Free Boundary Method Research Articles

Free Boundary Method for Coupled Problems of Gas–Solid Dynamics

Free Boundary Method for Coupled Problems of Gas–Solid Dynamics

Uncertainty over uncertainty in environmental policy adoption: Bayesian learning of unpredictable socioeconomic costs

The socioeconomic impact of pollution naturally comes with uncertainty due to, e.g., current new technological developments in emissions' abatement or demographic changes. On top of that, the trend of the future costs of the environmental damage is unknown: Will global warming dominate or technological advancements prevail? The truth is that we do not know which scenario will be realised and the scientific debate is still open. This paper captures those two layers of uncertainty by developing a real-options-like model in which a decision maker aims at adopting a once-and-for-all costly reduction in the current emissions rate, when the stochastic dynamics of the socioeconomic costs of pollution are subject to Brownian shocks and the drift is an unobservable random variable. By keeping track of the actual evolution of the costs, the decision maker is able to learn the unknown drift and to form a posterior dynamic belief of its true value. The resulting decision maker's timing problem boils down to a truly two-dimensional optimal stopping problem which we address via probabilistic free-boundary methods and a state-space transformation. We completely characterise the solution by showing that the optimal timing for implementing the emissions reduction policy is the first time that the learning process has become “decisive” enough; that is, when it exceeds a time-dependent percentage. This is given in terms of an endogenously determined threshold function, which solves uniquely a nonlinear integral equation. We numerically illustrate our results, discuss the implications of the optimal policy and also perform comparative statics to understand the role of the relevant model's parameters in the optimal policy.

An Alternative in H-Formulation to the Critical Current Model of HTS Conductors

A new concept of H-formulation is introduced to reformulate the self-consistent static current model which has been devised to evaluate the critical current of high-temperature superconducting conductors. On the certain collection of known ideas, the attempt is carried out playing to its strength of the vectorial field quantity as directly obtained from the discretized elements. Equipped with the free boundary method, the formulation is implemented not only for efficient modeling by reducing the air region, but also by enabling the background magnetic field to be an explicit term. To prove such an efficacy, a practical case is brought for benchmark test. Then, the result is discussed, in comparison with the previous method, supporting the idea as a potential measure to improve the numerical model of high-temperature superconductors.

Free boundary methods and non-scattering phenomena

We study a question arising in inverse scattering theory: given a penetrable obstacle, does there exist an incident wave that does not scatter? We show that every penetrable obstacle with real-analytic boundary admits such an incident wave. At zero frequency, we use quadrature domains to show that there are also obstacles with inward cusps having this property. In the converse direction, under a nonvanishing condition for the incident wave, we show that there is a dichotomy for boundary points of any penetrable obstacle having this property: either the boundary is regular, or the complement of the obstacle has to be very thin near the point. These facts are proved by invoking results from the theory of free boundary problems.

Numerical Simulation of the Recirculation Flow during the Supersonic Separation of Moving Bodies

The features of a flow generated during the injection of a supersonic small body (pellet) from the channel of a spherically blunted cylinder into a supersonic flow are studied. The cylinder’s diameter is 35 times larger than the diameter of the injected body. The numerical simulation is carried out using multilevel Cartesian grids with the local adaptation based on the wavelet analysis. The body’s motion is simulated by the free boundary method. The dynamics of the moving body’s interaction with the bow shock from the cylinder, the formation of the reverse flow region between the bodies, its evolution and disappearance, and the subsequent establishment of a stationary flow are studied. The decrease of the main body’s drag to 20% of the initial value is demonstrated.

Numerical Modeling of Wave Processes Accompanying Combustion of Inhomogeneously Distributed Composite Propellant

A mathematical model and a numerical method are proposed for studying the interior ballistic process related to combustion of inhomogeneously distributed propellant in the two-dimensional axisymmetric approximation. The gas–propellant mixture is modeled by a two-phase nonequilibrium heterogeneous medium consisting of a multicomponent gas phase of combustion products and a polydisperse solid phase of propellant granules. The mathematical model of nonequilibrium two-phase flow is based on the nonconservative Euler equations. A Godunov-type scheme with an approximate Riemann solver is developed for their solution. The propellant combustion is considered taking into account the motion of the projectile, which is modelled using the free-boundary method. Results are represented concerning the origin and evolution of the interior ballistic wave process proceeding during the combustion of an inhomogeneously distributed propellant charge and the motion of the projectile. A comparative analysis with the case of static (nonmoving) propellant is carried out.

