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NASA NTRSDate not supplied

Modified Epoxy Matrix Resins for Reduced Dependence on Redundant Fasteners in Secondary-Bonded Composite Structures

In structural bonds, the interface between adherend and adhesive is nearly two-dimensional making it susceptible to minute quantities of contamination that can lead to weak bonds. No methods currently exist to measure the strength of an adhesive bond nondestructively. For these reasons, regulating organizations such as the Federal Aviation Administration (FAA) often require redundant load paths in secondary-bonded, primary-structures to achieve certification for civil transport aircraft. To meet this requirement, manufacturers commonly install redundant fasteners (~105 fasteners in a twin-aisle airframe with composite fuselage and wings), which add tremendous complexity, time and cost, and significantly reduce performance of the airframe. Under NASA’s Convergent Aeronautics Solutions (CAS) Program, the Adhesive Free Bonding of Composites (AERoBOND) project is investigating reformulated aerospace epoxy matrix resins to enable reflow and diffusion of the resin at the interface during a secondary bonding/cure process. The mixing process eliminates the material discontinuity at the interface, thereby removing the dependence of bond performance on adhesion across a nearly two-dimensional boundary. The AERoBOND interface is similar to the interlaminar region formed during a co-cure process, so joint performance depends on the cohesive properties of the matrix resin, which can be measured, inspected non-destructively, and certified for flight with fewer redundant fasteners. This presentation describes the ongoing development of stoichiometrically-offset epoxy resins to control the degree of cure in composite joints and enable interfacial mixing during secondary bonding/curing. Details on the resin formulation, laminate fabrication, and mechanical testing will also be presented. The precracked mode-II fracture toughness measured from preliminary AERoBOND joints was similar for joints fabricated with conventional epoxy matrix materials.

thermoset, hardener, prepreg, assembly↗
NASA NTRS1988

Reaction kinetics between fiber and matrix

Interdiffusion and interdiffusion controlled intermediate phase formation in metal matrix composites can be of interest for the prolonged application of these systems at high temperatures. Methods are discussed that address the kinetics of interdiffusion in systems that exhibit solid solution intermixing or the formation of a third intermediate phase at the fiber/matrix interface, or both. The tungsten fiber reinforced niobium and tungsten fiber reinforced superalloy systems are employed as model systems for experimentation and discussion. In an effort to impede interdiffusion, the concept of ion implanted diffusion barriers have been examined. Preliminary results on the feasibility and effectiveness of ion implanted diffusion barriers are presented.

Kopp, M. W.↗
NASA NTRS1999

Ultrasonic Nondestructive Characterization of Adhesive Bonds

Adhesives and adhesive joints are widely used in various industrial applications to reduce weight and costs, and to increase reliability. For example, advances in aerospace technology have been made possible, in part, through the use of lightweight materials and weight-saving structural designs. Joints, in particular, have been and continue to be areas in which weight can be trimmed from an airframe through the use of novel attachment techniques. In order to save weight over traditional riveted designs, to avoid the introduction of stress concentrations associated with rivet holes, and to take full advantage of advanced composite materials, engineers and designers have been specifying an ever-increasing number of adhesively bonded joints for use on airframes. Nondestructive characterization for quality control and remaining life prediction has been a key enabling technology for the effective use of adhesive joints. Conventional linear ultrasonic techniques generally can only detect flaws (delamination, cracks, voids, etc) in the joint assembly. However, more important to structural reliability is the bond strength. Although strength, in principle, cannot be measured nondestructively, a slight change in material nonlinearity may indicate the onset of failure. Furthermore, microstructural variations due to aging or under-curing may also cause changes in the third order elastic constants, which are related to the ultrasonic nonlinear parameter of the polymer adhesive. It is therefore reasonable to anticipate a correlation between changes in the ultrasonic nonlinear acoustic parameter and the remaining bond strength. It has been observed that higher harmonics of the fundamental frequency are generated when an ultrasonic wave passes through a nonlinear material. It seems that such nonlinearity can be effectively used to characterize bond strength. Several theories have been developed to model this nonlinear effect. Based on a microscopic description of the nonlinear interface binding force, a quantitative method was presented. Recently, a comparison between the experimental and simulated results based on a similar theoretical model was presented. A through-transmission setup for water immersion mode-converted shear waves was used to analyze the ultrasonic nonlinear parameter of an adhesive bond. In addition, ultrasonic guided waves have been used to analyze adhesive or diffusion bonded joints. In this paper, the ultrasonic nonlinear parameter is used to characterize the curing state of a polymer/aluminum adhesive joint. Ultrasonic through-transmission tests were conducted on samples cured under various conditions. The magnitude of the second order harmonic was measured and the corresponding ultrasonic nonlinear parameter was evaluated. A fairly good correlation between the curing condition and the nonlinear parameter is observed. The results show that the nonlinear parameter might be used as a good indicator of the cure state for adhesive joints.

