Introduce every concept in the simplest setting and to maintain a level of treatment that is as rigorous as possible without being unnecessarily abstract.

Contains unique recent developments of various finite elements such as nonconforming, mixed, discontinuous, characteristic, and adaptive finite elements, along with their applications.

Describes unique recent applications of finite element methods to important fields such as multiphase flows in porous media and semiconductor modelling.

Treats the three major types of partial differential equations, i.e., elliptic, parabolic, and hyperbolic equations.

The text of the Persian poet Rum - - ?, written some eight centuries ago, and reproduced at the beginning of this book is still relevant to many of our pursuits of knowledge, not least of turbulence. The text illustrates the inability people have in seeing the whole thing, the 'big picture'. Everybody looks into the problem from his/her vi- point, and that leads to disagreement and controversy. If we could see the whole thing, our understanding would become complete and there would be no cont- versy. The turbulent motion of the atmosphere and oceans, at the heart of the observed general circulation, is undoubtedly very complex and dif?cult to understand in its entirety. Even 'bare' turbulence, without rotation and strati?cation whose effects are paramount in the atmosphere and oceans, still poses great fundamental ch- lenges for understanding after a century of research. Rotating strati?ed turbulence is a relatively new research topic. It is also far richer, exhibiting a host of distinct wave types interacting in a complicated and often subtle way with long-lived - herent structures such as jets or currents and vortices. All of this is tied together by basic ?uid-dynamical nonlinearity, and this gives rise to a multitude of phen- ena: spontaneous wave emission, wave-induced transport, both direct and inverse energy scale cascades, lateral and vertical anisotropy, fronts and transport barriers, anomalous transport in coherent vortices, and a very wide range of dynamical and thermodynamical instabilities.

This book is a continuation of 'Acoustic and Elastic Wave Fields in Geophysics, Part I' published in 2000. The second volume is dedicated to propagation of linear plane, spherical and cylindrical acoustic waves in different media. Chapter 1 is devoted to principles of geometric acoustic in plane wave approximation. The eikonal and transport equations are derived. Ray tracing and wavefront construction techniques are explained. Chapter 2 deals with dynamic properties of wave fields. The behavior of pressure and displacements amplitudes in zero approximation is analysed in two ways: using Poynting vector and solving the transport equation. This chapter contains several examples related to shadow zones and caustics. In Chapter 3 using the results of analysis of high-frequency wave kinematics and dynamics some fundamental aspects of Kirchhoff migration are described. Chapters 4 and 5 are devoted to propagation of plane waves in media with flat boundaries in the case of normal and oblique incidence. Special attention is paid to the case when an incident angle exceeds the critical angles. Formation of normal modes in the waveguide is discussed. Chapter 6 deals with a spherical wave reflection and refraction. The steepest descent method is introduced to describe the behavior of reflected, transmitted, head and evanescent waves. In Chapter 7 propagation of stationary and transient waves in a waveguide formed by a flat layer with low velocity are investigated. Normal modes and waves related to the branch points of integrands under consideration are studied. Dispersive properties of normal modes are discussed. Chapter 8 describes wave propagation inside cylinder in acoustic media. Several appendices are added to help the reader understand different aspects of mathematics used in the book.

This volume contains selected presentations of the "EUROMECH Colloquium 412 on LES of complex transitional and turbulent flows" held at the Munich University of Technology from 4 to 6 October 2000.

The articles focus on new developments in the field of large-eddy simulation of complex flows and are related to the topics: modelling and analysis of subgrid scales, numerical issues in LES cartesian grids for complex geometries, curvilinear and non-structured grids for complex geometries. DES and RANS-LES coupling, aircraft wake vortices, combustion and magnetohydrodynamics.

Progress has been made not only in understanding and modelling the dynamics of unresolved scales, but also in designing means that prevent the contamination of LES predictions by discretization errors. Progress is reported as well on the use of cartesian and curvilinear coordinates to compute flow in and around complex geometries and in the field of LES with unstructured grids. A chapter is dedicated to the detached-eddy simulation technique and its recent achievements and to the promising technique of coupling RANS and LES solutions in order to push the resolution-based Reynolds number limit of wall-resolving LES to higher values.

