冷冻外科中一维相变问题的数值解

用变时间步长法求解了冷冻外科手术时模拟生物组织中的一维相变问题。计算方法经与文献比较,简便可行。运用本文的计算方法计算并讨论了不同边界条件下,相变区的温度场分布和无量纲相界面降温速率随Ｓｔｅ数的变化,计算分析表明,相变区的温度基本呈线性分布,无量纲相界面降温速率随Ｓｔｅ数增大而降低,所得结论可对冷刀设计和冷冻外科手术提供帮助。     

The modelling of the freezing process in fine-grained porous media: Application to the frost heave estimation

The solution of the moving-boundary problem, related to heat- and mass-transfer processes in freezing, fine-grained, porous media under phase-transition conditions is presented. It is assumed that a freezing zone, characterized by a wide temperature range of phase transitions, is formed. Therefore a three-zone model is developed. The preservation of the term ∂L/∂t(L is the ice content) in the system of equations has made it possible to determine the ice distribution within the frozen and the freezing zones. For loamy soils the dependence of the freezing process on the characteristic parameters, the Stefan and Lewis numbers, was analyzed. It was found that increasing the enthalpy of phase transition, i.e., decreasing the Stefan number Ste, resulted in diminution of the frozen zone but, at the same time, the ice content within this zone increased. Intensification of the migration process, i.e., increasing the Lewis number Le, also led to diminution of the frozen zone, in which the ice content and, consequently, the total moisture (including ice) were increased. For large Lewis numbers the freezing zone was observed to decrease. When the water migration process is absent (Le = 0), the calculations, which were based on the described model show that in the course of freezing the redistribution takes place only between moisture and ice contents. The total moisture remains constant and equal to the initial water content. The theoretical conceptions and results derived from the analytical solution are in agreement with experimental findings. The presented model predicts the freezing process in porous media and satisfactorily reflects observed phenomena. The utilization of the considered problem solution to the prediction of the frost heave phenomenon in soils freezing processes shown that the calculated frost heave curve matches the experimental results very closely indicating that the model can well reproduce the frost heaving process associated with the freezing. Propagation of the freezing front in the test is predicted the experimental results with reasonable accuracy.     

Evolution of the Inner Liquid–Solid Interface During Metal Freezing

The influence of the inner interface initiation method on the interface shape (formation of the planar interface or the interface with the dendrites growing into the liquid metal) was studied both theoretically and experimentally. The results of numerical simulation of the process of heat removal from the metal, corresponding to different initiation methods, revealed the existence of different species of the inner interface. The interface modification during freezing arises from the inequality of temperature gradients on opposite sides of the interface, i.e., from imbalance of heat fluxes on the interphase boundary (Stefan problem). For indium point, the results of numerical simulation were confirmed experimentally.     

Control of the freezing interface morphology in solidification processes in the presence of natural convection

This paper presents a methodology for the solution of an inverse solidification design problem in the presence of natural convection. In particular, the boundary heat flux q₀ in the fixed mold wall, δΩ₀, is calculated such that a desired freezing front velocity and shape are obtained. As the front velocity together with the flux history q_ms on the solid side of the freezing front play a determinant role in the obtained cast structure, the potential applications of the proposed methods to the control of casting processes are enormous. The proposed technique consists of first solving a direct natural convection problem of the liquid phase in an a priori known shrinking cavity, Ω_L(t), before solving an ill-posed inverse design conduction problem in the solid phase in an a priori known growing region, Ω_S(t). The direct convection problem is used to evaluate the flux q_ml in the liquid side of the freezing front. A front tracking deforming finite element technique is employed. The flux q_ml can be used together with the Stefan condition to provide the freezing interface flux q_ms in the solid side of the front. As such, two boundary conditions (flux q_ms and freezing temperature θ_m) are especified along the (known) freezing interface δΩ_I(t). The developed design technique uses the adjoint method to calculate in L₂ the derivative of the cost functional, ∥θ_m – θ( x , t; q₀)∥, that expresses the square error between the calculated temperature θ( x , t; q₀) in the solid phase along δΩ_I(t) and the given melting temperature. The minimization of this cost functional is performed by the conjugate gradient method via the solutions of the direct, sensitivity and adjoint problems. A front tracking finite element technique is employed in this inverse analysis. Finally, an example is presented for the solidification of a superheated incompressible liquid aluminium, where the effects of natural convection in the moving interface shape are controlled with a proper adjustment of the cooling boundary conditions.     

