Poroelasticity theory models the dynamics of porous, fluid-saturated media. It was pioneered by Maurice Biot in the 1930s through the 1960s and has applications in several fields, including geophysics and modeling of in vivo bone. A wide variety of methods have been used to model poroelasticity, including finite difference, finite element, pseudospectral, and discontinuous Galerkin methods. In this work we use a Cartesian-grid high-resolution finite volume method to numerically solve Biot's equations in the time domain for orthotropic materials with the stiff relaxation source term in the equations incorporated using operator splitting. This class of finite volume method has several useful properties, including the ability to use wave limiters to reduce numerical artifacts in the solution, ease of incorporating material inhomogeneities, low memory overhead, and an explicit time-stepping approach. To the authors' knowledge, this is the first use of high-resolution finite volume methods to model poroelasticity. The solution code uses the clawpack finite volume method software, which also includes block-structured adaptive mesh refinement in its amrclaw variant. We present convergence results for known analytic plane wave solutions, achieving second-order convergence rates outside of the stiff regime of the system. Our convergence rates are degraded in the stiff regime, but we still achieve similar levels of error on the finest grids examined. We also demonstrate good agreement against other numerical results from the literature.

Abstract. 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. To aid in reproducibility, at the web site http://dx.doi.org/10.6084/m9.figshare.701483 we provide all of the code used to generate the results here.Key words. poroelastic, wave propagation, finite-volume, high-resolution, operator splitting, mapped grid, transverse solve, interface condition, cylindrical scatterer AMS subject classifications. 65M08, 74S10, 74F10, 74J10, 74L05, 74L15, 86-08 1. Introduction. Poroelasticity theory was developed by Maurice A. Biot to model the mechanics of a fluid-saturated porous medium. It models the medium in a homogenized fashion, with solid portion treated with linear elasticity, and the fluid with linearized compressible fluid dynamics combined with Darcy's law to relate its pressure gradient to its flow rate. Biot's work is summarized in his 1956 and 1962 papers [3, 4, 5], and Carcione also provides an excellent discussion of poroelasticity in chapter 7 of his book [9]. While it was originally developed to model fluid-saturated rock and soil, Biot theory has also found applications in modeling of in vivo bone [13,14,20] and underwater acoustics with a porous sea floor [7,21,22].Biot theory predicts three different families of propagating waves within a poroelastic medium. In order of decreasing speed, these are: fast P waves, where the fluid and solid parts of the medium move roughly parallel to the propagation direction -exactly parallel for an isotropic medium -and are typically in phase with each other; S waves, where the motion of the medium is transverse to the propagation direction; and slow P waves, where the motion is again roughly parallel to the wavevector but the fluid and solid typically move 180 degrees out of phase, so that the fluid is leaving a regi...

Defective bone mineralization has serious clinical manifestations, including deformities and fractures, but the regulation of this extracellular process is not fully understood. We have developed a mathematical model consisting of ordinary differential equations that describe collagen maturation, production and degradation of inhibitors, and mineral nucleation and growth. We examined the roles of individual processes in generating normal and abnormal mineralization patterns characterized using two outcome measures: mineralization lag time and degree of mineralization. Model parameters describing the formation of hydroxyapatite mineral on the nucleating centers most potently affected the degree of mineralization, while the parameters describing inhibitor homeostasis most effectively changed the mineralization lag time. Of interest, a parameter describing the rate of matrix maturation emerged as being capable of counter-intuitively increasing both the mineralization lag time and the degree of mineralization. We validated the accuracy of model predictions using known diseases of bone mineralization such as osteogenesis imperfecta and X-linked hypophosphatemia. The model successfully describes the highly nonlinear mineralization dynamics, which includes an initial lag phase when osteoid is present but no mineralization is evident, then fast primary mineralization, followed by secondary mineralization characterized by a continuous slow increase in bone mineral content. The developed model can potentially predict the function for a mutated protein based on the histology of pathologic bone samples from mineralization disorders of unknown etiology.

This paper deals with the inverse homogenization or dehomogenization problem of recovering geometric information about the structure of a twocomponent 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.

Estimating the parameters of an elastic or poroelastic medium from reflected or transmitted acoustic data is an important but difficult problem. Use of the Nelder-Mead simplex method to minimize an objective function measuring the discrepancy between some observable and its value calculated from a model for a trial set of parameters has been tried by several authors. In this paper, the difficulty with this direct approach, which is the existence of numerous local minima of the objective function, is documented for the in vitro experiment in which a specimen in a water tank is subject to an ultrasonic pulse. An indirect approach, based on the numerical solution of the equations for a set of 'effective' velocities and transmission coefficients, is then observed empirically to ameliorate the difficulties posed by the direct approach.