Mathematical Physics

The fields of computational fluid dynamics (CFD) and optimal shape design (OSD) have received considerable attention in the recent past, and are of practical importance for many engineering applications. This book deals with shape optimization problems for fluids, with the equations needed for their understanding (Euler and Navier Strokes, but also those for microfluids) and with the numerical simulation of these problems. It presents the state of the art in shape optimization for an extended range of applications involving fluid flows. Automatic differentiation, approximate gradients, unstructured mesh adaptation, multi-model configurations, and time-dependent problems are introduced, and their implementation into the industrial environments of aerospace and automobile equipment industry explained and illustrated. With the increases in the power of computers in industry since the first edition of this book, methods which were previously unfeasible have begun giving results, namely evolutionary algorithms, topological optimization methods, and level set algorithms. In this edition, these methods have been treated in separate chapters, but the book remains primarily one on differential shape optimization.

The idea of this book is to illustrate an interplay between distinct domains of mathematics. Firstly, this book provides an introduction to hyperbolic geometry, based on the Lorentz group PSO(1, d) and its Iwasawa decomposition, commutation relations and Haar measure, and on the hyperbolic Laplacian. The Lorentz group plays a role in relativistic space–time analogous to rotations in Euclidean space. Hyperbolic geometry is the geometry of the unit pseudo-sphere. The boundary of hyperbolic space is defined as the set of light rays. Special attention is given to the geodesic and horocyclic flows. This book presents hyperbolic geometry via special relativity to benefit from physical intuition. Secondly, this book introduces some basic notions of stochastic analysis: the Wiener process, Itô's stochastic integral and Itô calculus. The book studies linear stochastic differential equations on groups of matrices, and diffusion processes on homogeneous spaces. Spherical and hyperbolic Brownian motions, diffusions on stable leaves, and relativistic diffusion are constructed. Thirdly, quotients of hyperbolic space under a discrete group of isometries are introduced, and form the framework in which some elements of hyperbolic dynamics are presented, especially the ergodicity of the geodesic and horocyclic flows. An analysis is given of the chaotic behaviour of the geodesic flow, using stochastic analysis methods. The main result is Sinai's central limit theorem. Some related results (including a construction of the Wiener measure) which complete the expositions of hyperbolic geometry and stochastic calculus are given in the appendices.

This text focuses on mathematical and numerical techniques for the simulation of magnetohydrodynamic phenomena, with an emphasis on the magnetohydrodynamics of liquid metals, on two-fluid flows, and on a prototypical industrial application. The approach is a highly mathematical one, based on the rigorous analysis of the equations at hand, and a solid numerical analysis of the discretization methods. Up-to-date techniques, both on the theoretical side and the numerical side, are introduced to deal with the nonlinearities of the multifluid magnetohydrodynamics equations. At each stage of the exposition, examples of numerical simulations are provided, first on academic test cases to illustrate the approach, next on benchmarks well documented in the professional literature, and finally on real industrial cases. The simulation of aluminium electrolysis cells is used as a guideline throughout the book to motivate the study of a particular setting of the magnetohydrodynamics equations.

The heat equation is one of the three classical linear partial differential equations of second order that form the basis of any elementary introduction to the area of PDEs, and only recently has it come to be fairly well understood. This book provides a presentation of the mathematical theory of the nonlinear heat equation usually called the Porous Medium Equation (PME). This equation appears in a number of physical applications, such as to describe processes involving fluid flow, heat transfer, or diffusion. Other applications have been proposed in mathematical biology, lubrication, boundary layer theory, and other fields. Each chapter contains a detailed introduction and is supplied with a section of notes, providing comments, historical notes or recommended reading, and exercises.

Generalized dynamic thermoelasticity is a vital area of research in continuum mechanics, free of the classical paradox of infinite propagation speeds of thermal signals in Fourier‐type heat conduction. Besides that paradox, the classical dynamic thermoelasticity theory offers either unsatisfactory or poor descriptions of a solid's response to a fast transient loading (say, due to short laser pulses) or at low temperatures. Several models were developed and intensively studied over the past four decades, and this book is the first monograph on the subject since the 1970s, aiming to provide a point of reference in the field. It focuses on dynamic thermoelasticity governed by hyperbolic equations, and, in particular, on the two leading theories: that of Lord‐Shulman (with one relaxation time), and that of Green‐Lindsay (with two relaxation times). While the resulting field equations are linear partial differential ones, the complexity of theories is due to the coupling of mechanical with thermal fields. The book is concerned with the mathematical aspects of both theories — existence and uniqueness theorems, domain of influence theorems, convolutional variational principles — as well as with the methods for various initial/boundary value problems. In the latter respect, following the establishment of the central equation of thermoelasticity with finite wave speeds, there are extensive presentations of: the exact, aperiodic‐in‐time solutions of Green‐Lindsay theory; Kirchhoff‐type formulas and integral equations in Green‐Lindsay theory; thermoelastic polynomials; moving discontinuity surfaces; and time‐periodic solutions. This is followed by a chapter on physical aspects of generalized thermoelasticity, with a review of several applications. The book closes with a chapter on a nonlinear hyperbolic theory of a rigid heat conductor for which a number of asymptotic solutions are obtained using a method of weakly nonlinear geometric optics. The book is augmented by an extensive bibliography.

Radon showed how to write arbitrary functions in Rⁿ in terms of the characteristic functions (delta functions) of hyperplanes. This idea leads to various generalizations. For example, R can be replaced by a more general group and “plane” can be replaced by other types of geometric objects. All this is for the “nonparametric” Radon transform. For the parametric Radon transform, this book parametrizes the points of the geometric objects, leading to differential equations in the parameters because the Radon transform is overdetermined. Such equations were first studied by F. John. This book elaborates on them and puts them in a general framework.