Quak / Soomere Applied Wave Mathematics

Selected Topics in Solids, Fluids, and Mathematical Methods
1. Auflage 2009
ISBN: 978-3-642-00585-5
Verlag: Springer
Format: PDF
Kopierschutz: 1 - PDF Watermark

E-Book, Englisch, 471 Seiten

This edited volume consists of twelve contributions related to the EU Marie Curie Transfer of Knowledge Project Cooperation of Estonian and Norwegian Scienti c Centres within Mathematics and its Applications, CENS-CMA (2005-2009), - der contract MTKD-CT-2004-013909, which ?nanced exchange visits to and from CENS, the Centre for Nonlinear Studies at the Institute of Cybernetics of Tallinn University of Technology in Estonia. Seven contributions describe research highlights of CENS members, two the work of members of CMA, the Centre of Mathematics for Applications,Univ- sity of Oslo, Norway, as the partner institution of CENS in the Marie Curie project, and three the ?eld of work of foreign research fellows, who visited CENS as part of theproject. Thestructureofthebookre?ectsthedistributionofthetopicsaddressed: Part I Waves in Solids Part II Mesoscopic Theory Part III Exploiting the Dissipation Inequality Part IV Waves in Fluids Part V Mathematical Methods The papers are written in a tutorial style, intended for non-specialist researchers and students, where the authors communicate their own experiences in tackling a problem that is currently of interest in the scienti?c community. The goal was to produce a book, which highlights the importance of applied mathematics and which can be used for educational purposes, such as material for a course or a seminar. To ensure the scienti?c quality of the contributions, each paper was carefully - viewed by two international experts. Special thanks go to all authors and referees, without whom making this book would not have been possible.

Autoren/Hrsg.