Qu, Jianmin↗
NASA NTRS2011

Test-Anchored Vibration Response Predictions for an Acoustically Energized Curved Orthogrid Panel with Mounted Components

A rich body of vibroacoustic test data was recently generated at Marshall Space Flight Center for a curved orthogrid panel typical of launch vehicle skin structures. Several test article configurations were produced by adding component equipment of differing weights to the flight-like vehicle panel. The test data were used to anchor computational predictions of a variety of spatially distributed responses including acceleration, strain and component interface force. Transfer functions relating the responses to the input pressure field were generated from finite element based modal solutions and test-derived damping estimates. A diffuse acoustic field model was employed to describe the assumed correlation of phased input sound pressures across the energized panel. This application demonstrates the ability to quickly and accurately predict a variety of responses to acoustically energized skin panels with mounted components. Favorable comparisons between the measured and predicted responses were established. The validated models were used to examine vibration response sensitivities to relevant modeling parameters such as pressure patch density, mesh density, weight of the mounted component and model form. Convergence metrics include spectral densities and cumulative root-mean squared (RMS) functions for acceleration, velocity, displacement, strain and interface force. Minimum frequencies for response convergence were established as well as recommendations for modeling techniques, particularly in the early stages of a component design when accurate structural vibration requirements are needed relatively quickly. The results were compared with long-established guidelines for modeling accuracy of component-loaded panels. A theoretical basis for the Response/Pressure Transfer Function (RPTF) approach provides insight into trends observed in the response predictions and confirmed in the test data. The software modules developed for the RPTF method can be easily adapted for quick replacement of the diffuse acoustic field with other pressure field models; for example a turbulent boundary layer (TBL) model suitable for vehicle ascent. Wind tunnel tests have been proposed to anchor the predictions and provide new insight into modeling approaches for this type of environment. Finally, component vibration environments for design were developed from the measured and predicted responses and compared with those derived from traditional techniques such as Barrett scaling methods for unloaded and component-loaded panels.

Frady, Gregory P.↗
NASA NTRS2001

Space- and Ground-Based Crystal Growth Using a Baffle (CGB)

The composition of semiconductor crystals produced in space by conventional melt-growth processes (directional solidification and zone melting) is affected by minute levels of residual micro-acceleration, which causes natural convection. The residual acceleration has random magnitude, direction and frequency. Therefore, the velocity field in the melt is apriori unpredictable. As a result, the composition of the crystals grown in space can not be predicted and reproduced. The method for directional solidification with a submerged heater or a baffle was developed under NASA sponsorship. The disk-shaped baffle acts as a partition, creating a small melt zone at the solid-liquid interface. As a result, in ground based experiment the level of buoyancy-driven convection at the interface is significantly reduced. In several experiments with Te-doped GaSb, nearly diffusion controlled segregation was achieved.

Ostrogorsky, A. G.↗
NASA NTRS1998

Algebraic Nonoverlapping Domain Decomposition Methods for Stabilized FEM and FV Discretizations

We consider preconditioning methods for convection dominated fluid flow problems based on a nonoverlapping Schur complement domain decomposition procedure for arbitrary triangulated domains. The triangulation is first partitioned into a number of subdomains and interfaces which induce a natural 2 x 2 partitioning of the p.d.e. discretization matrix. We view the Schur complement induced by this partitioning as an algebraically derived coarse space approximation. This avoids the known difficulties associated with the direct formation of an effective coarse discretization for advection dominated equations. By considering various approximations of the block factorization of the 2 x 2 system, we have developed a family of robust preconditioning techniques. A computer code based on these ideas has been developed and tested on the IBM SP2 using MPI message passing protocol. A number of 2-D CFD calculations will be presented for both scalar advection-diffusion equations and the Euler equations discretized using stabilized finite element and finite volume methods. These results show very good scalability of the preconditioner for various discretizations as the number of processors is increased while the number of degrees of freedom per processor is fixed.

