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Introduction To Multiscale Mathematical Modeling [Kõva köide]

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(Univ Jean Monnet, France & Vilnius Univ, Lithuania)
  • Formaat: Hardback, 172 pages
  • Ilmumisaeg: 22-Jul-2022
  • Kirjastus: World Scientific Europe Ltd
  • ISBN-10: 1800612311
  • ISBN-13: 9781800612310
Teised raamatud teemal:
Introduction To Multiscale Mathematical ModelingSuurem pilt
  • Formaat: Hardback, 172 pages
  • Ilmumisaeg: 22-Jul-2022
  • Kirjastus: World Scientific Europe Ltd
  • ISBN-10: 1800612311
  • ISBN-13: 9781800612310
Teised raamatud teemal:
Püsilink: https://www.kriso.ee/db/9781800612310.html
Märksõnad:
"This book introduces the reader to multiscale mathematical modeling that starts from 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 allowing readers to analyze the limitations and accuracy of the method. The book is accessible to a wide range of readers: specialists in engineering, applied mathematicians working in the applications of materials science, biophysics and medicine"--

Based on a graduate course he has taught at universities in several countries, Panasenko introduces the main mathematical models that describe mechanical behavior at microscopic level of heterogeneous media and for blood flow in a network of vessels. He covers derivation and analysis of the main equations of mathematical physics, homogenization: from micro-scale to macro-scale: application to the mechanics of composite material, and dimension reduction and multiscale modeling for thin structures. Annotation ©2022 Ringgold, Inc., Portland, OR (protoview.com)

This Book Introduces The Reader To Multiscale Mathematical Modeling That Starts From 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 Allowing Readers To Analyze The Limitations And Accuracy Of The Method. The Book Is Accessible To A Wide Range Of Readers: Specialists In Engineering, Applied Mathematicians Working In Applied Mathematicians Working In The Applications Of Materials Science Including The Fields Of Biophysics And Medicine.

Preface v
About the Author vii
Chapter 1 Derivation of the Main Equations of Mathematical Physics
1(24)
1 Heat Equation
1(5)
2 Boundary/Initial/Interface Conditions
6(4)
3 Particular Cases: Generalizations
10(3)
3.1 Stationary equation
10(1)
3.2 Two-dimensional and one-dimensional models
11(2)
4 Elasticity Equation (Solid Mechanics)
13(8)
5 Navier-Stokes and Stokes Equations (Fluid Mechanics)
21(2)
6 Coupling of Different Models
23(2)
Chapter 2 Analysis of the Main Equations of Mathematical Physics
25(42)
1 Some Elements of Functional Analysis
25(8)
1.1 Vector spaces
25(1)
1.2 Normed spaces
26(3)
1.3 Inner product spaces
29(2)
1.4 Linear operators
31(2)
2 Sobolev Spaces
33(6)
2.1 Auxiliary spaces
33(2)
2.2 Sobolev space H1
35(4)
3 Poincare's Inequalities
39(4)
3.1 Poincare-Friedrichs inequality
39(1)
3.2 Poincare's inequality in a parallelepiped
40(3)
4 Stationary Conductivity Equation
43(4)
5 Stationary Elasticity Equation
47(2)
6 Stationary Stokes Equation
49(2)
7 Galerkin Method for the Heat Equation
51(4)
8 On the Finite Difference Method
55(12)
8.1 Approximation of the heat equation by an explicit finite difference scheme
55(6)
8.2 Generalizations: Formal spectral rule of stability (linear case)
61(4)
References
65(2)
Chapter 3 Homogenization: From Micro-scale to Macro-scale: Application to Mechanics of Composite Materials
67(34)
1 What is a Composite Material?
67(1)
2 From Micro to Macro
68(5)
3 Homogenization Techniques: Heat Equation (1-D case)
73(5)
4 Error Estimate
78(3)
5 Homogenization: Multiple Dimensions
81(12)
6 Error Estimate: Multiple Dimensions
93(3)
7 When the Equivalent Homogeneity Hypothesis is Wrong: Some Remarks on High-Contrast Media Homogenization
96(5)
7.1 Composite reinforced by highly conductive fibers
96(2)
7.2 High-contrast spectral problems
98(1)
References
99(2)
Chapter 4 Dimension Reduction and Multiscale Modeling for Thin Structures
101(24)
1 Dimension Reduction for the Poisson Equation in a Thin Rectangle: The Case of the Neumann Boundary Condition at the Lateral Boundary
101(3)
2 Asymptotic Coupling of Models of Different Dimensions: Method of Asymptotic Partial Decomposition of the Domain (MAPDD)
104(5)
3 Dimension Reduction for the Poisson Equation in a Thin Rectangle: Case of the Dirichlet Boundary Condition
109(4)
4 Dirichlet's Problem for Laplacian in a Thin Tube Structure
113(2)
5 Method of Asymptotic Partial Decomposition of Domain for a T-shaped Domain
115(2)
6 Method of Asymptotic Partial Decomposition of Domain for Flows in a Tube Structure (Applications in Hemodynamics)
117(8)
6.1 Tube structure: Graphs
117(2)
6.2 Formulation of the problem
119(1)
6.3 Partial asymptotic decomposition of the domain for the Stokes equation
120(3)
References
123(2)
Appendix A Diffusion Equation with Dirac-like Potential: Model of a Periodic Set of Small Cells in a Nutrient 125(24)
Appendix B Proof of Riesz-Frechet Representation Theorem 149(8)
Index 157