Explicit algebraic influence coefficients: two-dimensional aquifer model / Coefficients d'influence algébriques explicites: modèle d'aquifère bidimensionnel

Seismic oceanography can provide a two- or three-dimensional view of the water column thermocline structure at a vertical and horizontal resolution from the multi-channel seismic dataset. Several seismic imaging methods and techniques for seismic oceanography have been presented in previous research. In this study, we suggest a new formulation of the frequency-domain reverse-time migration method for seismic oceanography based on the analytic Green's function. For imaging thermocline structures in the water column from the seismic data, our proposed seismic reverse-time migration method uses the analytic Green's function for numerically calculating the forward- and backward-modeled wavefield rather than the wave propagation modeling in the conventional algorithm. The frequency-domain reverse-time migration with analytic Green's function does not require significant computational memory, resources, or a multifrontal direct solver to calculate the migration seismic images as like conventional reverse-time migration. The analytic Green's function in our reverse-time method makes it possible to provide a high-resolution seismic water column image with a meter-scale grid size, consisting of full-band frequency components for a modest cost and in a low-memory environment for computation. Our method was applied to multi-channel seismic data acquired in the Arctic Ocean and successfully constructed water column seismic images containing the oceanographic reflections caused by thermocline structures of the water mass. From the numerical test, we note that the oceanographic reflections of the migrated seismic images reflected the distribution of Arctic waters in a shallow depth and showed good correspondence with the anomalies of measured temperatures and calculated reflection coefficients from each XCDT profile. Our proposed method has been verified for field data application and accuracy of imaging performance.

An accurate solution of the wave equation at a fluid-solid interface requires a correct implementation of the boundary condition. Boundary conditions at fluid-solid interface require continuity of the normal component of particle velocity and traction, whereas the tangential components vanish. A main challenge is to model interface waves, namely, the Scholte and leaky Rayleigh waves. This study uses a nodal discontinuous Galerkin (dG) finite-element method with the medium discretized using an unstructured uniform triangular meshes. The natural boundary conditions in the dG method are implemented by (1) using an explicit upwind numerical flux and (2) by using an implicit penalty flux and setting the modulus of rigidity of the acoustic medium to zero. The accuracy of these methods is evaluated by comparing the numerical solutions with analytical ones, with source and receiver at and away from the interface. The study shows that the solutions obtained from the explicit and implicit boundary conditions provide the correct results. The stability of the dG scheme is determined by the numerical flux, which also implements the boundary conditions by unifying the numerical solution at shared edges of the elements in an energy stable manner.

<p style='text-indent:20px;'>In this paper, the chaotic oscillations of the initial-boundary value problem of linear hyperbolic partial differential equation (PDE) with variable coefficients are investigated, where both ends of boundary conditions are nonlinear implicit boundary conditions (IBCs). It separately considers that IBCs can be expressed by general nonlinear boundary conditions (NBCs) and cannot be expressed by explicit boundary conditions (EBCs). Finally, numerical examples verify the effectiveness of theoretical prediction.</p>

The determination of a gradient, or directional derivative, of a spatial variable is a common problem in many Earth science applications. For hydraulic heads, for example, the gradient defines the direction of the groundwater flow and is used in solving groundwater flow and transport equations. Kriging provides a methodology for estimating gradients directly from experimental data without the need to estimate the variable at the nodes of a regular grid (even when the data are widely and irregularly scattered) and without the need to use finite difference approximations. Kriging is more efficient than other methods of gradient estimation because it uses a probabilistic approach that gives an estimator that is unbiased and with minimum estimation variance. In addition, it provides a standard error of estimation, which is not available from deterministic methods. We extend the kriging formulation of gradient estimation to the universal kriging formulation to account for the frequently observed drift in spatial variables; this is almost always present at the regional scale, for example, for hydraulic head. We present an alternative approach to kriging the gradient that is simpler than previous formulations. This new formulation uses linear systems theory, which facilitates the inference of the required covariances as functions of the covariance of the primary variable whose gradient is to be estimated. An example of estimating the gradient of hydraulic head in an aquifer in southern Spain is presented to illustrate the methodology.

