A mass-conservative control volume-finite element method for solving Richards’ equation in heterogeneous porous media

We present a mass-conservative vertex-centred finite volume method for efficiently solving the mixed form of Richards’ equation in heterogeneous porous media. The spatial discretisation is particularly well-suited to heterogeneous media because it produces consistent flux approximations at quadrature points where material properties are continuous. Combined with the method of lines, the spatial discretisation gives a set of differential algebraic equations amenable to solution using higher-order implicit solvers. We investigate the solution of the mixed form using a Jacobian-free inexact Newton solver, which requires the solution of an extra variable for each node in the mesh compared to the pressure-head form. By exploiting the structure of the Jacobian for the mixed form, the size of the preconditioner is reduced to that for the pressure-head form, and there is minimal computational overhead for solving the mixed form. The proposed formulation is tested on two challenging test problems. The solutions from the new formulation offer conservation of mass at least one order of magnitude more accurate than a pressure head formulation, and the higher-order temporal integration significantly improves both the mass balance and computational efficiency of the solution.

Convexity, Classification, and Risk Bounds

Many of the classification algorithms developed in the machine learning literature, including the support vector machine and boosting, can be viewed as minimum contrast methods that minimize a convex surrogate of the 0–1 loss function. The convexity makes these algorithms computationally efficient. The use of a surrogate, however, has statistical consequences that must be balanced against the computational virtues of convexity. To study these issues, we provide a general quantitative relationship between the risk as assessed using the 0–1 loss and the risk as assessed using any nonnegative surrogate loss function. We show that this relationship gives nontrivial upper bounds on excess risk under the weakest possible condition on the loss function—that it satisfies a pointwise form of Fisher consistency for classification. The relationship is based on a simple variational transformation of the loss function that is easy to compute in many applications. We also present a refined version of this result in the case of low noise, and show that in this case, strictly convex loss functions lead to faster rates of convergence of the risk than would be implied by standard uniform convergence arguments. Finally, we present applications of our results to the estimation of convergence rates in function classes that are scaled convex hulls of a finite-dimensional base class, with a variety of commonly used loss functions.

A new control-volume finite-element scheme for heterogeneous porous media : application to the drying of softwood

An improved mesoscopic model is presented for simulating the drying of porous media. The aim of this model is to account for two scales simultaneously: the scale of the whole product and the scale of the heterogeneities of the porous medium. The innovation of this method is the utilization of a new mass-conservative scheme based on the Control-Volume Finite-Element (CV-FE) method that partitions the moisture content field over the individual sub-control volumes surrounding each node within the mesh. Although the new formulation has potential for application across a wide range of transport processes in heterogeneous porous media, the focus here is on applying the model to the drying of small sections of softwood consisting of several growth rings. The results conclude that, when compared to a previously published scheme, only the new mass-conservative formulation correctly captures the true moisture content evolution in the earlywood and latewood components of the growth rings during drying.

A Raman spectroscopic study of the different vanadate-groups in solid-state compounds, model case : mineral phases vésigniéite, BaCu3(VO4)2(OH)2, and volborthite, Cu3V2O7(OH)2•2H2O

Raman spectroscopy has been used to study vanadates in the solid state. The molecular structure of the vanadate minerals vésigniéite [BaCu3(VO4)2(OH)2] and volborthite [Cu3V2O7(OH)2·2H2O] have been studied by Raman spectroscopy and infrared spectroscopy. The spectra are related to the structure of the two minerals. The Raman spectrum of vésigniéite is characterized by two intense bands at 821 and 856 cm−1 assigned to ν1 (VO4)3− symmetric stretching modes. A series of infrared bands at 755, 787 and 899 cm−1 are assigned to the ν3 (VO4)3− antisymmetric stretching vibrational mode. Raman bands at 307 and 332 cm−1 and at 466 and 511 cm−1 are assigned to the ν2 and ν4 (VO4)3− bending modes. The Raman spectrum of volborthite is characterized by the strong band at 888 cm−1, assigned to the ν1 (VO3) symmetric stretching vibrations. Raman bands at 858 and 749 cm−1 are assigned to the ν3 (VO3) antisymmetric stretching vibrations; those at 814 cm−1 to the ν3 (VOV) antisymmetric vibrations; that at 508 cm−1 to the ν1 (VOV) symmetric stretching vibration and those at 442 and 476 cm−1 and 347 and 308 cm−1 to the ν4 (VO3) and ν2 (VO3) bending vibrations, respectively. The spectra of vésigniéite and volborthite are similar, especially in the region of skeletal vibrations, even though their crystal structures differ.