Numerical modelling of transient liquid phase bonding and other diffusion controlled phase changes

Diffusion in material of inhomogeneous composition can induce phase changes, even at a constant temperature. A transient liquid phase (TLP), in which a liquid layer is formed and subsequently solidifies, is one example of such an isothermal phase change. This phenomenon is exploited industrially in TLP bonding and sintering processes. Successful processing requires an understanding of the behaviour of the transient liquid layer in terms of both diffusion-controlled phase boundary migration and capillarity-driven flow. In this paper, a numerical model is presented for the simulation of diffusion-controlled dissolution and solidification in one dimension. The width of a liquid layer and time to solidification are studied for various bonding conditions. A novel approach is proposed, which generates results of a high precision even with coarse meshes and high interface velocities. The model is validated using experimental data from a variety of systems, including solid/solid diffusion couples.     

Numerical treatment of diffusional phase transformation through fully implicit control volume method

Abstract

Solid state, diffusion controlled phase transformation kinetics with a moving boundary has been quantified using a fully implicit, fixed grid, finite difference method based on the control volume approach. In a departure from the usual modeling techniques for phase change problems, the region undergoing phase change has also been considered as a control volume. A new equation for the interface flux balance has been obtained that minimises the mass balance error that normally plagues the numerical solution of moving boundary problems. The model has been validated with the calculated phase thickness based on binary equilibrium diagram and available experimental data in the literature for the Cu–Zn system and a good match has been obtained. The results obtained by the present formulation are compared with those obtained from the other models. In addition to the improved accuracy of the prediction because of elimination of the mass balance error, the proposed method has the usual advantages of a fully implicit scheme.     

A finite element model for liquid phase electroepitaxy

A finite element numerical simulation model for the liquid phase electroepitaxial growth process of gallium arsenide is presented. The basic equations obtained from the fundamental principles of electrodynamics of continua, the constitutive equations for the liquid and solid phases derived from a rational thermodynamic theory, and the associated interface and boundary conditions are presented for a two-dimensional axisymmetric growth cell configuration. The field equations are solved numerically by an adaptive finite element procedure. The effect of moving interfaces is taken into account. Numerical simulations are carried out for different convection levels by changing the value of the gravitational constant. Results show that convection has significant effect on the growth process under normal gravity conditions and results in thickness non-uniformity of the grown layers. The thickness non-uniformity leads to curved interfaces of growth and dissolution, which enhance convection.     

Boundary integral equation solution of moving boundary phase change problems

Boundary integral equation methods are presented for the solution of some two-dimensional phase change problems. Convection may enter through boundary conditions, but cannot be considered within phase boundaries. A general formulation based on space-time Green's functions is developed using the complete heat equation, followed by a simpler formulation using the Laplace equation. The latter is pursued and applied in detail. An elementary, noniterative system is constructed, featuring linear interpolation over elements on a polygonal boundary. Nodal values of the temperature gradient normal to a phase change boundary are produced directly in the numerical solution. The system performs well against basic analytical solutions, using these values in the interphase jump condition, with the simplest formulation of the surface normal at boundary vertices. Because the discretized surface changes automatically to fit the scale of the problem, the method appears to offer many of the advantages of moving mesh finite element methods. However, it only requires the manipulation of a surface mesh and solution for surface variables. In some applications, coarse meshes and very large time steps may be used, relative to those which would be required by fixed grid domain methods. Computations are also compared to original lab data, describing two-dimensional soil freezing with a time-dependent boundary condition. Agreement between simulated and measured histories is good.     

