Sharp Interface Formulation Research Articles

A level set approach to modelling diffusional phase transformations under finite strains with application to the formation of [formula omitted]

This paper presents a sharp interface formulation for modelling diffusional phase transformations. Grain boundary motion is, in accordance with diffusional phase transformation kinetics, determined by the amount of flux towards the interface and is formulated in a level set framework. This approach enables a computational efficiency that can be expected to be higher than what can be achieved with conventional phase field methods. Compatibility of the interfaces is obtained through an interface reconstruction process, in which the locations of triple junction points are also determined. To ensure local equilibrium and a continuous chemical potential across the interfaces, the chemical composition is prescribed at the phase interfaces. The presented model is used to study the growth of the intermetallic compound (IMC) Cu6Sn5 for a system with Sn electroplated on a Cu substrate. A finite strain formulation is incorporated into the model to investigate the effects of the volume change resulting from the IMC formation. In this formulation, the Cu and Sn phases are allowed to deform plastically. The numerical simulations demonstrate IMC growth rates in agreement with experimental measurements. Moreover, the IMC evolves into a scallop-like morphology, consistent with experimental observations.

Simplified conservative discretization of the Cahn-Hilliard-Navier-Stokes equations

In this paper we construct a novel discretization of the Cahn-Hilliard equation coupled with the Navier-Stokes equations. The Cahn-Hilliard equation models the separation of a binary mixture. We construct a very simple time integration scheme for simulating the Cahn-Hilliard equation, which is based on splitting the fourth-order equation into two second-order Helmholtz equations. We combine the Cahn-Hilliard equation with the Navier-Stokes equations to simulate phase separation in a two-phase fluid flow in two dimensions. The scheme conserves mass and momentum and exhibits consistency between mass and momentum, allowing it to be used with large density ratios. We introduce a novel discretization of the surface tension force from the phase-field variable that has finite support around the transition region. The model has a parameter that allows it to transition from a smoothed continuum surface force to a fully sharp interface formulation. We show that our method achieves second-order accuracy, and we compare our method to previous work in a variety of experiments.

A structure-preserving finite element method for the multi-phase Mullins–Sekerka problem with triple junctions

We consider a sharp interface formulation for the multi-phase Mullins–Sekerka flow. The flow is characterized by a network of curves evolving such that the total surface energy of the curves is reduced, while the areas of the enclosed phases are conserved. Making use of a variational formulation, we introduce a fully discrete finite element method. Our discretization features a parametric approximation of the moving interfaces that is independent of the discretization used for the equations in the bulk. The scheme can be shown to be unconditionally stable and to satisfy an exact volume conservation property. Moreover, an inherent tangential velocity for the vertices on the discrete curves leads to asymptotically equidistributed vertices, meaning no remeshing is necessary in practice. Several numerical examples, including a convergence experiment for the three-phase Mullins–Sekerka flow, demonstrate the capabilities of the introduced method.

Analysis of a diffuse interface method for the Stokes-Darcy coupled problem

We consider the interaction between a free flowing fluid and a porous medium flow, where the free flowing fluid is described using the time dependent Stokes equations, and the porous medium flow is described using Darcy’s law in the primal formulation. To solve this problem numerically, we use a diffuse interface approach, where the weak form of the coupled problem is written on an extended domain which contains both Stokes and Darcy regions. This is achieved using a phase-field function which equals one in the Stokes region and zero in the Darcy region, and smoothly transitions between these two values on a diffuse region of width (ϵ) around the Stokes-Darcy interface. We prove convergence of the diffuse interface formulation to the standard, sharp interface formulation, and derive rates of convergence. This is performed by deriving a priori error estimates for discretizations of the diffuse interface method, and by analyzing the modeling error of the diffuse interface approach at the continuous level. The convergence rates are also shown computationally in a numerical example.

