Regularity and mass conservation for discrete coagulation-fragmentation equations with diffusion

We present a new a-priori estimate for discrete coagulation fragmentation systems with size-dependent diffusion within a bounded, regular domain confined by homogeneous Neumann boundary conditions. Following from a duality argument, this a-priori estimate provides a global L2 bound on the mass density and was previously used, for instance, in the context of reaction-diffusion equations. In this paper we demonstrate two lines of applications for such an estimate: On the one hand, it enables to simplify parts of the known existence theory and allows to show existence of solutions for generalised models involving collision-induced, quadratic fragmentation terms for which the previous existence theory seems difficult to apply. On the other hand and most prominently, it proves mass conservation (and thus the absence of gelation) for almost all the coagulation coefficients for which mass conservation is known to hold true in the space homogeneous case.

Application of standard and refined heat balance integral methods to one-dimensional Stefan problems

The work in this paper concerns the study of conventional and refined heat balance integral methods for a number of phase change problems. These include standard test problems, both with one and two phase changes, which have exact solutions to enable us to test the accuracy of the approximate solutions. We also consider situations where no analytical solution is available and compare these to numerical solutions. It is popular to use a quadratic profile as an approximation of the temperature, but we show that a cubic profile, seldom considered in the literature, is far more accurate in most circumstances. In addition, the refined integral method can give greater improvement still and we develop a variation on this method which turns out to be optimal in some cases. We assess which integral method is better for various problems, showing that it is largely dependent on the specified boundary conditions.

An iterative Multiscale Finite-Volume algorithm converging to the exact solution

The multiscale finite volume (MsFV) method has been developed to efficiently solve large heterogeneous problems (elliptic or parabolic); it is usually employed for pressure equations and delivers conservative flux fields to be used in transport problems. The method essentially relies on the hypothesis that the (fine-scale) problem can be reasonably described by a set of local solutions coupled by a conservative global (coarse-scale) problem. In most cases, the boundary conditions assigned for the local problems are satisfactory and the approximate conservative fluxes provided by the method are accurate. In numerically challenging cases, however, a more accurate localization is required to obtain a good approximation of the fine-scale solution. In this paper we develop a procedure to iteratively improve the boundary conditions of the local problems. The algorithm relies on the data structure of the MsFV method and employs a Krylov-subspace projection method to obtain an unconditionally stable scheme and accelerate convergence. Two variants are considered: in the first, only the MsFV operator is used; in the second, the MsFV operator is combined in a two-step method with an operator derived from the problem solved to construct the conservative flux field. The resulting iterative MsFV algorithms allow arbitrary reduction of the solution error without compromising the construction of a conservative flux field, which is guaranteed at any iteration. Since it converges to the exact solution, the method can be regarded as a linear solver. In this context, the schemes proposed here can be viewed as preconditioned versions of the Generalized Minimal Residual method (GMRES), with a very peculiar characteristic that the residual on the coarse grid is zero at any iteration (thus conservative fluxes can be obtained).

Stoneley wave modeling in heterogeneous porous media with viscous pore fluids

We implemented Biot-type porous wave equations in a pseudo-spectral numerical modeling algorithm for the simulation of Stoneley waves in porous media. Fourier and Chebyshev methods are used to compute the spatial derivatives along the horizontal and vertical directions, respectively. To prevent from overly short time steps due to the small grid spacing at the top and bottom of the model as a consequence of the Chebyshev operator, the mesh is stretched in the vertical direction. As a large benefit, the Chebyshev operator allows for an explicit treatment of interfaces. Boundary conditions can be implemented with a characteristics approach. The characteristic variables are evaluated at zero viscosity. We use this approach to model seismic wave propagation at the interface between a fluid and a porous medium. Each medium is represented by a different mesh and the two meshes are connected through the above described characteristics domain-decomposition method. We show an experiment for sealed pore boundary conditions, where we first compare the numerical solution to an analytical solution. We then show the influence of heterogeneity and viscosity of the pore fluid on the propagation of the Stoneley wave and surface waves in general.

Weak approximations. A Malliavin calculus approach

We introduce a variation of the proof for weak approximations that issuitable for studying the densities of stochastic processes which areevaluations of the flow generated by a stochastic differential equation on a random variable that maybe anticipating. Our main assumption is that the process and the initial random variable have to be smooth in the Malliavin sense. Furthermore if the inverse of the Malliavin covariance matrix associated with the process under consideration is sufficiently integrable then approximations fordensities and distributions can also be achieved. We apply theseideas to the case of stochastic differential equations with boundaryconditions and the composition of two diffusions.