Barth, Timothy J.↗
NASA NTRS1985

Cloud Microphysics by Thermal Wave Methods

This experiments series is the first application of a low-gravity experimental technique to the study of cloud microphysics. The low-gravity environment is provided by the parabolic maneuver of NASA's KC-135 aircraft. The primary objective is to compare experimental observations of cloud droplet growth and evaporation in a convection free environment with a numerical model of the process. Beyond that, the work also involves the development and testing of low-gravity research techniques. In particular, passive methods of thermal control have been devised and used effectively. The study to date has shown that the method is particularly suitable for looking at interactions between adjoining portions of the cloud drop field and interactions of the drop field with a solid interface. After final analysis of the data, it is expected the results will shed light on the development of cloud droplet size spectra in natural clouds as well as the performance of certain types of cloud physics instrumentation, particularly continuous flow diffusion chambers and loud condensation nuclei counters.

Anderson, B. J.↗
NASA NTRS1999

High Order Approximations for Compressible Fluid Dynamics on Unstructured and Cartesian Meshes

The development of high-order accurate numerical discretization techniques for irregular domains and meshes is often cited as one of the remaining challenges facing the field of computational fluid dynamics. In structural mechanics, the advantages of high-order finite element approximation are widely recognized. This is especially true when high-order element approximation is combined with element refinement (h-p refinement). In computational fluid dynamics, high-order discretization methods are infrequently used in the computation of compressible fluid flow. The hyperbolic nature of the governing equations and the presence of solution discontinuities makes high-order accuracy difficult to achieve. Consequently, second-order accurate methods are still predominately used in industrial applications even though evidence suggests that high-order methods may offer a way to significantly improve the resolution and accuracy for these calculations. To address this important topic, a special course was jointly organized by the Applied Vehicle Technology Panel of NATO's Research and Technology Organization (RTO), the von Karman Institute for Fluid Dynamics, and the Numerical Aerospace Simulation Division at the NASA Ames Research Center. The NATO RTO sponsored course entitled "Higher Order Discretization Methods in Computational Fluid Dynamics" was held September 14-18, 1998 at the von Karman Institute for Fluid Dynamics in Belgium and September 21-25, 1998 at the NASA Ames Research Center in the United States. During this special course, lecturers from Europe and the United States gave a series of comprehensive lectures on advanced topics related to the high-order numerical discretization of partial differential equations with primary emphasis given to computational fluid dynamics (CFD). Additional consideration was given to topics in computational physics such as the high-order discretization of the Hamilton-Jacobi, Helmholtz, and elasticity equations. This volume consists of five articles prepared by the special course lecturers. These articles should be of particular relevance to those readers with an interest in numerical discretization techniques which generalize to very high-order accuracy. The articles of Professors Abgrall and Shu consider the mathematical formulation of high-order accurate finite volume schemes utilizing essentially non-oscillatory (ENO) and weighted essentially non-oscillatory (WENO) reconstruction together with upwind flux evaluation. These formulations are particularly effective in computing numerical solutions of conservation laws containing solution discontinuities. Careful attention is given by the authors to implementational issues and techniques for improving the overall efficiency of these methods. The article of Professor Cockburn discusses the discontinuous Galerkin finite element method. This method naturally extends to high-order accuracy and has an interpretation as a finite volume method. Cockburn addresses two important issues associated with the discontinuous Galerkin method: controlling spurious extrema near solution discontinuities via "limiting" and the extension to second order advective-diffusive equations (joint work with Shu). The articles of Dr. Henderson and Professor Schwab consider the mathematical formulation and implementation of the h-p finite element methods using hierarchical basis functions and adaptive mesh refinement. These methods are particularly useful in computing high-order accurate solutions containing perturbative layers and corner singularities. Additional flexibility is obtained using a mortar FEM technique whereby nonconforming elements are interfaced together. Numerous examples are given by Henderson applying the h-p FEM method to the simulation of turbulence and turbulence transition.