Free boundary method for calculating compressible viscous flows on unfitted meshes

We further develop the Free Boundary Method (FBM) for calculating compressible flow equations on Cartesian unfitted grids which has been recently proposed for the Euler model of inviscid equations. The method belongs to the class of immersed boundary methods. The effect of the solid surface is taken into account by means of the compensating flux applied on the element of solid surface in the cut cell. This flux is introduced to compensate losses in mass, momentum, and energy which occur when we virtually remove the solid element from the cut cell. The method is extended to the Navier-Stokes equations. We show how the viscous compensating flux should be introduced to assure the adequate solution in the flow domain, discuss its discrete approximation and propose the numerical method for solving the system of discrete equations. Calculation of the compensating flux requires a few data about the solid geometry in the cut cell, namely fluid volume fraction, area of the solid element and its unit normal, and baricentric coordinates of the fluid sub-element. All the data can be obtained from the level set method for setting the solid geometry. Numerical results concern testing the FBM on the solution of several benchmark viscous problems of aerodynamics.

Explicit solutions of the Bayes sequential estimation problem for a time-transformed exponential distribution

ABSTRACTA problem of Bayesian time-sequential estimation of an unknown parameter of a time-transformed exponential distribution is considered. It is supposed that the cost of observing the process is the sum of an increasing function of time and a linear function of the number of observations. Under some general assumptions concerning the loss function associated with the error of estimation, the optimal stopping time is derived using the free-boundary method. As an example, a solution of the problem considered is given in the case of a weighted precautionary loss function.

The effect of incident flow on a supersonic circumfluence of a blunt object

The influence of the inhomogeneity of a narrow wake with a reduced Mach number and total pressure values on the circumfluence about a blunt body (truncated cone) has been investigated. Two variants in the wake formation have been considered: behind the energy source and behind the moving body. A free boundary method was used for the flow simulation about moving bodies (a variant of the immersed boundary method). The dynamics of the moving body interaction with the head shock wave and formation of the reverse flow region have been studied. For consideration of these modes, it is shown that the presence of the wake before the front part of the body leads to a considerable decrease of wave resistance.

Free-boundary method for the numerical solution of gas-dynamic equations in domains with varying geometry

A novel method is proposed for numerical solution of gas-dynamic equations on stationary Cartesian grids in domains containing solid impermeable and, in the general case, moving inclusions (objects). The suggested technique is based on the immersed boundary method, in which the computational domain (including solid objects) is covered by a single Cartesian grid and the calculation is carried out by the shock-capturing method over all cells. Under this approach, the influence of the solid inclusions on the flow of the gas medium is simulated by the introduction of specially selected mass, momentum, and energy fluxes into the right-hand side of the equations. The currently developed methods for the solution of this class of problems are surveyed and the advantages of the proposed approach are discussed. The method is verified by calculating some test problems that admit analytical solutions and it is used to solve the problem of supersonic flow around a blunt body. The results are compared with the calculation findings based on the standard curvilinear grid tied to the geometry of the body.

A Fitted Finite Volume Method for Unit-linked Policy with Surrender Option

In this paper, fitted finite volume method is developed to solve a nonlinear degenerate Black-Scholes equation applied in the valuation of unit-linked policy with surrender option, based on the fitting idea in S. Wang [IMA J. Numer. Anal., 24 (2004), 699--720]. Unlike the conventional pricing method mentioned in [1] which is using the free boundary method to calibrate the valuation PDE, here we develop a power penalty method to solve numerically the linear complimentary problem in the variational inequality arising from the valuation of unit-linked policy with surrender option. With the degenerate boundary and non-smooth final condition, we will show that it is essential to refine the mesh to remain the convergence and super-convergence order.

A free boundary and an optimal shape problem of a floating body

The present article deals with a variational formulation of a floating body in a perfect fluid. Since the considered energy functional depends on both stream function and floating body, these two minimizations are realized in two separated steps. By the use of free boundary methods, existence and Lipschitz continuity of the minimizing stream function are proved. In addition, it is shown that the corresponding free surface has Holder boundary. The dependence on the floating body leads to an optimal shape problem which is treated by the use of Hausdorff convergence. Under the postulate of a bounded density parameter of the domain, the existence of an optimal floating body is shown.