Thermal phonon transport at length scales comparable to mean free paths is governed by the Boltzmann equation, which is challenging to solve due to its high dimensionality. Here, we present an analytical Green's function for the frequency-dependent, multidimensional Boltzmann equation under the relaxation-time approximation. The new analytical solution is valid from diffusive to ballistic transport regimes and rigorously includes frequency dependence of phonon properties. We demonstrate that our result enables simple closed-form solutions for a number of multidimensional problems for which the only prior solution methods have been computationally expensive numerical simulations.

Usually, the boundary value problems of the theory of elasticity are formulated with respect to displacements, and are reduced to the well-known Lame equations. Strains and stresses can be calculated from displacements as a solution to Lame's equation. Also known are the Beltrami Mitchell equations, which make it possible to formulate the boundary value problem of the theory of elasticity with respect to stresses. Currently, the boundary value problems of the theory of elasticity in stresses are studied in more detail in the two-dimensional case, and usually solved numerically with the introduction of the Airy stress function. But, the direct solution of boundary value problems of elasticity theory with respect to stresses requires further researches. This work, similarly to the boundary value problem in stresses, is devoted to the formulation and numerical solution of boundary value problems of the theory of elasticity with respect to deformations. The proposed boundary value problem consists of six Beltrami-Mitchell-type equations depending on strains and three equations of the equilibrium equation expressed with respect to deformations. As boundary conditions, in addition to the usual conditions for surface forces, three additional conditions are also introduced based on the equilibrium equations. The boundary value problem is considered in detail for a rectangular area. The discrete analogue of the boundary value problem is composed by the finite difference method. The convergence of difference schemes and an iterative method for their solution are studied. Software has been developed in the C++ environment for solving boundary value problems in the theory of elasticity and deformation. A number of boundary value problems on the deformation of a rectangular plate are solved numerically under various boundary conditions. The reliability of the obtained results is substantiated by comparing the numerical results, with the exact solution, as well as with the known solutions of the plate tension problems with parabolic and uniformly distributed edge loads.

Numerical validation of the volume penalization method in three-dimensional pseudo-spectral simulations

The parameters and boundary conditions which control the flow processes vary greatly over short distances and over time, and it is inevitable that flow variables such as the hydraulic head and the fluid’s velocity are spatially variable. It is reasonable to model them as SRFs, not only because of the spatial variability, but also due to our inability to model them in detail with only the limited number of measurements usually available. This chapter presents the basic principles and a few methods for modeling flow variables as SRFs, while showing how these moments reflect the physics of the flow and how to use them in applications. The starting point in our derivations will be the moments that characterize the medium variability, such as σ2γ and I γ, and our goal is to develop the SRF models of flow variables such as the head and the velocity in terms of the media parameters and boundary conditions. To drive this point home, let us consider Darcy’s law for isotropic hydraulic conductivity: where qi is the specific flux in the ith direction, K is the isotropic hydraulic conductivity, and H is the hydraulic head. For a spatially variable K, the flux qi is also spatially variable (with the exception of the one-dimensional flow case). An SRF model for K can be used to construct an SRF for qi through (4.1), with the aid of the flow equation. SRFs can be characterized through their low-order statistical moments, but preferably through their pdfs. For example, the hydraulic head will be defined by its expected value (H) and its variance σ2 H = ((H - ( H ) ) 2 ) , but also through its pdf fH(h) where fH(h) dh is the probability of having H in the vicinity dh of h. SRFs may or may not be stationary depending on the nature of the hydrogeological variables such as K and on the boundary conditions. For example, the head is always nonstationary, except when there is no flow.

