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## Über dieses Buch

This book presents a careful selection of the contributions presented at the Mathematical Methods in Engineering (MME10) International Symposium, held at the Polytechnic Institute of Coimbra- Engineering Institute of Coimbra (IPC/ISEC), Portugal, October 21-24, 2010.

The volume discusses recent developments about theoretical and applied mathematics toward the solution of engineering problems, thus covering a wide range of topics, such as: Automatic Control, Autonomous Systems, Computer Science, Dynamical Systems and Control, Electronics, Finance and Economics, Fluid Mechanics and Heat Transfer, Fractional Mathematics, Fractional Transforms and Their Applications, Fuzzy Sets and Systems, Image and Signal Analysis, Image Processing, Mechanics, Mechatronics, Motor Control and Human Movement Analysis, Nonlinear Dynamics, Partial Differential Equations, Robotics, Acoustics, Vibration and Control, and Wavelets.

## Inhaltsverzeichnis

### Mathematical Modeling for Software-in-the-Loop Prototyping of Automated Manufacturing Systems

Abstract
Nowadays automated manufacturing systems are designed as the complex interconnection of components belonging to different engineering domains. Actually high performances are required in order to satisfy market needs and standards. In this framework the validation via simulation plays a crucial role as it allows to verify the system during the design phase. Software-in-the-loop architectures represent a good practice to take into account also technological side-effects that represent a classical cause of long time-to-market or, in the worst case, to project failure. In this paper we present a mathematical simulator to be used within a software-in-the-loop prototyping system.
Claudio Bonivento, Matteo Cacciari, Andrea Paoli, Matteo Sartini

### A New Parallel Matrix Multiplication Algorithm for Wormhole-Routed All-Port 2D/3D Torus Networks

Abstract
A new matrix multiplication algorithm is proposed for massively parallel supercomputers with 2D/3D, all-port torus interconnection networks. The proposed algorithm is based on the traditional row-by-column multiplication matrix product model and employs a special routing pattern for better scalability. It compares favorably to the variants of Cannon’s and DNS algorithms since it allows matrices of the same size to be multiplied on a higher number of processors due to lower data communications overhead.
Cesur Baransel, Kayhan Imre, Harun Artuner

### 2.5D Acoustic Wave Propagation in Shallow Water Over an Irregular Seabed Using the Boundary Element Method

Abstract
In this paper a Boundary Element formulation, in the frequency domain, is used to investigate the 2.5D acoustic wave propagation in shallow water over an irregular seabed that is assumed to have a rigid bottom and a free surface.
The problem is solved using a model which incorporates Green’s functions that take into account the presence of flat surfaces. With this procedure only the irregular bottom and the vertical interface between regions of different depths are discretized. The model is implemented to obtain the 3D time domain pressure responses in a shallow water region with step or slope irregularities, originated by point pressure loads placed at different positions. Simulations are performed to identify wave propagation features that may help the assessment of the presence and shape of the bottom irregularities.
A. Pereira, A. Tadeu, L. Godinho, J. A. F. Santiago

### Towards Replacing Lyapunov’s “Direct” Method in Adaptive Control of Nonlinear Systems

Abstract
In adaptive nonlinear control Lyapunov’s 2nd or “Direct” method became a fundamental tool in control design due to the typical practical difficulties viz. a) most of the control problems do not have closed analytical solutions; b) from numerical calculations “well behaving within a finite period” the stability cannot be taken for granted. According to Lyapunov, guaranteeing negative time-derivative of the Lyapunov function by relatively simple estimations the stability of the solution can theoretically be guaranteed. However, finding an appropriate Lyapunov function to a given problem is rather an “art” that cannot algorithmically be automated. Adaptivity normally requires slow tuning of numerous model parameters. This process is sensitive to unknown external disturbances, and the tuning rule is determined by numerous other, more or less arbitrary “adaptive control parameters”. Furthermore, making the necessary estimations is a laborious, tedious work that normally results in “very strange conditions” to be met for guaranteeing stability of the solution. In the present paper the application of “Robust Fixed Point Transformations” is proposed instead of the Lyapunov technique. It can find the proper solution without any parameter tuning and depends on the setting only of three “adaptive control parameters”. As application example direct control of a “Single Input—Single Output (SISO)” system, and a novel version of the “Model Reference Adaptive Control (MRAC)” of a “Multiple Input—Multiple Output (MIMO)” system is presented. Since this method cannot automatically guarantee global stability, as a novelty, a possible adaptive tuning of one of the adaptive control parameters is proposed for SISO systems to keep the control within the local basin of attraction of the proper convergence. Its operation is presented via simulations at first time in this paper.
József K. Tar

