{
"type": "Article",
"authors": [
{
"type": "Person"
}
],
"identifiers": [],
"title": "Book reviews",
"meta": {},
"content": [
{
"type": "Figure",
"id": "fig1",
"licenses": [
{
"type": "CreativeWork",
"content": [
{
"type": "Paragraph",
"content": [
"All rights reserved."
]
}
]
}
],
"content": [
{
"type": "ImageObject",
"contentUrl": "MAG-177_Book-cover.png",
"mediaType": "image/png",
"meta": {
"inline": false
}
}
]
},
{
"type": "Paragraph",
"id": "p1",
"content": [
"Homogenization is a powerful tool used in the analysis of applied problems which have multiple scales and complex\nstructures. Broadly speaking, homogenization provides a basis to determine macroscopic (effective) equations for\nmaterials by using the properties of the material at the microscale. Oftentimes the structures to which homogenization\nis applied have or are assumed to have a periodic structure and the notion of two-scale convergence can play an\nimportant role in the analysis, but homogenization is not restricted to only this case and can be applied to more\ngeneral disordered (non-periodic) media. Here a more general framework of convergence such as ",
{
"type": "MathFragment",
"mathLanguage": "mathml",
"text": "G",
"meta": {
"altText": "G"
}
},
"-convergence may be\nused in the analysis. This research monograph serves as an introduction to homogenization theory, while at the same\ntime it explains how to use homogenization in applications in biology, physics, and engineering that will appeal to a\nwide audience."
]
},
{
"type": "Paragraph",
"id": "p2",
"content": [
"The book starts with an introductory chapter where important theory and notions needed for subsequent chapters are\nintroduced. This includes some fundamental functional analysis, important function spaces, and essential theorems\nregarding concepts such as strong and weak derivatives, the\ntrace theorem,\nthe\nLax–Milgram\ntheorem, and so on. This\nchapter also includes the geometric description of the porous medium that the authors propose to study. Starting with\nthe unit cell ",
{
"type": "MathFragment",
"mathLanguage": "mathml",
"text": "𝒴=]0,1[n",
"meta": {
"altText": "\\mathscr{Y}=\\mathopen{]}0,1\\mathclose{[}^{n}"
}
},
" where ",
{
"type": "MathFragment",
"mathLanguage": "mathml",
"text": "n=2,3",
"meta": {
"altText": "n=2,3"
}
},
", letting ",
{
"type": "MathFragment",
"mathLanguage": "mathml",
"text": "𝒴s",
"meta": {
"altText": "\\mathscr{Y}^{s}"
}
},
" (the solid part) be a subset of\n",
{
"type": "MathFragment",
"mathLanguage": "mathml",
"text": "𝒴¯",
"meta": {
"altText": "\\bar{\\mathscr{Y}}"
}
},
" and ",
{
"type": "MathFragment",
"mathLanguage": "mathml",
"text": "𝒴f=𝒴\\𝒴s",
"meta": {
"altText": "\\mathscr{Y}^{f}=\\mathscr{Y}\\backslash\\mathscr{Y}^{s}"
}
},
" (the fluid part), making the periodic\narrangement of ",
{
"type": "MathFragment",
"mathLanguage": "mathml",
"text": "𝒴s",
"meta": {
"altText": "\\mathscr{Y}^{s}"
}
},
" over ",
{
"type": "MathFragment",
"mathLanguage": "mathml",
"text": "ℝn",
"meta": {
"altText": "\\mathbb{R}^{n}"
}
},
", the authors outline the process for obtaining the domains\n",
{
"type": "MathFragment",
"mathLanguage": "mathml",
"text": "Ωsε",
"meta": {
"altText": "\\Omega_{s}^{\\varepsilon}"
}
},
" and ",
{
"type": "MathFragment",
"mathLanguage": "mathml",
"text": "Ωfε",
"meta": {
"altText": "\\Omega_{f}^{\\varepsilon}"
}
},
", which represent the solid and fluid parts of the porous\nmedium ",
{
"type": "MathFragment",
"mathLanguage": "mathml",
"text": "Ω=]0,L[n",
"meta": {
"altText": "\\Omega=\\mathopen{]}0,L\\mathclose{[}^{\\,n}"
}
},
"."
