Bifurcation for a sharp interface generation problem

The theory of sets of finite perimeter and BV functions in Wiener spaces, i.e., Banach spaces endowed with a Gaussian Borel probability measure γ, was initiated by Fukushima and Hino in [9, 10, 11], and has been further investigated in [12, 1, 2, 3]. The basic question one would like to consider is the research of infinite-dimensional analogues of the classical fine properties of BV functions and sets of finite perimeter in finite-dimensional spaces. The class of sets of finite Gaussian perimeter E in a Gaussian Banach space (X, γ) is defined by the integration by parts formula ˆ

Coastal areas are densely populated due to socioeconomic benefits and in turn also have a greater demand for fresh water. This ever-increasing demand for fresh water can be met by coastal aquifers, which act as large reservoirs of freshwater. Excessive and unmanaged pumping from coastal aquifer allows the salt water to flow inward encroaching on the voids created by the pumping of freshwater. This phenomenon is called saltwater intrusion. To stop the saltwater intrusion, an optimal pumping strategy needs to be adopted. Simulation models are generally linked with an optimization algorithm to develop an optimal pumping strategy for management of saltwater intrusion. Sharp interface based simulation models are often used which are computationally inexpensive but lacks in prediction accuracy, as it does not incorporate the effects of dispersion and diffusion. Density dependent simulation models include the effect of dispersion and diffusion, but have a very high computational budget in evaluating an optimal pumping strategy. To overcome above-mentioned limitation a new methodology is developed, where a density dependent model is used in conjunction with a sharp interface model to derive an optimal density ratio, such that interface obtained using this density ratio implicitly accommodated the effect of dispersion and diffusion in a sharp interface model. The performance of the developed methodology is evaluated for three hypothetical scenarios of saltwater intrusion. The performance evaluation results show the applicability of the methodology for management of saltwater intrusion while maximizing fresh water pumping in coastal aquifers.KeywordsGroundwater modellingCoastal aquifer managementPumping optimizationSaltwater intrusionDensity dependent modelSharp interface model

A two-phase CFD model for autogenous pressurization and draining of a cryogenic storage tank is presented using both the Sharp Interface and Volume-Of-Fluid (VOF) approaches for capturing the front and the associated interfacial heat, mass and momentum transfer between the liquid and vapor regions. Both models are validated against data provided by the Cryogenic Propellant Storage and Transfer (CPST) Engineering Development Unit (EDU) experiment1. The results of the autogenous pressurization are presented first, focusing on the phase change and turbulence effects on the tank pressure and temperature predictions. Both the Sharp Interface (SI-CFD) and VOF (VOF-CFD) multiphase models predict tank pressure during pressurization within 3% of the measured values. The sensitivity of key physical and numerical parameters of the problem are tested using the Sharp Interface model. Effects of the accommodation coefficient (AC), the computational grid structure, and turbulence are studied. The second part of this paper is devoted to validating the SI-VOF model, with an enhanced capability of moving the liquid-vapor interface, against the draining segment of the EDU experiment. The VOF-CFD model was also used to simulate tank draining and its results were compared with the results of the Sharp Interface model with the moving interface. Both models predict tank pressure during draining within 3.5% of the measured values. Pressure decrease rate is underpredicted by both models during the first 100 seconds of draining but matches the experimental rate for the rest of the simulation.

Managing saltwater intrusion using conjugate sharp interface and density dependent models linked with pumping optimization

Divergence-measure fields in L∞ over sets of finite perimeter are analyzed. A notion of normal traces over boundaries of sets of finite perimeter is introduced, and the Gauss-Green formula over sets of finite perimeter is established for divergence-measure fields in L∞. The normal trace introduced here over a class of surfaces of finite perimeter is shown to be the weak-star limit of the normal traces introduced in Chen & Frid [6] over the Lipschitz deformation surfaces, which implies their consistency. As a corollary, an extension theorem of divergence-measure fields in L∞ over sets of finite perimeter is also established. Then we apply the theory to the initial-boundary value problem of nonlinear hyperbolic conservation laws over sets of finite perimeter.

A number of models have been developed to simulate seawater intrusion in coastal aquifers, which differ in the accuracy level and computational demands, based on the approximation level of the application. In this paper, four seawater intrusion models are employed to calculate the optimal pumping rates in a coastal aquifer management problem. The first model considers both fluid flow and solute transport processes and assumes a variable-density transition zone between saltwater and freshwater. The implementation of the model in simulation-optimisation routines is impractical, due to the computational time required for the simulation. The second model neglects the dispersion mechanism and assumes a sharp interface between saltwater and freshwater. The sharp interface model is significantly faster than the variable density model, however, it may introduce errors in the estimation of the seawater intrusion extent. The remaining two models are modifications of the second model, which intent to correct the inaccuracies of the simplified sharp interface approximation. All four models are utilised to simulate an unconfined coastal aquifer with multiple pumping wells and an optimisation method is used to calculate the maximum allowed pumping rates. The optimisation results are then analysed, in order to examine if the three sharp interface models could provide feasible solutions in the area of the variable density optimum, which is considered as a benchmark solution.