Complexity due to physical mechanisms links the last two chapters. It is shown that LES constitutes the tool to analyse the physics of aircraft wake vortices during landing and takeoff. Its thorough understanding is a prerequisite for reliable predictions of the distance between consecutive landing airplanes. Subgrid combustion modelling for LES of single and two-phase reacting flows is demonstrated to have the potential to deal with finite-rate kinetics in high Reynolds numberflows of full-scale gas turbine engines. Fluctuating magnetic fields are more reliably predicted by LES when tensor-diffusivity rather than gradient-diffusion models are used. An encouraging result in the context of turbulence control by magnetic fields.

The symposiumwas motivatedby theincreasing need for modelling of material behaviourundervarious mechan icalconditions. This need is driven by the evolut ion ofcomputer capac ityand the resulting ability for engineers and scien tiststo address complexproblems . Reliable models formaterialbehaviour, including accurate numericalvalues of parameters ,are necessary for a continued beneficial development ofthe computational side of solid mechanics .High rate plasticity ,thermally assisted creep and phasetransformationsare only a fewexamplesof areas where more accurate modelsare needed. Experiments are necessary for the establishment ofmodels and parameters , and modified versionsof conventional test methods can make important contributions . Also modern optical methodsoffer a highpotentialfor futureexperimental development. Numerical simulations ofexperiments and so-called inverse modelling arealso frequentlyused techniques. The aim of the symposium was to bring together researchers with an interest in the areaofexperimental and computational aspects ofmaterial modelling for exchange and discussionofpromising methodsandresults. Abisko,a national park in the Swedish mountain district about 200 km north of the arctic circle and about one hourve dri from the airport ofKiruna,was chosen for the symposium. The tourist hotel in the park , overlookinga beautiful lake , offered a suitablevenue for the symposium. This environment with tracks for short walks (and long hikes),goals for small excursions and a hotel with restaurant and bar ve the ga delegatesmany opportunitiesto meet , socialiseand discuss during breaks and evenings.

The principal object of this volume is the creation of a mathematical theory of deformations for elastic anisotropic thermodynamic piezoelastic plates, beams and shells with variable thickness. The book is divided into two parts. The first part deals with problems related to the construction of refined theories (such as those of Richhof-Love, von Karman-A. Fioppl, and Reissner) and their equivalent new models (depending on arbitrary control functions). These are investigated by means of a new variational principle. Methods of reduction, containing regular processes of study of spatial problems, are also studied. Topics treated include problems of solvability, error estimations, convergence of processes in Sobolev spaces and construction of effective schemes of solutions of two-dimensional boundary value problems for systems of partial differential equations. The second part considers stable projective methods, using classical orthogonal polynomials and a new class of spline-functions as coordinate systems, and their numerical realizations for a design of one- and two- dimensional boundary value problems from the first part. These efficient methods increase the possibilities of classical finite-difference, exponential- fitted, variational-discrete and alternating-direction methods. Audience: This book will be of interest to researchers and graduate students whose work involves mechanics, analysis, numerics and computation, mathematical modelling and industrial mathematics, calculus of variations, and design engineering.

This collection is dedicated to the 70th jubilee of Yu. N. Savchenko, and presents experimental, theoretical, and numerical investigations written by an international group of well-known authors. The contributions solve very important problems of the high-speed hydrodynamics, such as supersonic motion in water, drag diminishing, dynamics and stability of supercavitating vehicles, water entry and hydrodynamic performances of hydrofoils, ventilated cavities after a disc and under the ship bottom.
The book is written for researches, scientists, engineers, and students interested in problems of hydromechanics."

All phenomena in nature are characterized by motion. Mechanics deals with the objective laws of mechanical motion of bodies, the simplest form of motion. In the study of a science of nature, mathematics plays an important role. Mechanics is the first science of nature which has been expressed in terms of mathematics, by considering various mathematical models, associated to phenomena of the surrounding nature. Thus, its development was influenced by the use of a strong mathematical tool. As it was already seen in the first two volumes of the present book, its guideline is precisely the mathematical model of mechanics. The classical models which we refer to are in fact models based on the Newtonian model of mechanics, that is on its five principles, i.e.: the inertia, the forces action, the action and reaction, the independence of the forces action and the initial conditions principle, respectively. Other models, e.g., the model of attraction forces between the particles of a discrete mechanical system, are part of the considered Newtonian model. Kepler's laws brilliantly verify this model in case of velocities much smaller then the light velocity in vacuum."

