Multi-scale Phenomena in Complex Fluids: Modeling, Analysis and Numerical Simulations (Series in Contemporary Applied Mathematics)

Contents......Page 8
Preface......Page 6
Abstract......Page 10
1.1 Hubbard model......Page 11
1.1.1 Hubbard model with no hopping......Page 15
1.1.2 Hubbard model without interaction......Page 17
1.2.1 Computable approximation of distribution operator P......Page 22
1.2.2 Algorithm......Page 26
1.2.3 Physical measurements......Page 27
1.3.1 Computable approximation of distribution operator P......Page 30
1.3.2 Algorithm......Page 33
1.3.3 Physical measurements......Page 39
2.1 Hubbard matrix......Page 40
2.2 Basic properties......Page 43
2.3 Matrix exponential B = et rK......Page 46
2.4 Eigenvalue distribution of M......Page 48
2.4.1 The case U = 0......Page 49
2.5 Condition number of M......Page 50
2.5.1 The case U = 0......Page 51
2.5.2 The case U = 0......Page 53
2.6 Condition number of M(k)......Page 56
3 Self-adaptive direct linear solvers......Page 58
3.1 Block cyclic reduction......Page 60
3.2 Block structural orthogonal factorization......Page 64
3.3 A hybrid method......Page 66
3.4 Self-adaptive reduction factor k......Page 67
3.6 Numerical experiments......Page 70
4 Preconditioned iterative linear solvers......Page 74
4.1 Iterative solvers and preconditioning......Page 75
4.2 Previous work......Page 78
4.3 Cholesky factorization......Page 79
4.4.1 IC......Page 82
4.4.2 Modified IC......Page 86
4.5 Robust incomplete Cholesky preconditioners......Page 87
4.5.1 RICl......Page 88
4.5.2 RIC2......Page 92
4.5.3 RIC3......Page 96
4.6.1 Moderately interacting systems......Page 101
4.6.2 Strongly interacting systems......Page 102
A.1 Rank-one updates......Page 106
A.2 Metropolis ratio and Green's function computations......Page 108
B.1 Algebraic identities......Page 110
B.2 Particle-hole transformation in DQMC......Page 112
B.3 Particle-hole transformation in the HQMC......Page 114
References......Page 115
1.1 Introduction......Page 120
1.2 Kirchhoff's theory of diffraction......Page 123
1.3 Huygens-Fresnel principle......Page 126
1.5 Focal spot size and resolution......Page 128
2.2 Rytovapproximation......Page 131
2.3 The extended Huygens-Fresnel principle......Page 132
2.4 Paraxial approximation......Page 133
3 The Wigner distribution......Page 138
4.1 White-noise scaling......Page 143
4.2 Markovian limit......Page 144
5.1 Paraxial waves......Page 147
5.1.1 Two-frequency radiative transfer equations......Page 149
5.1.2 The longitudinal and transverse cases......Page 150
5.2 Spherical waves......Page 151
5.2.1 Geometrical radiative transfer......Page 153
5.2.2 Spatial (frequency) spread and coherence bandwidth......Page 154
5.2.3 Small-scale asymptotics......Page 155
6.1 Spherical wave......Page 156
6.2 Paraxial wave......Page 158
6.3 Anomalous focal spot......Page 159
6.4 Duality and turbulence-induced aperture......Page 160
6.6 Broadband time reversal communications......Page 162
7.1 Imaging of phase objects......Page 164
7.2 Long-exposure imaging......Page 166
7.3 Short-exposure imaging......Page 169
7.4.1 Differential scattered field in clutter......Page 171
7.4.2 Imaging functions......Page 173
7.4.3 Numerical simulation with a Rician medium......Page 174
7.5 Coherent imaging in a Rayleigh medium......Page 175
References......Page 180
1 Introduction......Page 184
2.1 Homogenization theory for elliptic problems......Page 186
2.2 Homogenization for hyperbolic problems......Page 191
2.3 Convection of microstructure......Page 198
3 Numerical homogenization based on sampling techniques......Page 201
3.1 Convergence of the particle method......Page 204
3.2 Vortex methods for incompressible flows......Page 211
4 Numerical upscaling based on multiscale finite element methods......Page 212
4.1 Multiscale finite element methods for elliptic PDEs......Page 215
4.2 Error estimates (h < e)......Page 216
4.3 Error estimates (h > €)......Page 218
4.4 The over-sampling technique......Page 221
4.5 Performance and implementation issues......Page 223
4.6 Applications......Page 225
4.7 Brief overview of mixed finite element and finite volume element methods......Page 236
4.8 MsFEM using limited global information......Page 239
4.9 Analysis......Page 249
5 Multiscale finite element methods for nonlinear partial differential equations......Page 254
5.1 Multiscale finite volume element method (MsFVEM)......Page 256
5.2 Examples of V h 10......Page 257
5.3 Convergence of MsFEM for nonlinear partial differential equations......Page 258
5.4 Multiscale finite element methods for nonlinear parabolic equations......Page 259
5.5 Numerical results......Page 262
5.6 Generalizations of MsFEM and some remarks......Page 268
6 Multiscale simulations of two-phase immiscible flow in adaptive coordinate system......Page 270
6.1 Numerical averaging across streamlines......Page 275
6.2 N urnerical results......Page 278
7 Conclusions......Page 286
References......Page 287
1 Introduction......Page 295
2.2 Direct methods......Page 298
2.4 Dynamics......Page 299
2.5 Hamilton's principle......Page 300
2.5.2 Variation of the domain v.s. variation of the function......Page 301
2.6.1 Harmonic maps......Page 302
2.6.2 Liquid crystals......Page 303
2.6.3 Methods of penalty......Page 305
3 Navier-Stokes equation......Page 308
3.1.1 Existence of global weak solution......Page 309
3.1.4 Partial regularity......Page 311
4.1 Flow map and deformation tensor......Page 312
4.2 Force balance and Oldroyd-B systems......Page 313
4.3 Energetic variational formulation......Page 315
5.1 Ericksen-Leslie theory......Page 318
5.2 Existence and regularity......Page 321
6 Free interface motion in mixtures......Page 324
6.1 An energetic variational approach with phase field method......Page 326
6.2 Marangoni-Benard convection......Page 330
6.3 Mixtures involving liquid crystals......Page 332
7.2 The evolution of the magnetic field......Page 336
7.4 The linear momentum equation......Page 337
7.5 The dynamics of magnetic field lines......Page 338
References......Page 339
1 Introduction......Page 347
2 A primer for equilibrium thermodynamics......Page 348
3 Basics of statistical mechanics......Page 356
4 Equilibrium distribution of the end-toend vector in simple polymer models......Page 367
5 Kinetic theory for polymers......Page 370
5.1 Langevin equation......Page 373
5.2 System of constraints......Page 374
5.3 Bead-spring (Rouse) chain model......Page 375
References......Page 377