in failure during search. A more recent approach uses “random probing” with restarting to gain information less subject to sampling bias. To date, these approaches have not been carefully analysed experimentally. In the present work, several important findings are presented, including a better delineation of the class of events that is sampled, an analysis of the importance of informed choices at the beginning of search, and a demonstration that random probing identifies sources of global contention effectively even when these are not clearly demarcated. These experiments show how empirical analysis can clarify subtle issues in the analysis of heuristic procedures for difficult search problems.     

Steady-state solutions in a nonlinear pool boiling model

We consider a relatively simple model for pool boiling processes. This model involves only the temperature distribution within the heater and describes the heat exchange with the boiling fluid via a nonlinear boundary condition imposed on the fluid–heater interface. This results in a standard heat equation with a nonlinear Neumann boundary condition on part of the boundary. In this paper, we analyse the qualitative structure of steady-state solutions of this heat equation. It turns out that the model allows both multiple homogeneous and multiple heterogeneous solutions in certain regimes of the parameter space. The latter solutions originate from bifurcations on a certain branch of homogeneous solutions. We present a bifurcation analysis that reveals the multiple-solution structure in this mathematical model. In the numerical analysis a continuation algorithm is combined with the method of separation-of-variables and a Fourier collocation technique. For both the continuous and discrete problem a fundamental symmetry property is derived that implies multiplicity of heterogeneous solutions. Numerical simulations of this model problem predict phenomena that are consistent with laboratory observations for pool boiling processes.     

Multivariate rank statistics for testing randomness concerning some marginal distributions

In the present paper there is proposed a rank statistic for multivariate testing of randomness conceriing some marginal distributions. The asymptotic distribution of this statistic under hypothesis and “near” alternatives is treated.     

加肋双曲冷却塔的非线性稳定分析*

本文采用有限元数值分析方法对土木工程中的大型钢筋混凝土双曲冷却塔壳的非线性稳定问题作了具体的数值计算.文中为了讨论双曲塔壳在屈曲后的特性,采用了将普通的载荷增量迭代法与改进的R.C.弧长法相结合的混合算法.为了与实际工程相符合,文中除了将钢筋混凝土当作非均匀体以外,还考虑了塔自重、底部的离散支承、及环向加肋等因素的影响.     

Nonlinear analysis of thin rectangular plates on Winkler–Pasternak elastic foundations by DSC–HDQ methods

This article introduces a coupled methodology for the numerical solution of geometrically nonlinear static and dynamic problem of thin rectangular plates resting on elastic foundation. Winkler–Pasternak two-parameter foundation model is considered. Dynamic analogues Von Karman equations are used. The governing nonlinear partial differential equations of the plate are discretized in space and time domains using the discrete singular convolution (DSC) and harmonic differential quadrature (HDQ) methods, respectively. Two different realizations of singular kernels such as the regularized Shannon’s kernel (RSK) and Lagrange delta (LD) kernel are selected as singular convolution to illustrate the present DSC algorithm. The analysis provides for both clamped and simply supported plates with immovable inplane boundary conditions at the edges. Various types of dynamic loading, namely a step function, a sinusoidal pulse, an N-wave pulse, and a triangular load are investigated and the results are presented graphically. The effects of Winkler and Pasternak foundation parameters, influence of mass of foundation on the response have been investigated. In addition, the influence of damping on the dynamic analysis has been studied. The accuracy of the proposed DSC–HDQ coupled methodology is demonstrated by the numerical examples.     

The Cost Distribution of Queue-Mergesort, Optimal Mergesorts, and Power-of-2 Rules

Queue-mergesort is introduced by Golin and Sedgewick as an optimal variant of mergesorts in the worst case. In this paper, we present a complete analysis of the cost distribution of queue-mergesort, including the best, average, and variance cases. The asymptotic normality of its cost is also established under the uniform permutation model. We address the corresponding optimality problems and we show that if we fix the merging scheme then the optimal mergesort as far as the average number of comparisons is concerned is to divide as evenly as possible at each recursive stage (top-down mergesort). On the other hand, the variance of queue-mergesort reaches asymptotically the minimum value. We also characterize a class of mergesorts with the latter property. A comparative discussion is given on the probabilistic behaviors of top-down mergesort, bottom-up mergesort, and queue-mergesort. We derive an “invariance principle” for asymptotic linearity of divide-and-conquer recurrences based on general “power-of-2” rules of which the underlying dividing rule of queue-mergesort is a special case. These analyses reveal an interesting algorithmic feature for general power-of-2 rules.     

The determination of quasistationary distributions directly from the transition rates of an absorbing Markov chain

There are many stochastic systems arising in areas as diverse as wildlife management, chemical kinetics and reliability theory, which eventually “die out,” yet appear to be stationary over any reasonable time scale. The notion of a quasistationary distribution has proved to be a potent tool in modelling this behaviour. In finite-state systems the existence of a quasistationary distribution is guaranteed. However, in the infinite-state case this may not always be so and the question of whether or not quasistationary distributions exist requires delicate mathematical analysis. The purpose of this paper is to present simple conditions for the existence of quasistationary distributions for continuous-time Markov chains and to demonstrate how these can be applied in practice.     

Computational homogenization for heterogeneous cohesive layers incorporating size effects

A numerical framework is presented for the multiscale modelling of material layers that possess a both heterogeneous and microstructured mesostructure. The material layer is represented as a cohesive interface on the macro level, while on the meso level micromorphic representative volume elements are used. A computational homogenization approach for the cohesive material layers has been proposed to solve this nonlinear multiscale problem numerically. The present framework is particularly well suited for the modelling of damage and failure, as the micromorphic RVE is well known for its regularizing character. (© 2010 Wiley-VCH Verlag GmbH & Co. KGaA, Weinheim)