模拟生物组织冻结过程计算分析

以冷冻外科为应用背景,用变时间步长法数值求解模拟生物组织（土豆泥）中的一维冷冻过程,当冷刀壁面温度随时间线性变化时,可得到冷冻外科手术时模拟生物组织中各时刻的温度场分布和相变界面的移动速率。并进一步分析了冷刀壁面保持恒定降温速率时,不同介质初始温度和冷刀壁面初始温度对相变区发展过程的影响。     

Numerical implementaion of a freezing-front model for water-saturated media that includes the dependence of the phase-transition temperature on the pressure and concentration

A computational algorithm is constructed for solution of the problem of freezing of water-saturated porous media with account for the dependence of the temperature of the phase transition of pore moisture on the pressure and the concentration of a dissolved impurity. The mathematical model of the process considered is based on a generalized formulation of the well-known Stefan problem. Examples convey results of computation.     

Godunov Method for Stefan Problems with Enthalpy Formulations

A Stefan problem is a free boundary problem where a phase boundary moves as a function of time. In this article, we consider one-dimensional and two-dimensional enthalpy-formulated Stefan problems. The enthalpy formulation has the advantage that the governing equations stay the same, regardless of the material state (liquid or solid). Numerical solutions are obtained by implementing the Godunov method. Our simulation of the temperature distribution and interface position for the one-dimensional Stefan problem is validated against the exact solution, and the method is then applied to the two-dimensional Stefan problem with reference to cryosurgery, where extremely cold temperatures are applied to destroy cancer cells. The temperature distribution and interface position obtained provide important information to control the cryosurgery procedure.     

A Legendre spectral element method for the Stefan problem

In this paper we present a Legendre spectral element method for solution of multi-dimensional unsteady change-of-phase Stefan problems. The spectral element method is a high-order (p-type) finite element technique, in which the computational domain is broken up into general (curved) quadrilateral macroelements, and the solution, data and geometry are expanded within each element in terms of tensor-product Lagrangian interpolants. The discrete equations are generated by a Galerkin formulation followed by Gauss–Lobatto Legendre quadrature, for which it is shown that exponential convergence to smooth solutions is obtained as the polynomial order of fixed elements is increased. The spectral element equations are inverted by conjugate gradient iteration, in which the matrix-vector products are calculated efficiently using tensor-product sum-factorization. To solve the Stefan problem numerically, the heat equations in the liquid and solid phases are transformed to fixed domains applying an interface-local time-dependent immobilization transformation technique. The modified heat equations are discretized using finite differences in time, resulting at each time step in a Helmholtz equation in space that is solved using Legendre spectral element elliptic discretizations. The new interface position is then computed using a variationally consistent flux treatment along the phase boundary, and the solution is projected upon the corresponding updated mesh. The rapid convergence rate and stability of the method are discussed, and numerical results are presented for a one-dimensional Stefan problem using both a semi-implicit and a fully implicit time-stepping scheme. Finally, a two-dimensional Stefan problem with a complex phase boundary is solved using the semi-implicit scheme.     

The numerical solution of two generalizations of the neumann moving interface problem into two dimensions

We present numerical solutions for the following Stefan problems. The half-space z > 0 is initially filled with liquid at its fusion temperature. The boundary z = 0, taken to be the x axis, is maintained at a constant temperature, less than the fusion temperature, for x < 0. For x > 0, the first problem considers the case of an insulated boundary, and the second problem considers the case of the boundary maintained at the fusion temperature. This gives rise to a solid-liquid interface curved in the (x , z ) plane.     

