An inverse model for a free-boundary problem with a contact line: Steady case

This paper reformulates the two-phase solidification problem (i.e., the Stefan problem) as an inverse problem in which a cost functional is minimized with respect to the position of the interface and subject to PDE constraints. An advantage of this formulation is that it allows for a thermodynamically consistent treatment of the interface conditions in the presence of a contact point involving a third phase. It is argued that such an approach in fact represents a closure model for the original system and some of its key properties are investigated. We describe an efficient iterative solution method for the Stefan problem formulated in this way which uses shape differentiation and adjoint equations to determine the gradient of the cost functional. Performance of the proposed approach is illustrated with sample computations concerning 2D steady solidification phenomena.     

An Inverse Finite Element Method for Pure and Binary Solidification Problems

A 2D axisymmetric formulation for the solution of a directional solidification problem using an inverse finite-element method (IFEM) is presented. An algorithm developed by A. N. Alexandrou (Int. J. Numer. Methods Eng.28, 2383, 1989) has been modified and extended to include more general boundary conditions. The latter includes the explicit presence of an ampoule (with a complex shape) that contains the solid and the melt from which it is growing. Heat transfer between the ampoule and the external environment, time-dependent thermal boundary conditions, nonmonotonic temperature distributions, and species diffusion in the melt and crystal are also taken into account. Thus, our extended formulation encompasses a wider class of solidification problems than previous IFEM methods. Numerical experiments that illustrate the suitability of the extended IFEM are presented. In particular, we present a simulation of the directional solidification of zinc cadmium telluride using boundary conditions corresponding to an actual experiment scenario.     

Solidification microstructure selection of the peritectic Nd-Fe-B alloys

Bridgman directional solidification and laser remelting experiments were carried out on Nd_11.76Fe_82.36B_5.88 and Nd_13.5Fe_79.75B_6.75 alloys. Microstructure evolutions along with solidification parameters (temperature gradient G, growth velocity V and initial alloy composition C ₀) were investigated. A solidification microstructure selection map was established, based on the consideration of solidification characteristics of peritectic T₁ phase. In Bridgman directional solidification experiments, with the increasing growth velocities, the morphology of T₁ phase changed from plane front or faceted plane front to dendrites. In laser remelting experiments, a transition from primary γ-Fe dendrites to T₁ dendrites was found. Theoretical predictions are in good agreement with experimental results. Supported by the National Natural Science Foundation of China (Grant No. 50395100)     

The Three Dimensional Non-conforming Finite Element Solution of the Chapman–Ferraro Problem

We demonstrate the feasibility of using a non-conforming, piecewise harmonic finite element method on an unstructured grid in solving a magnetospheric physics problem. We use this approach to construct a global discrete model of the magnetic field of the magnetosphere that includes the effects of shielding currents at the outer boundary (the magnetopause). As in the approach of F. R. Toffolettoet al.(1994,Geophys. Res. Lett.21, 7) the internal magnetospheric field model is that of R. V. Hilmer and G.-H. Voigt (1995,J. Geophys. Res.) while the magnetopause shape is based on an empirically determined approximation (1997, J. Shueet al.,J. Geophys. Res.102, 9497). The results is a magnetic field model whose field lines are completely confined within the magnetosphere. The presented numerical results indicate that the discrete non-conforming finite element model is well-suited for magnetospheric field modeling.     

Data assimilation for the Navier–Stokes- equations

The well-posedness of the data assimilation problem for the Navier–Stokes-α equations on a bounded three-dimensional domain is investigated. The data assimilation procedures under consideration are the adjoint method of variational data assimilation (4D-Var) and the method of continuous data assimilation. Concerning the adjoint method the existence of optimal initial conditions with respect to an observation-dependent cost functional is proven, the optimizers are characterized by a first-order necessary condition involving the adjoint linearized Navier–Stokes-α equations and conditions for the uniqueness of the initial conditions are given. Well-posedness of the continuous data assimilation problem is proven and convergence rates in terms of observational resolution are provided.     