In this thesis, the author considers quantum gravity to investigate the mysterious origin of our universe and its mechanisms. He and his collaborators have greatly improved the analyticity of two models: causal dynamical triangulations (CDT) and n-DBI gravity, with the space-time foliation which is one common factor shared by these two separate models.

In the first part, the analytic method of coupling matters to CDT in 2-dimensional toy models is proposed to uncover the underlying mechanisms of the universe and to remove ambiguities remaining in CDT. As a result, the wave function of the 2-dimensional universe where matters are coupled is derived. The behavior of the wave function reveals that the Hausdorff dimension can be changed when the matter is non-unitary.

In the second part, the n-DBI gravity model is considered. The author mainly investigates two effects driven by the space-time foliation: the appearance of a new conserved charge in black holes and an extra scalar mode of the graviton. The former implies a breakdown of the black-hole uniqueness theorem while the latter does not show any pathological behavior.

"

One of the most important routes to chaos is the chaotic intermittency. However, there are many cases that do not agree with the classical theoretical predictions. In this book, an extended theory for intermittency in one-dimensional maps is presented. A new general methodology to evaluate the reinjection probability density function (RPD) is developed in Chapters 5 to 8. The key of this formulation is the introduction of a new function, called M(x), which is used to calculate the RPD function. The function M(x) depends on two integrals. This characteristic reduces the influence on the statistical fluctuations in the data series. Also, the function M(x) is easy to evaluate from the data series, even for a small number of numerical or experimental data. As a result, a more general form for the RPD is found; where the classical theory based on uniform reinjection is recovered as a particular case. The characteristic exponent traditionally used to characterize the intermittency type, is now a function depending on the whole map, not just on the local map. Also, a new analytical approach to obtain the RPD from the mathematical expression of the map is presented. In this way all cases of non standard intermittencies are included in the same frame work. This methodology is extended to evaluate the noisy reinjection probability density function (NRPD), the noisy probability of the laminar length and the noisy characteristic relation. This is an important difference with respect to the classical approach based on the Fokker-Plank equation or Renormalization Group theory, where the noise effect was usually considered just on the local Poincare map. Finally, in Chapter 9, a new scheme to evaluate the RPD function using the Perron-Frobenius operator is developed. Along the book examples of applications are described, which have shown very good agreement with numerical computations.

Introduction to Dynamical Systems and Geometric Mechanics provides a comprehensive tour of two fields that are intimately entwined: dynamical systems is the study of the behavior of physical systems that may be described by a set of nonlinear first-order ordinary differential equations in Euclidean space, whereas geometric mechanics explore similar systems that instead evolve on differentiable manifolds. The first part discusses the linearization and stability of trajectories and fixed points, invariant manifold theory, periodic orbits, Poincare maps, Floquet theory, the Poincare-Bendixson theorem, bifurcations, and chaos. The second part of the book begins with a self-contained chapter on differential geometry that introduces notions of manifolds, mappings, vector fields, the Jacobi-Lie bracket, and differential forms.

Designing new structural materials, extending lifetimes and guarding against fracture in service are among the preoccupations of engineers, and to deal with these they need to have command of the mechanics of material behaviour. The first volume of this two-volume work deals with elastic and elastoplastic behaviour; this second volume continues with viscoelasticity, damage, fracture (resistance to cracking) and contact mechanics. As in Volume I, the treatment starts from the active mechanisms on the microscopic scale and develops the laws of macroscopic behaviour. Chapter I deals with viscoplastic behaviour, as shown, for example, at low temperatures by the effects of oscillatory loads and at high temperatures by creep under steady load. Chapter 2 treats damage phenomena encountered in all materials - for example, metals, polymers, glasses, concretes - such as cavitation, fatigue and stress-corrosion cracking. Chapter 3 treats those concepts of fracture mechanics that are needed for the understanding of resistance to cracking and Chapter 4 completes the volume with a survey of the main concepts of contact mechanics. As with Volume I, each chapter has a set of exercises, either with solutions or with indications of how to attack the problem; and there are many explanatory diagrams and other illustrations.

