45 research outputs found

    Applied Mathematics to Mechanisms and Machines

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    This book brings together all 16 articles published in the Special Issue "Applied Mathematics to Mechanisms and Machines" of the MDPI Mathematics journal, in the section “Engineering Mathematics”. The subject matter covered by these works is varied, but they all have mechanisms as the object of study and mathematics as the basis of the methodology used. In fact, the synthesis, design and optimization of mechanisms, robotics, automotives, maintenance 4.0, machine vibrations, control, biomechanics and medical devices are among the topics covered in this book. This volume may be of interest to all who work in the field of mechanism and machine science and we hope that it will contribute to the development of both mechanical engineering and applied mathematics

    Mechanisms of natural and forced variability in the southern ocean

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    The Southern Ocean is an important regulator of global climate, and accurately predicting its future evolution under climate change constitutes a critical scientific challenge. Mesoscale eddies are key to the dynamics of the Southern Ocean, but the mechanisms and time scales of their natural and forced variability are not completely understood. Motivated by the dynamical analogy between the Antarctic Circumpolar Current and the tropospheric jet stream, the natural variability of eddymean flow interaction is studied by adapting a two-dimensional model of storm track variability to the oceanic case. It is found that eddies and the mean flow interact according to a predator-prey oscillatory relationship in both an idealised, eddy-resolving, channel configuration and the SOSE state estimate product of the Southern Ocean. The oscillatory nature of the dynamics reflects in the structure of the phase space diagrams, where quasi-periodic cycles with typical timescales of a few weeks can be observed. The simplified mathematical model qualitatively captures the statistical properties of the interaction well. The time scales of forced adjustment are investigated by means of an ensemble of wind step-change experiments run with the idealised channel configuration. It is found that the temperature response is driven largely, but not exclusively, by changes in the ocean’s circulation, with enhanced mixing also playing an important role. Circulation changes have a rich spatial structure, and vertical/meridional displacements of the residual overturning circulation cells have a large impact on the temperature response even though the channel is strongly eddy-compensated. The time scales of the response vary across the domain, and are set by the spin-up of baroclinic eddies. The results presented in this Thesis bring the fundamental mechanisms of eddy variability into clearer focus, and inform the interpretation of more realistic numerical simulations of the Southern Ocean

    Electronic Journal of Qualitative Theory of Differential Equations 2022

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    A variational multiscale computational framework for reaction-dominated thermo-chemo-mechanical process modeling in multi-constituent material systems

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    This dissertation develops a computational framework for modeling multi-constituent material systems characterized by the transport of reacting fluids through deformable solids, and their coupled, nonlinear, thermo-chemo-mechanical response in the reaction-dominated regime. This is accomplished through two major components of the work: (i) new robust variational multi-scale numerical methods that are consistently derived, and (ii) models for multi-physics processes in multi-constituent materials. New robust numerical methods are developed via the variational multiscale (VMS) framework. Through the concept of fine scales in VMS, unresolved physics are recovered and embedded at the coarse scale level, improving stability and accuracy of the method. Focus is placed on fine scales that do not vanish at element boundaries (so-called “edge bubbles”). Using edge bubbles and an explicit time integration algorithm, a VMS Discontinuous Galerkin (VMDG) method is derived for multi-domain problems in elastodynamics where different subdomains can be solved synchronously and concurrently with minimal sharing of information. In addition, a new VMS method is introduced for the reaction-dominated regime of the diffusion–reaction equation. The proposed fine-scale basis consists of enrichment functions that may be nonzero at element edges. The method captures sharp boundary and internal layers, suppresses spurious oscillations, and better satisfies the maximum principle as compared to other existing methods. A priori mathematical analysis of the stability and convergence of the method is presented, and optimal rates of convergence are verified numerically. The numerical methods developed in this work may be applied to many reaction-diffusion systems in mathematical models for coupled thermo-chemo-mechanical phenomena arising from different theoretical frameworks. Here, a model for thermo-chemo-mechanical response of open solid-fluid systems is presented in the context of mixture theory. Derivation starts from constituent-wise equations for balance of mass, momentum, and energy, accounting for energy in formation and breaking of chemical bonds. Interactions between different constituents are captured through interaction terms as per locally homogenized mixture theory. Satisfaction of the second law of thermodynamics is achieved by providing constitutive equations that guarantee non-negative entropy production. Resulting mathematical models yield transient diffusion-advection-reaction problems posed by systems of coupled, nonlinear, second-order partial differential equations (PDEs), whose solution require stable numerical methods. Several numerical studies are presented to highlight stability, accuracy, and other features of the newly developed variational multiscale methods and thermo-chemo-mechanical models. Tests involve hypothetical as well as realistic materials with boundary layers, advancing reaction fronts, chemical swelling, and fingering phenomena

    Advanced Strategies for Robot Manipulators

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    Amongst the robotic systems, robot manipulators have proven themselves to be of increasing importance and are widely adopted to substitute for human in repetitive and/or hazardous tasks. Modern manipulators are designed complicatedly and need to do more precise, crucial and critical tasks. So, the simple traditional control methods cannot be efficient, and advanced control strategies with considering special constraints are needed to establish. In spite of the fact that groundbreaking researches have been carried out in this realm until now, there are still many novel aspects which have to be explored