Barth, Timothy↗
NASA NTRS1994

CATLAC - Calibration and Validation Analysis Tool of Local Area Coverage for the SeaWiFS Mission

Calibration and validation Analysis Tool of Local Area Coverage (CATLAC) is an analysis package for selecting and graphically displaying Earth and space targets for calibration and validation activities on a polar orbiting satellite. The package is written in the Interactive Data Language (IDL) and includes a graphical user interface. Although it is designed specifically for the Sea-viewing Wide Field-of-view Sensor (SeaWiFS) mission, the package can be used for analysis on other Earth-viewing missions. An individual can use text or graphical methods in CATLAC to select Earth targets to be scanned by a satellite. Additional onboard calibration activities (such as observations of the moon, or solar irradiance from a solar diffuser), which use data recorder time, can also be specified. All information pertinent to the creation of a command schedule can be written to a file which is read by a command scheduler. The scheduler can be invoked and the Local Area Coverage (LAC) recording periods can be visually verified using CATLAC. The schedule can also be verified by examining record and error files written by the scheduler.

Woodward, Robert H.↗
NASA NTRSDate not supplied

Predicting Melt Properties Using Atomistic Simulations With A Highly Accurate Physically Informed Neural Network Interatomic Potential

The use of a recently developed machine learning (ML) interatomic potential for molecular dynamics simulations of aluminum melt properties will be presented. Such properties are critical for process modeling in additive manufacturing, including the melt pool size, solidification, and formation of solidification microstructures. Direct first-principles modeling of these processes is computationally prohibitive whereas simulations employing ML potentials combine the high accuracy of quantum-mechanical methods with high computational speeds. The physically-informed neural network (PINN) method used herein, integrates a high-dimensional regression implemented by an artificial neural network with a physics-based bond-order interatomic potential. PINN potentials can accurately reproduce many properties of aluminum in both crystalline-solid and liquid phases. We examine the accuracy of a PINN Al potential in predicting the density, self-diffusivity, viscosity, and the tension of the liquid surface and liquid-solid interfaces. Comparison with experimental data and ab initio molecular dynamics calculations shows very good agreement for all properties tested.

molecular dynamics↗
NASA NTRS1981

Sources of atmospheric methane - Measurements in rice paddies and a discussion

Field measurements of methane fluxes from rice paddies, fresh water lakes, and saltwater marshes have been made to infer estimates of the size of these sources of atmospheric methane. The rice-paddy measurements, the first of their kind, show that the principal means of methane escape is through the plants themselves as opposed to transport across the water-air interface via bubbles or molecular diffusion. Nitrogen-fertilized plants release much more methane than unfertilized plants but even these measured rates are only one fourth as large as those inferred earlier by Koyama (1963, 1964) and on which all global extrapolations have been based to date. Measured methane fluxes from lakes and marshes are also compared to similar earlier data and it is found that extant data and flux-measurement methods are insufficient for reliable global extrapolations.

Cicerone, R. J.↗
NASA NTRS1981

Fabrication of titanium multiwall Thermal Protection System (TPS) curved panel

A method is described for fabricating curved T1-6Al-4V multiwall panels having a single radius of curvature of 30.5 cm (12 inches). A panel 30.5 x 30.5 x 1.7 cm (12 x 12 x .688 inch) having 4 dimpled sheets, 5 undimpled sheets, and stepped side closures was designed, fabricated and delivered for evaluation. Tool design for skin and dimpled sheet forming are described as well as layup and liquid interface diffusion bonding.

Blair, W.↗
NASA NTRS2001

Remobilizing the Interface of Thermocapillary Driven Bubbles Retarded By the Adsorption of a Surfactant Impurity on the Bubble Surface