Microphone arrays have attracted a lot of interest over the last few decades since they have the potential to solve many important problems such as noise reduction/speech enhancement, source separation, dereverberation, spatial sound recording, and source localization/tracking, to name a few. However, the design and implementation of microphone arrays with beamforming algorithms is not a trivial task when it comes to processing broadband signals such as speech. Indeed, in most sensor arrangements, the beamformer output tends to have a frequency-dependent response. One exception, perhaps, is the family of differential microphone arrays (DMAs) who have the promise to form frequency-independent responses. Moreover, they have the potential to attain high directional gains with small and compact apertures. As a result, this type of microphone arrays has drawn much research and development attention recently. This book is intended to provide a systematic study of DMAs from a signal processing perspective. The primary objective is to develop a rigorous but yet simple theory

for the design, implementation, and performance analysis of DMAs. The theory includes some signal processing techniques for the design of commonly used first-order, second-order, third-order, and also the general "N"th-order DMAs. For each order, particular examples are given on how to form standard directional patterns such as the dipole, cardioid, supercardioid, hypercardioid, subcardioid, and quadrupole. The study demonstrates the performance of the different order DMAs in terms of beampattern, directivity factor, white noise gain, and gain for point sources. The inherent relationship between differential processing and adaptive beamforming is discussed, which provides a better understanding of DMAs and why they can achieve high directional gain. Finally, we show how to design DMAs that can be robust against white noise amplification.

The book focuses on the physical and mathematical foundations of model-based turbulence control: reduced-order modelling and control design in simulations and experiments. Leading experts provide elementary self-consistent descriptions of the main methods and outline the state of the art. Covered areas include optimization techniques, stability analysis, nonlinear reduced-order modelling, model-based control design as well as model-free and neural network approaches. The wake stabilization serves as unifying benchmark control problem.

This volume contains the Proceedings of the IUTAM Symposium held in Liverpool in 2002. It includes the articles presenting the results of recent work in mathematical modelling that covers the following areas of continuum mechanics and theoretical physics: *-Perturbation problems for partial differential equations and their applications in mechanics; * Analysis of singular fields; * Homogenisation theory in models of composite structures; * Mathematical models of cracks in solids; * Wave propagation, scattering; * Models of photonic and phononic band gap composite structures; * Advanced numerical techniques.

This book presents research advances in the field of Continuous Media with Microstructure and considers the three complementary pillars of mechanical sciences: theory, research and computational simulation. It focuses on the following problems: thermodynamic and mathematical modeling of materials with extensions of classical constitutive laws, single and multicomponent media including modern multifunctional materials, wave propagation, multiscale and multiphysics processes, phase transformations, and porous, granular and composite materials. The book presents the proceedings of the 2nd Conference on Continuous Media with Microstructure, which was held in 2015 in Lagow, Poland, in memory of Prof. Krzysztof Wilmanski.

Hard spheres and related objects (hard disks and mixtures of hard systems) are paradigmatic systems: indeed, they have served as a basis for the theoretical and numerical development of a number of fields, such as general liquids and fluids, amorphous solids, liquid crystals, colloids and granular matter, to name but a few. The present volume introduces and reviews some important basics and progress in the study of such systems. Their structure, thermodynamic properties, equations of state, as well as kinetic and transport properties are considered from different and complementary points of view. This book addresses graduate students, lecturers as well as researchers in statistical mechanics, physics of liquids, physical chemistry and chemical engineering.

