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Graeme W. Milton

Publications and source records attributed to Graeme W. Milton.

At least 19 recordsLinked to original sources
arXiv2026

Broadband quasistatic passive cloaking: bounds and limitations in the near-field regime

We consider here several aspects of the following challenging question: is it possible to use a passive cloak to make invisible a dielectric inclusion on a finite frequency interval in the quasistatic regime of Maxwell's equations for an observer close to the object? In this work, by considering the Dirichlet-to-Neumann (DtN) map, we not only answer negatively this question, but we go further and provide some quantitative bounds on this map that provide fundamental limits to both cloaking as well as approximate cloaking. These bounds involve the following physical parameters: the length and center of the frequency interval, the volume of the cloaking device, the volume of the obstacle, and the relative permittivity of the object. Our approach is based on two key tools: i) variational principles from the abstract theory of composites and ii) the analytic approach to deriving bounds from sum rules for passive systems. To use i), we prove a new representation theorem for the DtN map which allows us to interpret this map as an effective operator in the abstract theory of composites. One important consequence of this representation is that it allows one to incorporate the broad and deep results from the theory of composites, such as variational principles, and to apply the bounds derived from them to the DtN map. These results could be useful in other contexts other than cloaking. Next, to use ii), we show that the passivity assumption allows us to connect the DtN map (as function of the frequency) with two important classes of analytic functions, namely, Herglotz and Stieltjes functions. The sum rules for these functions, combined with the variational approach, allows us to derive new inequalities on the DtN map which impose fundamental limitations on passive cloaking, both exact and approximate, over a frequency interval. We consider both cases of lossy and lossless cloaks.

math-ph↗
arXiv2025

A continued fraction approximation for the effective elasticity tensor of two-dimensional polycrystals as a function of the crystal elasticity tensor

For two-dimensional polycrystals the effective elasticity tensor $C_*$ as a function $C_*(C_0)$ of the elasticity tensor $C_0$ of the constituent crystal is considered. It is shown that this function can be approximated by one with a continued fraction expansion resembling that associated with a class of microstructure known as sequential laminates. These are hierarchical microstructures defined inductively. Rank 0 sequential laminates are simply rotations of the pure crystal. Rank $j$ sequential laminates are obtained by laminating together, on a length scale much larger that the existing microstructure and with interfaces perpendicular to some direction $n_j$, rank $j-1$ sequential laminates with a rotation of the pure crystal. The continued fraction approximation for arbitrary polycrystal microstructures typically takes a more general form than that of sequential laminates, but has some free parameters. It is an open question as to whether these free parameters can always be adjusted so the continued fraction approximation matches exactly that of a sequential laminate. If so, one would have established that the elastic response of two-dimensional polycrystals can always be mimicked by that of sequential laminates. Our analysis carries over to the more general case where the strain is replaced by a field $E(x)$ that is the gradient of a vector potential $u(x)$, i.e. $E=\nabla u$ and the stress is replaced by a matrix valued field $J(x)$ that need not be symmetric but has zero divergence $\nabla\cdot J=0$. The tensor $L(x)$ entering the constitutive relation $J=L E$ is locally a rotation of the tensor $L_0$ of the pure crystal that need not have any special symmetries and has 16 independent tensor elements.

cond-mat.mtrl-sci↗
arXiv2025

Further fundamental bounds on the Hall effect in three-dimensional metamaterials

We consider the problem of bounding the effective nonreciprocal properties of metamaterials. Recently, significant progress was made by showing that this problem can be reduced to bounding an equivalent reciprocal one and applying a monotonicity argument. Here, we build upon this result and provide bounds incorporating additional information about the metamaterial. Specifically, in the two-phase case, we provide bounds for isotropic metamaterials. In the multiphase case, we provide bounds for uniaxial metamaterials that additionally incorporate the volume fractions. In both instances, the incorporated additional information significantly tightens the bounds. In the two-phase case, we evaluate the bounds by comparing them to the effective properties of hierarchical laminate microstructures. This comparison ultimately leads us to propose a set of conjectured bounds. While our discussion focuses on the Hall effect, our results are more broadly applicable to other nonreciprocal effects in so far as their mathematical description is equivalent. In particular, our bounds apply to the Faraday effect in the quasistatic regime and in the absence of losses and resonances.

physics.app-ph↗
arXiv2025

A rediscovery of stiff pentmodes. A comment on "High bulk modulus pentamodes: the three-dimensional metal water"