Thermocapillary migration is a method for moving bubbles in space in the absence of buoyancy. A temperature gradient is the continuous phase in which a bubble is situated, and the applied gradient impressed on the bubble surface causes one pole of the drop to be cooler than the opposite pole. As the surface tension is a decreasing function of temperature, the cooler pole pulls at the warmer pole, creating a flow that propels the bubble in the direction of the warmer fluid. A major impediment to the practical use of thermocapillary to direct the movement of bubbles in space is the fact that surfactant impurities, which are unavoidably present in the continuous phase, can significantly reduce the migration velocity. A surfactant impurity adsorbed onto the bubble interface is swept to the trailing end of the bubble. When bulk concentrations are low (which is the case with an impurity), diffusion of surfactant to the front end is slow relative to convection, and surfactant collects at the back end of the bubble. Collection at the back lowers the surface tension relative to the front end setting up a reverse tension gradient. (This can also be the case if kinetic desorption of surfactant at the back end of the bubble is much slower than convection.) For buoyancy driven bubble motions in the absence of a thermocapillarity, the tension gradient opposes the surface flow, and reduces the surface and terminal velocities (the interface becomes more solid-like and bubbles translate as solid particles). When thermocapillary forces are present, the reverse tension gradient set up by the surfactant accumulation reduces the temperature-induced tension gradient, and can decrease to near zero the bubble's thermocapillary velocity. The objective of our research is to develop a method for enhancing the thermocapillary migration of bubbles which have be retarded by the adsorption onto the bubble surface of a surfactant impurity. Our remobilization theory proposes to use surfactant molecules which kinetically rapidly exchange between the bulk and the surface and are at high bulk concentrations. Because the remobilizing surfactant is present at much higher concentrations than the impurity, it adsorbs to the bubble surface much faster than the impurity when the bubble is formed, and thereby prevents the impurity from adsorbing onto the surface. In addition, the rapid kinetic exchange and high bulk concentration maintain a saturated surface with uniform surface concentrations. This prevents retarding surface tension gradients and keeps the thermocapillary velocity high. In our reports over the first 2 years, we presented numerical simulations of the bubble motion and surfactant transport which verified theoretically the concept of remobilization, and the development of an apparatus to track and measure the velocity of rising bubbles in a glycerol/water surfactant solution. This year, we detail experimental observations of remobilization. Two polyethylene oxide surfactants were studied, C12E6 (CH3(CH2)11(OCH2)6OH) and C10E8 (CH3(CH2)4(OCH2CH2)8OH). Measurements of the kinetic exchange for these surfactants show that the one with the longer hydrophobe chain C12E6 has a lower rate of kinetic exchange. In addition, this surfactant is much less soluble in the glycerol/water mixture because of the shorter ethoxylate chain. As a result, we found that C12E6 had only a very limited ability to remobilize rising bubbles because of the limited kinetic exchange and reduced solubility. However, C10E8, with its higher solubility and more rapid exchange was found to dramatically remobilize rising bubbles. We also compared our theoretical calculations to the experimental measurements of velocity for both the non-remobilizing and remobilizing surfactants and found excellent agreement. We further observed that for C10E8 at high concentrations, which exceeded the critical micelle concentrations, additional remobilization was measured. In this case the rapid exchange of monomer between micelle and surfactant provides an additional mechanism for maintaining a uniform surface concentrations.

Palaparthi, Ravi↗
NASA NTRS2003

Measuring Diffusion of Liquids by Common-Path Interferometry

A method of observing the interdiffusion of a pair of miscible liquids is based on the use of a common-path interferometer (CPI) to measure the spatially varying gradient of the index refraction in the interfacial region in which the interdiffusion takes place. Assuming that the indices of refraction of the two liquids are different and that the gradient of the index of refraction of the liquid is proportional to the gradient in the relative concentrations of either liquid, the diffusivity of the pair of liquids can be calculated from the temporal variation of the spatial variation of the index of refraction. This method yields robust measurements and does not require precise knowledge of the indices of refraction of the pure liquids. Moreover, the CPI instrumentation is compact and is optomechanically robust by virtue of its common- path design. The two liquids are placed in a transparent rectangular parallelepiped test cell. Initially, the interface between the liquids is a horizontal plane, above which lies pure liquid 2 (the less-dense liquid) and below which lies pure liquid 1 (the denser liquid). The subsequent interdiffusion of the liquids gives rise to a gradient of concentration and a corresponding gradient of the index of refraction in a mixing layer. For the purpose of observing the interdiffusion, the test cell is placed in the test section of the CPI, in which a collimated, polarized beam of light from a low-power laser is projected horizontally through a region that contains the mixing layer.

Rashidnia, Nasser↗
NASA NTRS1996

Magneto-Hydrodynamic Damping of Convection During Vertical Bridgman-Stockbarger Growth of HgCdTe

In order to quantify the effects of convection on segregation, Hg(0.8)Cd(0.2)Te crystals were grown by the vertical Bridgman-Stockbarger method in the presence of an applied axial magnetic field of 50 kG. The influence of convection, by magneto-hydrodynamic damping, on mass transfer in the melt and segregation at the solid-liquid interface was investigated by measuring the axial and radial compositional variations in the grown samples. The reduction of convective mixing in the melt through the application of the magnetic field is found to decrease radial segregation to the diffusion-limited regime. It was also found that the suppression of the convective cell near the solid-liquid interface results in an increase in the slope of the diffusion-controlled solute boundary layer, which can lead to constitutional supercooling.

