In this paper, we prove the existence theorems of locbal or global classical solutions to Stefan problems with various kinetic conditions at the free boundary.
In this paper,the relaxation algorithm and two Uzawa type algorithms for solving discretized variational inequalities arising from the two-phase Stefan type problem are proposed.An analysis of their convergence is pre...In this paper,the relaxation algorithm and two Uzawa type algorithms for solving discretized variational inequalities arising from the two-phase Stefan type problem are proposed.An analysis of their convergence is presented and the upper bounds of the convergence rates are derived.Some numerical experiments are shown to demonstrate that for the second Uzawa algorithm which is an improved version of the first Uzawa algorithm,the convergence rate is uniformly bounded away from 1 if τh^-2 is kept bounded,where τ is the time step size and h the space mesh size.展开更多
One-dimensional non-Darcy flow in a semi-infinite porous media is investigated. We indicate that the non-Darcy relation which is usually determined from experimental results can always be described by a piecewise line...One-dimensional non-Darcy flow in a semi-infinite porous media is investigated. We indicate that the non-Darcy relation which is usually determined from experimental results can always be described by a piecewise linear function, and the problem can be equivalently transformed to a multiphase implicit Stefan problem. The novel feature of this Stefan problem is that the phases of the porous media are divided by hydraulic gradients, not the excess pore water pressures. Using the similarity transformation technique, an exact solution for the situation that the external load increases in proportion to the square root of time is developed. The study on the existence and uniqueness of the solution leads to the requirement of a group of inequalities. A similar Stefan problem considering constant surface seepage velocity is also investigated, and the solution, which we indicate to be uniquely existent under all conditions, is established. Meanwhile, the relation between our Stefan problem and the traditional multiphase Stefan problem is demonstrated. In the end, computational examples of the solution are presented and discussed. The solution provides a useful benchmark for verifying the accuracy of general approximate algorithms of Stefan problems, and it is also attractive in the context of inverse problem analysis.展开更多
A one-phase Stefan problem for the nonhomogeneous heat equation with the source term depending on an unknown parameter p(t) is considered. The existence and uniqueness of the solution (p, s, u) are also demonstrated.
A numerical analysis was carried out to study the moving boundary problem in the physical process of pulsed Nd-YAG laser surface melting prior to vaporization.The enthalpy method was applied to solve this two-phase ax...A numerical analysis was carried out to study the moving boundary problem in the physical process of pulsed Nd-YAG laser surface melting prior to vaporization.The enthalpy method was applied to solve this two-phase axisymmetrical mehing problem.Computational results of tempera ture fields were obtained,which provide useful information to practical lair treatment processing. The validity of enthalpy method in solving such problems is presented.展开更多
This paper considers the quasi-stationary Stefan problem:△u(x,t)=0 in space-time domain,u=0 and Vv +(?)u/(?)u=0 on the free boundary.Under the natural conditions the existence of classical solution locally in time is...This paper considers the quasi-stationary Stefan problem:△u(x,t)=0 in space-time domain,u=0 and Vv +(?)u/(?)u=0 on the free boundary.Under the natural conditions the existence of classical solution locally in time is proved bymaking use of the property of Frechet derivative operator and fixed point theorem. For thesake of simplicity only the one-phase problem is dealt with. In fact two-phase problem can bedealt with in a similar way with more complicated calculation.展开更多
In this paper,a meshless regularization method of fundamental solutions is proposed for a two-dimensional,two-phase linear inverse Stefan problem.The numerical implementation and analysis are challenging since one nee...In this paper,a meshless regularization method of fundamental solutions is proposed for a two-dimensional,two-phase linear inverse Stefan problem.The numerical implementation and analysis are challenging since one needs to handle composite materials in higher dimensions.Furthermore,the inverse Stefan problem is ill-posed since small errors in the input data cause large errors in the desired output solution.Therefore,regularization is necessary in order to obtain a stable solution.Numerical results for several benchmark test examples are presented and discussed.展开更多
A Stefan problem with nonlinear boundary flux and internal convection of a material are considered. The existence, uniqueness and continuous dependence of globally weak solution of this problem are obtained. This pape...A Stefan problem with nonlinear boundary flux and internal convection of a material are considered. The existence, uniqueness and continuous dependence of globally weak solution of this problem are obtained. This paper extends the results of Fahuai Yi and T.M.Shih, relaxes restrictions that does not be to accord with reality very much on internal convection and boundary conditions in their articles.展开更多
