The efficient numerical solution of the one‐phase linear inverse Stefan and Cauchy–Stefan problems is a delicate task owing to the problems' susceptibility to the perturbation of the given data. In ...this context, heuristic a posteriori error indicators are constructed for such inverse problems with noisy data in two dimensions (2D). Given a fixed computational effort, the estimator controls the error due to discretization by the method of fundamental solution (MFS). It is accomplished through two mean‐driven double‐filtering algorithms. Numerical results substantiate the effectiveness of the algorithms.
We formulate and solve a free target optimal Brownian stopping problem from a given distribution while the target distribution is free and is conditioned to satisfy a given density height constraint. ...The free target optimization problem exhibits monotonicity, from which a remarkable universality follows, in the sense that the optimal target is independent of its Lagrangian cost type. In particular, the solutions to this optimization problem generate solutions to both unstable and stable type of the Stefan problem, where the former stands for freezing of supercooled fluid (St1) and the latter for ice melting (St2). This unified approach to both types of the Stefan problem is new. In particular we obtain global-time existence and weak-strong uniqueness for the ill-posed freezing problem (St1), for a given initial data and for a well-prepared class of initial domains generated from the initial data.
We study a non-classical one phase Stefan problem with a particular control function which depends on the evolution of the temperature at the fixed face x=0 and, we assume a Neumann boundary ...condition and an over specified Robin condition at the fixed face. Under certain restrictions on the data an explicit similarity type solution is given. Moreover, we determine the free boundary and one unknown thermal coefficient, or two unknown thermal coefficients depending on whether the direct or inverse Stefan problem is considered.
In this paper, we propose Meshfree Generalized Multiscale Finite Element Method with a partially explicit time scheme for solving the nonlinear Stefan problem with high-contrast inclusions. The ...hybrid approach of a partially explicit learning is demonstrated. The neural network is trained to generate temperature values at specific nodes at a coarse scale at each time layer. The solution values at the remaining nodes are calculated using an explicit time scheme. Numerical results for a two-dimensional problem with the heat stabilizers are presented.
A modeling framework is developed to perform two- and three-dimensional simulations of ice accretion over solid bodies in a wet air flow. The PoliMIce (Politecnico di Milano Ice accretion software) ...library provides a general interface allowing different aerodynamic and ice accretion software to communicate. The built-in ice accretion engine moves from the well-known Myers approach and it includes state-of-the-art ice formation models. The ice accretion engine implements a fully three-dimensional representation of the two-phase flow over the solid body, accounting for both rime and glaze ice formation. As an improvement over the reference model, a parabolic temperature profile is assumed to guarantee the consistency with respect to the wall boundary conditions. Moreover, the mass balance is generalized to conserve the liquid fraction at the interface between the glaze and the rime ice types. Numerical simulations are presented regarding in-flight ice accretion over two-dimensional airfoils and three-dimensional straight- and swept-wings. The CFD open-source software OpenFOAM was used to compute the aerodynamic field and the droplet trajectories. Simulation results compare fairly well with available experiments on ice accretion.
•One-phase cylindrical Stefan problem with size-dependent thermal conductivity and convection is considered.•Time dependent temperature and Newton type temperature boundary conditions are subjected ...at the outer surface of the body.•Heat Balance Integral Method is applied to obtain the numerical result of the problem.•Convection produced pronounced impact on propagation rate of the melting interface.•It is observed that size-dependent thermal conductivity produced pronounced effect on melting process.
The melting of a phase change material is the most applicable process in thermal energy storage system to capture heat transfer phenomena arising in a class of moving boundary problem. Demand of present technology motivates researchers to develop new theories and techniques for thermal management of a material. Experimental work on melting of a material may be difficult and development of robust theoretical formulation in cylindrical geometry with convection is critical. While there is already available study on cylindrical moving boundary problem, but still insufficient modeling of a size-dependent thermal conductivity and convection effect is not addressed properly, which is being considered in this paper and is expected to improve the previous understanding. In this work, a one-dimensional moving boundary problem with size-dependent heat conductivity and convection effect is analyzed in cylindrical geometry. In the mathematical model, we have considered a time-dependent temperature boundary condition which later assumed in periodic form, and a convective boundary condition at the outer surface of the body. The numerical result of the problem is obtained successfully via heat-balance integral method. Our numerical result is compared with a previous work and found in good acceptance. From mathematical framework, it is found that convection delayed melting process. With a size-no independent thermal conductivity, the rate of moving front decreases more in comparison to the fixed thermal conductivity.
The mathematical model describing the dynamics of closed contact heating which involves vaporization of the metal when instantaneous explosion of micro-asperity occurs is presented through a Stefan ...type problem. The temperature field for metallic vaporization zone is introduced as heat resistance that decreases linearity. Temperature fields for liquid and solid phases of the metal described by spherical heat equations with nonlinear thermal coefficients and Joule heat source have to be determined as well as the free boundaries. Joule heating component depends on space and time variable when alternating current is considered. Solution method of the problem based on similarity variable transformation is applied which enables us to reduce the problem to an ordinary differential equations and nonlinear integral equations. Existence and uniqueness of the integral equations are proved by using fixed point theorem in Banach space.
•The mathematical model describing the dynamics of closed contact heating.•Two phase Stefan problem for the spherical heat equation analyzed.•Nonlinear thermal coefficients and Joule heat source.•Equivalent ordinary differential problem, giving rise to nonlinear integral equations.
This study presents a mathematical model that describes the heat process during the vaporization of metal at electrical contacts. The arc erosion model for the micro-asperity in the liquid state is ...established using a one-phase Stefan problem. Two problems involving Joule heating effects on the generalized heat equation are considered. The solution of these two problems is based on the similarity principle, which allows for the reduction of the problem to ordinary differential equations. The existence and uniqueness of the solutions are proven using fixed point theory in a Banach space. Additionally, a solution for the particular case of constant thermal coefficients is provided.
•Mathematical model of the heat process during the vaporization of metal at electrical contacts.•Determination of the temperature solution in liquid phase and free boundary describing location of the melting temperature.•The similarity solution of the one-phase nonlinear Stefan problem with temperature boundary condition.•Proof of existence and uniqueness of a solution using the theory of fixed points in a Banach space.
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