دانلود کتاب Multiscale Problems: Theory, Numerical Approximation and Applications

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Contents\nPreface\nAlain Damlamian An Introduction to Periodic Homogenization\n 1 Introduction\n 2 The main ideas of Homogenization\n The three steps of Homogenization\n 3 The model problem and three theoretical methods\n 3.1 The multiple-scale expansion method\n 3.2 The oscillating test functions method\n 3.2.1 The proof of Theorem 3.4\n 3.2.2 Convergence of the energy\n 3.3 The two-scale convergence method\n References\nAlain Damlamian The Periodic Unfolding Method in Homogenization\n 1 Introduction\n 2 Unfolding in Lp-spaces\n 2.1 The unfolding operator T\n 2.2 The averaging operator U\n 2.3 The connection with two-scale convergence\n 2.4 The local average operator M\n 3 Unfolding and gradients\n 4 Periodic unfolding and the standard homogenization problem\n 4.1 The model problem and the standard homogenization result\n 4.2 The Unfolding result: the case of strong convergence of the right-hand side\n 4.3 Proof of Theorem 4.3\n 4.4 The convergence of the energy and its consequences\n 4.5 Some corrector results and error estimates\n 4.6 The case of weak convergence of the right-hand side\n 5 Periodic unfolding and multiscales\n 6 Further developments\n References\nGabriel Nguetseng and Lazarus Signing Deterministic Homogenization of Stationary Navier-Stokes Type Equations\n 1 Introduction\n 2 Periodic homogenization of stationary Navier-Stokes type equations\n 2.1 Preliminaries\n 2.2 A global homogenization theorem\n 2.3 Macroscopic homogenized equations\n 3 General deterministic homogenization of stationary Navier-Stokes type equations\n 3.1 Preliminaries and statement of the homogenization problem\n 3.2 A global homogenization theorem\n 3.3 Macroscopic homogenized equations\n 3.4 Some concrete examples\n 4 Homogenization of the stationary Navier- Stokes equations in periodic porous media\n 4.1 Preliminaries\n 4.2 Homogenization results\n References\nPatricia Donato Homogenization of a Class of Imperfect Transmission Problems\n 1 Introduction\n 2 Setting of the problem and main results\n 3 Some preliminary results\n 4 A priori estimates\n 5 A class of suitable test functions\n 5.1 The test functions in the reference cell Y\n 5.2 The test functions in\n 6 Proofs of Theorems 2.1 and 2.2\n 6.1 Identification of 1 + 2\n 6.2 Identification of 1 and 2 for -1 < < 1\n 6.3 Identification of u2\n 7 Proof of Theorem 2.4 (case > 1)\n 7.1 A priori estimates\n 7.2 Identification of 1\n 7.3 Identification of 2\n References\nGeorges Griso Decompositions of Displacements of Thin Structures\n 1 Introduction\n 2 The main theorem\n 2.1 Poincar´e-Wirtinger’s inequality in an open bounded set star-shaped with respect to a ball\n 2.2 Distances between a displacement and the space of the rigid body displacements\n 3 Decomposition of curved rod displacements\n 3.1 Notations\n 3.2 Elementary displacements and decomposition\n 4 Decomposition of shell displacements\n 4.1 Notations and preliminary\n 4.2 Elementary displacements and decompositions\n References\nGeorges Griso Decomposition of Rods Deformations. Asymptotic Behavior of Nonlinear Elastic Rods\n 1 Introduction\n 2 The Theorem of Geometric Rigidity in a star-shaped domain and two nonlinear Korn’s inequalities\n 3 Decomposition of the deformations of a straight rod. Estimates\n 3.1 The geometry\n 3.2 The elementary deformations\n 3.3 The main theorem\n 3.4 Two nonlinear Korn’s inequalities for rods\n 4 Case ||dist( , SO(3))||L2( ) ~ 2\n 5 Asymptotic behavior of an elastic rod\n 5.1 Assumption on the forces\n 5.2 Limit model in the case = 2\n 5.3 Solutions of the nonlinear minimization Problem (5.16)\n 6 Appendix\n References\nDominique Blanchard Junction of a Periodic Family of Rods