The seasonal change in depths of the frozen and thawed soils within their active layer is reduced to a moving boundary problem, which describes the dynamics of the total ice content using an independent mass balance e...The seasonal change in depths of the frozen and thawed soils within their active layer is reduced to a moving boundary problem, which describes the dynamics of the total ice content using an independent mass balance equation and treats the soil frost/thaw depths as moving (sharp) interfaces governed by some Stefan-type moving boundary conditions, and hence simultaneously describes the liquid water and solid ice states as well as the positions of the frost/thaw depths in soil. An adaptive mesh method for the moving boundary problem is adopted to solve the relevant equations and to determine frost/thaw depths, water content and temperature distribution. A series of sensitivity experiments by the numerical model under the periodic sinusoidal upper boundary condition for temperature are conducted to validate the model, and to investigate the effects of the model soil thickness, ground surface temperature, annual amplitude of ground surface temperature and thermal conductivity on frost/thaw depths and soil temperature. The simulated frost/thaw depths by the model with a periodical change of the upper boundary condition have the same period as that of the upper boundary condition, which shows that it can simulate the frost/thaw depths reasonably for a periodical forcing.展开更多
We investigate a blowup problem of a reaction-advection-diffusion equa-tion with double free boundaries and aim to use the dynamics of such a problem to describe the heat transfer and temperature change of a chemical ...We investigate a blowup problem of a reaction-advection-diffusion equa-tion with double free boundaries and aim to use the dynamics of such a problem to describe the heat transfer and temperature change of a chemical reaction in advective environment with the free boundary representing the spreading front of the heat.We study the influence of the advection on the blowup properties of the solutions and con-clude that large advection is not favorable for blowup.Moreover,we give the decay estimates of solutions and the two free boundaries converge to a finite limit for small initial data.展开更多
We critically compare the practicality and accuracy of numerical approximations of phase field models and sharp interface models of solidification.Here we focus on Stefan problems,and their quasi-static variants,with ...We critically compare the practicality and accuracy of numerical approximations of phase field models and sharp interface models of solidification.Here we focus on Stefan problems,and their quasi-static variants,with applications to crystal growth.New approaches with a high mesh quality for the parametric approximations of the resulting free boundary problems and new stable discretizations of the anisotropic phase field system are taken into account in a comparison involving benchmark problems based on exact solutions of the free boundary problem.展开更多
文摘In this paper, we prove the existence theorems of locbal or global classical solutions to Stefan problems with various kinetic conditions at the free boundary.
基金supported by the National Natural Science Foundation (10871179) of China
文摘In this paper,the relaxation algorithm and two Uzawa type algorithms for solving discretized variational inequalities arising from the two-phase Stefan type problem are proposed.An analysis of their convergence is presented and the upper bounds of the convergence rates are derived.Some numerical experiments are shown to demonstrate that for the second Uzawa algorithm which is an improved version of the first Uzawa algorithm,the convergence rate is uniformly bounded away from 1 if τh^-2 is kept bounded,where τ is the time step size and h the space mesh size.
基金supported by the Fundamental Research Funds for the Central Universities(Grant 2015XKMS014)
文摘One-dimensional non-Darcy flow in a semi-infinite porous media is investigated. We indicate that the non-Darcy relation which is usually determined from experimental results can always be described by a piecewise linear function, and the problem can be equivalently transformed to a multiphase implicit Stefan problem. The novel feature of this Stefan problem is that the phases of the porous media are divided by hydraulic gradients, not the excess pore water pressures. Using the similarity transformation technique, an exact solution for the situation that the external load increases in proportion to the square root of time is developed. The study on the existence and uniqueness of the solution leads to the requirement of a group of inequalities. A similar Stefan problem considering constant surface seepage velocity is also investigated, and the solution, which we indicate to be uniquely existent under all conditions, is established. Meanwhile, the relation between our Stefan problem and the traditional multiphase Stefan problem is demonstrated. In the end, computational examples of the solution are presented and discussed. The solution provides a useful benchmark for verifying the accuracy of general approximate algorithms of Stefan problems, and it is also attractive in the context of inverse problem analysis.