Computation of singular solutions to the Helmholtz equation with high order accuracy

The study of electrical signals associated with groundwater flow is a powerful method to determine, in a non-invasive way, the distribution of hydraulic heads in aquifers, hence the distribution of the hydraulic conductivity and storativity. Experimental hydraulic heads and electrical (self-potential, SP) signals associated with a pumping test in an unconfined aquifer were reproduced with a numerical model based on the finite-difference method. For simplicity, we assumed axial symmetry around the pumping well. We assessed the equivalent hydraulic conductivity and specific storage of the aquifer by fitting the piezometric levels obtained in the course of the pumping against synthetic hydraulic heads produced by the model. The current coupling coefficient is obtained from the best fit between the experimental and modelled SP signals at the end of the pumping phase. Then, keeping the previously determined hydraulic parameters, the relaxation of both the hydraulic heads and the SP signals were modelled and found to be consistent with the measured values. The data show almost linear relationships between the hydraulic heads in the aquifer and the electrical signals recorded at the ground surface. These results show that SP signals allow monitoring subsurface flow in the course of pumping experiments.

Sensitivity analysis is an important component of many model activities in hydrology. Numerous studies have been conducted in calculating various sensitivities. Most of these sensitivity analysis focus on the sensitivity of state variables (e.g., hydraulic head) to parameters representing medium properties such as hydraulic conductivity or prescribed values such as constant head or flux at boundaries, while few studies address the sensitivity of the state variables to some shape parameters or design parameters that control the model domain. Instead, these shape parameters are typically assumed to be known in the model. In this study, based on the flow equation, we derive the equation (and its associated initial and boundary conditions) for sensitivity of hydraulic head to shape parameters using continuous sensitivity equation (CSE) approach. These sensitivity equations can be solved numerically in general or analytically in some simplified cases. The approach has been demonstrated through two examples and the results are compared favorably to those from analytical solutions or numerical finite difference methods with perturbed model domains, while numerical shortcomings of the finite difference method are avoided.

Calculating the unknown coupled multiphysics fields from the known boundary condition is of great significance in computational physics. Existing classical algorithms are usually time consuming and resource demanding. Explosive growth in deep learning (DL) has provided a substitutive way to accelerate the calculation process by fully making use of the parallel computing capability of the graphics processing unit. In this work, a cascaded DL framework is presented to compute the coupled multiphysics fields related to electrical potential, temperature, velocity, etc. The whole framework is comprised of an input module and several Fourier modules, which achieves to acquire the coupled fields from the explicit boundary conditions. Compared with conventional networks, the Fourier network emerges consistent average error among different resolutions which saves training costs. As a result, a fully trained network can obtain the expected physics fields in real time, offering rosy prospects for practical scenarios.

This paper extends the generating function approach to the state constrained linear quadratic optimal control problem by utilizing penalties. The original constrained problem is approximated by an unconstrained problem with a novel penalty function by which the latter problem can be solved via generating function method based on Taylor series approximation. In view of this, the optimal solution can be given as the state feedback control with explicit boundary conditions and pre-computed coefficients. This structure is useful in the repetitive on-line computation for optimal solutions satisfying a number of different boundary conditions. Examples demonstrate the effectiveness of the developed method.

A new integral equation-based nonoverlapping domain decomposition method (NDDM) is proposed for analyzing electromagnetic (EM) scattering from electrically large PEC objects whose MLFMA models are too large for the user’s computer to accommodate. The integral equation is first built on the entire PEC surface, and the entire surface is partitioned into some nonoverlapping subsurfaces. The RWG functions are used to expand the surface current on the entire surface. Then, each full-RWG function across some boundary curve is split into two half-RWG functions whose unknown coefficients are constrained by the boundary current continuity (using the explicit boundary condition). The convergence of the outer-iterative scheme of the proposed NDDM is investigated theoretically and demonstrated to be very good, and hence a very large problem can be effectively solved in an ordinary computer. Some numerical examples are provided to demonstrate the correctness and robustness of the proposed method, and to compare the proposed method with the existing overlapping DDMs having similar attributes.

Finite difference schemes on quasi-uniform grids for BVPs on infinite intervals