Three-dimensional axisymmetric model for convection in laser-melted pools

Abstract

A three-dimensional axisymmetric model of the fluid flow and heat transfer in a laser-melted pool is developed. The model corresponds to the limiting case when the scanning velocity is small compared with the recirculating velocity. This model is also valid for spot welding. Non-dimensional forms of the governing equations are derived, from which four dimensionless parameters are obtained: the Marangoni number, the Prandtl number, the dimensionless melting temperature, and the radiation factor. Their effects and significance are discussed, and numerical solutions are obtained. The position and shape of the solid/liquid interface are obtained by an iterative scheme. The quantitative effects of the dimensionless parameters on pool shape are presented. In the presence of the flow field, the heat transfer becomes convection dominated. The effect of convection on isotherms within the molten pool is discussed, and experimental results are presented.

MST/535     

Numerical investigation of supersonic flow past blunt bodies of intricate shape at an angle of attack and slip angle

A supersonic flow of viscous homogeneous gas past blunt bodies of intricate shape at an angle of attack and slip angle is investigated numerically within the model of complete three-dimensional viscous shock layer using the time relaxation method. The main regularities are studied of the general structure of flow and of the distribution of pressure and heat flux along the surface. An analysis is performed of their dependence on the shape of the body, angle of attack and slip angle, Mach and Reynolds numbers, and on other determining parameters of the problem. The accuracy and range of validity of a number of approximate approaches to the solution of the problem are estimated.     

Laws of vapor bubble growth in the superheated liquid volume (thermal growth scheme)

The mathematical formulation of the heat-input-controlled vapor bubble growth in an infinite volume of uniformly heated liquid is described. Using the dimensional theory, the structure of the solution was analyzed qualitatively. A historical survey of theoretical works devoted to the considered problem is presented. Asymptotic solutions are obtained and studied systematically. The results of the complete analytical solution of the problem and formulas for the calculation of the bubble growth rate in the whole domain of possible variations in regime parameters are presented. The conclusion is made that the influence of permeability of the interface has a significant effect on the bubble growth rate. It is shown that the Plesset-Zwick formula, which is commonly accepted in computational practice, is not applicable at both small and large Jakob numbers and its good agreement with the experiment is determined to a large extent by a combination of the imperfectness of the theoretical analysis and the experimental error. The conclusion is made that, for many liquids, the ultimately achievable value of the dimensionless superheating parameter (Stefan number) can exceed unity. In this case, the regularities in the bubble growth acquire some features unexplored to date.     

The propagation and basal solidification of two-dimensional and axisymmetric viscous gravity currents

The effect of basal solidification on viscous gravity currents is analysed using continuum models. A Stefan condition for basal solidification is incorporated into the Navier-Stokes equations. A simplified version of this model is determined in the lubrication and large-Bond-number limit. Asymptotic solutions are obtained in three parameter régimes. (i) A similarity solution is possible in the following cases: the two-dimensional problem when volume per unit length (V) is proportional to time (t) raised to the power 7/4(V = qt ^7/4) and the Julian number (v ³ g ² /q ⁴ ) is large, where v is kinematic viscosity, q is a constant of proportionality and g is the acceleration due to gravity; the axisymmetric problem when volume is proportional to time raised to the power 3 (V = Qt ³) and the dimensionless group vg/Q is large, where Q is a constant of proportionality. In both cases, the front is found to depend on time raised to the power 5/4, as it does in the absence of solidification, but the constant of proportionality satisfies a modified system of equations. (ii) In the case of large Stefan number and small modified Peclet number (Pe ² 1, where Pe is the Peclet number and is the aspect ratio), asymptotic and numerical methods are combined to produce the most revealing results. The temperature of the fluid approaches the melting point over a short time-scale. Over the long time-scale, the solid/liquid interface is determined from the conduction of latent heat into the solid. Strong coupling is observed with the basal solidification modifying the flow at leading order. The solidification may retard and eventually arrest the front motion long before complete phase change has taken place. (iii) In the case of constant volume and large modified Peclet number (Pe ² 1), similarity solutions are found for the solidification at the base of the gravity current on the short time-scale. The coupling is weak on this time-scale with the solidification being dependent on the front position but not influencing the fluid motion at leading order. Over the long time-scale, the drop completely solidifies. Analytical solutions are not obtained on this time-scale, but scalings are deduced.     