Watring, D. A.↗
NASA NTRS1976

Fabrication process development of SiC/superalloy composite sheet for exhaust system components

A chemical compatibility study was conducted between SiC filament and the following P/M matrix alloys: Waspaloy, Hastelloy-X, NiCrAlY, Ha-188, S-57, FeCrAlY, and Incoloy 800. None of the couples demonstrated sufficient chemical compatibility to withstand the minimum HIP consolidation temperatures (996 C) or intended application temperature of the composite (982 C). However, Waspaloy, Haynes 188, and Hastelloy-X were the least reactive with SiC of the candidate alloys. Chemical vapor deposited tungsten was shown to be an effective diffusion barrier between the superalloy matrix and SiC filament providing a defect-free coating of sufficient thickness. However, the coating breaks down when the tungsten is converted into intermetallic compounds by interdiffusion with matrix constituents. Waspaloy was demonstrated to be the most effective matrix alloy candidate in contact with the CVD tungsten barrier because of its relatively low growth rate constant of the intermediate compound and the lack of formation of Kirkendall voids at the matrix-barrier interface. Fabrication methods were developed for producing panels of uniaxial and angle ply composites utilizing CVD tungsten coated filament.

Cornie, J. A.↗
NASA NTRS1998

Numerical Modeling of Physical Vapor Transport in Contactless Crystal Growth Geometry

Growth from the vapor under conditions of limited contact with the walls of the growth ampoule is beneficial for the quality of the growing crystal due to reduced stress and contamination which may be caused by interactions with the growth container. The technique may be of a particular interest for studies on crystal growth under microgravity conditions: elimination of some factors affecting the crystal quality may make interpretation of space-conducted processes more conclusive and meaningful. For that reason, and as a part of our continuing studies on 'contactless' growth technique, we have developed a computational model of crystal growth process in such system. The theoretical model was built, and simulations were performed using the commercial computational fluid dynamics code, (CFD) ACE. The code uses an implicit finite volume formulation with a gray discrete ordinate method radiation model which accounts for the diffuse absorption and reflection of radiation throughout the furnace. The three-dimensional model computes the heat transfer through the crystal, quartz, and gas both inside and outside the ampoule, and mass transport from the source to the crystal and the sink. The heat transport mechanisms by conduction, natural convection, and radiation, and mass transport by diffusion and convection are modeled simultaneously and include the heat of the phase transition at the solid-vapor interfaces. As the thermal boundary condition, temperature profile along the walls of the furnace is used. For different thermal profiles and furnace and ampoule dimensions, the crystal growth rate and development of the crystal-vapor and source-vapor interfaces (change of the interface shape and location with time) are obtained. Super/under-saturation in the ampoule is determined and critical factors determining the 'contactless' growth conditions are identified and discussed. The relative importance of the ampoule dimensions and geometry, the furnace dimensions and its temperature, and the properties of the grown material are analyzed. The results of the simulations are compared with related experimental results on growth of CdTe, CdZnTe, ZnTe, PbTe, and PbSnTe crystals by this technique.

Palosz, W.↗
NASA NTRS1982

Finite element analysis of the effect of a non-planar solid-liquid interface on the lateral solute segregation during unidirectional solidification

The effect of solid-liquid interface shape on lateral solute segregation during steady-state unidirectional solidification of a binary mixture is calculated under the assumption of no convection in the liquid. A finite element technique is employed to compute the concentration field in the liquid and the lateral segregation in the solid with a curved boundary between the liquid and solid phases. The computational model is constructed assuming knowledge of the solid-liquid interface shape; no attempt is made to relate this shape to the thermal field. The influence of interface curvature on the lateral compositional variation is investigated over a range of system parameters including diffusivity, growth speed, distribution coefficient, and geometric factors of the system. In the limiting case of a slightly nonplanar interface, numerical results from the finite element technique are in good agreement with the analytical solutions of Coriell and Sekerka obtained by using linear theory. For the general case of highly non-planar interface shapes, the linear theory fails and the concentration field in the liquid as well as the lateral solute segregation in the solid can be calculated by using the finite element method.

Carlson, F. M.↗