An adaptive fixed mesh method for modelling and simulating of unsteady two‐phase flows with sharp physical interfaces

We present in this paper a new computational method for simulation of two‐phase flow problems with moving boundaries and sharp physical interfaces. An adaptive interface‐capturing technique (ICT) of the Eulerian type is developed for capturing the motion of the interfaces (free surfaces) in an unsteady flow state. The adaptive method is mainly based on the relative boundary conditions of the zero pressure head, at which the interface is corresponding to a free surface boundary. The definition of the free surface boundary condition is used as a marker for identifying the position of the interface (free surface) in the two‐phase flow problems. An initial‐value‐problem (IVP) partial differential equation (PDE) is derived from the dynamic conditions of the interface, and it is designed to govern the motion of the interface in time. In this adaptive technique, the Navier–Stokes equations written for two incompressible fluids together with the IVP are solved numerically over the flow domain. An adaptive mass conservation algorithm is constructed to govern the continuum of the fluid. The finite element method (FEM) is used for the spatial discretization and a fully coupled implicit time integration method is applied for the advancement in time. FE‐stabilization techniques are added to the standard formulation of the discretization, which possess good stability and accuracy properties for the numerical solution. The adaptive technique is tested in simulation of some numerical examples. With the test problems presented here, we demonstrated that the adaptive technique is a simple tool for modelling and computation of complex motion of sharp physical interfaces in convection–advection‐dominated flow problems. We also demonstrated that the IVP and the evolution of the interface function are coupled explicitly and implicitly to the system of the computed unknowns in the flow domain. Copyright © 2005 John Wiley & Sons, Ltd.     

A cell-by-cell thermally driven mushy cell tracking algorithm for phase-change problems

The finite volume based numerical approach is used to simulate phase-change processes including natural convection. This approach is based on a cell-by-cell, thermally driven mushy cell tracking equation, developed in Part I [20], to trace the front at which phase-change occurs. A mushy cell is a specialized cell where the interface between liquid and solid phases is located. In this paper, the mushy cell tracking equation and the associated boundary condition around the mushy cells are derived in a general manner and shown to have the same form as that used in Part I. The SIMPLE algorithm is adopted to solve the flow, including pressure field, as well, in the liquid phase and a conjugate gradient method is used when solving the system of discretized equations. To reduce computational time, an acceleration technique, based on a justified quasi-steady state assumption, is adopted. The proposed numerical method is applied to simulate the solidification and melting of Tin with natural convection. The numerical predictions are compared well with the available experimental data and previously published numerical results. Specifically, these comparisons demonstrate that the proposed methodology is capable of predicating the location of moving fronts and the temperature distributions for phase-change processes with natural convection.     

An analysis of solidification problems by the boundary element method

The problem of interest in this paper is the calculation of the motion of the solid–liquid interface and the time-dependent temperature field during solidification of a pure metal. An iterative implicit algorithm has been developed for this purpose using the boundary element method (BEM) with time-dependent Green's functions and convolution integrals. The BEM approach requires discretization of only the surface of the solidifying body. Thus, the numerical method closely follows the physics of the problems and is intuitively very appealing. The formulation and the numerical scheme presented here are general and can be applied to a broad range of moving boundary problems. Emphasis is given to two-dimensional problems. Comparison with existing semi-analytical solutions and other numerical solutions from the literature reveals that the method is fast, accurate and without major time step limitations.     

A fixed grid method for the numerical solution of phase change problems

A fixed grid method using an updated iterative implicit scheme is developed to solve one-dimensional phase change problems. The temperature field is deduced from the resolution of the governing equations whose discretization takes into account the discontinuous variation of the temperature derivative at the phase change front. At each iteration an updated position of the moving front is found from the resolution of the energy conservation at the solid-liquid interface. The accuracy of the proposed numerical method has been checked on three test problems.     