Analysis and modeling of heat and mass transfer and adsorption isotherms of the aqueous extracts vacuum contact drying

Vacuum contact drying is the most merit drying technology to ensure the quality and throughput of traditional Chinese medicine aqueous extracts. To study the heat and mass transfer of colloidal semi-solid porous media during vacuum contact drying, the sharp interface formulation was established to simulate the thin layer (4 mm) Lonicera japonica Thunb. aqueous extracts vacuum drying at high-temperature (110 and 120 ℃) and low-pressure (10, 15 and 20 kPa). And mathematical models were used to investigated drying kinetics and adsorption isotherms. Results show that the sharp interface model can predict final temperature and moisture ratio with the error is less than 20 %. The models of viscosity variation, specific heat capacity, heat capacity, effective thermal conductivity and heat transfer coefficient of boiling under different conditions were investigated to provide reference for subsequent research. When the vacuum drying superheat is between 50 ℃ and 75 ℃, the boiling heat transfer coefficient is among 6 to 9 W/ (m2·K). The Page is the most suitable semi-empirical model to describe the water migration law of the aqueous extracts. The adsorption isotherms of the extract powder were investigated by experiment and the Peleg is the most suitable model.

A sharp-interface mesoscopic model for dendritic growth

The grain envelope model (GEM) describes the growth of envelopes of dendritic crystal grains during solidification. Numerically the growing envelopes are usually tracked using an interface capturing method employing a phase field equation on a fixed grid. Such an approach describes the envelope as a diffuse interface, which can lead to numerical artefacts that are possibly detrimental. In this work, we present a sharp-interface formulation of the GEM that eliminates such artefacts and can thus track the envelope with high accuracy. The new formulation uses an adaptive meshless discretization method to solve the diffusion in the liquid around the grains. We use the ability of the meshless method to operate on scattered nodes to accurately describe the interface, i.e., the envelope. The proposed algorithm combines parametric surface reconstruction, meshless discretization of parametric surfaces, global solution construction procedure and partial differential operator approximation using monomials as basis functions. The approach is demonstrated on a two-dimensional h-adaptive solution of diffusive growth of dendrites and assessed by comparison to a conventional diffuse-interface fixed-grid GEM.

On a marching level‐set method for extended discontinuous Galerkin methods for incompressible two‐phase flows: Application to two‐dimensional settings

AbstractIn this work a solver for two‐dimensional, instationary two‐phase flows on the basis of the extended discontinuous Galerkin (extended DG/XDG) method is presented. The XDG method adapts the approximation space conformal to the position of the interface. This allows a subcell accurate representation of the incompressible Navier‐Stokes equations in their sharp interface formulation. The interface is described as the zero set of a signed‐distance level‐set function and discretized by a standard DG method. For the interface, resp. level‐set, evolution an extension velocity field is used and a two‐staged algorithm is presented for its construction on a narrow‐band. On the cut‐cells a monolithic elliptic extension velocity method is adapted and a fast‐marching procedure on the neighboring cells. The spatial discretization is based on a symmetric interior penalty method and for the temporal discretization a moving interface approach is adapted. A cell agglomeration technique is utilized for handling small cut‐cells and topology changes during the interface motion. The method is validated against a wide range of typical two‐phase surface tension driven flow phenomena in a 2D setting including capillary waves, an oscillating droplet and the rising bubble benchmark.

Scarf for Lifshitz

Polarization of a vacuum as well as of dispersive and dissipative dielectric media with piece-wise and smooth inhomogeneities is studied with the goal to clarify the question of renormalizability of diverging electromagnetic stress-energy tensor. First, the stress tensor is computed with the Lifshitz approach to London forces in the non-retarded limit, which after the substraction of the leading free space ultraviolet divergencies still retains the divergencies associated with the presence of sharp boundaries between piece-wise inhomogeneities. We call these contributions finite because they become renormalized after a sharp interface is replaced with a dielectric permittivity changing according to a smooth function of spatial coordinates. In addition, such a smoothed out interface exhibits new subleading ultraviolet divergencies that appear due to its internal structure. To systematically deal with the polarization of inhomogeneous media, the Hadamard expansion is applied to single out both finite and subleading contributions and to unequivocally demonstrate incomplete renormalizability of the Lifshitz theory. The above approach also allows us to reveal the nature of surface tension, which proves to be purely quantum mechanical. The deduced theory of surface tension and its calculations for real dielectric media are favorably compared to the available experimental data. While the sharp interface limit recovers the classical boundary conditions for the electric field and uncovers the origin of the apparent local divergencies of the renormalized stresses in the sharp interface formulation previously pointed out in the literature, the problem of surface tension proves to be of a distinguished limit type because the sharp interface formulation loses the information about the internal structure of an interface and hence cannot explain the origin of surface tension.

