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Graeme W. Milton

Publications and source records attributed to Graeme W. Milton.

At least 37 records · Page 2Linked to original sources
arXiv2021

Planar polycrystals with extremal bulk and shear moduli

Here we consider the possible bulk and shear moduli of planar polycrystals built from a single crystal in various orientations. Previous work gave a complete characterization for crystals with orthotropic symmetry. Specifically, bounds were derived separately on the effective bulk and shear moduli, thus confining the effective moduli to lie within a rectangle in the (bulk, shear) plane. It was established that every point in this rectangle could be realized by an appropriate hierarchical laminate microgeometry, with the crystal taking different orientations in the layers, and the layers themselves being in different orientations. The bounds are easily extended to crystals with no special symmetry, but the path to constructing microgeometries that achieve every point in the rectangle defined by the bounds is considerably more difficult. We show that the two corners of the box having minimum bulk modulus are always attained by hierarchical laminates. For the other two corners we present algorithms for generating hierarchical laminates that attain them. Numerical evidence strongly suggests that the corner having maximum bulk and maximum shear modulus is always attained. For the remaining corner, with maximum bulk modulus and minimum shear modulus, it is not yet clear whether the algorithm always succeeds, and hence whether all points in the rectangle are always attained. The microstructures we use are hierarchical laminate geometries that at their core have a self-similar microstructure, in the sense that the microstructure on one length scale is a rotation and rescaling of that on a smaller length scale.

math.AP↗
arXiv2021

An extremal problem arising in the dynamics of two-phase materials that directly reveals information about the internal geometry

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.

math-ph↗
arXiv2020

Further comments on Mark Stockman's article "Criterion for Negative Refraction with Low Optical Losses from a Fundamental Principle of Causality"

We clarify the claims and errors in the paper "Criterion for Negative Refraction with Low Optical Losses from a Fundamental Principle of Causality" by Mark Stockman. Contrary to the central assertion in that paper, simple examples consistent with the basic inequality which Stockman discovered show that it is possible to have negative refraction and low loss in an arbitrarily large frequency window. Further examination of the paper reveals additional errors that invalidate his argument that active materials cannot have low loss and negative refraction in a frequency window. Also, we point out that for active materials non-analyticity of the electrical permittivity in the upper half complex frequency plane does not necessarily imply noncausality, as Stockman infers.

physics.optics↗
arXiv2020

Some open problems in the theory of composites

A selection of open problems in the theory of composites is presented. Particular attention is drawn to the question of whether two-dimensional, two-phase, composites with general geometries have the same set of possible effective tensors as those of hierarchical laminates. Other questions involve the conductivity and elasticity of composites. Finally, some future directions for wave and other equations are mentioned.

math.AP↗
arXiv2020

A unifying perspective on linear continuum equations prevalent in science. Part I: Canonical forms for static, steady, and quasistatic 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 I we elucidate the form for many static, steady, and quasistatic equations.

math-ph↗
arXiv2020

A unifying perspective on linear continuum equations prevalent in science. Part III: Canonical forms for dynamic equations with moduli that may, or may not, vary with time

Enlarging on Parts I and II we write more equations in the desired format of the extended abstract theory of composites. We focus on a multitude of full dynamic equations, including equations where the medium is moving or otherwise changing in time. The motivation is that results and methods in the theory of composites then extend to these equations.

math-ph↗
arXiv2020

A unifying perspective on linear continuum equations prevalent in science. Part VI: rapidly converging series expansions for their solution

We obtain rapidly convergent series expansions of resolvents of operators taking the form ${\bf A}=Γ_1{\bf B}Γ_1$ where $Γ_1({\bf k})$ is a projection that acts locally in Fourier space and ${\bf B}({\bf x})$ is an operator that acts locally in real space. Such resolvents arise naturally when one wants to solve any of the large class of linear physical equations surveyed in Parts I, II, III, and IV that can be reformulated as problems in the extended abstract theory of composites. We show how the information about the spectrum of ${\bf A}$ can be used to greatly improve the convergence rate.