### Fractional Particle Swarm Optimization

Abstract
The paper addresses new perspective of the PSO including a fractional block. The local gain is replaced by one of fractional order considering several previous positions of the PSO particles. The algorithm is evaluated for several well known test functions and the relationship between the fractional order and the convergence of the algorithm is observed. The fractional order influences directly the algorithm convergence rate demonstrating a large potential for developments.
E. J. Solteiro Pires, J. A. Tenreiro Machado, P. B. de Moura Oliveira

### Toward the Concept of Robot Society: A Multi-Robot SLAM Case Study

Abstract
Over time, biological societies such as humans, ants or bees have shown us the advantages inherent to the collective work. It is based on such results that many researchers have been trying to successfully develop new approaches in Multi-Robot Systems. Nevertheless, several assumptions need to be assured for collective work to emerge. In this paper, it is presented the significance and the advantages of cooperation in the different societies bridging the gap to the concept of robot society. In order to compare the advantages of cooperative robots, it is considered essential the development of computational simulation based on the robotic cooperation in unstructured environments. Hence, a Multi-Robot Simultaneous Localization and Mapping (SLAM) using Rao-Blackwellized particle filter is implemented in a simulation environment developed in the Player/ Stage platform for robot and sensor applications.
Micael S. Couceiro, Andria R. Lopes, N. M. Fonseca Ferreira, Anabela G. Ferreira, Rui Rocha

### Generalized State-Space Modeling for m Level Diode-Clamped Multilevel Converters

Abstract
Multilevel power converter structures have been introduced as the solutions for high power high voltage applications and also for grid interface connection of renewable energy sources systems, where they have several advantages, namely low distortion voltages and currents, low switching losses resulting in higher efficiency. As a consequence of the increasing interest on multilevel converter applications, accurate models of these power converters are essential for computer simulation studies. This paper presents a systematic modeling approach suitable to obtain generalized state-space models for m level diode-clamped multilevel converters supplying AC loads. In particular, for m = 5, the proposed model is compared to the corresponding model using general purpose Simulink blocks and SimPowerSystems toolbox.
Miguel Chaves, Elmano Margato, J. Fernando Silva, Sónia F. Pinto

### Modelling Codfish Drying: Comparison Between Artificial Neural Network, Diffusive and Semi-Empirical Models

Abstract
Convective drying is of prime importance in the food conservation industry and has been constantly studied and improved to obtain products with higher quality and lower processing time. In this work, three different models were used to perform the codfish drying simulation: artificial neural network (ANN), diffusive and semi-empirical models. The simulation results were compared for the following experimental conditions: drying air temperature of 20 °C, air velocities of 2 and 3 m/s and drying air relative humidities comprise between 55 and 65 %. The simulations showed good results for the semi-empirical and ANN models, requiring improvements to the diffusion model.
CN Boeri, FJ Neto da Silva, JAF Ferreira

### Stability of Matrix Differential Equations with Commuting Matrix Constant Coefficients

Abstract
Sufficient conditions for the asymptotic stability of systems of first order linear differential equations with commuting matrix constant coefficients is studied. Stability criterion in terms of blocks is presented. Inertia of a block circulant matrix is obtained.
Fernando Martins, Edgar Pereira, M. A. Facas Vicente, José Vitória

### Identification of Material Thermophysical Parameters with Regard to Optimum Location of Sensors

Abstract
As a practical example illustrating the considerations presented in the paper the thermal processes proceeding in a system casting-mould are considered.
The casting is made from Fe-C alloy (cast iron) and the austenite and eutectic latent heats of this material should be identified. To estimate these parameters the knowledge of temperature history at the points selected from the domain considered is necessary. The location of sensors should assure the best conditions of identification process.
So, the algorithm of optimum location of sensors basing on the D-optimality criterion is presented, while the inverse problem is solved using the gradient method.
Ewa Majchrzak, Jerzy Mendakiewicz