with a Plate in Elasticity\n 1 Introduction\n 2 Linear isotropic elasticity\n 3 Junction of a periodic family of rods with a 3D plate\n 3.1 The geometry\n 3.2 The equations in\n 3.3 Local decomposition and a priori estimates\n 3.4 Definitions of U , R and u and estimates\n 3.5 Assumption on the forces and global estimates\n 3.6 Unfolding operator\n 3.7 Weak convergences of the unfolded fields\n 3.8 Relations between the limit fields\n 3.8.1 Limit displacement in +\n 3.8.2 Limit on the unfolded strain in +\n 3.8.3 Limit of the unfolded stress\n 3.8.4 Limit kinematic conditions\n 3.9 Unfolded formulation\n 3.10 Passing to the limit\n 3.10.1 Derivation of u0 in terms of U0 and elimination of u0 in the ij\n 3.10.2 Derivation of R3\n 3.10.3 Derivation of the problem for the bending in the continuum of rods\n 3.10.4 Derivation of the coupled problem for the compression in the continuum rods and the plate\n 4 Junction of a periodic family of rods with a thin plate\n 4.1 The equations in\n 4.2 Decomposition of the displacement in -\n 4.3 Global estimates in term of the total elastic energy\n 4.4 Assumption on the forces and global estimates\n 4.5 Unfolding operators\n 4.6 Weal convergences of the unfolded fields\n 4.7 Relations between the limit fields\n 4.7.1 Limit displacement in +\n 4.7.2 Limit displacement in -\n 4.7.3 Limit on the unfolded stress in +\n 4.7.4 Limit on the unfolded strain in -\n 4.7.5 Limit of the unfolded stress in -\n 4.7.6 Limit kinematic conditions\n 4.8 Unfolded formulation\n 4.9 Passing to the limit\n 4.9.1 The uncoupled “membrane” problem in the plate\n 4.9.2 The coupled problem for the bending of the rods and the plate\n References\nBernadette Miara Multi-scale Modelling of New Composites: Theory and Numerical Simulation\n 1 Introduction\n 1.1 Periodic structures and unfolding operator\n 1.2 Extension to perforated domain\n 2 Bianisotropic electromagnetic composites with memory\n 2.1 Maxwell equations\n 2.1.1 Existence theorem\n 2.1.2 Conservation law\n 2.2 Heterogeneous material and limit homogeneous model\n 2.3 Stability of the constitutive law\n 2.4 Frequency formulation\n 2.4.1 Heterogeneous material and homogenization\n 2.5 Numerical simulations\n 2.6 Conclusions\n 3 Piezoelectric composites\n 3.1 A tool for the design of a bio-material\n 3.1.1 Description of the structure and constitutive laws\n 3.1.2 Limit problem\n 3.2 Extension to perforated shells\n 3.2.1 Geometry of a thin shell and Koiter model\n 3.2.2 Modelling of piezoelectric Koiter shell\n 3.2.3 Description of the structure\n 3.2.4 Limit model\n 3.3 Numerical simulation\n 3.4 Conclusions\n 4 Strongly heterogeneous elastic composites — a sensitivity analysis\n 4.1 Propagation of elastic waves — static problem\n 4.2 The composite structure and the scaling of the data\n 4.3 Limit homogeneous problem\n 4.3.1 Distribution of acoustic band-gaps\n 4.4 Sensitivity analysis\n 4.4.1 Sensitivity analysis with 2 design variables\n 4.4.2 Sensitivity analysis with 4 design variables\n 4.5 Numerical simulation of band gaps in 2-dimension\n 5 Conclusion\n 6 Appendix. Sketch of the proofs\n 6.1 Electromagnetic composites\n 6.2 Piezoelectric composites\n 6.3 Strongly heterogeneous elastic composites\n Acknowlegments\n References\nAssyr Abdulle A Priori and a Posteriori Error Analysis for Numerical Homogenization: A Unified Framework\n 1 Introduction\n 2 Model problem and homogenization\n 3 Coupling macro and micro FE methods the FE-HMM\n 4 Preliminaries\n 4.1 Standard FEM with numerical quadrature\n 4.2 Energy equivalence and coercivity\n 4.3 Micro problem, coupling condition and micro error\n 4.4 Multiscale flux for non-conforming FEs and a posteriori estimates\n 4.5 Reformulation of the FE-HMM\n 5 A priori error analysis\n 6 A posteriori error analysis\n Acknowledgments\n References