Hybrid Method for Tokamak MHD Equilibrium ConfigurationReconstruction

A hybrid method for tokamak MHD equilibrium configuration reconstructionis proposed and employed in the modified EFIT code. This method uses thefree boundary tokamak equilibrium configuration reconstruction algorithmwith one boundary point fixed. The results show that the position of thefixed point has explicit effects on the reconstructed divertorconfigurations. In particular, the separatrix of the reconstructeddivertor configuration precisely passes the required position when thehybrid method is used in the reconstruction. The profiles of plasmaparameters such as pressure and safety factor for reconstructed HL-2Atokamak configurations with the hybrid and the free boundary methods arecompared. The possibility for applications of the method to swing theseparatrix strike point on the divertor target plate is discussed.

Convergence analysis of trial free boundary methods for the two-dimensional filtration problem

In this paper, we present a scheme of convergence analysis of trial free boundary methods for the two-dimensional filtration (or dam) problem. For the purpose we present a new variational principle of the filtration problem. This variational principle is defined on the set of admissible domains (candidates of the solution) in the dam. Under mild assumptions on the configuration of the dam, we may assume that all admissible domains are mapped from the unit disk by conformal mappings. Thus, proving convergence of trial free boundaries is reduced to proving convergence of the conformal mappings on the unit disk, and it is done using a method in the theory of minimal surfaces. Numerical examples are given.

Some recent methods for partial differential equations of divergence form

Some recent methods for solving second-order nonlinear partial differential equations of divergence form and related nonlinear problems are surveyed. These methods include entropy methods via the theory of divergence-measure fields for hyperbolic conservation laws, kinetic methods via kinetic formulations for degenerate parabolichyperbolic equations, and free-boundary methods via free-boundary iterations for multidimensional transonic shocks for nonlinear equation of mixed elliptic-hyperbolic type. Some recent trends in this direction are also discussed.

Redox front propagation and banding modalities

Oxygenated waters flowing through a reduced sandstone cause propagating redox fronts. Mathematical reaction-transport models of these fronts studied here show a number of nonlinear phenomena including one-parameter families of constant-velocity fronts, decelerating fronts and two types of front instabilities leading to pattern formation. These redox front phenomena are examples of nonlinear wave propagation and self-organization. Redox fronts in nature are economically important because they can trap accumulations of metallic ores. Furthermore, they are but one example of a wider class of water-rock interaction systems rich in nonlinear reaction-transport phenomena. Analytical results are presented on conversation law and free boundary methods to study the velocity and profiles of the waves. A new model of the Ostwald supersaturation-depletion cycle is presented that incorporates features of Liesegang banding not predicted by other formulations: these include finite band widths and continuous undulatory as well as discrete banding.

MHD-like and kinetic equilibria of Extrap, multipoles and related configurations

Two-dimensional, collision free, z-independent equilibria of a high- beta plasma in an external magnetic field are studied, using distributions of the form f(H,pz). The charge-neutral approximation then makes the macroscopic moments and the electrostatic potential functions of the magnetic vector potential A only, whereby these quantities assume constant values along the magnetic field lines, and the treatment becomes essentially one-dimensional with the use of flux coordinates. MHD-like equations of simple form for both particle species are obtained, including both an ambipolar electric field, arbitrarily large gyro excursions and lines with zero magnetic field component in the (r, theta ) plane. These equations provide useful interrelations between the macroscopic quantities that may be of value for interpretation of experimental results. A parametrized subclass of distributions gives a flexible tool for constructing solutions corresponding to given macroscopic profiles, and an expression for the smallest obtainable pinch radius is derived. An alternative to free-boundary methods for computing the magnetic field is presented. Intended applications concern linear and large aspect ratio toroidal systems of z pinch and multipole type.

Numerical computation of certain three-dimensional stellar structures using a semidiscrete pseudospectral method

Numerical method for solving three-dimensional stellar structure problems, formulated as free boundary problems for a mildly nonlinear elliptic partial differential equation, is developed by a combination of Newton-Raphson procedure, trial free boundary method, a pseudospectral Galerkin type method (for angular dependence) and finite differences (for radial dependence). Implementation of the method on a computer is discussed. The method is applied to the computation of structure of a critical, polytropic primary in a synchronous binary system. Forn=3.0, the results are found to be in reasonable agreement with those obtained previously by an expansion method. This expansion method is not capable of computing the structure of the other interesting polytrope withn=1.5 which presented no difficulties to the present method.

Electrophoretic Interaction Studies by the Stable-Flow Free-Boundary Method

A new method is described for rapidly mixing and unmixing (separating) components in free solution, enabling studies to be carried out on interactions of the components during their time of contact (presently as short as a few seconds). This method combines a multilayer, stable-flow fluid system with one (or more) transversely acting. force fields, commonly an electric field, and is applicable to small molecules, large molecules, and cells.