### Mathematical Modeling of Heat and Mass Transfer in Domain of Solidifying Alloy

Abstract
In the paper the mathematical model, numerical algorithm and example of cylindrical casting solidification are presented. In particular the casting made from Cu-Zn alloy is considered. It is assumed that the temperature corresponding to the beginning of solidification is time-dependent and it is a function of temporary alloy component concentration. The course of macrosegregation has been modeled using the mass balances in the set of control volumes resulting from a domain discretization. The balances have been constructed in different ways, in particular under the assumption of instant equalization of alloy chemical constitution (a lever arm rule), next the Scheil model (e.g. Sczygiol 2000, Publ Czest Univ Techn Monographs, 71) has been used and finally the broken line model (Curran et al. 1980, Appl Math Modelling, 4, 398–400) has been taken into account. On a stage of numerical algorithm construction the boundary element method has been used in the variant called BEM using discretization in time (Curran et al. 1980, Appl Math Modelling, 4, 398–400; Sichert 1989, Technischen Fakultat der Universitat Erlangen; Szopa 1999, Publ. of the Silesian Univ. of Techn, 54) supplemented by the alternating phase truncation procedure (Majchrzak and B.Mochnacki 1995, Engineering Analysis with Boundary Elements, 16, 99–121; Lara 2003, Application of generalized FDM in numerical modelling of moving boundary problems, Doctoral Theses, Czestochowa).
Bohdan Mochnacki, Ewa Majchrzak

### Total Variation Approach to Density Reconstruction from X-Ray Radiograph Tomography

Abstract
For cone beam x-ray radiographic tomography, density reconstruction for 1-dimensional objects can be performed by Abel transform inversion. For 2-dimensional cylindrical objects, we propose to divide the object into small blocks, in each block the density is viewed as a constant. The projection operator corresponds to a matrix. Density reconstruction leads to solve a linear algebraic equation system. To deal with its ill conditioning, we use Total Variation regularization. Numerical experiments show that TV regularization gives correct recovery of object edges, while the density contrast may be changed in some smooth parts.
Suhua Wei, Guiping Zhao

### The Improvement of Total Variation Based Image Restoration Method and Its Application

Abstract
Total variation based image restoration method was first proposed by Rudin Osher and Fatermi in 1992. The images resulting from its application are usually piecewise constant, and have sometimes undesirable staircasing effect. To reduce this effect, we propose an improved model by combining the advantages of total variation and H 1 regularization. The new model substantially reduces the staircase effect, while preserving sharp edges. This model can be used in image reconstruction, it has advantages of keeping edges and recovering smooth region’s value. We give 1D and 2D experimental results to show the efficiency of the proposed model.
Suhua Wei, Guiping Zhao

### Adaptive System for Control of Active Ankle-Foot Orthosis and Gait Analysis

Abstract
The main aim of this research is the development of an autonomous adaptive system for actuation, data acquisition and control of active ankle-foot orthosis. In this paper the design of a control unit composed by microcontroller, driver and sensor system, and its application to the actuation and position of the foot orthotic segment is presented. The research work combines hardware and software design of the intelligent control device with graphical interface for representation and analysis of the data acquired during human motion. The dynamic system simulation is done in Matlab Simulink and SimMechanics.
A laboratory model of the proposed system was implemented to demonstrate its autonomy and verify experimentally its functionality.
The proposed control device can be used in several applications involving human motion analysis and control of different types of orthoses or functional electrical stimulation used for gait correction.
Ivanka Veneva, Nuno Ferreira

### Vector Ellipsoidal Harmonics Structure Peculiarities and Limitations

Abstract
The theory of scalar ellipsoidal harmonics was introduced by Lamé in 1837, more than half a century after Laplace introduced his theory of spherical harmonics in 1782. It is amazing that the relative theory of vector spherical harmonics was demonstrated as late as 1935 by Hansen.The appearance of a corresponding theory for vector ellipsoidal harmonics was resisting until 2009, for the very simple reason that such a theory can not exist, at least not in such a nice form as the theory of their spherical counterparts. The intrinsic difficulties of the ellipsoidal coordinate system, the fine and symmetric structure that is encoded in the anisotropic character of the ellipsoidal geometry, as well as the necessary generalizations and limitations that are needed are discussed in this presentation. Furthermore, the new analytical techniques that are suggested through the introduction of vector ellipsoidal harmonics are also demonstrated via special examples.
George Dassios