Pumping well management in coastal aquifers required to account for the saltwater intrusion problem. The prevention saltwater contamination of pumping wells should be considered along with the objective of maximum groundwater withdrawal. Saltwater intrusion constraint can be based on (1) sharp interface model (2) density-dependent transport model. Sharp interface models are preferable in the case of limited computation cost available and density-dependent transport models are preferable for accuracy. The correction factor introduced to account for the density-dependent dispersion by Pool and Carrera (Water Resour Res 47(5):W05506, 2011) vastly improves the sharp interface solution. In this present study, the application of the modified sharp interface solution based on the density-dependent correction factor for the pumping optimization is demonstrated for a regional scale aquifer in Nellore, Andhra Pradesh, India. The proposed optimization model sought to maximize the total pumping and minimize the landward toe intrusion from the sea.

<p>In coastal areas, seawater intrusion is a main driver of groundwater salinization and numerical models are widely used to support sustainable groundwater management. Sharp interface models, in which mixing between freshwater and seawater is not explicitly simulated, have fast run times which enable the implementation of parameter estimation and uncertainty analysis. These are essential steps for decision-support modeling, however their implementation in sharp interface models has remained limited. Few guidelines exist regarding which observations to use, and what processing and weighting strategies to employ. We developed a data assimilation framework for a regional, sharp interface model designed for management purposes. We built a sharp interface model for an island aquifer using the SWI2 package for MODFLOW. We then extracted freshwater head observations from shallow wells, pumping wells and deep open wells, and observations of the seawater-freshwater interface from deep open wells, time-domain electromagnetic (TDEM) and electrical resistivity tomography (ERT) surveys. After quantification of measurement uncertainties, parameter estimation was conducted with PEST and a data worth analysis was carried out using a linear approach. Model residuals provided insight on the potential of different observation groups to constrain parameter estimation. The data worth analysis provided insight on these groups’ importance in reducing the uncertainty of model forecasts. Overall a satisfying fit was obtained between simulated and observed data, but observations from deep open wells were biased. While observations from deep open wells and geophysical surveys had a low signal-to-noise ratio, parameter estimation effectively reduced predictive uncertainty. Interface observations, especially from geophysical surveys, were essential to reduce the uncertainty of model forecasts. The use of different types of observations is discussed and recommendations are provided for future data collection strategies in coastal aquifers. This framework was developed in the Magdalen Islands (Quebec, Canada) and could be carried out more systematically for sharp interface seawater intrusion modeling.</p>

Morphology: from sharp interface to phase field models

In pumping optimization of coastal aquifers, the evaluation of the objective function and constraints using density-dependent models is overwhelmed by complex and time-consuming numerical simulations. To address those cases where the available density-dependent model runs are very limited, due to excessive computational burden, an efficient optimization strategy is developed. The proposed methodology uses an efficient sharp interface model jointly with a complex density-dependent model in an evolutionary optimization algorithm. While most evaluations are based on the sharp interface model, the density-dependent model is selectively called to evaluate promising solutions and to improve the predictions of the sharp interface model through the adaptive modification of the saltwater-freshwater density ratio. The method is tested for pumping optimization problems in confined and unconfined coastal aquifers with multiple pumping wells. The optimal solutions are compared to those obtained by density-dependent as well as by sharp interface optimization alone. Under a very restrictive computational budget, the best feasible solution is attained in less than 25 density-dependent model runs for two optimization problems of 10 and 20 decision variables. The results indicate that this optimization method leads to good feasible solutions and that an improved estimation of optimal pumping rates can be achieved within a limited computational budget. The method could also stand as an efficient preliminary exploration of the optimal search space, to provide good feasible starting points for the implementation of more comprehensive methods of coastal aquifer management.

We analyze a class of weakly differentiable vector fieldsF: ℝn→ ℝnwith the property thatF∈L∞and divFis a (signed) Radon measure. These fields are calledbounded divergence‐measure fields. The primary focus of our investigation is to introduce a suitable notion of the normal trace of any divergence‐measure fieldFover the boundary of an arbitrary set of finite perimeter that ensures the validity of the Gauss‐Green theorem. To achieve this, we first establish a fundamental approximation theorem which states that, given a (signed) Radon measure μ that is absolutely continuous with respect to ℋ︁N− 1on ℝN, any set of finite perimeter can be approximated by a family of sets with smooth boundary essentially from the measure‐theoretic interior of the set with respect to the measure ||μ||, the total variation measure. We employ this approximation theorem to derive the normal trace ofFon the boundary of any set of finite perimeterEas the limit of the normal traces ofFon the boundaries of the approximate sets with smooth boundary so that the Gauss‐Green theorem forFholds onE. With these results, we analyze the Cauchy flux that is bounded by a nonnegative Radon measure over any oriented surface (i.e., an (N− 1)‐dimensional surface that is a part of the boundary of a set of finite perimeter) and thereby develop a general mathematical formulation of the physical principle of the balance law through the Cauchy flux. Finally, we apply this framework to the derivation of systems of balance laws with measure‐valued source terms from the formulation of the balance law. This framework also allows the recovery of Cauchy entropy flux through the Lax entropy inequality for entropy solutions of hyperbolic conservation laws. © 2008 Wiley Periodicals, Inc.