3D Analysis of Crystal/Melt Interface Shape and Marangoni Flow Instability in Solidifying Liquid Bridges

Solidification of gallium (Pr=0.02) in liquid bridges in zero-gravity conditions is investigated by numerical solutions of the three-dimensional and time-dependent flow-field equations. A single region (continuum) formulation based on the enthalpy method is adopted to model the phase-change problem. This paper analyzes the influence of the azimuthally asymmetric and steady first bifurcation of the Marangoni flow on the shape of the solid/melt interface during the crystal growth process. The numerical results show that this interface is distorted in the azimuthal direction. The distortion is related to the sinusoidal three-dimensional temperature disturbances due to the instability of the Marangoni flow. The three-dimensional flow field organization, related to the wave number, changes during the solidification process; this behavior is explained according to the variation of the aspect ratio of the solidifying liquid bridge. A correlation law is found for the azimuthal wave number of the instability as function of the melt zone aspect ratio.     

Directional cellular growth of Al-2?wt% Li bulk samples

Al-2 wt% Li alloy was prepared using metals of 99.99% high purity in the vacuum atmosphere. The bulk samples were directionally solidified upward with a constant growth rate, V, (∼8.30 μm/s) and different temperature gradients, G, (3.11–6.06 K/mm) and also with a constant G (6.06 K/mm) and different V (8.3–164.70 μm/s) in the directional solidification apparatus. The cellular spacings, λ, were measured from both transverse and longitudinal section of the specimens and expressed as functions of solidification processing parameters, G and V, by using a linear regression analysis. The effects of the G and V on λ, were investigated. The experimental results were compared with the current theoretical and numerical models, and similar previous experimental results.     

T1–T2 Correlation Spectra Obtained Using a Fast Two-Dimensional Laplace Inversion

Spin relaxation is a sensitive probe of molecular structure and dynamics. Correlation of relaxation time constants, such as T₁ and T₂, conceptually similar to the conventional multidimensional spectroscopy, have been difficult to determine primarily due to the absense of an efficient multidimensional Laplace inversion program. We demonstrate the use of a novel computer algorithm for fast two-dimensional inverse Laplace transformation to obtain T₁–T₂ correlation functions. The algorithm efficiently performs a least-squares fit on two-dimensional data with a nonnegativity constraint. We use a regularization method to find a balance between the residual fitting errors and the known noise amplitude, thus producing a result that is found to be stable in the presence of noise. This algorithm can be extended to include functional forms other than exponential kernels. We demonstrate the performance of the algorithm at different signal-to-noise ratios and with different T₁–T₂ spectral characteristics using several brine-saturated rock samples.     

Free Surface and Interface Thermodynamics of Liquid Nickel in Contact with Alumina

Sessile drop experiments of Ni and Ni(2at.%Al) were conducted under controlled working conditions, at 1500°C, P(O₂) 10^–9 Torr. It is shown that Al and oxygen atoms engaged in the capillary driven mass transport at the interface have a significant impact on the surface/interface thermodynamics. The surface energy of liquid Ni determined from experiments in which Ni comes into contact with Al₂O₃ is significantly lower than that of high purity Ni, due to the segregation of Al. The free energy of segregation of Al to the free surface of Ni ( G _S) was found to range from –164 to –152 kJ/mol, indicating a relatively strong tendency for segregation of Al to the free surface of Ni(Al). It is proposed that an Al(O)-rich liquid layer forms adjacent to the Ni-Al₂O₃ interface, which improves interfacial adhesion. In the Ni(Al)-Al₂O₃ system, an increase in the Al content of the alloy leads to the improvement of both wetting and adhesion of the alloy on the ceramic, correlating with the improvement in the interface strength after solidification.     

Directional solidification of Ti–49 at.%Al alloy

Ti–49Al (at.%) alloy was directionally solidified in Bridgman-type directional solidification furnace. The specimens were directionally solidified under an argon atmosphere with the different growth rate (V=5–30 μm/s) at a constant temperature gradient (G=12.1 K/mm), and with the different temperature gradient (G=2.8–12.1 K/mm) at a constant growth rate (V=10 μm/s). The dendritic spacings (λ ₁) were measured from both transverse and longitudinal sections of the specimens. The dependence of λ ₁ on the growth rate (V) and temperature gradient (G) were determined by using linear regression analysis. According to the experimental results, the value of λ ₁ decreases with the increase of values of V and G. The experimental results were compared with the current theoretical and numerical models, and similar previous experimental results.     