### Casualties Distribution in Human and Natural Hazards

Abstract
Catastrophic events, such as wars and terrorist attacks, big tornadoes and hurricanes, huge earthquakes, tsunamis, floods, and landslides, are always accompanied by a large number of casualties. The size distribution of these casualties have separately been shown to follow approximate power law (PL) distributions. In this paper, we analyze the number of victims of catastrophic phenomena, in particular, terrorism, and find double PL behavior. This means that the data set is better approximated by two PLs instead of one. We have plotted the two PL parameters corresponding to all terrorist events occurred in every year, from 1980 to 2010. We observe an interesting pattern in the chart, where the lines, that connect each pair of points defining the double PLs, are roughly aligned to each other.
Carla M. A. Pinto, A. Mendes Lopes, J. A. Tenreiro Machado

### Optimization of Quadruped Robot Locomotion Gaits Through a Genetic Algorithm

Abstract
During the last years research and development on legged robots has grown steadily. Legged systems present major advantages when compared with “traditional” vehicles, allowing locomotion in terrain inaccessible to vehicles with wheels and tracks. However, its energy consumption still lag being these vehicles, existing several aspects that need to be improved and optimized. One of them regards the parameters values that these machines should adopt to minimize the energy consumption. Due to the large number of parameters involved in this optimization process, one way to achieve meaningful results is using evolutionary strategies. Genetic Algorithms are a way to “imitate nature” replicating the process that nature designed for the generation and evolution of species. The objective of this paper is to present a genetic algorithm, running over a simulation application of legged robots, which allows the optimization of several parameters of a quadruped robot model, for distinct locomotion gaits.
Manuel F. Silva

### Analysis of an Incomplete Information System Using the Rough Set Theory

Abstract
In this paper it is applied a Rough Set approach that takes into account an incomplete information system to study the steady-state security of an electric power system. The Rough Set Theory has been conceived as a tool to conceptualize, organize and analyze various types of data, in particular, to deal with inexact, uncertain or vague knowledge. The knowledge acquisition process is a complex task, since the experts have difficulty to explain how to solve a specified problem. So, an incomplete set of relevant information may arise. The study presents a systematic approach to transform examples in a reduced set of rules. These rules can be used successfully to avoid security problems and provides a deeper insight into the influence of parameters on the steady-state system performance.
C. I. Faustino Agreira, M. M. Travassos Valdez, C. M. Machado Ferreira, F. P. Maciel Barbosa

### Chain Drives Modelling Using Kinematic Constraints and Revolute Clearance Joints Formulations

Abstract
Based on Multibody Dynamics two different formulations for modelling chain drive mechanisms are presented in this work: (i) one in which the revolute joints are considered as ideal joints, modelled as kinematic constraints; (ii) and another in which the kinematic constraints are removed and replaced by a pair of forces representing the contact between the connected bodies, i.e., modelled using the revolute clearance joint formulation. When the chain drive components’ connections are modelled as kinematic joints, the integration of the equations of motion lead to constraint violations that grow to a point at which the chain seems to start vibrating with a very high frequency and ends up disintegrating, even when the Baumgarte stabilization method is used. This problem is, however, eliminated when the interaction between the chain drive components is modelled using the revolute clearance joint formulation, since any constraint violation is exhibited as the number of kinematic constraints used in the multibody model is kept to a minimum.
Cândida Pereira, Jorge Ambrósio

### Jacobi Polynomials and Some Related Functions

Abstract
The classical Jacobi orthogonal polynomials (especially their special case—the Legendre polynomials) appear as the solutions of some problems of mathematical physics. In the contribution we deal with some relations connecting generalized Legendre polynomials of a certain type and the classical Jacobi polynomials orthogonal with respect to two different special weight functions. We also point out relations between the classical Legendre polynomials, the associated Legendre functions of the first kind, the Legendre functions of the first kind and the generalized g-Legendre functions obtained by Mirevski et al. using fractional calculus.