    Heat Transfer

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    Over the past few decades there has been a prolific increase in research and development in area of heat transfer, heat exchangers and their associated technologies. This book is a collection of current research in the above mentioned areas and describes modelling, numerical methods, simulation and information technology with modern ideas and methods to analyse and enhance heat transfer for single and multiphase systems. The topics considered include various basic concepts of heat transfer, the fundamental modes of heat transfer (namely conduction, convection and radiation), thermophysical properties, computational methodologies, control, stabilization and optimization problems, condensation, boiling and freezing, with many real-world problems and important modern applications. The book is divided in four sections : "Inverse, Stabilization and Optimization Problems", "Numerical Methods and Calculations", "Heat Transfer in Mini/Micro Systems", "Energy Transfer and Solid Materials", and each section discusses various issues, methods and applications in accordance with the subjects. The combination of fundamental approach with many important practical applications of current interest will make this book of interest to researchers, scientists, engineers and graduate students in many disciplines, who make use of mathematical modelling, inverse problems, implementation of recently developed numerical methods in this multidisciplinary field as well as to experimental and theoretical researchers in the field of heat and mass transfer

    Partial Differential Equations in Ecology

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    Partial differential equations (PDEs) have been used in theoretical ecology research for more than eighty years. Nowadays, along with a variety of different mathematical techniques, they remain as an efficient, widely used modelling framework; as a matter of fact, the range of PDE applications has even become broader. This volume presents a collection of case studies where applications range from bacterial systems to population dynamics of human riots

    Nonlinear Control of Unmanned Aerial Vehicles : Systems With an Attitude

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    This thesis deals with the general problem of controlling rigid-body systems through space, with a special focus on unmanned aerial vehicles (UAVs). Several promising UAV control algorithms have been developed over the past decades, enabling truly astounding feats of agility when combined with modern sensing technologies. However, these control algorithms typically come without global stability guarantees when implemented with estimation algorithms. Such control systems work well most of the time, but when introducing the UAVs more widely in society, it becomes paramount to prove that stability is ensured regardless of how the control system is initialized.The main motivation of the research lies in providing such (almost) global stability guarantees for an entire UAV control system. We develop algorithms that are implementable in practice and for which (almost) all initial errors result in perfect tracking of a reference trajectory. In doing so, both the tracking and the estimation errors are shown to be bounded in time along (almost) all solutions of the closed-loop system. In other words, if the initialization is sound and the initial errors are small, they will remain small and decrease in time, and even if the initial errors are large, they will not increase with time.As the field of UAV control is mature, this thesis starts by reviewing some of the most promising approaches to date in Part I. The ambition is to clarify how various controllers are related, provide intuition, and demonstrate how they work in practice. These ideas subsequently form the foundation on which a new result is derived, referred to as a nonlinear filtered output feedback. This represents a diametrically different approach to the control system synthesis. Instead of a disjoint controller/estimator design, the proposed method is comprised of two controller/estimator pairs, which when combined through a special interconnection term yields a system with favorable stability properties.While the first part of the thesis deals with theoretical controller design,Part II concerns application examples, demonstrating how the theory can solve challenging problems in modern society. In particular, we consider the problem of circumnavigation for search and rescue missions and show how UAVs can gather data from radioactive sites to estimate radiation intensity

    Numerical Solution of Optimal Control Problems with Explicit and Implicit Switches

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    This dissertation deals with the efficient numerical solution of switched optimal control problems whose dynamics may coincidentally be affected by both explicit and implicit switches. A framework is being developed for this purpose, in which both problem classes are uniformly converted into a mixed–integer optimal control problem with combinatorial constraints. Recent research results relate this problem class to a continuous optimal control problem with vanishing constraints, which in turn represents a considerable subclass of an optimal control problem with equilibrium constraints. In this thesis, this connection forms the foundation for a numerical treatment. We employ numerical algorithms that are based on a direct collocation approach and require, in particular, a highly accurate determination of the switching structure of the original problem. Due to the fact that the switching structure is a priori unknown in general, our approach aims to identify it successively. During this process, a sequence of nonlinear programs, which are derived by applying discretization schemes to optimal control problems, is solved approximatively. After each iteration, the discretization grid is updated according to the currently estimated switching structure. Besides a precise determination of the switching structure, it is of central importance to estimate the global error that occurs when optimal control problems are solved numerically. Again, we focus on certain direct collocation discretization schemes and analyze error contributions of individual discretization intervals. For this purpose, we exploit a relationship between discrete adjoints and the Lagrange multipliers associated with those nonlinear programs that arise from the collocation transcription process. This relationship can be derived with the help of a functional analytic framework and by interrelating collocation methods and Petrov–Galerkin finite element methods. In analogy to the dual-weighted residual methodology for Galerkin methods, which is well–known in the partial differential equation community, we then derive goal–oriented global error estimators. Based on those error estimators, we present mesh refinement strategies that allow for an equilibration and an efficient reduction of the global error. In doing so we note that the grid adaption processes with respect to both switching structure detection and global error reduction get along with each other. This allows us to distill an iterative solution framework. Usually, individual state and control components have the same polynomial degree if they originate from a collocation discretization scheme. Due to the special role which some control components have in the proposed solution framework it is desirable to allow varying polynomial degrees. This results in implementation problems, which can be solved by means of clever structure exploitation techniques and a suitable permutation of variables and equations. The resulting algorithm was developed in parallel to this work and implemented in a software package. The presented methods are implemented and evaluated on the basis of several benchmark problems. Furthermore, their applicability and efficiency is demonstrated. With regard to a future embedding of the described methods in an online optimal control context and the associated real-time requirements, an extension of the well–known multi–level iteration schemes is proposed. This approach is based on the trapezoidal rule and, compared to a full evaluation of the involved Jacobians, it significantly reduces the computational costs in case of sparse data matrices
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