Elena Cherkaev

Inverse homogenization for evaluation of effective properties of a mixture

The paper deals with indirect evaluation of the effective thermal or hydraulic conductivity of a random mixture of two different materials from the known effective complex permittivity of the same mixture. The method is based on deriving information about the microstructure of the composite from measurements of its effective properties; we call this approach inverse homogenization. This structural information is contained in the spectral measure in the Stieltjes representation of the effective complex permittivity. The spectral measure can be reconstructed from effective measurements and used to estimate other effective properties of the same material. We introduce S-equivalence of the geometric structures corresponding to the same spectral measure, and show that the microstructures of different mixtures can be distinguished by the homogenized measurements up to the introduced equivalence. We show that the identification problem for the spectral function has a unique solution, however, the problem is extremely ill-posed. Several stabilization techniques are discussed such as quadratically constrained minimization and reconstruction in the class of functions of bounded variation. The approach is applicable to porous media, biological materials, artificial composites and other heterogeneous materials in which the scale of microstructure is much smaller than the wavelength of the electromagnetic signal.

Principal compliance and robust optimal design

The paper addresses a problem of robust optimal design of elastic structures when the loading is unknown and only an integral constraint for the loading is given. We propose to minimize the principal compliance of the domain equal to the maximum of the stored energy over all admissible loadings. The principal compliance is the maximal compliance under the extreme, worst possible loading. The robust optimal design is formulated as a min-max problem for the energy stored in the structure. The maximum of the energy is chosen over the constrained class of loadings, while the minimum is taken over the design parameters. It is shown that the problem for the extreme loading can be reduced to an elasticity problem with mixed nonlinear boundary conditions; the last problem may have multiple solutions. The optimization with respect to the designed structure takes into account the possible multiplicity of extreme loadings and divides resources (reinforced material) to equally resist all of them. Continuous change of the loading constraint causes bifurcation of the solution of the optimization problem. It is shown that an invariance of the constraints under a symmetry transformation leads to a symmetry of the optimal design. Examples of optimal design are investigated; symmetries and bifurcations of the solutions are revealed.

Reconstruction of spectral function from effective permittivity of a composite material using rational function approximations

The paper deals with the problem of reconstruction of microstructural information from known effective complex permittivity of a composite material. A numerical method for recovering geometric information from measurements of frequency dependent effective complex permittivity is developed based on Stieltjes analytic representation of the effective permittivity tensor of a two-component mixture. We derive the Stieltjes representation for the effective permittivity of the medium using the eigenfunction expansion of the solution of a boundary-value problem. The spectral function in this representation contains all information about the microgeometry of the mixture. A discrete approximation of the spectral measure is derived from a rational (Pade) approximation followed by its partial fractions decomposition. The approach is based on the least squares minimization with regularization constraints provided by the spectral properties of the operator. The method is applied to calculation of volume fractions of the components in a mixture of two materials in a Bruggeman effective medium analytic model which has a continuous spectral density and to analytical models of two-phase composites with coated cylindrical and ellipsoidal inclusions. The numerical results of reconstruction of spectral measure for a mixture of silver and silicon dioxide and a composite of magnesium and magnesium fluoride show good agreement between theoretical and predicted values. The approach is applicable to geological materials, biocomposites, porous media, etc.

Stieltjes representation of the 3D Bruggeman effective medium and Padé approximation

The paper deals with Bruggeman effective medium approximation (EMA) which is often used to model effective complex permittivity of a two-phase composite. We derive the Stieltjes integral representation of the 3D Bruggeman effective medium and use constrained Pade approximation method introduced in [39] to numerically reconstruct the spectral density function in this representation from the effective complex permittivity known in a range of frequencies. The problem of reconstruction of the Stieltjes integral representation arises in inverse homogenization problem where information about the spectral function recovered from the effective properties of the composite, is used to characterize its geometric structure. We present two different proofs of the Stieltjes analytical representation for the effective complex permittivity in the 3D Bruggeman effective medium model: one proof is based on direct calculation, the other one is the derivation of the representation using Stieltjes inversion formula. We show that the continuous spectral density in the integral representation for the Bruggeman EMA model can be efficiently approximated by a rational function. A rational approximation of the spectral density is obtained from the solution of a constrained minimization problem followed by the partial fractions decomposition. We show results of numerical rational approximation of Bruggeman continuous spectral density and use these results for estimation of fractions of components in a composite from simulated effective permittivity of the medium. The volume fractions of the constituents in the composite calculated from the recovered spectral function show good agreement between theoretical and predicted values.

Electrical impedance spectroscopy as a potential tool for recovering bone porosity

This paper deals with the recovery of porosity of bone from measurements of its effective electrical properties. The microstructural information is contained in the spectral measure in the Stieltjes representation of the bone effective complex permittivity or complex conductivity and can be recovered from the measurements over a range of frequencies. The problem of reconstruction of the spectral measure is very ill-posed and requires the use of regularization techniques. We apply the method to the effective electrical properties of cancellous bone numerically calculated using micro-CT images of human vertebrae. The presented method is based on an analytical approach and does not rely on correlation analysis nor on any a priori model of the bone micro-architecture. However the method requires a priori knowledge of the properties of the bone constituents (trabecular tissue and bone marrow). These properties vary from patient to patient. To address this issue, a sensitivity analysis of the technique was performed. Normally distributed random noise was added to the data to simulate uncertainty in the properties of the constituents and possible experimental errors in measurements of the effective properties. The values of porosity calculated from effective complex conductivity are in good agreement with the true values of bone porosity even assuming high level errors in the estimation of the bone components. These results prove the future potential of electrical impedance spectroscopy for in vivo monitoring of level and treatment of osteoporosis.