Effect of growth rate and Mg content on dendrite tip characteristics of Al–Cu–Mg ternary alloys

An experimental analysis is presented to correlate the secondary dendrite arm spacing λ ₂ and dendrite tip radius R with growth rate V and Mg content C _0-Mg of Al–Cu–Mg ternary alloys. Under constant temperature gradient G (4.84±0.13 K mm⁻¹), a series of directional solidification experiments were performed at five different growth rates V (16.7–83.3 μm/s) and five different Mg contents C _0-Mg in Al–5 wt.% Cu–(0.5–5) wt.% Mg alloys. Solid–liquid interface was investigated from the longitudinal sections of the quenched samples, and λ ₂ and R were measured on the dendrite tips. The dependencies of λ ₂ and R on V and C _0-Mg were determined. The experimental results showed that the values of λ ₂ and R decrease as V and C _0-Mg increase at a constant G. The present exponent values related to V are found to be slightly lower than the values of the theoretical models and previous experimental works; however, C _0-Mg exponent values are found to be much lower than the theoretical models and previous experimental works. The ratio of the secondary dendrite arm spacing to the dendrite tip radius is 2.09±0.15, in good agreement with the scaling law. At a constant C _0-Mg, the values of VR ² were found to slightly increase with the ascending V. However, as C _0-Mg increases, the values of VR ² decrease.     

Aluminum integral foams with tailored density profile by adapted blowing agents

The goal of the present work is the variation of the structure of aluminum integral foams regarding the thickness of the integral solid skin as well as the density profile. A modified die casting process, namely integral foam molding, is used in which an aluminum melt and blowing agent particles (magnesium hydride MgH₂) are injected in a permanent steel mold. The high solidification rates at the cooled walls of the mold lead to the formation of a solid skin. In the inner region, hydrogen is released by thermal decomposition of MgH₂ particles. Thus, the pore formation takes place parallel to the continuing solidification of the melt. The thickness of the solid skin and the density profile of the core strongly depend on the interplay between solidification velocity and kinetics of hydrogen release. By varying the melt and blowing agent properties, the structure of integral foams can be systematically changed to meet the requirements of the desired field of application of the produced component.     

Unconditionally Stable Methods for Hamilton–Jacobi Equations

We present new numerical methods for constructing approximate solutions to the Cauchy problem for Hamilton–Jacobi equations of the form u_t+H(D_xu)=0. The methods are based on dimensional splitting and front tracking for solving the associated (non-strictly hyperbolic) system of conservation laws p_t+D_xH(p)=0, where p=D_xu. In particular, our methods depend heavily on a front tracking method for one-dimensional scalar conservation laws with discontinuous coefficients. The proposed methods are unconditionally stable in the sense that the time step is not limited by the space discretization and they can be viewed as “large-time-step” Godunov-type (or front tracking) methods. We present several numerical examples illustrating the main features of the proposed methods. We also compare our methods with several methods from the literature.     

Transparent Boundary Conditions for Split-Step Padé Approximations of the One-Way Helmholtz Equation

In this paper, we generalize the nonlocal discrete transparent boundary condition introduced by F. Schmidt and P. Deuflhard (1995, Comput. Math. Appl.29, 53–76) and by F. Schmidt and D. Yevick (1997, J. Comput. Phys.134, 96–107) to propagation methods based on arbitrary Padé approximations of the two-dimensional one-way Helmholtz equation. Our approach leads to a recursive formula for the coefficients appearing in the nonlocal condition, which then yields an unconditionally stable propagation method.     

