In this work we develop a high-resolution mapped-grid finite volume method code to model wave propagation in two dimensions in systems of multiple orthotropic poroelastic media and/or fluids, with curved interfaces between different media. We use a unified formulation to simplify modeling of the various interface conditions---open pores, imperfect hydraulic contact, or sealed pores---that may exist between such media. Our numerical code is based on the clawpack framework, but in order to obtain correct results at a material interface we use a modified transverse Riemann solution scheme, and at such interfaces are forced to drop the second-order correction term typical of high-resolution finite volume methods. We verify our code against analytical solutions for reflection and transmission of waves at a material interface, and for scattering of an acoustic wave train around an isotropic poroelastic cylinder. For reflection and transmission at a flat interface, we achieve second-order convergence in the 1-norm and first-order in the max-norm; for the cylindrical scatterer, the highly distorted grid mapping degrades performance but we still achieve convergence at a reduced rate. We also simulate an acoustic pulse striking a simplified model of a human femur bone as an example of the capabilities of the code.
The scattering of a plane wave incident obliquely upon an infinite poroelastic cylinder immersed in inviscid fluid is investigated in this paper. Convergence analysis of the series expansion of the solutions for various interface conditions is conducted and it provides a priori estimates on number of terms necessary for achieving a desired accuracy. In contrast to the existing results in the literature, we consider viscous pore fluid and arbitrary interface discharge efficiency [Formula: see text]. Moreover, the approach presented here does not require any restriction on the viscodynamic operator of the poroelastic equations and hence it can handle general cases beyond the dissipation models proposed by Biot and by Johnson, Koplik and Dashen. The back scattering form function is then calculated from the coefficients of the series solution. Numerical results with various incident angles and interface discharge efficiencies are also presented in this paper.
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.
This is the first part of the review article which focuses on theory and applications of Herglotz-Nevanlinna functions in material sciences. It starts with the definition of scalar valued Herglotz-Nevanlinna functions and explains in detail the theorems that are pertinent to applications, followed by a short overview of the matrix-valued and operator-valued versions of these functions and the properties that carry over from scalar cases. The theory is complemented by some applications from electromagnetics that are related to the sum rules. More applications of Herglotz Nevanlinnna functions in material sciences can be found in Part II.
This paper investigates existence of the nonstandard Pade approximants introduced by Cherkaev and Zhang in J. Comp. Phys. 2009 for approximating the spectral function of composites from effective properties at different frequencies. The spectral functions contain microstructure information. Since this reconstruction problem is ill-posed [9], the well-performed Pade approach is noteworthy and requires further investigations. In this paper, we validate the assumption that the effective dielectric component of interest can be approximated by Pade approximants whose denominator has nonzero power one term. We refer to this as the nonstandard Pade approximant, in contrast to the standard approximants with nonzero constant terms. For composites whose spectral function assumes infinitely many different values, the proof is carried by using classic results for Stieltjes functions. For those with spectral functions having only finitely many different values, we prove the results by utilizing a special product decomposition of the coefficient matrix of the Pade system. The results in this paper can be considered as an extension of the Pade theory for Stieltjes functions whose spectral function take infinitely many different values to those taking only finitely many values. In the literature, the latter is usually excluded from the definition of Stieltjes functions because they correspond to rational functions, hence convergence of their Pade approximants is trivial. However, from an inverse problem point of view, our main concern is the existence of the nonstandard Pade approximants, rather than their convergence. The results in this paper provide a mathematical foundation for applying the Pade approach for reconstructing the spectral functions of composites whose microstructure is not a priori known.
In this paper, we show that the permeability of a porous material (Tartar (1980)) and that of a bubbly fluid (Lipton and Avellaneda. Proc. R. Soc. Edinburgh Sect. A: Math . 114 (1990), 71–79) are limiting cases of the complexified version of the two-fluid models posed in Lipton and Avellaneda ( Proc. R. Soc. Edinburgh Sect. A: Math . 114 (1990), 71–79). We assume the viscosity of the inclusion fluid is $z\mu _1$ and the viscosity of the hosting fluid is $\mu _1\in \mathbb {R}^{+}$ , $z\in \mathbb {C}$ . The proof is carried out by the construction of solutions for large $|z|$ and small $|z|$ with an iteration process similar to the one used in Bruno and Leo ( Arch. Ration. Mech. Anal . 121 (1993), 303–338) and Golden and Papanicolaou ( Commun. Math. Phys . 90 (1983), 473–491) and the analytic continuation. Moreover, we also show that for a fixed microstructure, the permeabilities of these three cases share the same integral representation formula (3.17) with different values of contrast parameter $s:=1/(z-1)$ , as long as $s$ is outside the interval $\left [-\frac {2E_2^{2}}{1+2E_2^{2}},-\frac {1}{1+2E_1^{2}}\right ]$ , where the positive constants $E_1$ and $E_2$ are the extension constants that depend only on the geometry of the periodic pore space of the material.
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