We bring attention to the fact that the claim of Brambilla et.al. [Extreme Mechanics Letters 74 (2025) 102267; arXiv:2406.14502] of discovering a novel design for pentamode materials is incorrect. Back in 2016 Briane, Harutyunyan and myself [Mathematics and Mechanics of Complex Systems 5 (2016) 41--94; arXiv:1606.03305] designed a class of stiff pentamodes, that include the high bulk modulus pentamodes of Brambilla et.al. Our design generalized to three-dimensions, and to full anisotropy, the main aspects of a two-dimensional construction of Sigmund [Journal of the Mechanics and Physics of Solids 48 (2000) 397--428]. It is emphasized that the in depth analysis of Brambilla et.al. goes well beyond our brief treatment.

physics.app-ph↗
arXiv2025

Limits to the Hall effect and other nonreciprocal effects in three-dimensional metamaterials

Metamaterials can exhibit exotic nonreciprocal properties, yet corresponding fundamental limits and design blueprints achieving them are largely unexplored. Here, we derive comprehensive bounds on the effective nonreciprocal properties of three-dimensional metamaterials and identify microstructures achieving or approaching these bounds. We assume that the underlying equations are equivalent to the conductivity problem in a weak applied magnetic field. While we focus on the Hall effect, our results are more generally applicable, particularly to the Faraday effect in the quasistatic regime and in the absence of losses and resonances. Our bounds yield three important implications: First, the effective Hall mobility of a metamaterial cannot be larger than the largest Hall mobility among the constituent materials. Second, under additional conditions, the effective Verdet constant cannot be enhanced either. Third, for diagonal Hall tensor components, the optimal values are achieved either by one of the pure phases or a rank-1 laminate formed from them, provided that the Hall coefficients of all phases have the same sign. Our work elucidates the limits of nonreciprocal metamaterials and identifies key prerequisites for obtaining exotic phenomena such as sign-inversions and enhancements. Several extensions appear within reach, for example to the Faraday effect in metamaterials exhibiting plasmonic resonances.

physics.app-ph↗
arXiv2025

Bounds on the Uniaxial Effective Complex Permittivity of Two-phase Composites and Optimal or Near Optimal Microstructures

Electromagnetic materials with a uniaxial effective permittivity tensor, characterized by its transverse ($ε_\perp$) and axial ($ε_\parallel$) components, play a central role in the design of advanced photonic and electromagnetic materials including hyperbolic metamaterials, and biological imaging platforms. Tight bounds on the complex effective permittivity of such metamaterials are critical for predicting and optimizing their macroscopic electromagnetic response. While rigorous tight bounds exist for isotropic two-phase composites, corresponding results for uniaxial composites remain relatively unexplored. In this work, we systematically investigate the attainable range of $ε_\perp$ and $ε_\parallel$ in the quasistatic regime for two-phase metamaterials with isotropic homogeneous phases. By analyzing known microgeometries and constructing hierarchical laminates (HLs), we demonstrate that the classical bounds on $ε_\perp$ are not optimal. We conjecture improved bounds based on numerically fitted circular arcs derived from convex hulls of $ε_\perp$ values obtained from HLs, and we identify optimal rank-4 HL structures that achieve all points on the conjectured bounds. Additionally, we quantify the correlation between $ε_\perp$ and $ε_\parallel$ for fixed volume fractions, and propose a design algorithm to construct HL microstructures achieving prescribed values of $ε_\perp$. Leveraging the Cherkaev-Gibiansky transformation and the translation method, we extend recent techniques developed for isotropic composites by Kern-Miller-Milton to derive translation bounds on the uniaxial complex effective permittivity tensor. Finally, bounds on the sensitivity of the effective permittivity tensor of low-loss composites are obtained and their optimality is shown in two-dimensions.

physics.optics↗
arXiv2025

Complete characterization of symmetric Kubo-Ando operator means satisfying Molnár's weak associativity

We provide a complete characterization of a subclass of weakly associative means of positive operators in the class of symmetric Kubo-Ando means. This class, which includes the geometric mean, was first introduced and studied in L. Molnár, ``Characterizations of certain means of positive operators," Linear Algebra Appl. 567 (2019) 143-166, where he gives a characterization of this subclass (which we call the Molnár class of means) in terms of the properties of their representing operator monotone functions. Molnár's paper leaves open the problem of determining if the geometric mean is the only such mean in that subclass. Here we give a negative answer to this question by constructing an order-preserving bijection between this class and a class of real measurable odd periodic functions bounded in absolute value by $1/2$. Each member of the latter class defines a Molnar mean by an explicit exponential-integral representation. From this we are able to understand the order structure of the Molnár class and construct several infinite families of explicit examples of Molnár means that are not the geometric mean. Our analysis also shows how to modify Molnár's original characterization so that the geometric mean is the only one satisfying the requisite set of properties.