math-ph↗
arXiv2020

A unifying perspective on linear continuum equations prevalent in physics. Part V: resolvents; bounds on their spectrum; and their Stieltjes integral representations when the operator is not selfadjoint

We consider resolvents of operators taking the form ${\bf A}=Γ_1{\bf B}Γ_1$ where $Γ_1({\bf k})$ is a projection that acts locally in Fourier space and ${\bf B}({\bf x})$ is an operator that acts locally in real space. Such resolvents arise naturally when one wants to solve any of the large class of linear physical equations surveyed in Parts I, II, III, and IV that can be reformulated as problems in the extended abstract theory of composites. We review how $Q^*$-convex operators can be used to bound the spectrum of ${\bf A}$. Then, based on the Cherkaev-Gibiansky transformation and subsequent developments, that we reformulate, we obtain for non-Hermitian ${\bf B}$ a Stieltjes type integral representation for the resolvent $(z_0{\bf I}-{\bf A})^{-1}$. The representation holds in the half plane $\Re(e^{i\vartheta}z_0)>c$, where $\vartheta$ and $c$ are such that $c{\bf I}-[e^{i\vartheta}{\bf B}+e^{-i\vartheta}{\bf B}^\dagger]$ is positive definite (and coercive).

math-ph↗
arXiv2020

Inclusions of general shapes having constant field inside the core and non-elliptical neutral coated inclusions with anisotropic conductivity

For certain shapes of inclusions embedded in a body, the field inside the inclusion is uniform for some boundary condition. We provide a construction scheme for inclusions of general shapes having such a uniformity property in two dimensions based on the conformal mapping technique for the potential problem. Using this complex analysis method, we also design non-elliptical neutral coated inclusions with anisotropic conductivity. Neutral coated inclusions do not perturb a background uniform field when they are inserted into a homogeneous matrix. Although coated inclusions of various shapes are neutral to a single field, only concentric ellipses or confocal ellipsoids can be neutral to all uniform fields. This paper presents our work relating to the construction of non-elliptical coated inclusions with anisotropic conductivity in two dimensions that are neutral to all uniform fields, where the assignment of the flux condition on the boundary of the core depends on the applied background field. Using these neutral inclusions, we obtain cylindrical neutral inclusions in three dimensions, with no flux applied to the boundary of the core and with the anisotropic conductivity function of the shell given in accordance with the background uniform field.

physics.class-ph↗
arXiv2020

Substitution of subspace collections with nonorthogonal subspaces to accelerate Fast Fourier Transform methods applied to conducting composites

We show the power of the algebra of subspace collections developed in Chapter 7 of the book "Extending the Theory of Composites to Other Areas of Science (edited by Milton, 2016). Specifically we accelerate the Fast Fourier Transform schemes of Moulinec and Suquet and Eyre and Milton (1994, 1998) for computing the fields and effective tensor in a conducting periodic medium by substituting a subspace collection with nonorthogonal subspaces inside one with orthogonal subspaces. This can be done when the effective conductivity as a function of the conductivity $σ_1$ of the inclusion phase (with the matrix phase conductivity set to $1$) has its singularities confined to an interval $[-β,-α]$ of the negative real $σ_1$ axis. Numerical results of Moulinec and Suquet show accelerated convergence for the model example of a square array of squares at $25\%$ volume fraction. For other problems we show how $Q^*_C$-convex functions can be used to restrict the region where singularities of the effective tensor as a function of the component tensors might be found.

math.NA↗
arXiv2019

On the forces that cable webs under tension can support and how to design cable webs to channel stresses