The extended discontinuous Galerkin method adapted for moving contact line problems via the generalized Navier boundary condition

AbstractIn this work, an extended discontinuous Galerkin (extended DG/XDG also called unfitted DG) solver for two‐dimensional flow problems exhibiting moving contact lines is presented. The generalized Navier boundary condition is employed within the XDG discretization for the handling of the moving contact lines. The spatial discretization is based on a symmetric interior penalty method and the numerical treatment of the surface tension force is done via the Laplace–Beltrami formulation. The XDG method adapts the approximation space conformal to the position of the interface and allows a sub‐cell accurate representation within the sharp interface formulation. The interface is described as the zero set of a signed‐distance level‐set function and discretized by a standard DG method. No adaption of the level‐set evolution algorithm is needed for the extension to moving contact line problems. The developed solver is validated against typical two‐dimensional contact line driven flow phenomena including droplet simulations on a wall and the two‐phase Couette flow.

A semi-implicit conservative sharp-interface method for liquid-solid phase transition

The solidification of an undercooled liquid is physically unstable. The dominating instability modes are affected by both the evolving temperature field in the solid and liquid phases, and characteristics of the phase interface such as the curvature and the propagation velocity. To capture the instability mode, therefore both the temperature field and the interface have to be represented accurately in a numerical model of the phase-change process. In this work, we develop conservative interface exchange terms for a sharp-interface formulation of liquid-solid phase transition. Conservation at the interface is maintained by explicit formulation of interface fluxes into both solid and liquid phases. We propose a semi-implicit level-set formulation to evolve the phase interface. A new formulation for the interface surface in a cut cell is derived, which includes the Stefan condition. We achieve low numerical dissipation by an explicit third-order Runge-Kutta scheme for time discretization, and a novel WENO-like (Weighted Essentially Non-Oscillatory) interface-gradient reconstruction. This distinguishes our level-set based sharp-interface model from previous level-set based approaches, which rely on finite-difference based interface treatment, and thus do not ensure discrete conservation at the interface. The flux terms in our approach take into account surface-tension and kinetic effects on the interface temperature (Gibbs-Thomson relation). The Stefan condition provides a relation between interface fluxes of mass and energy, and the interface-propagation velocity. Computational efficiency is maintained by a multiresolution approach for local mesh adaptation, and an adaptive local time-stepping scheme.We present one- and two-dimensional simulation results for the growth of a planar solidification front and a single parabolic dendrite affected by surface tension. The results agree well with experimental and analytical reference data, showing that the model is capable to capture both stable (planar) and unstable (dendritic-like) growth processes in the heat-diffusion dominated regime. The convergence order for successively finer meshes in the one-dimensional case is one for the interface location and the temperature field, outperforming previously reported level-set based approaches. We present numerical data of a growing crystal with four-fold symmetry. Our results indicate that the artificial dissipation of the underlying numerical scheme affects its capability to reproduce consistently physical tip-splitting instabilities. The proposed low-dissipation scheme is able to resolve such instabilities. Finally, we demonstrate the capability of the method to simulate multiple growing crystals with anisotropic surface-tension and kinetic effects.

Development of General Finite Differences for complex geometries using a sharp interface formulation

We apply sharp interface boundary conditions to collocated meshfree approximations. Using the sharp interface formulation with the General Finite Difference (GFD) shape functions, we solve several test cases on meshfree grids with uniform and variable resolutions. Numerical results for the 2D Poisson equation indicate that the sharp interface formulation performs as well as the traditional collocated boundary approach with respect to both the convergence rate and accuracy. The analytic lid driven cavity results indicate that our discretization of a semi-implicit approximate projection scheme is indeed consistent and spatially converges at a second order rate under diffusive time stepping and at a first order rate under convective time stepping due to the first order in time splitting error. To resolve sharp gradients, variable resolution meshfree grids are considered for the classic lid driven cavity problem and for the uniform flow over a cylinder problem. Comparison to appropriate data sets indicate that our approach accurately estimates the various flow parameters using small cloud sizes of 13 neighbors while using high clustering ratios.