math.FA↗
arXiv2024

Rapidly convergent series expansions for a class of resolvents

Following advances in the abstract theory of composites, we develop rapidly converging series expansions about $z=\infty$ for the resolvent ${\bf R}(z)=[z{\bf I}-{\bf P}^\dagger{\bf Q}{\bf P}]^{-1}$ where ${\bf Q}$ is an orthogonal projection and ${\bf P}$ is such that ${\bf P}{\bf P}^\dagger$ is an orthogonal projection. It is assumed that the spectrum of ${\bf P}^\dagger{\bf Q}{\bf P}$ lies within the interval $[z^-,z^+]$ for some known $z^+\leq 1$ and $z^-\geq 0$ and that the actions of the projections ${\bf Q}$ and ${\bf P}{\bf P}^\dagger$ are easy to compute. The series converges in the entire $z$-plane excluding the cut $[z^-,z^+]$. It is obtained using subspace substitution, where the desired resolvent is tied to a resolvent in a larger space and ${\bf Q}$ gets replaced by a projection $\underline{\bf Q}$ that is no longer orthogonal. When $z$ is real the rate of convergence of the new method matches that of the conjugate gradient method.

math.NA↗
arXiv2023

Tight Bounds on the Effective Complex Permittivity of Isotropic Composites and Related Problems

Almost four decades ago, Bergman and Milton independently showed that the isotropic effective electric permittivity of a two-phase composite material with a given volume fraction is constrained to lie within lens-shaped regions in the complex plane that are bounded by two circular arcs. An implication of particular significance is a set of limits to the maximum and minimum absorption of an isotropic composite material at a given frequency. Here, after giving a short summary of the underlying theory, we show that the bound corresponding to one of the circular arcs is at least almost optimal by introducing a certain class of hierarchical laminates. In regard to the second arc, we show that a tighter bound can be derived using variational methods. This tighter bound is optimal as it corresponds to assemblages of doubly coated spheres, which can be easily approximated by more realistic microstructures. We briefly discuss the implications for related problems, including bounds on the complex polarizability.

physics.app-ph↗
arXiv2023

Bounds on the Quality-factor of Two-phase Quasi-static Metamaterial Resonators and Optimal Microstructure Designs

Material resonances are fundamentally important in the field of nano-photonics and optics. So it is of great interest to know what are the limits to which they can be tuned. The bandwidth of the resonances in materials is an important feature which is commonly characterized by using the quality (Q) factor. We present bounds on the quality factor of two-phase quasi-static metamaterial resonators evaluated at a given resonant frequency by introducing an alternative definition for the Q-factor in terms of the complex effective permittivity of the composite material. Optimal metamaterial microstrcuture designs achieving points on these bounds are presented. The most interesting optimal microstructure, is a limiting case of doubly coated ellipsoids that attains points on the lower bound. We also obtain bounds on Q for three dimensional, isotropic, and fixed volume fraction two-phase quasi-static metamaterials. Some almost optimal isotropic microstructure geometries are identified.

physics.optics↗
arXiv2023

Determining the volume fraction in 2-phase composites and bodies using time varying applied fields

A body $Θ$ containing two phases, which may form a periodic composite with microstructure much smaller that the body, or which may have structure on a length scale comparable to the body, is subjected to slowly time varying boundary conditions that would produce an approximate uniform field in $Θ$ were it filled with homogeneous material. Here slowly time varying means that the wavelengths and attenuation lengths of waves at the frequencies associated with the time variation are much larger than the size of $Θ$, so that we can make a quasistatic approximation. At least one of the two phase does not have an instantaneous response but rather depends on fields at prior times. The fields may be those associated with electricity, magnetism, fluid flow in porous media, or antiplane elasticity. We find, subject to these approximations, that the time variation of the boundary conditions can be designed so boundary measurements at a specific time $t=t_0$ exactly yield the volume fractions of the phases, independent of the detailed geometric configuration of the phases. Moreover, for specially tailored time variations, the volume fraction can be exactly determined frommeasurements at any time $t$, not just at the specific time $t=t_0$. We also show how time varying boundary conditions, not oscillating at the single frequency $ω_0$, can be designed to exactly retrieve the response at $ω_0$.

math-ph↗
arXiv2022

A unifying perspective on linear continuum equations prevalent in science. Part II: Canonical forms for time-harmonic equations

Following some past advances, we reformulate a large class of linear continuum science equations in the format of the extended abstract theory of composites so that we can apply this theory to better understand and efficiently solve those equations. Here in part II we elucidate the form for many time-harmonic equations that do not involve higher order gradients.