Quak, Ewald

Soomere, Tarmo

Inhaltsverzeichnis

1;Preface;6
2;Contents;7
3;List of Contributors;9
4;CENS, CMA and the CENS-CMA Project;11
4.1;CENS 1999-2009;11
4.2;CMA;14
4.3;The CENS-CMA Project;15
5;Part I Waves in Solids;17
5.1;Overview;18
5.2;Deformation Waves in Solids;21
5.2.1;Introduction;21
5.2.1.1;General ideas;21
5.2.1.2;Notes from history;23
5.2.1.3;Description of what follows;23
5.2.2;Basic theory;23
5.2.3;Advanced theories;26
5.2.3.1;General ideas;26
5.2.3.2;Separation of macro- and microstructure;27
5.2.3.3;Balance of pseudomomentum;29
5.2.3.4;Internal variables;30
5.2.4;Model governing equations;31
5.2.4.1;Basic linear theory;31
5.2.4.2;Wave hierarchy;32
5.2.4.3;Nonlinearities;33
5.2.4.4;One-wave models;34
5.2.5;Final remarks;36
5.2.6;References;37
5.3;The Perturbation Technique for Wave Interaction in Prestressed Material;39
5.3.1;Introduction;39
5.3.2;Prelude;40
5.3.3;Basic relations in continuum mechanics;42
5.3.3.1;Coordinate systems;43
5.3.3.2;Conservation laws;43
5.3.3.3;The constitutive equation;44
5.3.3.4;Initial and boundary conditions;46
5.3.3.5;Compatibility conditions;46
5.3.4;Governing equations;47
5.3.5;The perturbation technique;49
5.3.5.1;The prestressed state;50
5.3.5.2;Counterpropagating waves;51
5.3.6;Harmonic waves;54
5.3.7;Nondestructive characterization of plane strain;56
5.3.8;Conclusions;60
5.3.9;References;60
5.4;Waves in Inhomogeneous Solids;62
5.4.1;Introduction;62
5.4.1.1;Governing equations;63
5.4.2;The wave-propagation algorithm;64
5.4.2.1;Averaged quantities;65
5.4.2.2;Numerical fluxes;65
5.4.2.3;Second-order corrections;67
5.4.2.4;The conservative wave propagation algorithm;67
5.4.3;Excess quantities and numerical fluxes;68
5.4.3.1;Excess quantities at the boundaries between cells;69
5.4.4;One-dimensional waves in periodic media;70
5.4.5;One-dimensional weakly nonlinear waves in periodic media;72
5.4.6;One-dimensional linear waves in laminates;74
5.4.7;Nonlinear elastic waves in laminates under impact loading;76
5.4.7.1;Comparison with experimental data;79
5.4.8;Waves in functionally graded materials;83
5.4.9;Concluding remarks;85
5.4.10;References;86
6;Part II Mesoscopic Theory;89
6.1;Overview;90
6.1.1;References;93
6.2;Dynamics of Internal Variables from the Mesoscopic Background for the Example of Liquid Crystals and Ferrofluids;94
6.2.1;Introduction to liquid crystals;94
6.2.1.1;Some properties of liquid crystals;94
6.2.2;Mesoscopic theory of complex materials;98
6.2.2.1;Complex materials;98
6.2.2.2;Examples of internal structure;99
6.2.2.3;The mesoscopic concept;102
6.2.2.4;Mesoscopic balance equations;104
6.2.3;Mesoscopic theory of uniaxial liquid crystals;104
6.2.3.1;Mesoscopic balance equations;106
6.2.3.2;Macroscopic balance equations;107
6.2.3.3;Macroscopic constitutive quantities;108
6.2.3.4;Order parameters;109
6.2.3.5;Differential equation for the distribution function and for the alignment tensors;110
6.2.3.6;Example of a closed differential equation for the second order alignment tensor;111
6.2.3.7;Landau theory of phase transitions as a special case;114
6.2.3.8;A remark on constitutive theory and the Second Law of Thermodynamics;116
6.2.3.9;A set of differential equations for the moments and a second order differential equation for the alignment tensor;117
6.2.4;Application of the mesoscopic theory to dipolar media ;121
6.2.4.1;Orientation distribution function and alignment tensors;121
6.2.4.2;Exploitation of the balance of spin;122
6.2.4.3;Equation of motion for the magnetization;123
6.2.4.4;Summary;126
6.2.5;Summary of the mesoscopic theory;126
6.2.6;References;127
6.3;Towards a Description of Twist Waves in Mesoscopic Continuum Physics;131
6.3.1;Introduction;131
6.3.2;Mesoscopic Continuum Physics;133
6.3.2.1;Generalization of vector fields to the mesoscopic space;133
6.3.2.2;Mesoscopic balances and a transport theorem;135
6.3.3;Orientation waves;137
6.3.3.1;Twist waves in classical macroscopic theory;137
6.3.3.2;Twist waves in mesoscopic theory;140
6.3.4;Comparison;143
6.3.4.1;Mesoscopic mass density, orientation distribution function and macroscopic director;144
6.3.4.2;Macroscopic balance equations;144
6.3.5;Conclusions and outlook;147
6.3.6;References;148
7;Part III Exploiting the Dissipation Inequality;150
7.1;Overview;151
7.1.1;References;153
7.2;Weakly Nonlocal Non-equilibrium Thermodynamics -- Variational Principles and Second Law;154
