Modified HPMs Inspired by Homotopy Continuation Methods

Nonlinear differential equations have applications in the modelling area for a broad variety of phenomena and physical processes; having applications for all areas in science and engineering. At the present time, the homotopy perturbation method (HPM) is amply used to solve in an approximate or exact manner such nonlinear differential equations. This method has found wide acceptance for its versatility and ease of use. The origin of the HPM is found in the coupling of homotopy methods with perturbation methods. Homotopy methods are a well established research area with applications, in particular, an applied branch of such methods are the homotopy continuation methods, which are employed on the numerical solution of nonlinear algebraic equation systems. Therefore, this paper presents two modified versions of standard HPM method inspired in homotopy continuation methods. Both modified HPMs deal with nonlinearities distribution of the nonlinear differential equation. Besides, we will use a calcium-induced calcium released mechanism model as study case to test the proposed techniques. Finally, results will be discussed and possible research lines will be proposed using this work as a starting point

Fixed-Term Homotopy

A new tool for the solution of nonlinear differential equations is presented. The Fixed-Term Homotopy (FTH) delivers a high precision representation of the nonlinear differential equation using only a few linear algebraic terms. In addition to this tool, a procedure based on Laplace-Padé to deal with the truncate power series resulting from the FTH method is also proposed. In order to assess the benefits of this proposal, two nonlinear problems are solved and compared against other semianalytic methods. The obtained results show that FTH is a power tool capable of generating highly accurate solutions compared with other methods of literature

Piezoelectric Digital Vibration Absorbers for Multimodal Vibration Mitigation of Complex Mechanical Structures

Engineering structures are becoming lighter and more complex to accommodate the ever-increasing demand for performance and to comply with stringent environmental regulations. This trend comes with several challenges, one of which is the increased susceptibility to high-amplitude vibrations. These vibrations can be detrimental to structural performance and lifetime, and may sometimes even threaten safety. Passive and active vibration reduction techniques can provide a solution to this issue. Among the possibilities, piezoelectric damping is an attractive option, due to its compact and lightweight character, its reduced cost and its tunability. This technique uses the ability of a piezoelectric transducer to transform part of its mechanical energy into electrical energy. The converted energy can then be dissipated by connecting a shunt circuit to the transducer. However, the difficulty of realizing such circuits limits the broad applicability of piezoelectric shunting. This doctoral thesis investigates the potential of replacing the electrical circuit comprising classical components such as resistors and inductors by a digital unit and a current source, thereby creating a digital vibration absorber (DVA). Virtually any circuit can be emulated with a digital controller, providing this approach with an extreme versatility for vibration mitigation of complex mechanical structures. In this regard, the DVA is first analyzed in terms of power consumption and stability of the controlled system. Then, effective and easy-to-use tuning approaches for the control of multiple structural modes either with passive electrical circuits or a DVA are proposed, namely a passivity-based tuning of shunt circuits, a modal-based synthesis of electrical networks interconnecting multiple piezoelectric transducers, and a numerical norm-homotopy optimization resulting in an all-equal-peak design. These techniques are eventually applied and adapted to real-life structures with potentially complex dynamics. Specifically, effective vibration mitigation is demonstrated on structures exhibiting nonlinear behaviors and high modal density

SCEE 2008 book of abstracts : the 7th International Conference on Scientific Computing in Electrical Engineering (SCEE 2008), September 28 – October 3, 2008, Helsinki University of Technology, Espoo, Finland

This report contains abstracts of presentations given at the SCEE 2008 conference.reviewe

Applications of dynamical systems with symmetry

This thesis examines the application of symmetric dynamical systems theory to two areas in applied mathematics: weakly coupled oscillators with symmetry, and bifurcations in flame front equations. After a general introduction in the first chapter, chapter 2 develops a theoretical framework for the study of identical oscillators with arbitrary symmetry group under an assumption of weak coupling. It focusses on networks with 'all to all' Sn coupling. The structure imposed by the symmetry on the phase space for weakly coupled oscillators with Sn, Zn or Dn symmetries is discussed, and the interaction of internal symmetries and network symmetries is shown to cause decoupling under certain conditions. Chapter 3 discusses what this implies for generic dynamical behaviour of coupled oscillator systems, and concentrates on application to small numbers of oscillators (three or four). We find strong restrictions on bifurcations, and structurally stable heteroclinic cycles. Following this, chapter 4 reports on experimental results from electronic oscillator systems and relates it to results in chapter 3. In a forced oscillator system, breakdown of regular motion is observed to occur through break up of tori followed by a symmetric bifurcation of chaotic attractors to fully symmetric chaos. Chapter 5 discusses reduction of a system of identical coupled oscillators to phase equations in a weakly coupled limit, considering them as weakly dissipative Hamiltonian oscillators with very weakly coupling. This provides a derivation of example phase equations discussed in chapter 2. Applications are shown for two van der Pol-Duffing oscillators in the case of a twin-well potential. Finally, we turn our attention to the Kuramoto-Sivashinsky equation. Chapter 6 starts by discussing flame front equations in general, and non-linear models in particular. The Kuramoto-Sivashinsky equation on a rectangular domain with simple boundary conditions is found to be an example of a large class of systems whose linear behaviour gives rise to arbitrarily high order mode interactions. Chapter 7 presents computation of some of these mode interactions using competerised Liapunov-Schmidt reduction onto the kernel of the linearisation, and investigates the bifurcation diagrams in two parameters

Non-linear dynamics and power systems

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