Continuously deforming finite elements for the solution of parabolic problems,with and without phase change

A number of transport problems are complicated by the presence of physically important transition zones where quantities exhibit steep gradients and special numerical care is required. When the location of such a transition zone changes as the solution evolves through time, use of a deforming numerical mesh is appropriate in order to preserve the proper numerical features both within the transition zone and at its boundaries. A general finite element solution method is described wherein the elements are allowed to deform continuously, and the effects of this deformation are accounted for exactly. The method is based on the Galerkin approximation in space, and uses finite difference approximations for the time derivatives. In the absence of element deformation, the method reduces to the conventional Galerkin formulation. The method is applied to the two-phase Stefan problem associated with the melting and solidification of A substance. The interface between the solid and liquid phase form an internal moving boundary, and latent heat effects are accounted for in the associated boundary condition. By allowing continuous mesh deformation, as dictated by this boundary condition, the moving boundary always lies on element boundaries. This circumvents the difficulties inherent in interpolation of parameters and dependent variables across regions where those quantities change abruptly. Basis functions based on Hermite polynomials are used, to allow exact specification of the flux-latent heat balance condition at the phase boundary. Analytic solutions for special cases provide tests of the method.     

Treatment of contact separation in Eulerian high‐speed multimaterial dynamic simulations

A frictionless contact separation treatment in a sharp‐interface Eulerian framework is presented to handle the general situation of high‐speed impact and separation of materials. The algorithm has been developed for an established Eulerian‐based Cartesian grid multimaterial flow code in which the interfaces are tracked in a sharp manner using a standard narrow‐band level set approach. Boundary conditions have been applied using a modified ghost fluid method for elasto‐plastic materials. The sharp‐interface treatment maintains the distinct interacting interfaces without smearing the contact zone while also removing the difficulties associated with Lagrangian moving mesh entities in contact‐separation situations. The algorithm has been tested and verified against experimental and numerical results for three different problems in the high strain rate regime, which involve contact, separation and sliding of materials. Copyright © 2014 John Wiley & Sons, Ltd.     

Finite Element Simulation of Low Velocity Impact Damage on an Aeronautical Carbon Composite Structure

Low velocity barely visible impact damage (BVID) in laminated carbon composite structures has a major importance for aeronautical industries. This contribution leads with the development of finite element models to simulate the initiation and the propagation of internal damage inside a carbon composite structure due by a low velocity impact. Composite plates made from liquid resin infusion process (LRI) have been subjected to low energy impacts (around 25 J) using a drop weight machine. In the experimental procedure, the internal damage is evaluated using an infrared thermographic camera while the indentation depth of the face is measured by optical measurement technique. In a first time we developed a robust model using homogenised shells based on degenerated tri-dimensional brick elements and in a second time we decided to modelize the whole stacking sequence of homogeneous layers and cohesive interlaminar interfaces in order to compare and validate the obtained results. Both layer and interface damage initiation and propagation models based on the Hashin and the Benzeggagh-Kenane criteria have been used for the numerical simulations. Comparison of numerical results and experiments has shown the accuracy of the proposed models.     

Modeling and Simulation of the Microstructure Evolution during a Cooling of Immiscible Alloys in the Miscibility Gap

The microstructure development during a cooling period of alloys being immiscible in the liquid state such as Al-Pb or Al-Bi has gained renewed scientific and technical interest during the last decades.Experiments have been performed to investigate the phase transformation kinetics in the liquid miscibility gap and numerical models have been developed to simulate and analyze the solidification process.The recently developed computational modeling techniques can,to some extent,be applied to describe the decomposition,the spatial phase separation and the microstructure evolution during a cooling period of an immiscible alloy through the miscibility gap.This article overviews the researches in this field.     

A numerical simulation for two-dimensional moving boundary problems with a mushy zone

For materials such as alloy, organic phase-change materials and many others, the change of phases may take place over a temperature range. This leads to phase-change problems with the mushy zone in which the solid and liquid phases coexist. The present study introduces a numerical method combining the Laplace transform technique and the control volume method to solve two-dimensional phase-change problems with the mushy zone. The hybrid numerical method involves the control volume formulation for the space domain and the Laplace transform technique for the time domain. The Taylor's series approximation is applied to linearize nonlinear terms in the governing equation. The transfinite mapping method is used to generate control-volume meshes in each region. The growth of the mushy zone is unknown a priori and is predicted by using the least-square iteration scheme. It will be found that the present hybrid numerical method can be efficiently applied to solve two-dimensional phase-change problems with a mushy zone.     

