Bounds for effective parameters of heterogeneous media by analytic continuation

Golden, Kenneth M.; Papanicolaou, George

doi:10.1007/bf01216179

Cited by 262 publications

(320 citation statements)

References 19 publications

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“…Here · denotes ensemble averaging over the probability distribution defining the random medium, and E 0 = E 0 e 1 for example, where e 1 is a unit vector in the x−direction [16]. Equivalently, we find φ satisfying…”

Section: Mathematical Methodsmentioning

confidence: 99%

“…Consider conduction in two phase composites [8,10,16,17], where E and J are the electric and current density fields satisfying J = σE, ∇· J = 0 and ∇× E = 0, and σ is the local conductivity. For a stationary random medium in 2D or 3D with component conductivities σ 1 and σ 2 , σ = σ 1 χ 1 + σ 2 χ 2 , where χ 1 = 1 in medium one and is 0 otherwise, with χ 2 = 1 − χ 1 .…”

Section: Mathematical Methodsmentioning

confidence: 99%

“…The key to the analytic continuation method is the Stieltjes integral representation [14][15][16][17] …”

Section: Mathematical Methodsmentioning

confidence: 99%

“…Stieltjes integral representations for the bulk transport coefficients such as σ * incorporate the two phase mixture geometry in a spectral measure µ of G [16]. For discrete media such as the random resistor network (RRN) [11], G is a real-symmetric random matrix and the spectral measure µ as well as the electric field E are given explicitly in terms of its eigenvalues and eigenvectors [10,17].…”

Section: Introductionmentioning

confidence: 99%

“…The operator G arises in the analytic continuation method [14][15][16] for studying transport in two phase composites. Stieltjes integral representations for the bulk transport coefficients such as σ * incorporate the two phase mixture geometry in a spectral measure µ of G [16].…”

Section: Introductionmentioning

confidence: 99%

See 4 more Smart Citations

Anderson Transition for Classical Transport in Composite Materials

2017

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The Anderson transition in solids and optics is a wave phenomenon where disorder induces localization of the wavefunctions. We find here that the hallmarks of the Anderson transition are exhibited by classical transport at a percolation threshold -without wave interference or scattering effects. As long range order or connectedness develops, the eigenvalue statistics of a key random matrix governing transport crossover toward universal statistics of the Gaussian orthogonal ensemble, and the field eigenvectors delocalize. The transition is examined in resistor networks and sea ice structures.

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Section: Mathematical Methodsmentioning

confidence: 99%

Section: Mathematical Methodsmentioning

confidence: 99%

“…The key to the analytic continuation method is the Stieltjes integral representation [14][15][16][17] …”

Section: Mathematical Methodsmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

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Anderson Transition for Classical Transport in Composite Materials

2017

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An Extremal Problem Arising in the Dynamics of Two‐Phase Materials That Directly Reveals Information about the Internal Geometry

Mattei

Milton

Putinar

2022

Comm Pure Appl Math

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In two‐phase materials, each phase having a non‐local response in time, it has been found that for some driving fields the response somehow untangles at specific times, and allows one to directly infer useful information about the geometry of the material, such as the volume fractions of the phases. Motivated by this, and to obtain an algorithm for designing appropriate driving fields, we find approximate, measure independent, linear relations between the values that Markov functions take at a given set of possibly complex points, not belonging to the interval [‐1,1] where the measure is supported. The problem is reduced to simply one of polynomial approximation of a given function on the interval [‐1,1] and, to simplify the analysis, Chebyshev approximation is used. This allows one to obtain explicit estimates of the error of the approximation, in terms of the number of points and the minimum distance of the points to the interval [‐1,1]. Assuming this minimum distance is bounded below by a number greater than 1/2, the error converges exponentially to zero as the number of points is increased. Approximate linear relations are also obtained that incorporate a set of moments of the measure. In the context of the motivating problem, the analysis also yields bounds on the response at any particular time for any driving field, and allows one to estimate the response at a given frequency using an appropriately designed driving field that effectively is turned on only for a fixed interval of time. The approximation extends directly to Markov‐type functions with a positive semidefinite operator valued measure, and this has applications to determining the shape of an inclusion in a body from boundary flux measurements at a specific time, when the time‐dependent boundary potentials are suitably tailored. © 2022 Wiley Periodicals, Inc.

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Representations for the conductivity functions of multicomponent composites

Milton

Golden

1990

Comm Pure Appl Math

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The erective conductivity a * of a multicomponent composite material is considered. Integral representations for a* treated as a holomorphic function on a polydisk with values in a half-plane are analyzed. A representation for a* is introduced which is symmetric in the component conductivities and for which the moments of the positive measure in the integral are directly related to the coefficients in a perturbation expansion of a* around a homogeneous medium. This second feature, which is important for obtaining bounds on a*, was previously available only in the two-component case. In addition, a bound valid for any holomorphic function of the above type is proven.

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Bounds for effective parameters of heterogeneous media by analytic continuation

Cited by 262 publications

References 19 publications

Anderson Transition for Classical Transport in Composite Materials

Anderson Transition for Classical Transport in Composite Materials

An Extremal Problem Arising in the Dynamics of Two‐Phase Materials That Directly Reveals Information about the Internal Geometry

Representations for the conductivity functions of multicomponent composites

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