文摘A one-phase Stefan problem for the nonhomogeneous heat equation with the source term depending on an unknown parameter p(t) is considered. The existence and uniqueness of the solution (p, s, u) are also demonstrated.
基金the National Natural Science Foundation of China and the Chinese Academy of Sciences
文摘A numerical analysis was carried out to study the moving boundary problem in the physical process of pulsed Nd-YAG laser surface melting prior to vaporization.The enthalpy method was applied to solve this two-phase axisymmetrical mehing problem.Computational results of tempera ture fields were obtained,which provide useful information to practical lair treatment processing. The validity of enthalpy method in solving such problems is presented.
文摘This paper considers the quasi-stationary Stefan problem:△u(x,t)=0 in space-time domain,u=0 and Vv +(?)u/(?)u=0 on the free boundary.Under the natural conditions the existence of classical solution locally in time is proved bymaking use of the property of Frechet derivative operator and fixed point theorem. For thesake of simplicity only the one-phase problem is dealt with. In fact two-phase problem can bedealt with in a similar way with more complicated calculation.
基金T.Reeve would like to acknowledge the financial support received from the EPSRC.
文摘In this paper,a meshless regularization method of fundamental solutions is proposed for a two-dimensional,two-phase linear inverse Stefan problem.The numerical implementation and analysis are challenging since one needs to handle composite materials in higher dimensions.Furthermore,the inverse Stefan problem is ill-posed since small errors in the input data cause large errors in the desired output solution.Therefore,regularization is necessary in order to obtain a stable solution.Numerical results for several benchmark test examples are presented and discussed.
基金Supported by National Natural Science Foundation of China (90410011)the Natural Science Foundation of the Education Department of Anhui Province (2005KJ316ZC).
文摘A Stefan problem with nonlinear boundary flux and internal convection of a material are considered. The existence, uniqueness and continuous dependence of globally weak solution of this problem are obtained. This paper extends the results of Fahuai Yi and T.M.Shih, relaxes restrictions that does not be to accord with reality very much on internal convection and boundary conditions in their articles.
基金the National Basic Research Program(Grant No.2005CB321703)the Knowledge Innovation Project of Chinese Academy of Sciences(Grant Nos.KZCX2-yw-126-2,KZCX2-yw-217)the Chinese Coordinated Observation and Prediction of the Earth System project(Grant No.GYHY20070605)
文摘The seasonal change in depths of the frozen and thawed soils within their active layer is reduced to a moving boundary problem, which describes the dynamics of the total ice content using an independent mass balance equation and treats the soil frost/thaw depths as moving (sharp) interfaces governed by some Stefan-type moving boundary conditions, and hence simultaneously describes the liquid water and solid ice states as well as the positions of the frost/thaw depths in soil. An adaptive mesh method for the moving boundary problem is adopted to solve the relevant equations and to determine frost/thaw depths, water content and temperature distribution. A series of sensitivity experiments by the numerical model under the periodic sinusoidal upper boundary condition for temperature are conducted to validate the model, and to investigate the effects of the model soil thickness, ground surface temperature, annual amplitude of ground surface temperature and thermal conductivity on frost/thaw depths and soil temperature. The simulated frost/thaw depths by the model with a periodical change of the upper boundary condition have the same period as that of the upper boundary condition, which shows that it can simulate the frost/thaw depths reasonably for a periodical forcing.
基金supported by Natural Science Foundation of China(No.11901238)Natural Science Foundation of Shandong Province(No.ZR2019MA063).
文摘We investigate a blowup problem of a reaction-advection-diffusion equa-tion with double free boundaries and aim to use the dynamics of such a problem to describe the heat transfer and temperature change of a chemical reaction in advective environment with the free boundary representing the spreading front of the heat.We study the influence of the advection on the blowup properties of the solutions and con-clude that large advection is not favorable for blowup.Moreover,we give the decay estimates of solutions and the two free boundaries converge to a finite limit for small initial data.
文摘We critically compare the practicality and accuracy of numerical approximations of phase field models and sharp interface models of solidification.Here we focus on Stefan problems,and their quasi-static variants,with applications to crystal growth.New approaches with a high mesh quality for the parametric approximations of the resulting free boundary problems and new stable discretizations of the anisotropic phase field system are taken into account in a comparison involving benchmark problems based on exact solutions of the free boundary problem.
