ON THE THREE-DIMENSIONAL BLASCHKE-LEBESGUE PROBLEM

The width of a closed convex subset of n-dimensional Euclidean space is the distance between two parallel supporting hyperplanes. The Blaschke-Lebesgue problem consists of minimizing the volume in the class of convex sets of fixed constant width and is still open in dimension n >= 3. In this paper we describe a necessary condition that the minimizer of the Blaschke-Lebesgue must satisfy in dimension n = 3: we prove that the smooth components of the boundary of the minimizer have their smaller principal curvature constant and therefore are either spherical caps or pieces of tubes (canal surfaces).

GLOBALIZATION OF TWISTED PARTIAL ACTIONS

Let A be a unital ring which is a product of possibly infinitely many indecomposable rings. We establish a criteria for the existence of a globalization for a given twisted partial action of a group on A. If the globalization exists, it is unique up to a certain equivalence relation and, moreover, the crossed product corresponding to the twisted partial action is Morita equivalent to that corresponding to its globalization. For arbitrary unital rings the globalization problem is reduced to an extendibility property of the multipliers involved in the twisted partial action.

An alternative to the Pythagorean rule? Reevaluating Problem 1 of cuneiform tablet BM 34 568

The first problem of the Seleucid mathematical cuneiform tablet BM 34 568 calculates the diagonal of a rectangle from its sides without resorting to the Pythagorean rule. For this reason, it has been a source of discussion among specialists ever since its first publication. but so far no consensus in relation to its mathematical meaning has been attained. This paper presents two new interpretations of the scribe`s procedure. based on the assumption that he was able to reduce the problem to a standard Mesopotamian question about reciprocal numbers. These new interpretations are then linked to interpretations of the Old Babylonian tablet Plimpton 322 and to the presence of Pythagorean triples in the contexts of Old Babylonian and Hellenistic mathematics. (C) 2007 Elsevier Inc. All rights reserved.

Inverse analysis FOR two-dimensional structures using the boundary element method

Inverse analysis is currently an important subject of study in several fields of science and engineering. The identification of physical and geometric parameters using experimental measurements is required in many applications. In this work a boundary element formulation to identify boundary and interface values as well as material properties is proposed. In particular the proposed formulation is dedicated to identifying material parameters when a cohesive crack model is assumed for 2D problems. A computer code is developed and implemented using the BEM multi-region technique and regularisation methods to perform the inverse analysis. Several examples are shown to demonstrate the efficiency of the proposed model. (C) 2010 Elsevier Ltd. All rights reserved,

A hybrid Particle Swarm Optimization - Simplex algorithm (PSOS) for structural damage identification

This study proposes a new PSOS-model based damage identification procedure using frequency domain data. The formulation of the objective function for the minimization problem is based on the Frequency Response Functions (FRFs) of the system. A novel strategy for the control of the Particle Swarm Optimization (PSO) parameters based on the Nelder-Mead algorithm (Simplex method) is presented; consequently, the convergence of the PSOS becomes independent of the heuristic constants and its stability and confidence are enhanced. The formulated hybrid method performs better in different benchmark functions than the Simulated Annealing (SA) and the basic PSO (PSO(b)). Two damage identification problems, taking into consideration the effects of noisy and incomplete data, were studied: first, a 10-bar truss and second, a cracked free-free beam, both modeled with finite elements. In these cases, the damage location and extent were successfully determined. Finally, a non-linear oscillator (Duffing oscillator) was identified by PSOS providing good results. (C) 2009 Elsevier Ltd. All rights reserved

A STUDY OF PENALTY FORMULATIONS USED IN THE NUMERICAL APPROXIMATION OF A RADIALLY SYMMETRIC ELASTICITY PROBLEM

We consider a class of two-dimensional problems in classical linear elasticity for which material overlapping occurs in the absence of singularities. Of course, material overlapping is not physically realistic, and one possible way to prevent it uses a constrained minimization theory. In this theory, a minimization problem consists of minimizing the total potential energy of a linear elastic body subject to the constraint that the deformation field must be locally invertible. Here, we use an interior and an exterior penalty formulation of the minimization problem together with both a standard finite element method and classical nonlinear programming techniques to compute the minimizers. We compare both formulations by solving a plane problem numerically in the context of the constrained minimization theory. The problem has a closed-form solution, which is used to validate the numerical results. This solution is regular everywhere, including the boundary. In particular, we show numerical results which indicate that, for a fixed finite element mesh, the sequences of numerical solutions obtained with both the interior and the exterior penalty formulations converge to the same limit function as the penalization is enforced. This limit function yields an approximate deformation field to the plane problem that is locally invertible at all points in the domain. As the mesh is refined, this field converges to the exact solution of the plane problem.