A continuation method for rigid‐cohesive fracture in a discontinuous Galerkin finite element setting

An energy minimization formulation of initially rigid cohesive fracture is introduced within a discontinuous Galerkin finite element setting with Nitsche flux. The finite element discretization is directly applied to an energy functional, whose term representing the energy stored in the interfaces is nondifferentiable at the origin. Unlike finite element implementations of extrinsic cohesive models that do not operate directly on the energy potential, activation of interfaces happens automatically when a certain level of stress encoded in the interface potential is reached. Thus, numerical issues associated with an external activation criterion observed in the previous literature are effectively avoided. Use of the Nitsche flux avoids the introduction of Lagrange multipliers as additional unknowns. Implicit time stepping is performed using the Newmark scheme, for which a dynamic potential is developed to properly incorporate momentum. A continuation strategy is employed for the treatment of nondifferentiability and the resulting sequence of smooth nonconvex problems is solved using the trust region minimization algorithm. Robustness of the proposed method and its capabilities in modeling quasistatic and dynamic problems are shown through several numerical examples.     

Numerical prediction of elasto-plastic behaviour of interpenetrating phase composites by EFGM

Interpenetrating Phase Composites (IPCs) can be defined as multiphase materials in which each phase is three-dimensionally interconnected throughout the structure. No phase can be distinguished from the other based on the states of isolation and continuity; however both the phases contribute to the strengthening and improvement of the composite. The tensile and compressive yield and ultimate strengths of IPCs are much higher than a similar particulate composite due to their interpenetrating structure. This behaviour has been numerically simulated using element free Galerkin method. Ramberg–Osgood material model has been used to model the elasto-plastic behaviour of the composite. A progressive damage model has been used to simulate the failure mechanism of each phase. Three types of models have been proposed based on the treatment of the interface. The ultimate strength and the yield strength of IPC are obtained. The ultimate strength and the yield strength of the IPC depend largely on the properties, volume fraction and interpenetration of the constituent phases. The results of the present simulations are found in good agreement with the experimental results.     

Numerical simulation of macrosegregation of indium phase in rapidly solidified Al-In hypermonotectic sheets

Abstract

Numerical simulation and experimental methods have been adopted to study the macrosegregation of indium phase in rapidly solidified Al-In hypermonotectic sheets. Based on the particle separation method, a numerical model reflecting the actual solidification process has been developed to simulate the precipitating, coarsening, and moving process of secondary phase droplets under conditions of variable temperature, thermophysical properties, and supersaturation. In this model, the droplets are divided into various size classes according to their radius and a stochastic collection equation is used to deal with collisions, so the length of computation is reduced sharply, which makes application of the particle separation method possible. Simulation and experimental results show that indium particle distribution is uniform when the specimen thickness is less than ~4 mm. With increasing specimen thickness above 4 mm, indium particle distribution becomes more and more inhomogeneous. Good agreement between calculated and experimental results verifies that the model established in the present work has adequate predictive ability.     

Numerical modeling and simulation of a diffusion-controlled liquid–solid phase change in polycrystalline solids

A fully implicit two-dimensional moving-mesh finite element simulation model was developed to study the influence of grain boundaries in polycrystalline solids on diffusion-controlled liquid–solid transition during transient liquid phase (TLP) bonding. The new model, which was developed without the non-trivial symmetry assumption in existing numerical models for the process, was found to conserve solute and its calculated solutions were unconditionally stable and in good agreement with experimental results. Contrary to the assumption that increased grain boundary diffusion coefficient would significantly accelerate the rate of liquid–solid interface migration, numerical calculations and experimental verification showed that enhanced intergranular diffusivity had a minimal effect on the time required to achieve complete diffusion-induced solidification in cast superalloys. The results indicate that reducing the number of grain boundaries in structural alloys through directional solidification casting techniques did not constitute a disincentive to efficient application of TLP bonding to this class of materials.     