In this study, we present a phase-field model that describes the process of intercalation of Li ions into a layer of an amorphous solid such as amorphous silicon (a-Si). The governing equations couple a viscous Cahn–Hilliard-Reaction model with elasticity in the framework of the Cahn–Larché system. We discuss the parameter settings and flux conditions at the free boundary that lead to the formation of phase boundaries having a sharp gradient in lithium ion concentration between the initial state of the solid layer and the intercalated region. We carry out a matched asymptotic analysis to derive the corresponding sharp-interface model that also takes into account the dynamics of triple points where the sharp interface intersects the free boundary of the Si layer. We numerically compare the interface motion predicted by the sharp-interface model with the long-time dynamics of the phase-field model.

We present a new approach for modeling strongly anisotropic crystal and epitaxial growth using regularized, anisotropic Cahn-Hilliard-type equations as a model for the growth and coarsening of thin films. When the surface anisotropy is sufficiently strong, sharp corners form and unregularized anisotropic Cahn-Hilliard equations become ill-posed. Our models contain a high order Willmore regularization to remove the ill posedness at the corners. A key feature of our approach is the development of a new formulation in which the interface thickness is independent of crystallographic orientation. In our previous work, we have provided matched asymptotic analysis to show the convergence of our diffuse interface model to the analytical sharp interface model. In previous models there was no such convergence to sharp interface model when the Willmore energy was considered. We present 2D numerical results using an adaptive, nonlinear multigrid finite-difference method. In particular, we find excellent agreement between the computed shapes using the Cahn-Hilliard approach, with a finite but small Willmore regularization, with dynamical numerical simulations of a sharp interface model. The equilibrium shapes from our diffuse model are compared with an analytical sharp-interface theory recently developed by Spencer [1] at the corners, and there is excellent match. Away from the corners there is an excellent agreement between the diffuse model and the classical Wulff shape. Finally, in order to model the misfit and displacement strains, we add the elastic energy and corresponding forces to our diffuse model. We analyze numerically the effect of elastic stress on the corner regularization in terms of two parameters: one parameter that describes the relative strength of the elastic energy to surface energy and the second that characteristics the strength of the surface energy anisotropy. Adding elastic energy modifies the equilibrium shape and in particular affects the shape of the corners. We can predict different Qdot shapes, such as pyramids and domes, based on the strength of the elastic interactions.

Chen, Torres and Ziemer ([9], 2009) proved the validity of generalized Gauss–Green formulas and obtained the existence of interior and exterior normal traces for essentially bounded divergence measure fields on sets of finite perimeter using an approximation theory through sets with a smooth boundary. However, it is known that the proof of a crucial approximation lemma contained a gap. Taking inspiration from a previous work of Chen and Torres ([7], 2005) and exploiting ideas of Vol’pert ([29], 1985) for essentially bounded fields with components of bounded variation, we present here a direct proof of generalized Gauss–Green formulas for essentially bounded divergence measure fields on sets of finite perimeter which includes the existence and essential boundedness of the normal traces. Our approach appears to be simpler since it does not require any special approximation theory for the domains and it relies only on the Leibniz rule for divergence measure fields. This freedom allows one to localize the constructions and to derive more general statements in a natural way.

We introduce a useful and relatively easy to check condition, local simplicity, which provides significantly more structure to sets of finite perimeter, while not being too restrictive. Local simplicity holds for minimizers in a wide variety of variational problems in materials science, biology, image processing, oncology, and other fields. We prove several regularity and structural properties of locally simple sets and their boundaries, including a vital decomposition theorem that in our setting strengthens the conclusion of theorems of H. Federer and L. Ambrosio, V. Caselles, S. Masnou, and J.-M. Morel. We establish strong connections between topology and sets of finite perimeter, so that ordinary notions of openness, closedness, and connectedness may be readily used in the finite perimeter setting.We apply these results to an image reconstruction procedure from image processing, _L_1 TV-minimization. The density ratio bounds computed to establish local simplicity are themselves of practical importance, as they provide concrete, easy to compute criteria to check simulations against.