The random barrier-crossing problem

This paper addresses the time-variant reliability analysis of structures with random resistance or random system parameters. It deals with the problem of a random load process crossing a random barrier level. The implications of approximating the arrival rate of the first overload by an ensemble-crossing rate are studied. The error involved in this so-called ""ensemble-crossing rate"" approximation is described in terms of load process and barrier distribution parameters, and in terms of the number of load cycles. Existing results are reviewed, and significant improvements involving load process bandwidth, mean-crossing frequency and time are presented. The paper shows that the ensemble-crossing rate approximation can be accurate enough for problems where load process variance is large in comparison to barrier variance, but especially when the number of load cycles is small. This includes important practical applications like random vibration due to impact loadings and earthquake loading. Two application examples are presented, one involving earthquake loading and one involving a frame structure subject to wind and snow loadings. (C) 2007 Elsevier Ltd. All rights reserved.

Leak detection by inverse transient analysis in an experimental PVC pipe system

Leakage reduction in water supply systems and distribution networks has been an increasingly important issue in the water industry since leaks and ruptures result in major physical and economic losses. Hydraulic transient solvers can be used in the system operational diagnosis, namely for leak detection purposes, due to their capability to describe the dynamic behaviour of the systems and to provide substantial amounts of data. In this research work, the association of hydraulic transient analysis with an optimisation model, through inverse transient analysis (ITA), has been used for leak detection and its location in an experimental facility containing PVC pipes. Observed transient pressure data have been used for testing ITA. A key factor for the success of the leak detection technique used is the accurate calibration of the transient solver, namely adequate boundary conditions and the description of energy dissipation effects since PVC pipes are characterised by a viscoelastic mechanical response. Results have shown that leaks were located with an accuracy between 4-15% of the total length of the pipeline, depending on the discretisation of the system model.

A simple and effective inverse projection scheme for void distribution control in topology optimization

The ability to control both the minimum size of holes and the minimum size of structural members are essential requirements in the topology optimization design process for manufacturing. This paper addresses both requirements by means of a unified approach involving mesh-independent projection techniques. An inverse projection is developed to control the minimum hole size while a standard direct projection scheme is used to control the minimum length of structural members. In addition, a heuristic scheme combining both contrasting requirements simultaneously is discussed. Two topology optimization implementations are contributed: one in which the projection (either inverse or direct) is used at each iteration; and the other in which a two-phase scheme is explored. In the first phase, the compliance minimization is carried out without any projection until convergence. In the second phase, the chosen projection scheme is applied iteratively until a solution is obtained while satisfying either the minimum member size or minimum hole size. Examples demonstrate the various features of the projection-based techniques presented.

Minimizing total tardiness in a stochastic single machine scheduling problem using approximate dynamic programming

This paper addresses the non-preemptive single machine scheduling problem to minimize total tardiness. We are interested in the online version of this problem, where orders arrive at the system at random times. Jobs have to be scheduled without knowledge of what jobs will come afterwards. The processing times and the due dates become known when the order is placed. The order release date occurs only at the beginning of periodic intervals. A customized approximate dynamic programming method is introduced for this problem. The authors also present numerical experiments that assess the reliability of the new approach and show that it performs better than a myopic policy.

Scatter search for a real-life heterogeneous fleet vehicle routing problem with time windows and split deliveries in Brazil

In this paper, we consider a real-life heterogeneous fleet vehicle routing problem with time windows and split deliveries that occurs in a major Brazilian retail group. A single depot attends 519 stores of the group distributed in 11 Brazilian states. To find good solutions to this problem, we propose heuristics as initial solutions and a scatter search (SS) approach. Next, the produced solutions are compared with the routes actually covered by the company. Our results show that the total distribution cost can be reduced significantly when such methods are used. Experimental testing with benchmark instances is used to assess the merit of our proposed procedure. (C) 2008 Published by Elsevier B.V.