]
},
{
"type": "Paragraph",
"id": "p3",
"content": [
"After discussing\nthe geometry, several important homogenization notions are introduced, including\nthe following:"
]
},
{
"type": "Paragraph",
"content": []
},
{
"type": "List",
"items": [
{
"type": "ListItem",
"content": [
{
"type": "Paragraph",
"id": "S0.I1.i1.p1",
"content": [
"Two-scale convergence, namely:"
]
},
{
"type": "Paragraph",
"content": [
{
"type": "Emphasis",
"content": [
"The sequence ",
{
"type": "MathFragment",
"mathLanguage": "mathml",
"text": "{wε}⊂L2(Ω)",
"meta": {
"altText": "\\{w^{\\varepsilon}\\}\\subset L^{2}(\\Omega)"
}
},
" is said to two-scale converge to a limit ",
{
"type": "MathFragment",
"mathLanguage": "mathml",
"text": "w∈L2(Ω×𝒴)",
"meta": {
"altText": "w\\in L^{2}(\\Omega\\times\\mathscr{Y})"
}
},
" if\nfor any ",
{
"type": "MathFragment",
"mathLanguage": "mathml",
"text": "σ∈C∞(Ω;C#∞(𝒴))",
"meta": {
"altText": "\\sigma\\in C^{\\infty}(\\Omega;C_{\\#}^{\\infty}(\\mathscr{Y}))"
}
},
" one has"
]
}
]
},
{
"type": "MathBlock",
"id": "S0.Ex1",
"mathLanguage": "mathml",
"text": "limε→0∫Ωwε(x)σ(x,xε)𝑑x=∫Ω∫𝒴w(x,y)σ(x,y)𝑑y𝑑x.",
"meta": {
"altText": "\\qquad\\!\\!\\lim_{\\varepsilon\\rightarrow 0}\\int_{\\Omega}w^{\\varepsilon}(x)\\sigma\\Bigl(x,\\frac{x}{\\varepsilon}\\Bigr)\\,dx=\\int_{\\Omega}\\int_{\\mathscr{Y}}w(x,y)\\sigma(x,y)\\,dy\\,dx."
}
},
{
"type": "Paragraph",
"content": [
"Here ",
{
"type": "MathFragment",
"mathLanguage": "mathml",
"text": "#",
"meta": {
"altText": "\\#"
}
},
" denotes unit cube periodicity."
]
}
]
},
{
"type": "ListItem",
"content": [
{
"type": "Paragraph",
"content": [
"A homogenized equation for a boundary value problem with unknown ",
{
"type": "MathFragment",
"mathLanguage": "mathml",
"text": "𝐮(𝐱)",
"meta": {
"altText": "\\mathbf{u}({\\mathbf{x}})"
}
},
" and an asymptotic solution in powers of ",
{
"type": "MathFragment",
"mathLanguage": "mathml",
"text": "ε→0",
"meta": {
"altText": "\\varepsilon\\rightarrow 0"
}
},
",\nnamely:"
]
},
{
"type": "MathBlock",
"id": "S0.Ex2",
"mathLanguage": "mathml",
"text": "𝐮(𝐱)=𝐮0(𝐱)+ε1𝐮1(𝐱,𝐲)+ε2𝐮2(𝐱,𝐲)+⋯, where 𝐲=𝐱ε.",
"meta": {
"altText": "\\qquad\\!\\!\\mathbf{u}(\\mathbf{x})=\\mathbf{u}^{0}(\\mathbf{x})+\\varepsilon^{1}\\mathbf{u}^{1}(\\mathbf{x},\\mathbf{y})+\\varepsilon^{2}\\mathbf{u}^{2}(\\mathbf{x},\\mathbf{y})+\\cdots\\,,\\textrm{\\ \\ where\\ }\\mathbf{y}=\\frac{\\mathbf{x}}{\\varepsilon}."