This volume collects the edited and reviewed contribution presented in the 7th iTi Conference in Bertinoro, covering fundamental and applied aspects in turbulence. In the spirit of the iTi conference, the volume is produced after the conference so that the authors had the opportunity to incorporate comments and discussions raised during the meeting. In the present book, the contributions have been structured according to the topics: I Theory II Wall bounded flows III Pipe flow IV Modelling V Experiments VII Miscellaneous topics

Analytical Heat Transfer explains how to analyze and solve conduction, convection, and radiation heat transfer problems. It enables students to tackle complex engineering heat transfer problems prevalent in practice. Covering heat transfer in high-speed flows and unsteady highly turbulent flows, the book also discusses enhanced heat transfer in channels, heat transfer in rotating channels, numerical modeling for turbulent flow heat transfer, and thermally developing heat transfer in a circular tube. The second edition features new content on Duhamel's superposition method, Green's function method for transient heat conduction, finite-difference method for steady state and transient heat conduction in cylindrical coordinates, and laminar mixed convection. It includes two new chapters on laminar-to-turbulent transitional heat transfer and turbulent flow heat transfer enhancement, in addition to end-of-chapter problems. The book bridges the gap between basic heat transfer undergraduate courses and advanced heat transfer graduate courses for a single semester of intermediate heat transfer, advanced conduction/radiation heat transfer, or convection heat transfer. Features: Focuses on analyzing and solving classic heat transfer problems in conduction, convection, and radiation Covers 2-D and 3-D view factor evaluation, combined radiation with conduction and/or convection, and gas radiation optically thin and optically thick limits Features updated content and new chapters on mass and heat transfer analogy, thermally developing heat transfer in a circular tube, laminar-turbulent transitional heat transfer, unsteady highly turbulent flows, enhanced heat transfer in channels, heat transfer in rotating channels, and numerical modeling for turbulent flow heat transfer Provides step-by-step mathematical formula derivations, analytical solution procedures, and demonstration examples Includes end-of-chapter problems with an accompanying Solutions Manual for instructors This book is ideal for undergraduate and graduate students studying basic heat transfer and advanced heat transfer.

This monograph gives a comprehensive description of the relationship and connections between kinetic theory and fluid dynamics, mainly for a time-independent problem in a general domain. Ambiguities in this relationship are clarified, and the incompleteness of classical fluid dynamics in describing the behavior of a gas in the continuum limita "recently reported as the ghost effecta "is also discussed. The approach used in this work engages an audience of theoretical physicists, applied mathematicians, and engineers.

By a systematic asymptotic analysis, fluid-dynamic-type equations and their associated boundary conditions that take into account the weak effect of gas rarefaction are derived from the Boltzmann system. Comprehensive information on the Knudsen-layer correction is also obtained. Equations and their boundary conditions are carefully classified depending on the physical context of problems. Applications are presented to various physically interesting phenomena, including flows induced by temperature fields, evaporation and condensation problems, examples of the ghost effect, and bifurcation of flows.

Key features:

* many applications and physical models of practical interest

* experimental works such as the Knudsen compressor are examined to supplement theory

* engineers will not be overwhelmed by sophisticated mathematical techniques

* mathematicians will benefit from clarity of definitions and precise physical descriptions given in mathematical terms

* appendices collect key derivations and formulas, important to the practitioner, but not easily found in the literature

Kinetic Theory and Fluid Dynamics serves as a bridge for those working indifferent communities where kinetic theory or fluid dynamics is important: graduate students, researchers and practitioners in theoretical physics, applied mathematics, and various branches of engineering. The work can be used in graduate-level courses in fluid dynamics, gas dynamics, and kinetic theory; some parts of the text can be used in advanced undergraduate courses.

Why writing a book about a specialized task of the large topic of complex systems? And who will read it? The answer is simple: The fascination for a didactically valuable point of view, the elegance of a closed concept and the lack of a comprehensive disquisition.

The fascinating part is that field equations can have localized solutions exhibiting the typical characteristics of particles. Regarding the field equations this book focuses on, the field phenomenon of localized solutions can be described in the context of a particle formalism, which leads to a set of ordinary differential equations covering the time evolution of the position and the velocity of each particle. Moreover, starting from these particle dynamics and making the transition to many body systems, one considers typical phenomena of many body systems as shock waves and phase transitions, which themselves can be described as field phenomena. Such transitions between different level of modelling are well known from conservative systems, where localized solutions of quantum field theory lead to the mechanisms of elementary particle interaction and from this to field equations describing the properties of matter. However, in dissipative systems such transitions have not been considered yet, which is adjusted by the presented book. The elegance of a closed concept starts with the observation of self-organized current filaments in a semiconductor gas discharge system. These filaments move on random paths and exhibit certain particle features like scattering or the formation of bound states. Neither the reasons for the propagation of the filaments nor the laws of the interaction between the filaments can be registered by direct observations. Therefore a model is established, which is phenomenological in the first instance due to the complexity of the experimental system. This model allows to understand the existence of localized structures, their mechanisms of movement, and their interaction, at least, on a qualitative level. But this model is also the starting point for developing a data analysis method that enables the detection of movement and interaction mechanisms of the investigated localized solutions. The topic is rounded of by applying the data analysis to real experimental data and comparing the experimental observations to the predictions of the model.