A pseudo-spectral method for the simulation of poro-elastic seismic wave propagation in 2D polar coordinates using domain decomposition

We present a novel numerical approach for the comprehensive, flexible, and accurate simulation of poro-elastic wave propagation in 2D polar coordinates. An important application of this method and its extensions will be the modeling of complex seismic wave phenomena in fluid-filled boreholes, which represents a major, and as of yet largely unresolved, computational problem in exploration geophysics. In view of this, we consider a numerical mesh, which can be arbitrarily heterogeneous, consisting of two or more concentric rings representing the fluid in the center and the surrounding porous medium. The spatial discretization is based on a Chebyshev expansion in the radial direction and a Fourier expansion in the azimuthal direction and a Runge-Kutta integration scheme for the time evolution. A domain decomposition method is used to match the fluid-solid boundary conditions based on the method of characteristics. This multi-domain approach allows for significant reductions of the number of grid points in the azimuthal direction for the inner grid domain and thus for corresponding increases of the time step and enhancements of computational efficiency. The viability and accuracy of the proposed method has been rigorously tested and verified through comparisons with analytical solutions as well as with the results obtained with a corresponding, previously published, and independently bench-marked solution for 2D Cartesian coordinates. Finally, the proposed numerical solution also satisfies the reciprocity theorem, which indicates that the inherent singularity associated with the origin of the polar coordinate system is adequately handled.

Quantitative comparison of simulations of seismic wave propagation in heterogeneous poro-elastic media involving fluid-solid interfaces and in equivalent visco-elastic solids

There is increasing evidence to suggest that the presence of mesoscopic heterogeneities constitutes the predominant attenuation mechanism at seismic frequencies. As a consequence, centimeter-scale perturbations of the subsurface physical properties should be taken into account for seismic modeling whenever detailed and accurate responses of the target structures are desired. This is, however, computationally prohibitive since extremely small grid spacings would be necessary. A convenient way to circumvent this problem is to use an upscaling procedure to replace the heterogeneous porous media by equivalent visco-elastic solids. In this work, we solve Biot's equations of motion to perform numerical simulations of seismic wave propagation through porous media containing mesoscopic heterogeneities. We then use an upscaling procedure to replace the heterogeneous poro-elastic regions by homogeneous equivalent visco-elastic solids and repeat the simulations using visco-elastic equations of motion. We find that, despite the equivalent attenuation behavior of the heterogeneous poro-elastic medium and the equivalent visco-elastic solid, the seismograms may differ due to diverging boundary conditions at fluid-solid interfaces, where there exist additional options for the poro-elastic case. In particular, we observe that the seismograms agree for closed-pore boundary conditions, but differ significantly for open-pore boundary conditions. This is an interesting result, which has potentially important implications for wave-equation-based algorithms in exploration geophysics involving fluid-solid interfaces, such as, for example, wave field decomposition.

Impacto de la circulación ecuatorial en la zona mínima de oxógeno presente en el norte del Ecosistema de la Corriente del Humboldt

El Norte del Ecosistema de la Corriente Humboldt (NECH) constituye una de las mayores zonas de afloramiento, localizada en el borde oriental del Pacifico Sur, la cual presenta características particulares, entre las que destacan una alta producción primaria y la presencia de una de las zonas mínimas de oxigeno (ZMO) más intensas en el océano abierto. La ZMO presente en esta zona es producto de la alta demanda de oxígeno durante la remineralización de la materia orgánica, el largo tiempo de residencia de sus aguas y su poca ventilación. En el presente estudio nos enfocamos en estudiar la influencia de cambios en la ventilación en la ZMO del NECH, tomando como las principales fuentes de aporte de oxígeno en esta zona a la Corriente Sub-superficial Ecuatorial (EUC) y las Contracorrientes Sub-superficiales del Sur (SSCCs) o también conocidas como los Jets de Tsuchiya. Utilizamos el modelo acoplado físico-biogeoquímico ROMS-PISCES, para observar la sensibilidad de la ZMO a diferentes condiciones de la circulación ecuatorial provenientes de dos modelos oceánicos de circulación general (SODA y MERCATOR). Los resultados muestran que el flujo de oxígeno a los 88ºW disminuye latitudinalmente de la EUC a los SSCCs; además, se observa que la ZMO desaparece de los 4ºN - 4ºS en la simulación que presenta una circulación más intensa (RPSoda) por lo que se puede concluir que una intensificación de la circulación ecuatorial afectaría principalmente a la zona ecuatorial y no frente a Perú, debido a que una mayor ventilación sería compensada con un mayor consumo de oxigeno durante la remineralización, producto de una alta productividad generada por un mayor flujo de nutrientes.