In many applications of Structural Engineering the following question arises: given a set of forces $\mathbf{f}_1,\mathbf{f}_2,\dots,\mathbf{f}_N$ applied at prescribed points $\mathbf{x}_1,\mathbf{x}_2,\dots,\mathbf{x}_N$, under what constraints on the forces does there exist a truss structure (or wire web) with all elements under tension that supports these forces? Here we provide answer to such a question for any configuration of the terminal points $\mathbf{x}_1,\mathbf{x}_2,\dots,\mathbf{x}_N$ in the two- and three-dimensional case. Specifically, the existence of a web is guaranteed by a necessary and sufficient condition on the loading which corresponds to a finite dimensional linear programming problem. In two-dimensions we show that any such web can be replaced by one in which there are at most $P$ elementary loops, where elementary means the loop cannot be subdivided into subloops, and where $P$ is the number of forces $\mathbf{f}_1,\mathbf{f}_2,\dots,\mathbf{f}_N$ applied at points strictly within the convex hull of $\mathbf{x}_1,\mathbf{x}_2,\dots,\mathbf{x}_N$. In three-dimensions we show that, by slightly perturbing $\mathbf{f}_1,\mathbf{f}_2,\dots,\mathbf{f}_N$, there exists a uniloadable web supporting this loading. Uniloadable means it supports this loading and all positive multiples of it, but not any other loading. Uniloadable webs provide a mechanism for distributing stress in desired ways.

math-ph↗
arXiv2018

Exact relations for Green's functions in linear PDE and boundary field equalities: a generalization of conservation laws

Many physics problems have $J(x)=L(x)E(x)+h(x)$, source $h(x)$, fields $E$,$J$ satisfying differential constraints, symbolized by $E\in\cal E$,$J\in\cal J$ where $\cal E$,$\cal J$ are orthogonal spaces. If $L(x)$ takes values in certain nonlinear manifolds $\cal M$, and coercivity, boundedness hold, then the Green's function satisfies exact identities. We also link Green's functions of different problems. The analysis, based on the theory of exact relations for composites, does not assume microscale variations in $L(x)$, and allows for other equations, such as for waves in lossy media. For bodies $Ω$, in which $L(x)\in{\cal M}$, the Dirichlet-to-Neumann map satisfies boundary field equalities. These generalize the notion of conservation laws: the constraints on the fields inside $Ω$ give identities satisfied by the boundary fields, and provide extra constraints on the interior fields. A consequence is this: if a matrix valued field $Q(x)$ with $\nabla\cdot Q=0$ takes values in a set $\cal B$ (independent of $x$) that lies on a nonlinear manifold, we find conditions on the manifold, and on $\cal B$, that with appropriate conditions on the boundary fluxes $q(x)=n(x)\cdot Q(x)$ (where $n(x)$ is the outwards normal to $\partialΩ$) force $Q(x)$ within $Ω$ to take values in a subspace $\cal D$. This forces $q(x)$ to take values in $n(x)\cdot\cal D$. We find there are additional divergence free fields inside $Ω$ that in turn generate additional boundary field equalities. There exist partial Null-Lagrangians, functionals $F(w,\nabla w)$ of a vector potential $w$ and its gradient, that act as null-Lagrangians when $\nabla w$ is constrained for $x\inΩ$ to take values in certain sets $\cal A$, of appropriate non-linear manifolds, and when $w$ satisfies appropriate boundary conditions. The extension to certain non-linear minimization problems is also sketched.

math.AP↗
arXiv2018

A new route to finding bounds on the generalized spectrum of many physical operators

Here we obtain bounds on the spectrum of that operator whose inverse, when it exists, gives the Green's function. We consider the wide of physical problems that can be cast in a form where a constitutive equation ${\bf J}({\bf x})={\bf L}({\bf x}){\bf E}({\bf x})-{\bf h}({\bf x})$ with a source term ${\bf h}({\bf x})$ holds for all ${\bf x}$ in some domain $Ω$, and relates fields ${\bf E}$ and ${\bf J}$ that satisfy appropriate differential constraints, symbolized by ${\bf E}\in\cal{E}_Ω$ and ${\bf J}\in\cal{J}_Ω$ where $\cal{E}_Ω$ and $\cal{J}_Ω$ are orthogonal spaces that span the space $\cal{H}_Ω$ of square-integrable fields in which ${\bf h}$ lies. Boundedness and coercivity conditions on the moduli ${\bf L}({\bf x})$ ensure there exists a unique ${\bf E}$ for any given ${\bf h}$, i.e., ${\bf E}={\bf G}_Ω{\bf h}$ which then establishes the existence of the Green's function ${\bf G}_Ω$. We show that the coercivity condition is guaranteed to hold if weaker conditions, involving generalized quasiconvex functions, are satisfied. The advantage is that these weaker conditions are easier to verify, and for multiphase materials they can be independent of the geometry of the phases. For ${\bf L}({\bf x} )$ depending linearly on a vector of parameters ${\bf z}=(z_1, z_2,\ldots, z_n)$, we obtain constraints on ${\bf z}$ that ensure the Green's function exists, and hence which provide bounds on the spectrum.