math-ph↗
arXiv2022

Limit analysis of strut nets

Truss structures composed of members that work exclusively in tension or in compression appear in several problems of science and engineering, e.g., in the study of the resisting mechanisms of masonry structures, as well as in the design of spider web-inspired web structures. This work generalizes previous results on the existence of cable webs that are able to support assigned sets of nodal forces under tension. We extend such a problem to the limit analysis of compression-only 'strut nets' subjected to fixed and variable nodal loads. These systems provide discrete element models of masonry bodies, which lie inside the polygon/polyhedron with vertices at the points of application of the given forces ('underlying masonry structures'). It is assumed that fixed nodal forces are combined with variable forces growing proportionally to a scalar multiplier (load multiplier), and that the supporting strut net is subjected to kinematic constraints at given nodal positions.

physics.class-ph↗
arXiv2022

An Energy Conserving Mechanism for Temporal Metasurfaces

Changing the microstructure properties of a space-time metamaterial while a wave is propagating through it, in general requires addition or removal of energy, which can be of exponential form depending on the type of modulation. This limits the realization and application of space-time metamaterials. We resolve this issue by introducing a novel mechanism of conserving energy at temporal metasurfaces in a non-linear setting. The idea is first demonstrated by considering a wave-packet propagating in a discrete medium of 1-d chain of springs and masses, where using our energy conserving mechanism we show that the spring stiffness can be incremented at several time interfaces and the energy will still be conserved. We then consider an interesting application of time-reversed imaging in 1-d and 2-d spring-mass systems with a wave packet traveling in the homogenized regime. Our numerical simulations show that, in 1-d, when the wave packet hits the time-interface two sets of waves are generated, one traveling forward in time and the other traveling backward. The time-reversed waves re-converge at the location of the source and we observe its regeneration. In 2-d, we use more complicated initial shapes and, even then, we observe regeneration of the original image or source. Thus, we achieve time-reversed imaging with conservation of energy in a non-linear system. The energy conserving mechanism can be easily extended to continuum media.

physics.class-ph↗
arXiv2022

The obstacle problem in masonry structures and cable nets

We consider the problem of finding a net that supports prescribed forces applied at prescribed points, yet avoids certain obstacles, with all the elements of the net under compression (strut net) or under tension (cable web). In the case of masonry structures, for instance, this consists in finding a strut net that supports the forces, is contained within the physical structure, and avoids regions that may be not accessible due, for instance, to the presence of holes. We solve such a problem in the two-dimensional case, where the prescribed forces are applied at the vertices of a convex polygon, and we treat the cases of both single and multiple obstacles. By approximating the obstacles by polygonal regions, the task reduces to identifying the feasible domain in a linear programming problem. For a single obstacle we show how the region $Γ$ available to the obstacle can be enlarged as much as possible in the sense that there is no other strut net, having a region $Γ_1$ available to the obstacle with $Γ_1 \subsetΓ$. The case where some of the forces are reactive, unprescribed but reacting to the other prescribed forces, is also treated. It again reduces to identifying the feasible domain in a linear programming problem. Finally, one may allow a subset of the reactive forces to each act not at a prescribed point, but rather at any point on a prescribed line segment. Then the task reduces to identifying the feasible domain in a quadratic programming problem.

math.OC↗
arXiv2022

A possible explanation of dark matter and dark energy involving a vector torsion field

A simple gravitational model with torsion is studied, and it is suggested that it could explain the dark matter and dark energy in the universe. It can be reinterpreted as a model using the Einstein gravitational equations where spacetime has regions filled with a perfect fluid with negative energy (pressure) and positive mass density, other regions containing an anisotropic substance that in the rest frame (where the momentum is zero) has negative mass density and a uniaxial stress tensor, and possibly other "luminal" regions where there is no rest frame. The torsion vector field is inhomogeneous throughout spacetime, and possibly turbulent. Numerical simulations should reveal whether or not the equations are consistent with cosmological observations of dark matter and dark energy.

gr-qc↗
arXiv2021

A unifying perspective on linear continuum equations prevalent in physics. Part VII: Boundary value and scattering problems

We consider simply connected bodies or regions of finite extent in space or space-time and write conservation laws associated with the equations in Parts I-IV. We review earlier work where, for elliptic equations,the boundary value problem is reformulated as a problem in the abstract theory of composites and the associated effective operator is equated with the Dirichlet-to-Neumann map that governs the response of the body. The dielectric polarizability problem and acoustic and electromagnetic scattering by an inclusion are formulated as problems in the extended abstract theory of composites. The scattering response can be determined from appropriate integrals over the inclusion.

math-ph↗