Fractals in Engineering: Theoretical Aspects and Numerical Approximations (SEMA SIMAI Springer Series, 8)

✍ Scribed by Maria Rosaria Lancia (editor), Anna Rozanova-Pierrat (editor)

• Publisher: Springer
• Year: 2021
• Tongue: English
• Leaves: 179
• Category: Library

✦ Synopsis

Fractal structures or geometries currently play a key role in all models for natural and industrial processes that exhibit the formation of rough surfaces and interfaces. Computer simulations, analytical theories and experiments have led to significant advances in modeling these phenomena across wild media. Many problems coming from engineering, physics or biology are characterized by both the presence of different temporal and spatial scales and the presence of contacts among different components through (irregular) interfaces that often connect media with different characteristics. This work is devoted to collecting new results on fractal applications in engineering from both theoretical and numerical perspectives. The book is addressed to researchers in the field.

✦ Table of Contents

Preface
Contents
Editors and Contributors
About the Editors
Contributors
A Numerical Approach to a Nonlinear Diffusion Model for Self-Organized Criticality Phenomena
1 Introduction
2 The Basic Model
3 Numerical Approximation on a Fixed Grid
4 Approximation on a Synchronized Family of Grids
5 Numerical Tests
References
Approximation of 3D Stokes Flows in Fractal Domains
1 Introduction
2 Preliminaries
3 Existence and Uniqueness Results
4 Regularity in Weighted Sobolev Spaces
5 Mean Shear Stress
6 Numerical Approximation
7 Numerical Simulations
References
∞-Laplacian Obstacle Problems in Fractal Domains
1 Introduction
2 Fractal Domains, Approximating Domains and Fibers
3 Setting and Asymptotic Behavior
4 Uniqueness and Perspectives
5 Uniform and Error Estimates
References
Discretization of the Koch Snowflake Domain with Boundary and Interior Energies
1 Introduction
2 Dirichlet Form on the Koch Snowflake
3 Dirichlet Form on the Snowflake Domain
4 Inductive Mesh Construction and Discrete Energy Forms
5 Numerical Results
5.1 Algorithm and Implementation
5.2 The Eigenvalue Counting Function
5.3 Eigenvectors and Eigenvalues in the Low Eigenvalue Regime
5.4 Localization in the High Eigenvalue Regime
6 A Landscape Approach to High Frequency Localization
References
On the Dimension of the Sierpinski Gasket in l2
1 Introduction
2 Invariant Sets
2.1 Infinite Dimensional Sierpinski Gasket
2.2 Hausdorff Dimension of Invariant Sets
2.3 N-Dimensional Simplices
3 Invariant Measures
References
On the External Approximation of Sobolev Spaces by M-Convergence
1 Introduction
2 Sobolev Space Approximations
3 The M-Convergence Result
4 Proof of Lemma 1
5 Proof of Lemma 2
6 Comments
References
Generalization of Rellich–Kondrachov Theorem and Trace Compactness for Fractal Boundaries
1 Introduction
2 Sobolev Extension Domains
3 Trace on the Boundary and Green Formulas
3.1 Framework of d-Sets and Markov's Local Inequality
3.2 General Framework of Closed Subsets of Rn
3.3 Integration by Parts and the Green Formula
4 Sobolev Admissible Domains and the Generalization of the Rellich–Kondrachov Theorem
5 Compactness of the Trace
6 Application to the Poisson Boundary Valued and Spectral Problems
References