Generalized Rayleigh-Lamb equation

The dynamics of the vapor-liquid interface when a solid particle heated to a high temperature comes into contact with a cold liquid is analyzed. The generalized Rayleigh-Lamb equation taking into account temperature changes in the liquid and the vapor, the saturation vapor pressure change associated with it, and the emergence of the mass flux due to liquid evaporation at the interface is deduced. At medium parameters far from the critical point, the generalized Rayleigh-Lamb equation reduces to its well-known form. Around the critical point, the difference between the obtained equation and arising modes of changes of the vapor-liquid interface can be rather considerable. Unlike the well-known Stefan problem, far from the critical liquid temperature, the phase transition temperature and, respectively, the saturation vapor pressure change with the vapor-liquid interface movement, which yields various modes of dynamics of the vapor cavity. In the special case of small variations from the stationary mode, an analytical solution for the problem on a small change in the vapor cavity radius with time was developed.     

Shape identification for water–ice interface within the cylindrical capsule in cold storage system by inverse heat transfer method

In this study, the inverse heat transfer method is applied to shape identification for the ice layer within the cylindrical capsule in cold storage system. The approach is constructed by combining the curvilinear grid generation scheme, the direct problem solver, the conjugate gradient optimization method, and the redistribution method. According to the practical condition of freezing ice, shape identification for the water–ice interface based on the data of the outer surface temperature is attempted. Results show that the profile of the water–ice interface is possible to be identified by using the inverse heat transfer approach and the accuracy of the ice shape identification is dependent on the uncertainty of the outer surface temperature data, the Biot number, the thickness of the ice layer, and the geometric configuration as well.     

A comparative study of Domain Embedding Methods for regularized solutions of inverse Stefan problems

In this paper, various Domain Embedding Methods (DEMs) for an inverse Stefan problem are presented and compared. These DEMs extend the moving boundary domain to a larger, but simple and fixed domain. The original unknown interface position is then replaced by a new unknown, which can be a boundary temperature or heat flux, or an internal heat source. In this way, the non-linear identification problem is transformed into a linear one in the enlarged domain. Using different physical quantities as the new unknown leads to different DEMs. They are analysed from various points of view (accuracy, efficiency, etc.) through two test problems, by a comparison with a common Front-Tracking Method (FTM). The first test has a smooth temperature field and the second one has some singularities. The advantage of the DEMs in solving the inverse problem and in computing the corresponding direct mapping is shown. In the direct problem, high-order accurate schemes could be obtained more easily with the DEMs than with the FTM. In the inverse problem, an iterative regularization and a Tikhonov regularization have been employed. For the FTM, the iterative regularization is not efficient—the solution oscillates when the data are noisy. As for the Tikhonov regularization, it requests special care to choose an adequate penalty term. In contrast, both the regularizations give good results with all the considered DEMs, except for the second test problem at the beginning (t=0⁺) when the value of the heat flux and the heat source tends to ∞. Slightly different regularization effects have been obtained when using different DEMs. Finally, an automatic choice of the optimal regularization parameter is also discussed, using data with different noise levels. We propose the use of the curve of the residual norm against the regularization parameter. © 1997 John Wiley & Sons, Ltd.     