}
}
]
}
],
"order": "Unordered"
},
{
"type": "Paragraph",
"id": "p5",
"content": [
"The introductory chapter ends with a discussion of two-scale convergence with time\ndependence,\nand potential and solenoidal fields."
]
},
{
"type": "Paragraph",
"content": [
"With the stage set for subsequent chapters, the authors then move on to discuss a range of applications of\nhomogenization theory. These cover the technique applied to soft tissue (the authors note that soft tissue does not\nhave a periodic structure, but there is a scale separation), and include applications of homogenization pertaining to\nthe following topics:"
]
},
{
"type": "Paragraph",
"content": []
},
{
"type": "List",
"items": [
{
"type": "ListItem",
"content": [
{
"type": "Paragraph",
"id": "S0.I2.i1.p1",
"content": [
"acoustics\nin porous media,"
]
}
]
},
{
"type": "ListItem",
"content": [
{
"type": "Paragraph",
"id": "S0.I2.i2.p1",
"content": [
"wet\nionic piezoelectric bone,"
]
}
]
},
{
"type": "ListItem",
"content": [
{
"type": "Paragraph",
"id": "S0.I2.i3.p1",
"content": [
"viscoelasticity\nwith contact friction between the phases,"
]
}
]
},
{
"type": "ListItem",
"content": [
{
"type": "Paragraph",
"id": "S0.I2.i4.p1",
"content": [
"acoustics\nin random microstructure (in this chapter the notion of stochastic two-scale limits as well as compactness properties of the convergence are discussed),"
]
}
]
},
{
"type": "ListItem",
"content": [
{
"type": "Paragraph",
"id": "S0.I2.i5.p1",
"content": [
"bone\ntissue modeled as a periodic two-phase material composed of a viscoelastic solid matrix filled with a non-Newtonian fluid (representing bone marrow),"
]
}
]
},
{
"type": "ListItem",
"content": [
{
"type": "Paragraph",
"id": "S0.I2.i6.p1",
"content": [
"homogenization\nof viscoelastic flows (the notion of ",
{
"type": "MathFragment",
"mathLanguage": "mathml",
"text": "G",
"meta": {
"altText": "G"
}
},
"-convergence is introduced when this application is discussed), "
]
}
]
},
{
"type": "ListItem",
"content": [
{
"type": "Paragraph",
"id": "S0.I2.i7.p1",
"content": [
"multiscale\nFEM for the modeling of cancellous bone."
]
}
]
}
],
"order": "Unordered"
},
{
"type": "Paragraph",
"id": "p7",
"content": [
"In summary, this textbook provides a well-planned introduction to the theory and applications of homogenization. The\napplications discussed provide a strong motivation for further study of the topic. As stated by the authors, the book\nis a result of years of research collaborations. As such, it would serve as a great reference for researchers\nincluding those such as applied mathematicians, engineers, and geophysicists. It could also serve as a textbook for\na course or courses in homogenization theory or a special graduate seminar course. A motivated student could also use\nthe book for self-study. The bibliography contains over 400 references and provides a good basis for further reading."
]
},
{
"type": "Paragraph",
"id": "p8",
"content": [
"Robert P. Gilbert, Ana Vasilic, Sandra Klinge, Alex Panchenko and Klaus Hackl, ",
{
"type": "Emphasis",
"content": [
"Applications of Homogenization Theory to the Study of Mineralized Tissue"
]
},
". Chapman & Hall, 2020, 297 pages, Hardback ISBN 978-1-584-88791-1, Paperback ISBN 978-0-367-71372-0, eBook ISBN 978-0-4291-4338-0.\n"
]
},
{
"type": "Paragraph",
"id": "authorinfo",
"content": [
"\nMichael Shoushani is an associate professor of mathematics at Western Connecticut State University. His interests include inverse and transmission problems in poroelastic media.\n",
{
"type": "Link",
"target": "mailto:shoushanim@wcsu.edu",
"content": [
"shoushanim@wcsu.edu"
]
}
]
}
]
}