A Separation Principle for the Continuous-Time LQ-Problem With Markovian Jump Parameters

In this paper, we devise a separation principle for the finite horizon quadratic optimal control problem of continuous-time Markovian jump linear systems driven by a Wiener process and with partial observations. We assume that the output variable and the jump parameters are available to the controller. It is desired to design a dynamic Markovian jump controller such that the closed loop system minimizes the quadratic functional cost of the system over a finite horizon period of time. As in the case with no jumps, we show that an optimal controller can be obtained from two coupled Riccati differential equations, one associated to the optimal control problem when the state variable is available, and the other one associated to the optimal filtering problem. This is a separation principle for the finite horizon quadratic optimal control problem for continuous-time Markovian jump linear systems. For the case in which the matrices are all time-invariant we analyze the asymptotic behavior of the solution of the derived interconnected Riccati differential equations to the solution of the associated set of coupled algebraic Riccati equations as well as the mean square stabilizing property of this limiting solution. When there is only one mode of operation our results coincide with the traditional ones for the LQG control of continuous-time linear systems.

On the Filtering Problem for Continuous-Time Markov Jump Linear Systems with no Observation of the Markov Chain

We consider in this paper the optimal stationary dynamic linear filtering problem for continuous-time linear systems subject to Markovian jumps in the parameters (LSMJP) and additive noise (Wiener process). It is assumed that only an output of the system is available and therefore the values of the jump parameter are not accessible. It is a well known fact that in this setting the optimal nonlinear filter is infinite dimensional, which makes the linear filtering a natural numerically, treatable choice. The goal is to design a dynamic linear filter such that the closed loop system is mean square stable and minimizes the stationary expected value of the mean square estimation error. It is shown that an explicit analytical solution to this optimal filtering problem is obtained from the stationary solution associated to a certain Riccati equation. It is also shown that the problem can be formulated using a linear matrix inequalities (LMI) approach, which can be extended to consider convex polytopic uncertainties on the parameters of the possible modes of operation of the system and on the transition rate matrix of the Markov process. As far as the authors are aware of this is the first time that this stationary filtering problem (exact and robust versions) for LSMJP with no knowledge of the Markov jump parameters is considered in the literature. Finally, we illustrate the results with an example.

New simple and efficient heuristics for the uncapacitated single allocation hub location problem

Hub-and-spoke networks are widely studied in the area of location theory. They arise in several contexts, including passenger airlines, postal and parcel delivery, and computer and telecommunication networks. Hub location problems usually involve three simultaneous decisions to be made: the optimal number of hub nodes, their locations and the allocation of the non-hub nodes to the hubs. In the uncapacitated single allocation hub location problem (USAHLP) hub nodes have no capacity constraints and non-hub nodes must be assigned to only one hub. In this paper, we propose three variants of a simple and efficient multi-start tabu search heuristic as well as a two-stage integrated tabu search heuristic to solve this problem. With multi-start heuristics, several different initial solutions are constructed and then improved by tabu search, while in the two-stage integrated heuristic tabu search is applied to improve both the locational and allocational part of the problem. Computational experiments using typical benchmark problems (Civil Aeronautics Board (CAB) and Australian Post (AP) data sets) as well as new and modified instances show that our approaches consistently return the optimal or best-known results in very short CPU times, thus allowing the possibility of efficiently solving larger instances of the USAHLP than those found in the literature. We also report the integer optimal solutions for all 80 CAB data set instances and the 12 AP instances up to 100 nodes, as well as for the corresponding new generated AP instances with reduced fixed costs. Published by Elsevier Ltd.

Determining the effect of grain size and maximum induction upon coercive field of electrical steels

Although theoretical models have already been proposed, experimental data is still lacking to quantify the influence of grain size upon coercivity of electrical steels. Some authors consider a linear inverse proportionality, while others suggest a square root inverse proportionality. Results also differ with regard to the slope of the reciprocal of grain size-coercive field relation for a given material. This paper discusses two aspects of the problem: the maximum induction used for determining coercive force and the possible effect of lurking variables such as the grain size distribution breadth and crystallographic texture. Electrical steel sheets containing 0.7% Si, 0.3% Al and 24 ppm C were cold-rolled and annealed in order to produce different grain sizes (ranging from 20 to 150 mu m). Coercive field was measured along the rolling direction and found to depend linearly on reciprocal of grain size with a slope of approximately 0.9 (A/m)mm at 1.0 T induction. A general relation for coercive field as a function of grain size and maximum induction was established, yielding an average absolute error below 4%. Through measurement of B(50) and image analysis of micrographs, the effects of crystallographic texture and grain size distribution breadth were qualitatively discussed. (C) 2011 Elsevier B.V. All rights reserved.