Introduction To Multiscale Mathematical Modeling
- Length: 170 pages
- Edition: 1
- Language: English
- Publisher: Wspc (Europe)
- Publication Date: 2022-06-30
- ISBN-10: 1800612311
- ISBN-13: 9781800612310
- Sales Rank: #0 (See Top 100 Books)
This book introduces the reader to multiscale mathematical modeling that starts by describing a physical process at the microscopic level, and is followed by the macroscopic description of that process. There are two preliminary chapters introducing the main equations of mathematical physics and serves as revision of all of the necessary mathematical notions needed to navigate the domain of multiscale research. The author gives a rigorous presentation of the tools of mathematical modeling, as well as an evaluation of the errors of the method. This allows readers to analyze the limitations and accuracy of the method. The book is accessible to a wide range of readers, from specialists in engineering to applied mathematicians working in the applications of materials science, biophysics and medicine.
Contents Preface About the Author Chapter 1. Derivation of the Main Equations of Mathematical Physics 1. Heat Equation 2. Boundary/Initial/Interface Conditions 3. Particular Cases: Generalizations 3.1. Stationary equation 3.2. Two-dimensional and one-dimensional models 4. Elasticity Equation (Solid Mechanics) 5. Navier–Stokes and Stokes Equations (Fluid Mechanics) 6. Coupling of Different Models Chapter 2. Analysis of the Main Equations of Mathematical Physics 1. Some Elements of Functional Analysis 1.1. Vector spaces 1.2. Normedspaces 1.3. Inner product spaces 1.4. Linear operators 2. Sobolev Spaces 2.1. Auxiliary spaces 2.2. Sobolev space H1 3. Poincaré’s Inequalities 3.1. Poincaré–Friedrichs inequality 3.2. Poincaré’s inequality in a parallelepiped 4. Stationary Conductivity Equation 5. Stationary Elasticity Equation 6. Stationary Stokes Equation 7. Galerkin Method for the Heat Equation 8. On the Finite Difference Method 8.1. Approximation of the heat equation by an explicit finite difference scheme 8.1.1. Stability of the difference scheme 8.2. Generalizations: Formal spectral rule of stability (linear case) References Chapter 3. Homogenization: From Micro-scale to Macro-scale: Application to Mechanics of Composite Materials 1. What is a Composite Material? 2. From Micro to Macro 3. Homogenization Techniques: Heat Equation (1-D case) 4. Error Estimate 5. Homogenization: Multiple Dimensions 6. Error Estimate: Multiple Dimensions 7. When the Equivalent Homogeneity Hypothesis is Wrong: Some Remarks on High-Contrast Media Homogenization 7.1. Composite reinforced by highly conductive fibers 7.2. High-contrast spectral problems References Chapter 4. Dimension Reduction and Multiscale Modeling for Thin Structures 1. Dimension Reduction for the Poisson Equation in a Thin Rectangle: The Case of the Neumann Boundary Condition at the Lateral Boundary 2. Asymptotic Coupling of Models of Different Dimensions: Method of Asymptotic Partial Decomposition of the Domain (MAPDD) 3. Dimension Reduction for the Poisson Equation in a Thin Rectangle: Case of the Dirichlet Boundary Condition 4. Dirichlet’s Problem for Laplacian in a Thin Tube Structure 5. Method of Asymptotic Partial Decomposition of Domain for a T-shaped Domain 6. Method of Asymptotic Partial Decomposition of Domain for Flows in a Tube Structure (Applications in Hemodynamics) 6.1. Tube structure: Graphs 6.2. Formulation of the problem 6.3. Partial asymptotic decomposition of the domain for the Stokes equation References Appendix A. Diffusion Equation with Dirac-like Potential: Model of a Periodic Set of Small Cells in a Nutrient 1. On the Approximation of Dirac’s Potential 2. Periodic Dirac-like Potential 2.1. Counterexample of the convergence of the solution for problem (A.15), (A.16) to thesolution for the homogenized equation (A.19), (A.20) for d = 3, εω1/6 ≫ 1 3. Numerical Tests in 2-D 4. Numerical Tests in 3-D 5. Partial Homogenization Appendix B. Proof of Riesz–Frechet Representation Theorem Index
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