### Descartes Rule of Signs and Linear Programming

Abstract
Let $$\Delta=\{(x,y):x+y=1, x,y \geq 0\}$$ be the 1-simplex and for $$m\geq 2$$ consider the (binary) form
$$F(x,y)=u_n x^n +u_0 y^n -\sum_{\begin{array}{c} i, j\geq 1 \\ i+j=n\end{array}} u_i x^i y^j.$$
Using linear programming and a little known refinement of Descartes’ rule of signs due to Laguerre, it is shown that if all $$u_i\geq 0$$ and F is nonzero and nonnegative on $$\Delta,$$ then it assumes there exactly one global minimum. The investigation is motivated by a question concerning sum of squares representation.
Carla Fidalgo, Alexander Kovačec

### Multidimensional Markov Chains Usage in the Radio Resource Management Problem

Abstract
This paper presents an analytical model that evaluates the performance of the Maximum Packing channel allocation technique on linear and planar cellular systems. The main innovations introduced by this model are the deterministic identification of the system space-state SMP and its application to a multi-dimensional Markov chain.
The model was thereafter applied to several cellular systems with different characteristics: number of cells, number of channels, interference constraints and offered traffic values. Simulation results have validated the model and shown that Maximum Packing technique provides the best performance among all the available algorithms.
Victor D. N. Santos, N. M. Fonseca Ferreira, F. Moita

### A Manufacturing Scheduling Approach by Combining Simulation Technique with the Hodgson’s Algorithm

Abstract
The main objective of this paper consists on presenting an application of The Hodgson’s manufacturing scheduling algorithm applied on a production system model, through simulation technique. The simulation, executed on Arena, is applied to a model of a production environment for producing three new products, by using five different components. The components are processed on several work centres, which include different machines, organized on a job shop environment. The Hodgson’s algorithm is applied on a particular machine of the job shop, which consists on a bottleneck, and the main objective consists on minimizing the total number of tardy jobs on that bottleneck machine.
Telmo Pinto, Leonilde Varela

### Long Time Numerical Approximation of Coherent-Structure Solutions of the Cubic Schrödinger Equation

Abstract
The purpose of this work is to determine suitable numerical methods to simulate the evolution of coherent structures for the cubic nonlinear Schrödinger equation with Dirichlet boundary conditions on a finite one-dimensional interval. We consider different time integrators, some of them preserving one or two invariants of the problem. We show that the preservation of these invariants is essential for a good long time simulation.
I. Alonso-Mallo, A. Durán, N. Reguera

### A Statistical Approach for Tuning the Windowed Fourier Transform

Abstract
A time frequency analysis is used in many fields for studying signals with a time-varying spectral content. The windowed Fourier transform is one of the most used time-frequency representations. In order to use this technique several parameters must be defined, including the type, the length and the overlap of the windows. For tuning the windowed Fourier transform a new method based on the information theory is presented. Several tests with robotic signals illustrate the appropriateness of the proposed method.
Miguel F. M. Lima, J. A. Tenreiro Machado

### Can People With High Physical Movement Restrictions Access to Any Computer? The CaNWII Tool

Abstract
The potential of the common webcam, allied to the technology of the command of the well known Nintendo’s game console, the WII, enlarge the possibilities of new ways to interact with computers. The presented work describe one of those ways, an accessibility tool to people with very restrict physical movements. The CaNWII tool allows an easy and robust way to interact with any computer.
N. Rodrigues, N. Martins, J. Barbosa

### A Proposal for Detection and Estimation of Golf Putting

Abstract
This study presents an experimental research design of a PhD work, studying the effects of the variability in the performance of the Golf putting. The instruments used to analyze the putting were two digital cameras to detect the relevant dynamic objects (i.e., ball and putter) and a biaxial accelerometer to confirm the exact moment at which the putter hits the ball. To synchronize the instruments, it was used a trigger. The ball’s trajectory and the putting movement were automatically analyzed based on visual detection and parameter estimation. The kinematic analysis of the putting was performed using the Matlab software, to determine the amplitude, velocity and acceleration of the players’ gestures. We concluded that the Golf putting action parameters are divided into different stages (i.e., backswing, downswing and follow-through) and that those can be useful to investigate the effects of variability in this movement.
Gonçalo Dias, J. Miguel A. Luz, Micael S. Couceiro, Carlos M Figueiredo, Nuno Ferreira, Pedro Iglésias, Rui Mendes, Maria Castro, Orlando Fernandes

### Analysis of Electricity Market Prices Using Multidimensional Scaling

Abstract
This paper studies the impact of the energy upon electricity markets using Multidimensional Scaling (MDS). Data from major energy and electricity markets is considered. Several maps produced by MDS are presented and discussed revealing that this method is useful for understanding the correlation between them. Furthermore, the results help electricity markets agents hedging against Market Clearing Price (MCP) volatility.