Homogenization Applications to the Biological and Physical Sciences

Homogenization is a fairly new, yet deep field of mathematics which is used as a powerful tool for analysis of applied problems which involve multiple scales. Generally, homogenization is utilized as a modeling procedure to describe processes in complex structures. Applications of Homogenization Theory to the Study of Mineralized Tissue functions as an introduction to the theory of homogenization. At the same time, the book explains how to apply the theory to various application problems in biology, physics and engineering. The authors are experts in the field and collaborated to create this book which is a useful research monograph for applied mathematicians, engineers and geophysicists. As for students and instructors, this book is a well-rounded and comprehensive text on the topic of homogenization for graduate level courses or special mathematics classes. Features: Covers applications in both geophysics and biology. Includes recent results not found in classical books on the topic Focuses on evolutionary kinds of problems; there is little overlap with books dealing with variational methods and T-convergence Includes new results where the G-limits have different structures from the initial operators

Introductory Remarks Some Functional Spaces Variational Formulation Geometry of Two Phase Composite Two-scale Convergence Method The Concept of a Homogenized Equation Two-Scale convergence with time dependence Potential and Solenoidal Fields The Homogenization Technique Applied to Soft Tissue Homogenization of Soft Tissue Galerkin approximations Derivation of the effective equation of U0 Acoustics in Porous Media Introduction Diphasic Macroscopic Behavior Well-posedness for problem (3.2.49 and 3.2.55) The slightly compressible di-phasic behavior Wet Ionic, Piezo-electric Bone Introduction Wet bone with ionic interaction Homogenization using Formal Power Series Wet bone without ionic interaction Electrodynamics Visco-elasticity and Contact Friction Between the Phases Kelvin-Voigt Material Rigid Particles in a Visco-elastic Medium Equations of motion and contact conditions Two-scale expansions and formal homogenization Model case I: Linear contract conditions Model case II: Quadratic contract conditions Model case III: Power type contact condition Acoustics in a Random Microstructure Introduction Stochastic Two-scale limits Periodic Approximation Non-Newtonian Interstitial Fluid The Slightly Compressible Polymer. Microscale Problem A Priori Estimates Two-Scale System Description of the effective stress Effective equations Multiscale FEM for the modeling of cancellous bone Concept of the multiscale FEM Microscale: Modeling of the RVE and calculation of the effective material properties Macroscale: Simulation of the ultrasonic test Simplified version of the RVE and comparison with the experimental results Anisotropy of cancellous bone Investigation of the influence of reflection on the attenuation of cancellous bone Determination of the geometry of the RVE for cancellous bone by using the effective complex shear modulus G-convergence and Homogenization of Viscoelastic Flows Introduction Main definitions. Corrector operators for G-convergence A scalar elliptic equation in divergence form Homogenization of two-phase visco-elastic flows with time-varying interface Main theorem and outline of the proof Corrector operators and oscillating test functions Inertial terms in the momentum balance equation Effective deviatoric stress. Proof of the main theorem Fluid-structure interaction Biot Type Models for Bone Mechanics Bone Rigidity Anisotropic Biot Systems The Case of a non-Newtonian Interstitial Fluid Some Time-Dependent Solutions to the Biot System Creation of RVE for Bone Microstructure The RVE Model Reformulation as a Graves-like scheme Absorbring boundary condition-perfectly matched layer Discretized systems Bone Growth and Adaptive Elasticity The Model Scalings of Unknowns Asymptotic Solutions Further Reading

Robert P. Gilbert is a Unidel Professor of Mathematics at the University of Delaware and has authored numerous papers and articles for journals and conferences. He is also a founding editor of Complex Variables and Applicable Analysis and serves on many editorial boards. His current research involves the area of inverse problems, homogenization and the flow of viscous fluids. Ana Vasilic is an associate professor of mathematics at Northern New Mexico College. She has contributed to mathematical publications such as Mathematical and Computer Modeling and Applicable Analysis. Her research interests include applied analysis, partial differential equations, homogenization and multiscale problems in porous media. Sandra Klinge is an assistant professor in computational mechanics at Technische Universitat, Dortmund, Germany. She has written several articles for science journals and contributed to many books. Homogenization, modeling of polymers and multiscale modeling are among her research interests. Alex Panchenko is a professor of mathematics at Washington State University and has written for several publications, many of which are collaborations with Robert Gilbert. These include journals such as SIAM Journal Math. Analysis and Mathematical and Computer Modeling. Klaus Hackl is a professor of mechanics at Ruhr-Universitat Bochum, Germany and has made many scholarly contributions to various journals. His research interests include continuum mechanics, numerical mechanics, modeling of materials and multiscale problems.