Three phase flow dynamics in tumor growth

Existing tumor models generally consider only a single pressure for all the cell phases. Here, a three-fluid model originally proposed by the authors is further developed to allow for different pressures in the host cells (HC), the tumor cells (TC) and the interstitial fluid (IF) phases. Unlike traditional mixture theory models, this model developed within the thermodynamically constrained averaging theory contains all the necessary interfaces. Appropriate constitutive relationships for the pressure difference among the three fluid phases are introduced with respect to their relative wettability and fluid–fluid interfacial tensions, resulting in a more realistic modeling of cell adhesion and invasion. Five different tumor cases are studied by changing the interfacial tension between the three liquid phases, adhesion and dynamic viscosity. Since these parameters govern the relative velocities of the fluid phases and the adhesion of the phases to the extracellular matrix significant changes in tumor growth are observed. High interfacial tensions at the TC–IF and TC–HC interface support the lateral displacement of the healthy tissue in favor of a rapid growth of the malignant mass, with a relevant amount of HC which cannot be pushed out by TC and remain in place. On the other hand, lower TC–IF and TC–HC interfacial tensions tend to originate a more compact and dense tumor mass with a slower growth rate of the overall size. This novel computational model emphasizes the importance of characterizing the TC–HC interfacial properties to properly predict the temporal and spatial pattern evolution of tumor.     

A model of phase stability,microstructure and properties during leaching of portland cement binders

A new model of 3D cement paste microstructure development is described and used to simulate the influence of leaching on hydrated cement pastes. In contrast to recent leaching models that have used empirical rules for phase dissolution, this model uses continual thermodynamic speciation and phase stability calculations to guide the microstructural changes that happen throughout hydration and subsequent exposure to low-pH solutions. This novel aspect of the model enables it to predict not only the well-known phase instability of calcium hydroxide at the onset of leaching, but also the detailed compositional and volumetric changes of C–S–H gel and other calcium, aluminate, and sulfate phases. Besides tracking the compositional and microstructural changes, we use the evolving microstructure as input to calculate changes in the relative diffusivity and effective Young’s modulus of the binder using established finite difference and finite element models. The results are broadly consistent with previous experimental and modeling investigations of leaching. In particular, the leaching process can be roughly divided into initial, intermediate, and final stages, each of which has distinct degradation characteristics and consequences for mechanical and transport properties. The thermodynamic basis of the model makes it readily extensible to simulate a wide range of cementitious materials and degradation phenomena, so we discuss its potential as a virtual microprobe for use with continuum-scale service life models of concrete elements.     

A coupled BEM‐flexibility force method for bending analysis of internally supported plates

This paper presents a new method for the analysis of plates in bending with internal supports. The proposed method can be regarded as an extension of the well‐known force method (the flexibility matrix method) in the matrix analysis of structures. The solution is performed through two phases: the released plate phase, in which the plate is released from all internal supports and solved using the Boundary Element Method (BEM). The effect of internal supports is considered in the second phase, where a series of unit virtual loads is placed instead of the unknown redundant reactions at internal supports. The flexibility matrix is formed and compatibility of deformations at the locations of internal supports is satisfied. Hence, the corresponding system of equations is solved for the unknown redundant forces at internal supports. The final solution of the problem consists of the summation of two phases: the released plate phase and the cases of virtual unit loads phase. An efficient solution algorithm is developed to solve both phases simultaneously. The main advantages of the present formulation are: (1) the present formulation increases the versatility of the BEM as it allows the re‐usability of standard BEM codes for solution of plates in bending to be used in solving problems having internal supports, with even no modifications; and (2) the two solution phases are completely uncoupled; therefore it is easy to trace behaviour of the plate due to failure of one or more of the internal supports without re‐analysis. Several numerical examples are analysed. The results are compared to those of analytical and finite element models to demonstrate the accuracy and the validity of the present formulation. The present formulation is used also to study the differences between the finite element and boundary element modelling for building slabs. Copyright © 2002 John Wiley & Sons, Ltd.