Frost heave and phase front instability in freezing soils

The main equations and conditions at the phase transition front are presented for a generalized model of secondary frost heave in freezing fine-grained soils. The analytical criterion for the stability/instability of the freezing phase front in porous media is derived. This criterion is obtained for the occurrence of the frost heave process by using the perturbation method in a two-dimensional, coupled heat and mass transfer model. This model assumes that the non-instantaneous crystallization process takes place in the kinetic zone, and that the rate of crystallization is a function of supercooling. This corresponds to the Arrhenius form equation and agrees with experimental investigations. The perturbation analysis of the freezing front shows that the stability criterion depends upon 1) the Stefan and Peclet numbers, 2) a parameter describing the phase transition kinetics and also 3) dimensionless parameters which characterize the frost heave process. Employing Fourier synthesis, actual front shape evolution is calculated. It is seen that the front displays a periodic morphology whose scale is essentially unrelated to that of the initial (starting) perturbation. The effect of the non-instantaneous kinetics on the front shape evolution is described. As is shown in results, the kinetics has a stabilizing effect and, in this case, the perturbations grow more slowly. The theoretical stability/instability conditions as predicted from the derived criterion were found to be in agreement with experimental investigations of the formation of soil cryogenic structure in the freezing process. On the basis of the asymptotic solution the engineering approach for the calculation of the heave rate and maximal frost penetration depth values — main characteristics for design and construction in cold regions, is presented. The good agreement between calculated values and experimental data is observed.     

Three-Dimensional Micromechanical Modeling of Cord-Rubber Composites

A three-dimensional micromechanical model was developed to investigate the load-deformation characteristics of cord-rubber composites. A finite-element model that integrates a solid rubber element and a twisted cord element which takes into account coupling effects of various deformations is developed to investigate the influence of cord shape on the load-deformation characteristics. The finite-element model developed was validated by comparing the results with those from a solid three-dimensional finite-element analysis. Numerical results of deformations and stress distributions are presented to illustrate the influence of cord shape, cord-rubber anisotropy, and rubber thickness. The results presented illustrate that cord shape and rubber thickness surrounding the cord have a strong effect on the values of deformations and interface stress distributions.     

An enthalpy formulation based on an arbitrarily deforming mesh for solution of the Stefan problem

Traditionally schemes for dealing with the Stefan phase change problem are separated into fixed grif or front tracking (deforming grid) schemes. A standard fixed grid scheme is to use an enthalpy formulation and track the movement of the phase front via a liquid fraction variable. In this paper, an enthalpy formulation is applied on a continuously deforming finite element grid. This approach results in a general numerical scheme that incorporates both front tracking and fixed grid schemes. It is shown how on appropriate setting of the grid velocity a fixed or deforming grid solution can be generated from the general scheme. In addition an approximate front tracking scheme is developed which can produce accurate non-oscillatory predictions at a computational cost close to an efficient fixed grid scheme. The versatility of the general scheme and the approximate front tracking scheme are demonstrated on solution of a number of Stefan problems in both one and two dimensions.     

Relevance of liquid state to solid state properties

We outline in this talk the beginning of a new programme to study physical properties of crystalline solids. It is based on considering the latter, a broken symmetry phase, in terms of the higher symmetry liquid phase. The solid is a calculable perturbation on the fluid. This is exactly opposite to the standard approach which relates mechanical properties to the behaviour of defects (mainly dislocations) etc., in an otherwise perfect crystalline solid. However, most other broken symmetry phases (e.g. ferromagnets) are discussed starting from a symmetric Hamiltonian or a free energy functional, and earlier work by one of the authors shows that the liquid-solid transition is well described, qualitatively and quantitatively, by this approach. On the other hand, defect theories of melting have a long record of nonsuccess. In the first part of the talk, the density wave theory of freezing will be outlined, and it will be shown how properties such as Debye Waller factor, entropy change of freezing etc. can be calculated with no or one free parameter. The problem of calculating shear elastic constants and dislocation core structures as well as energies in terms only of observable liquid state properties will be set up, and results presented. The method will be contrasted with zero temperature ‘atomistic’ models which obscure the essential dependence on structure and flounder in a mass of detail. The concluding part will describe further proposed applications, some suggestive experimental results extant in the literature, and some speculations. Only a summary is presented.