There is, I think, something ethereal about i —the square root of minus one. I remember first hearing about it at school. It seemed an odd beast at that time—an intruder hovering on the edge of reality.

Usually familiarity dulls this sense of the bizarre, but in the case of i it was the reverse: over the years the sense of its surreal nature intensified. It seemed that it was impossible to write mathematics that described the real world in …

I and i

Preface 1. Introduction 2. Conservation laws and differential equations 3. Characteristics and Riemann problems for linear hyperbolic equations 4. Finite-volume methods 5. Introduction to the CLAWPACK software 6. High resolution methods 7. Boundary conditions and ghost cells 8. Convergence, accuracy, and stability 9. Variable-coefficient linear equations 10. Other approaches to high resolution 11. Nonlinear scalar conservation laws 12. Finite-volume methods for nonlinear scalar conservation laws 13. Nonlinear systems of conservation laws 14. Gas dynamics and the Euler equations 15. Finite-volume methods for nonlinear systems 16. Some nonclassical hyperbolic problems 17. Source terms and balance laws 18. Multidimensional hyperbolic problems 19. Multidimensional numerical methods 20. Multidimensional scalar equations 21. Multidimensional systems 22. Elastic waves 23. Finite-volume methods on quadrilateral grids Bibliography Index.

https://depts.washington.edu/clawpack/book2/sample.pdf

Finite Volume Methods for Hyperbolic Problems

to be done in this area. Face recognition is a problem that scarcely clamoured for attention before the computer age but, having surfaced, has involved a wide range of techniques and has attracted the attention of some fine minds (David Mumford was a Fields Medallist in 1974). This singular application of mathematical modelling to a messy applied problem of obvious utility and importance but with no unique solution is a pretty one to share with students: perhaps, returning to the source of our opening quotation, we may invert Duncan's earlier observation, 'There is an art to find the mind's construction in the face!'.

Linear and nonlinear waves, by G. B. Whitham. Pp.636. £50. 1999. ISBN 0 471 35942 4 (Wiley).

This paper presents a review of the principal developments in functionally graded materials (FGMs) with an emphasis on the recent work published since 2000. Diverse areas relevant to various aspects of theory and applications of FGM are reflected in this paper. They include homogenization of particulate FGM, heat transfer issues, stress, stability and dynamic analyses, testing, manufacturing and design, applications, and fracture. The critical areas where further research is needed for a successful implementation of FGM in design are outlined in the conclusions. DOI: 10.1115/1.2777164

https://web.mst.edu/~vbirman/papers/Modeling%20and%20Analysis%20of%20FGM_Review_2007.pdf

Modeling and Analysis of Functionally Graded Materials and Structures

http://users.auth.gr/~akonsta/2011_Gradient%20elasticity%20in%20statics%20and%20dynamics.pdf

Gradient elasticity in statics and dynamics: An overview of formulations, length scale identification procedures, finite element implementations and new results

The dispersive effects due to the presence of microstructure in solids are studied. The basic mathematical model is derived following Mindlin's theory. In the one-dimensional case the governing equations of a linear system are presented. An approximation using the slaving principle indicates a hierarchy of waves. The corresponding dispersion relations are compared with each other. The choice between the models can be made on the basis of physical effects described by dispersion relations.

Waves in microstructured materials and dispersion

The propagation of stress waves in functionally graded materials (FGMs) is studied numerically by means of the composite wave-propagation algorithm. Two distinct models of FGMs are considered: (i) a multilayered metal–ceramic composite with averaged properties within layers; (ii) randomly embedded ceramic particles in a metal matrix with prescribed volume fraction. The numerical simulation demonstrates the applicability of that algorithm to the modelling of FGMs without any averaging procedure. The analysis based on simulation shows significant differences in the stress wave characteristics for the distinct models that can be used for optimizing the response of such structures to impact loading.

http://www.cs.ioc.ee/~berez/texts/11.ejms-03.pdf

Numerical simulation of two-dimensional wave propagation in functionally graded materials

Dynamic degrees of freedom and internal variables are treatedin a uniform way. The unification is achieved by means of the introduction of a dual internal variable. This duality provides the corresponding evolution equations depending on whether the Onsager–Casimir reciprocity relations are satisfied or not.

http://www.cs.ioc.ee/~berez/texts/46.VanBerezovskiEngelbrecht08.pdf

Internal Variables and Dynamic Degrees of Freedom

1Institute of Cybernetics at Tallinn University of Technolo gy, Akadeemia tee 21, 12618 Tallinn, Estonia 2Dept. of Theoretical Physics, Wigner RCP, HAS, H-1121 Budap est, Konkoly Thege Miklós út. 29-33, Hungary and 3 Dept. of Energy Engineering, Budapest University of Techno logy and Economics, H-1111, Budapest, Bertalan Lajos út. 4 -6, Hungary and Montavid Thermodynamic Research Group arkadi.berezovski@cs.ioc.ee

Internal Variables in Thermoelasticity

This book shows the advanced methods of numerical simulation of waves and fronts propagation in inhomogeneous solids and introduces related important ideas associated with the application of numerical methods for these problems. Great care has been taken throughout the book to seek a balance between the thermomechanical analysis and numerical techniques. It is suitable for advanced undergraduate and graduate courses in continuum mechanics and engineering. Necessary prerequisites for this text are basic continuum mechanics and thermodynamics. Some elementary knowledge of numerical methods for partial differential equations is also preferable.

/pdf/numerical-simulation-of-waves-and-fronts-in-inhomogeneous-1v0llnqe9o.pdf

Arkadi Berezovski

Papers

Waves in microstructured materials and dispersion

Numerical simulation of two-dimensional wave propagation in functionally graded materials

Internal Variables and Dynamic Degrees of Freedom

Internal Variables in Thermoelasticity

Numerical Simulation of Waves and Fronts in Inhomogeneous Solids