Padé approximations for identification of air bubble volume from temperature- or frequency-dependent permittivity of a two-component mixture

The article presents a numerical method developed for identification of information about structural parameters of a two-component mixture from effective complex permittivity measurements. The identification is based on the reconstruction of the spectral function in the analytic Stieltjes representation of the effective permittivity using Padé approximation. The spectral function contains all information about the microgeometry of the mixture, it is used to calculate volume fractions of the components in the mixture. Padé approximation is derived from a constrained minimization problem. Numerical results of recovering volume fraction of air in mixtures of air prolate and oblate spheroidal inclusions in water and in ethanol show good agreement of theoretical and predicted values. The proposed method can be used for estimating volume fractions and other structural parameters using the effective complex permittivity of two-component composite materials.

Dehomogenization: reconstruction of moments of the spectral measure of the composite

This paper deals with the inverse homogenization or dehomogenization problem of recovering geometric information about the structure of a two-component composite medium from the effective complex permittivity of the composite. The approach is based on the reconstruction of moments of the spectral measure in the Stieltjes analytic representation of the effective property. The moments of the spectral measure are linked to n-point correlation functions of the structure of the composite and thus contain information about the microgeometry. We show that the moments can be uniquely recovered from the measurements of the effective property in a range of frequencies. Two methods of numerical reconstruction of the moments are developed and analyzed. One method, which is referred to as a direct method of moment reconstruction, is based on the solution of the Vandermonde system arising in series expansion of the Stieltjes integral. The second, indirect, method reformulates the problem and reduces it to the problem of reconstruction of the spectral function. This last problem is ill-posed and requires regularization. We show that even though the reconstructed spectral function can be quite sensitive to the choice of the regularization scheme, the moments of the spectral functions can be stably reconstructed. The applicability of these two methods in terms of the choice of data points is also discussed in this paper.

Coupling of the effective properties of a random mixture through the reconstructed spectral representation

The spectral representation of the effective complex permittivity of a two-component composite medium is used to develop an approach to coupling of various effective properties of a random mixture. The spectral function contains all information about the microstructure, hence providing a coupling link between the various effective properties of the same composite material. It is demonstrated that the representation can be reconstructed from measurements of one effective property and used then to evaluate other properties of the same material. The reconstruction problem is very ill-posed and requires regularization. Several numerical examples of reconstruction of the spectral function from broadband measurements of the effective complex permittivity and the measurements of the effective thermal conductivity are shown. The approach can be used for indirect estimation of the thermal conductivity (or other properties) of the medium from broadband measurements of the effective complex permittivity.

Spectral analysis and connectivity of porous microstructures in bone

Cancellous bone is a porous composite of calcified tissue interspersed with soft marrow. Sea ice is also a porous composite, consisting of pure ice with brine, air, and salt inclusions. Interestingly, the microstructures of bone and sea ice exhibit notable similarities. In recent years, we have developed mathematical and experimental techniques for imaging and characterizing the brine microstructure of sea ice, such as its volume fraction and connectivity, as well as a range of theoretical approaches for studying fluid, thermal, and electromagnetic transport in sea ice. Here we explore the application of our sea ice techniques to investigate trabecular bone. For example, percolation theory that quantifies brine connectivity and its thermal evolution can also help assess the impact of osteoporosis on trabecular structure. Central to our approach is the spectral measure of a composite material, which contains detailed information about the mixture geometry, and can be used in powerful integral representations to compute the effective properties. The spectral measure is obtained from the eigenvalues and eigenvectors of a self-adjoint operator determined exclusively by the composite microgeometry. Here we compute the spectral measures for discretizations of images of healthy and osteoporotic bone. The measures are used to compute the effective electromagnetic properties of the bone specimens. These data are then inverted to reconstruct the porosity of the original specimens, with excellent agreement.

IDENTIFICATION OF BONE MICROSTRUCTURE FROM EFFECTIVE COMPLEX MODULUS

This work deals with the problem of reconstruction of bone structure from measurements of its effective mechanical properties. We propose a novel method of calculation of bone porosity from measured effective complex modulus. Bone is modelled as a medium with a microstructure composed of trabecular bone (elastic component) and bone marrow (viscoelastic component). We model bone as a cylinder subjected to torsion and assume that the effective complex modulus can be measured as a result of experiment. The analytical representation of the effective complex mod- ulus of the two-component composite material is exploited to recover information about porosity of the bone. The microstructural information is contained in the spectral measure in the Stieltjes representation of the effective complex modulus and can be recovered from the measurements over a range of frequencies. The problem of reconstruction of the spectral measure is very ill-posed and requires regularization. To verify the approach we apply it to analytically and numerically simulated response of a cylinder subjected to torsion assuming that it is filled with a composite material with known (laminated) microstructure. The values of porosity calculated from the effective shear modulus are in good agreement with the model values.