A comprehensive publication covering the interesting topic of localized solutions in reaction diffusion systems in its width and its relation to the well known phenomena of spirals and patterns does not yet exist, and this is the third reason for writing this book. Although the book focuses on a specific experimental system the model equations are as simple as possible so that the discussed methods should be adaptable to a large class of systems showing particle-like structures.

Therefore, this book should attract not only the experienced scientist, who is interested in self-organization phenomena, but also the student, who would like to understand the investigation of a complex system on the basis of a continuous description.

Althoughtheprinciplesofoperationofhelicalscrewmachines, ascompressors or expanders, have been well known for more than 100 years, it is only during the past 30 years that these machines have become widely used. The main reasons for the long period before they were adopted were their relatively poor e?ciency and the high cost of manufacturing their rotors. Two main developments led to a solution to these di?culties. The ?rst of these was the introduction of the asymmetric rotor pro?le in 1973. This reduced the bl- hole area, which was the main source of internal leakage by approximately 90%, and thereby raised the thermodynamic e?ciency of these machines, to roughly the same level as that of traditional reciprocating compressors. The second was the introduction of precise thread milling machine tools at - proximately the same time. This made it possible to manufacture items of complex shape, such as the rotors, both accurately and cheaply. From then on, as a result of their ever improving e?ciencies, high rel- bility and compact form, screw compressors have taken an increasing share of the compressor market, especially in the ?elds of compressed air production, and refrigeration and air conditioning, and today, a substantial proportion of compressors manufactured for industry are of this type. Despite, the now wide usage of screw compressors and the publication of many scienti?c papers on their development, only a handful of textbooks have been published to date, which give a rigorous exposition of the principles of their operation and none of these are in English

This book presents a unified hierarchical formulation of theories for three-dimensional continua, two-dimensional shells, one-dimensional rods, and zero-dimensional points. It allows readers with varying backgrounds easy access to fundamental understanding of these powerful Cosserat theories.

This book reports on the latest developments in computational fluid dynamics and turbulence modeling, with a special emphasis on hybrid RANS-LES methods and their industrial applications. It gathers the proceedings of the Sixth Symposium on Hybrid RANS-LES Methods, held on September 26-28 in Strasbourg, France. The different chapters covers a wealth of topics such as flow control, aero-acoustics, aero-elasticity and CFD-based multidisciplinary optimization. Further topics include wall-modelled Large Eddy Simulation (WMLES), embedded LES, Lattice-Bolzman methods, turbulence-resolving applications and comparisons between LES, hybrid RANS-LES and URANS methods. The book addresses academic researchers, graduate students, industrial engineers, as well as industrial R&D managers and consultants dealing with turbulence modelling, simulation and measurement, and with multidisciplinary applications of computational fluid dynamics.

This book provides an introduction to turbulence in vortex systems, and to turbulence theory for incompressible flow described in terms of the vorticity field. It is the author's hope that by the end of the book the reader will believe that these subjects are identical, and constitute a special case of fairly standard statistical mechanics, with both equilibrium and non-equilibrium aspects. The author's main goal is to relate turbulence to statistical mechanics. The book is organized as follows: the first three chapters constitute a fairly standard introduction to homogeneous turbulence in incompressible flow; a quick review of fluid mechanics; a summary of the appropriate Fourier theory; a summary of Kolmogorov's theory of the inertial range. The next four chapters present the statistical theory of vortex notion, and the vortex dynamics of turbulence. The book ends with the major conclusion that turbulence can no longer be viewed as incomprehensible. This book will be appropriate for professionals in the fields of applied mathematics, mechanical engineering, or physics, as well as graduate students in these noted areas.