A pseudo-spectral method for simulating of poro-elastic seismic wave propagation in complex borehole environments

We present a novel numerical approach for the comprehensive, flexible, and accurate simulation of poro-elastic wave propagation in cylindrical coordinates. An important application of this method is the modeling of complex seismic wave phenomena in fluid-filled boreholes, which represents a major, and as of yet largely unresolved, computational problem in exploration geophysics. In view of this, we consider a numerical mesh consisting of three concentric domains representing the borehole fluid in the center, the borehole casing and the surrounding porous formation. The spatial discretization is based on a Chebyshev expansion in the radial direction, Fourier expansions in the other directions, and a Runge-Kutta integration scheme for the time evolution. A domain decomposition method based on the method of characteristics is used to match the boundary conditions at the fluid/porous-solid and porous-solid/porous-solid interfaces. The viability and accuracy of the proposed method has been tested and verified in 2D polar coordinates through comparisons with analytical solutions as well as with the results obtained with a corresponding, previously published, and independently benchmarked solution for 2D Cartesian coordinates. The proposed numerical solution also satisfies the reciprocity theorem, which indicates that the inherent singularity associated with the origin of the polar coordinate system is handled adequately.

On The Spot Damage Detection Methodology for Highway Bridges, July 2010

Vibration-based damage identification (VBDI) techniques have been developed in part to address the problems associated with an aging civil infrastructure. To assess the potential of VBDI as it applies to highway bridges in Iowa, three applications of VBDI techniques were considered in this study: numerical simulation, laboratory structures, and field structures. VBDI techniques were found to be highly capable of locating and quantifying damage in numerical simulations. These same techniques were found to be accurate in locating various types of damage in a laboratory setting with actual structures. Although there is the potential for these techniques to quantify damage in a laboratory setting, the ability of the methods to quantify low-level damage in the laboratory is not robust. When applying these techniques to an actual bridge, it was found that some traditional applications of VBDI methods are capable of describing the global behavior of the structure but are most likely not suited for the identification of typical damage scenarios found in civil infrastructure. Measurement noise, boundary conditions, complications due to substructures and multiple material types, and transducer sensitivity make it very difficult for present VBDI techniques to identify, much less quantify, highly localized damage (such as small cracks and minor changes in thickness). However, while investigating VBDI techniques in the field, it was found that if the frequency-domain response of the structure can be generated from operating traffic load, the structural response can be animated and used to develop a holistic view of the bridge’s response to various automobile loadings. By animating the response of a field bridge, concrete cracking (in the abutment and deck) was correlated with structural motion and problem frequencies (i.e., those that cause significant torsion or tension-compression at beam ends) were identified. Furthermore, a frequency-domain study of operational traffic was used to identify both common and extreme frequencies for a given structure and loading. Common traffic frequencies can be compared to problem frequencies so that cost-effective, preventative solutions (either structural or usage-based) can be developed for a wide range of IDOT bridges. Further work should (1) perfect the process of collecting high-quality operational frequency response data; (2) expand and simplify the process of correlating frequency response animations with damage; and (3) develop efficient, economical, preemptive solutions to common damage types.

Reply to Comment on 'Two-finger selection theory in the Saffman-Taylor problem

We clarify the meaning of the results of Phys. Rev. E 60, R5013 (1999). We discuss the use and implications of periodic boundary conditions, as opposed to rigid-wall ones. We briefly argue that the solutions of the paper above are physically relevant as part of a more general issue, namely the possible generalization to dynamics, of the microscopic solvability scenario of selection.

Monte Carlo simulation of kilovolt electron transport in solids

A Monte Carlo procedure to simulate the penetration and energy loss of low¿energy electron beams through solids is presented. Elastic collisions are described by using the method of partial waves for the screened Coulomb field of the nucleus. The atomic charge density is approximated by an analytical expression with parameters determined from the Dirac¿Hartree¿Fock¿Slater self¿consistent density obtained under Wigner¿Seitz boundary conditions in order to account for solid¿state effects; exchange effects are also accounted for by an energy¿dependent local correction. Elastic differential cross sections are then easily computed by combining the WKB and Born approximations to evaluate the phase shifts. Inelastic collisions are treated on the basis of a generalized oscillator strength model which gives inelastic mean free paths and stopping powers in good agreement with experimental data. This scattering model is accurate in the energy range from a few hundred eV up to about 50 keV. The reliability of the simulation method is analyzed by comparing simulation results and experimental data from backscattering and transmission measurements.