A phase-field model for fluid–structure interaction

In this paper, we develop a novel phase-field model for fluid–structure interaction (FSI), that is capable to handle very large deformations as well as topology changes like contact of the solid to a wall. The model is based on a fully Eulerian description of the velocity field in both, the fluid and the elastic domain. Viscous and elastic stresses in the Navier–Stokes equations are restricted to the corresponding domains by multiplication with their characteristic functions. The solid is described as a hyperelastic neo-Hookean material and the elastic stress is obtained by solving an additional Oldroyd-B – like equation. Thermodynamically consistent forces are derived by energy variation. The convergence of the derived equations to the traditional sharp interface formulation of fluid–structure interaction is shown by matched asymptotic analysis. The model is evaluated in a challenging benchmark scenario of an elastic body traversing a fluid channel. A comparison to reference values from Arbitrary Lagrangian Eulerian (ALE) simulations shows very good agreement. We highlight some distinct advantages of the new model, like the avoidance of re-triangulations and the stable inclusion of surface tension. Further, we demonstrate how simple it is to include contact dynamics into the model, by simulating a ball bouncing off a wall. We extend this scenario to include adhesion of the ball, which to our knowledge, cannot be simulated with any other FSI model. While we have restricted simulations to fluid–structure interaction, the model is capable to simulate any combination of viscous fluids, visco-elastic fluids and elastic solids.

Gradient flow dynamics of two-phase biomembranes: Sharp interface variational formulation and finite element approximation

A finite element method for the evolution of a two-phase membrane in a sharp interface formulation is introduced. The evolution equations are given as an $L^2$--gradient flow of an energy involving an elastic bending energy and a line energy. In the two phases Helfrich-type evolution equations are prescribed, and on the interface, an evolving curve on an evolving surface, highly nonlinear boundary conditions have to hold. Here we consider both $C^0$-- and $C^1$--matching conditions for the surface at the interface. A new weak formulation is introduced, allowing for a stable semidiscrete parametric finite element approximation of the governing equations. In addition, we show existence and uniqueness for a fully discrete version of the scheme. Numerical simulations demonstrate that the approach can deal with a multitude of geometries. In particular, the paper shows the first computations based on a sharp interface description, which are not restricted to the axisymmetric case.

On the explicit resolution of the mushy zone in the modelling of the continuous casting of alloys

• Dual boundary immobilization technique for determining the mushy zone in the continuous casting of alloys. • Analytical behaviour of the mushliquid and solidmush interfaces at the start of phase change. • Analytical behaviour of the mushliquid and solidmush interfaces at phase depletion. • Method implemented in Comsol Multiphysics. In the continuous casting of alloys, it is well-known that the mushy zone is decisive for the final properties of the casting. Most numerical models for the process use enthalpy-based methods on fixed grids which determine the extent the mushy zone implicitly. Here, on the other hand, we develop a methodology for explicitly resolving the geometrical extent of the mushy zone; this involves a sharp-interface formulation to solve a dual moving boundary problem to locate the solidus and liquidus isotherms. The results compare favourably with those from enthalpy-based methods, and the advantages of our approach with respect to future multiphysics calculations are discussed.

A sharp-interface level-set method for analysis of Marangoni effect on microdroplet evaporation

A level-set method is presented for computation of microdroplet evaporation including not only the effects of heat and mass transfer, phase change and contact line dynamics but also the Marangoni effect, which is a key parameter affecting the internal flow of the droplet and the particle deposition pattern. A sharp-interface formulation of the Marangoni force is derived and tested for two-phase Marangoni convection in a cavity. The computed results show good convergence in both the liquid and gas regions and are in excellent agreement with the analytical solutions. The level-set formulation is applied to microdroplet evaporation on a solid surface to investigate the Marangoni effect.