math-ph↗
arXiv2018

Approximating the effective tensor as a function of the component tensors in two-dimensional composites of two anisotropic phases

A conducting two-dimensional periodic composite of two anisotropic phases with anisotropic, not necessarily symmetric, conductivity tensors is considered. By finding approximate representations for the relevant operators, an approximation formula is derived for the effective matrix valued conductivity as a function of the two matrix valued conductivity tensors of the phases. This approximation should converge to the exact effective conductivity function as the number of basis fields tends to infinity. Using the approximations for the relevant operators one can also directly obtain approximations, with the same geometry, for the effective tensors of coupled field problems, including elasticity, piezoelectricity, and thermoelectricity. To avoid technical complications we assume that the phase geometry is symmetric under reflection about one of the centerlines of the unit cell.

math-ph↗
arXiv2018

Near optimal pentamodes as a tool for guiding stress while minimizing compliance in $3d$-printed materials: a complete solution to the weak $G$-closure problem for $3d$-printed materials

For a composite containing one isotropic elastic material, with positive Lame moduli, and void, with the elastic material occupying a prescribed volume fraction $f$, and with the composite being subject to an average stress, ${ σ}^0$, Gibiansky, Cherkaev, and Allaire provided a sharp lower bound $W_f({ σ}^0)$ on the minimum compliance energy $σ^0:ε^0$, in which ${ ε}^0$ is the average strain. Here we show these bounds also provide sharp bounds on the possible $({ σ}^0,{ ε}^0)$-pairs that can coexist in such composites, and thus solve the weak $G$-closure problem for $3d$-printed materials. The materials we use to achieve the extremal $(σ^0,ε^0)$-pairs are denoted as near optimal pentamodes. We also consider two-phase composites containing this isotropic elasticity material and a rigid phase with the elastic material occupying a prescribed volume fraction $f$, and with the composite being subject to an average strain, $ε^0$. For such composites, Allaire and Kohn provided a sharp lower bound $\widetilde{W}_f({ ε}^0)$ on the minimum elastic energy $σ^0:ε^0$. We show that these bounds also provide sharp bounds on the possible $({ σ}^0,{ ε}^0)$-pairs that can coexist in such composites of the elastic and rigid phases, and thus solve the weak $G$-closure problem in this case too. The materials we use to achieve these extremal $({ σ}^0,{ ε}^0)$-pairs are denoted as near optimal unimodes.

math.AP↗
arXiv2018

Convergence of iterative methods based on Neumann series for composite materials: theory and practice

Iterative Fast Fourier Transform methods are useful for calculating the fields in composite materials and their macroscopic response. By iterating back and forth until convergence, the differential constraints are satisfied in Fourier space, and the constitutive law in real space. The methods correspond to series expansions of appropriate operators and to series expansions for the effective tensor as a function of the component moduli. It is shown that the singularity structure of this function can shed much light on the convergence properties of the iterative Fast Fourier Transform methods. We look at a model example of a square array of conducting square inclusions for which there is an exact formula for the effective conductivity (Obnosov). Theoretically some of the methods converge when the inclusions have zero or even negative conductivity. However, the numerics do not always confirm this extended range of convergence and show that accuracy is lost after relatively few iterations. There is little point in iterating beyond this. Accuracy improves when the grid size is reduced, showing that the discrepancy is linked to the discretization. Finally, it is shown that none of the three iterative schemes investigated over-performs the others for all possible microstructures and all contrasts.

math.NA↗