Berezinskii-Kosterlitz-Thouless phase transition in systems with exotic symmetries

A Berezinksii-Kosterlitz-Thouless phase transition in systems with the exceptional symmetry groups G=E _6,7,8, G ₂, and F ₂ is studied. The critical exponents and the exponents of the logarithmic corrections to the correlation functions at the transition point are found by the renormalization-group method. It is shown that for G=A, D, and E the critical exponents can be expressed in terms of the Coxeter numbers h _G (or the values of the Casimir operator in the adjoint representation K ₂ ^G ). Pis’ma Zh. éksp. Teor. Fiz. 63, No. 9, 743–747 (10 May 1996)     

A New Two-Dimensional Pulse Sequence for T2* Measurements of Protons in C Isotopomers

A new two-dimensional pulse sequence for T₂* measurement of protons directly coupled to ¹³C spins is proposed. The sequence measures the tranverse relaxation time of heteronuclear proton single-quantum coherence under conditions of free precession and is therefore well suited to evaluate relaxation losses of proton magnetization during preparation delays of heteronuclear pulse experiments in analytical NMR. The relevant part of the pulse sequence can be inserted as a “building block” into any direct or inverse detecting H,C correlation pulse sequence if proton spin–spin relaxation is to be investigated. In this contribution, the building block is inserted into a HETCOR as well as into a HMQC pulse sequence. Experimental results for the HETCOR-based sequence are given.     

Aerodynamic design optimization by using a continuous adjoint method

This paper presents the fundamentals of a continuous adjoint method and the applications of this method to the aerodynamic design optimization of both external and internal flows.General formulation of the continuous adjoint equations and the corresponding boundary conditions are derived.With the adjoint method,the complete gradient information needed in the design optimization can be obtained by solving the governing flow equations and the corresponding adjoint equations only once for each cost function,regardless of the number of design parameters.An inverse design of airfoil is firstly performed to study the accuracy of the adjoint gradient and the effectiveness of the adjoint method as an inverse design method.Then the method is used to perform a series of single and multiple point design optimization problems involving the drag reduction of airfoil,wing,and wing-body configuration,and the aerodynamic performance improvement of turbine and compressor blade rows.The results demonstrate that the continuous adjoint method can efficiently and significantly improve the aerodynamic performance of the design in a shape optimization problem.     

层状共晶定向凝固

本文从理论和实验两方面研究层状共晶定向凝固。理论上将层状共晶定向凝固与胞晶列定向凝固及Hele-Shaw胞中粘性指发展进行类比,给出层状共晶稳态凝固条件下溶质扩散、热传输与界面张力效应之间的耦合系,导出λ²ν=consi标度关系。实验上考察了规则Al-Al₂Cu共晶和非规则Al-Si共晶定向凝固,证实耦合关系存在。还讨论了两种共晶凝固界面特征和热传输的影响。 关键词：     

Relaxation Behavior of Laser-Polarized Xe Gas: Size Dependency and Wall Effect of the T1 Relaxation Time in Glass and Gelatin Bulbs

Size dependency of the relaxation time T₁ was measured for laser-polarized ¹²⁹Xe gas encapsulated in different sized cavities made by glass bulbs or gelatin capsules. The use of laser-polarized gas enhances the sensitivity a great deal, making it possible to measure the longer ¹²⁹Xe relaxation time in quite a short time. The size dependency is analyzed on the basis of the kinetic theory of gases and a relationship is derived in which the relaxation rate is connected with the square inverse of the diameter of the cavity. Such an analysis provides a novel parameter which denotes the wall effect on the relaxation rate when a gas molecule collides with the surface once in a second. The relaxation time of ¹²⁹Xe gas is also dependent on the material which forms the cavity. This dependency is large and the relaxation study using polarized ¹²⁹Xe gas is expected to offer important information about the state of the matter of the cavity wall.     

Pressure effects in the planar solidification of a finite slab: convective cooling and prescribed heat flux boundary conditions

Summary The classical Stefan problem assumes a fixed melting temperature. However, when the solid phase is the one with lower density (e.g., water) the solidification of the system causes an overall volume increase that is often contrasted by the container walls. In that case the growing pressure determines a continuous lowering of the freezing point, and the temperature field as well as the interface motion are strongly affected. This paper is concerned with these aspects of the problem; the planar solidification of a slab of finite thickness, contrasted by an opposing elastic force, is numerically simulated. The effects of two different boundary conditions are analysed. When the solidification is driven by convective cooling, the continuous advancement of the melting front is replaced by an asymptotic behaviour, until thermal equilibrium is attained. When the boundary condition is specified in terms of a prescribed heat flow, the melting front velocity is slowed down by a growing adverse temperature gradient. The influence of various parameters on the process is presented and discussed.