Bifurcations and chaos in single-roll natural convection with low Prandtl number

Convective flows of a small Prandtl number fluid contained in a two-dimensional cavity subject to a lateral thermal gradient are numerically studied by using different techniques. The aspect ratio (length to height) is kept at around 2. This value is found optimal to make the flow most unstable while keeping the basic single-roll structure. Two cases of thermal boundary conditions on the horizontal plates are considered: perfectly conducting and adiabatic. For increasing Rayleigh numbers we find a transition from steady flow to periodic oscillations through a supercritical Hopf bifurcation that maintains the centrosymmetry of the basic circulation. For a Rayleigh number of about ten times that of the Hopf bifurcation the system initiates a complex scenario of bifurcations. In the conductive case these include a quasiperiodic route to chaos. In the adiabatic one the dynamics is dominated by the interaction of two Neimark-Sacker bifurcations of the basic periodic solutions, leading to the stable coexistence of three incommensurate frequencies, and finally to chaos. In all cases, the complex time-dependent behavior does not break the basic, single-roll structure.

Revisiting the S-index for soil physical quality and its use in Brazil

The S-index was introduced in 2004 in a publication by A.R. Dexter. S was proposed as an indicator of soil physical quality. A critical value delimiting soils with rich and poor physical quality was proposed. At present, Brazil is world leader in citations of Dexter's publication. In this publication the S-theory is mathematically revisited and extended. It is shown that S is mathematically correlated to bulk density and total porosity. As an absolute indicator, the value of S alone has proven to be incapable of predicting soil physical quality. The critical value does not always hold under boundary conditions described in the literature. This is to be expected because S is a static parameter, therefore implicitly unable to describe dynamic processes. As a relative indicator of soil physical quality, the S-index has no additional value over bulk density or total porosity. Therefore, in the opinion of the author, the fact that bulk density or total porosity are much more easily determined than the water retention curve for obtaining S disqualifies S as an advantageous indicator of relative soil physical quality. Among the several equations available for the fitting of water retention curves, the Groenevelt-Grant equation is preferable for use with S since one of its parameters and S are linearly correlated. Since efforts in soil physics research have the purpose of describing dynamic processes, it is the author's opinion that these efforts should shift towards mechanistic soil physics as opposed to the search for empirical correlations like S which, at present, represents far more than its reasonable share of soil physics in Brazil.

Accurate and efficient simulation of multiphase flow in a heterogeneous reservoir by using error estimate and control in the multiscale finite-volume framework

The multiscale finite-volume (MSFV) method is designed to reduce the computational cost of elliptic and parabolic problems with highly heterogeneous anisotropic coefficients. The reduction is achieved by splitting the original global problem into a set of local problems (with approximate local boundary conditions) coupled by a coarse global problem. It has been shown recently that the numerical errors in MSFV results can be reduced systematically with an iterative procedure that provides a conservative velocity field after any iteration step. The iterative MSFV (i-MSFV) method can be obtained with an improved (smoothed) multiscale solution to enhance the localization conditions, with a Krylov subspace method [e.g., the generalized-minimal-residual (GMRES) algorithm] preconditioned by the MSFV system, or with a combination of both. In a multiphase-flow system, a balance between accuracy and computational efficiency should be achieved by finding a minimum number of i-MSFV iterations (on pressure), which is necessary to achieve the desired accuracy in the saturation solution. In this work, we extend the i-MSFV method to sequential implicit simulation of time-dependent problems. To control the error of the coupled saturation/pressure system, we analyze the transport error caused by an approximate velocity field. We then propose an error-control strategy on the basis of the residual of the pressure equation. At the beginning of simulation, the pressure solution is iterated until a specified accuracy is achieved. To minimize the number of iterations in a multiphase-flow problem, the solution at the previous timestep is used to improve the localization assumption at the current timestep. Additional iterations are used only when the residual becomes larger than a specified threshold value. Numerical results show that only a few iterations on average are necessary to improve the MSFV results significantly, even for very challenging problems. Therefore, the proposed adaptive strategy yields efficient and accurate simulation of multiphase flow in heterogeneous porous media.