Saltwater Intrusion in Coastal Aquifers: A Primary Case Study along the Adriatic Coast Investigated within a Probabilistic Framework

Environmentally sensitive areas along coastlines may be adversely affected by saltwater intrusion (SI), a condition which can be worsened by extensive groundwater extraction. Given the uncertainty of problem parameters, the risk of contamination of the vegetation capture zone needs to be cast in a probabilistic framework. In order to exemplify real situations existing along the Adriatic coast of Emilia-Romagna, a case study involving a pinewood strip and a well field drawing freshwater from an unconfined coastal aquifer was examined. On the basis of a widely adopted sharp interface formulation, key hydrogeological problem parameters were modeled as random variables, and a global sensitivity analysis was carried out to determine their influence on the position of the interface. This analysis utilized an efficient model reduction technique based on Polynomial Chaos Expansion. The risk that saltwater intrusion affects coastal vegetation was then evaluated via a two-step procedure by computing the probability that (i) the leading edge of the saltwater wedge reaches the sensitive area in the horizontal plane, and (ii) the freshwater/saltwater interface reaches the capture zone. The influence of the design parameters of the well field on the overall probability of contamination was investigated, revealing the primary role of the pumping discharge in the examined configuration.

Investigation of keyhole plume and molten pool based on a three-dimensional dynamic model with sharp interface formulation

Laser keyhole welding is a complicated multi-phases, multi-physics process, especially when assisting gases are involved. A three-dimensional transient model is developed to investigate the dynamics of keyhole, together with the vapour plume and molten pool, in a self-consistent manner. The model features the utilization of sharp interface method for accurate consideration of the complex surface phenomena on the keyhole wall and a comprehensive hydrodynamic calculation for both the vapour plume and molten pool. The model is validated against experiments and the simulation results are discussed. It is found that the interplay of the multiple reflections and the plume attenuation due to particle absorption/scattering is crucial for the laser absorption intensity and hence the temperature on the keyhole wall, and the keyhole wall temperature distribution has profound influences on the fluid flow and temperature/species distributions in both the molten pool and keyhole plume.

Phase-field modeling of dry snow metamorphism

Snow on the ground is a complex three-dimensional porous medium consisting of an ice matrix formed by sintered snow crystals and a pore space filled with air and water vapor. If a temperature gradient is imposed on the snow, a water vapor gradient in the pore space is induced and the snow microstructure changes due to diffusion, sublimation, and resublimation: the snow metamorphoses. The snow microstructure, in turn, determines macroscopic snow properties such as the thermal conductivity of a snowpack. We develop a phase-field model for snow metamorphism that operates on natural snow microstructures as observed by computed x-ray microtomography. The model takes into account heat and mass diffusion within the ice matrix and pore space, as well as phase changes at the ice-air interfaces. Its construction is inspired by phase-field models for alloy solidification, which allows us to relate the phase-field to a sharp-interface formulation of the problem without performing formal matched asymptotics. To overcome the computational difficulties created by the large difference between diffusional and interface-migration time scales, we introduce a method for accelerating the numerical simulations that formally amounts to reducing the heat- and mass-diffusion coefficients while maintaining the correct interface velocities. The model is validated by simulations for simple one- and two-dimensional test cases. Furthermore, we perform qualitative metamorphism simulations on natural snow structures to demonstrate the potential of the approach.

Phase field modeling of fracture and stress-induced phase transitions

We present a continuum theory to describe elastically induced phase transitions between coherent solid phases. In the limit of vanishing elastic constants in one of the phases, the model can be used to describe fracture on the basis of the late stage of the Asaro-Tiller-Grinfeld instability. Starting from a sharp interface formulation we derive the elastic equations and the dissipative interface kinetics. We develop a phase field model to simulate these processes numerically; in the sharp interface limit, it reproduces the desired equations of motion and boundary conditions. We perform large scale simulations of fracture processes to eliminate finite-size effects and compare the results to a recently developed sharp interface method. Details of the numerical simulations are explained, and the generalization to multiphase simulations is presented.

Phase-field modeling of eutectic growth.

A phase-field model of eutectic growth is proposed in terms of a free energy F, which is a functional of a liquid-solid order parameter psi, and a conserved concentration field c. The model is shown to recover the important features of a eutectic phase diagram and to reduce to the standard sharp-interface formulation of nonequilibrium growth. It is successfully applied to the study of directional solidification when the solid phase is a single or two phase state. The crystallization of a eutectic compound under isothermal conditions is also considered. For that process, the transformed volume fraction and psi-field structure factor, both measured during numerical simulations, closely match theoretical predictions. Three possible growth mechanisms are also identified: diffusion-limited growth, lamellar growth, and spinodal decomposition.