7.2.1;Introduction;154
7.2.2;Second law and weakly nonlocal constitutive spaces ;156
7.2.3;Thermodynamic evolution of internal variables;157
7.2.3.1;First order nonlocality -- relaxation;157
7.2.3.2;Second order nonlocality -- the Ginzburg-Landau equation;159
7.2.3.3;Dual internal variables -- Hamiltonian structure;161
7.2.4;Classical Irreversible Thermodynamics;169
7.2.5;One component fluids -- second order nonlocal in the density;171
7.2.5.1;Fluid mechanics in general;171
7.2.5.2;Schrödinger-Madelung fluid ;176
7.2.6;Summary and outlook;178
7.2.7;Appendix -- Farkas's lemma and some of its consequences;180
7.2.7.1;Affine Farkas's lemma;181
7.2.7.2;Liu's theorem;181
7.2.8;References;183
8;Part IV Waves in Fluids;188
8.1;Overview;189
8.1.1;Surface waves at the cutting edge of research and applications;189
8.1.2;References;192
8.2;Long Ship Waves in Shallow Water Bodies;193
8.2.1;Introduction;193
8.2.2;Linear ship wakes;196
8.2.2.1;Kelvin wedge;196
8.2.2.2;Navigational speeds;201
8.2.2.3;Distribution of wave heights and periods;203
8.2.3;Patterns of wakes from fast ferries;205
8.2.3.1;Changes in the ship wave pattern;205
8.2.3.2;Realistic spatial patterns of ship waves;209
8.2.3.3;Ship waves at a fixed point;211
8.2.4;Contribution of ship wakes to local hydrodynamics;215
8.2.4.1;Parameters of wind waves in semi-enclosed seas;215
8.2.4.2;Ship wakes versus wind waves;216
8.2.4.3;Ship wakes and the coast;220
8.2.4.4;Excessive near-bottom velocities and impulse loads;222
8.2.5;Conclusions;224
8.2.6;References;225
8.3;Modelling of Ship Waves from High-speed Vessels;229
8.3.1;Waves generated by ships;230
8.3.1.1;Governing equations and results from linear wave theory;230
8.3.1.2;Steady ship wakes and ship wave parameters;233
8.3.1.3;Wave making resistance to steady ship motion;236
8.3.1.4;Ship wake patterns in deep and shallow water;239
8.3.1.5;Why is wake wash from high-speed vessels a problem?;245
8.3.2;Properties of long wave model equations;246
8.3.2.1;Approximations for shallow water flow;246
8.3.2.2;Comparison between different model equations;248
8.3.3;Numerical modelling of ship waves;252
8.3.3.1;Numerical models based on Boussinesq-type equations;253
8.3.3.2;Ship representation by a moving pressure disturbance;254
8.3.3.3;The influence of the dispersive and nonlinear components;256
8.3.4;Concluding Remarks;260
8.3.5;References;260
8.4;New Trends in the Analytical Theory of Long Sea Wave Runup;264
8.4.1;Introduction;264
8.4.2;Basic equations and parameters;266
8.4.3; Method of solution: hodograph transformation;268
8.4.4;Linear approximation of nonlinear long wave runup;271
8.4.5;The relation between linear and nonlinear runup properties;275
8.4.6;Runup of solitary waves;280
8.4.7; Runup of periodic waves;288
8.4.8; Conclusion;293
8.4.9;References;294
9;Part V Mathematical Methods;296
9.1;Overview;297
9.2;The Pseudospectral Method and Discrete Spectral Analysis;299
9.2.1;Introduction;299
9.2.2;The model equations;301
9.2.3;The pseudospectral method;303
9.2.3.1;Approximation of space derivatives;303
9.2.3.2;The discrete Fourier transform;304
9.2.3.3;The essence of the pseudospectral method;305
9.2.3.4;The pseudospectral method and different equation types;307
9.2.3.5;Filtering and other practical tips;310
9.2.4;Discrete spectral analysis;313
9.2.4.1;Spectral amplitudes and spectral densities;313
9.2.4.2;Cumulative spectrum and time averaged normalised spectral densities;315
9.2.4.3;Applications;316
9.2.5;Conclusions;324
9.2.6;References;327
9.3;Foundations of Finite Element Methods for Wave Equations of Maxwell Type;332
9.3.1;Introduction;332
9.3.2;Analysis of the finite element method for waves;334
9.3.2.1;Linear wave equations;334
9.3.2.2;Convergence theory for linear equations;337
9.3.2.3;Consistency;342
9.3.2.4;Eigenvalue approximation;346
9.3.3;Construction of finite element spaces;351
9.3.3.1;Algebra;351
9.3.3.2;Differential geometry;361
9.3.3.3;Finite elements on cellular complexes;372
9.3.4;Conclusion;387
9.3.5;References;388
9.4;An Introduction to the Theory of Scalar Conservation Laws with Spatially Discontinuous Flux Functions;391
9.4.1;Introduction;391
9.4.2;The Riemann problem;394
9.4.2.1;Existence of a solution;395
9.4.2.2;Vanishing viscosity and smoothing;413
9.4.3;The Cauchy problem;419
9.4.3.1;A model equation;420
9.4.4;Uniqueness of entropy solutions;447
9.4.5;References;459
10;Index;461