Development of a Volume of Fluid Method for Computing Interfacial Incompressible Fluid Flows

This study is aimed to develop a volume of fluid (VOF) method to capture the free surface flow. The incompressible two-phase flow is computed by second-order Adams-Bashforth algorithm with a uniform staggered Cartesian grid arrangement. The tangent of hyperbola for interface capturing (THINC) scheme and weighted linear interface calculation (WLIC) based geometrical reconstruction procedure have been implemented in the operator-splitting method for the VOF method. The proposed VOF method preserves mass well, and the interface normal vector can be easily estimated from the level set (LS) function. The LS function, which is a continuous signed distance function around the interface, is represented by solving the re-initialization equation. Numerical results using the present scheme are compared with experimental data and other numerical results in the Rayleigh-Taylor instability, dam-break flow, travelling solitary wave, Kelvin-Helmholtz instability, rising bubble and merging bubble problems. We also present numerical results in detail between computations made with the proposed VOF method and computations made with the conventional LS method.     

An Implicit Scheme for Moving Walls and Multi-Material Interfaces in Weakly Compressible Materials

We propose a numerical method for the simulation of flows from weakly compressible to low Mach regimes in domains with moving boundaries. Non-miscible weakly compressible materials separated by an interface are included as well. The scheme is fully implicit and it exploits the relaxation all-speed scheme introduced in [1]. We consider media with significantly different physical properties and constitutive laws, as fluids and hyperelastic solids. The proposed numerical scheme is fully Eulerian and it is the same for all materials. We present numerical validations by simulating weakly compressible fluid/fluid, solid/solid and solid/fluid interactions.     

Multiphysic Two-Phase Flow Lattice Boltzmann: Droplets with Realistic Representation of the Interface

Free energy lattice Boltzmann methods are well suited for the simulation of two phase flow problems. The model for the interface is based on well understood physical grounds. In most cases a numerical interface is used instead of the physical one because of lattice resolution limitations. In this paper we present a framework where we can both follow the droplet behavior in a coarse scale and solve the interface in a fine scale simultaneously. We apply the method for the simulation of a droplet using an interface to diameter size ratio of 1 to 280. In a second simulation, a small droplet coalesces with a 42 times larger droplet producing on it only a small capillary wave that propagates and dissipates.     

Weak Galerkin and Continuous Galerkin Coupled Finite Element Methods for the Stokes-Darcy Interface Problem

We consider a model of coupled free and porous media flow governed by Stokes equation and Darcy's law with the Beavers-Joseph-Saffman interface condition. In this paper, we propose a new numerical approach for the Stokes-Darcy system. The approach employs the classical finite element method for the Darcy region and the weak Galerkin finite element method for the Stokes region. We construct corresponding discrete scheme and prove its well-posedness. The estimates for the corresponding numerical approximation are derived. Finally, we present some numerical experiments to validate the efficiency of the approach for solving this problem.     

A Conservative Modification to the Ghost Fluid Method for Compressible Multiphase Flows

A conservative modification to the ghost fluid method (GFM) is developed for compressible multiphase flows. The motivation is to eliminate or reduce the conservation error of the GFM without affecting its performance. We track the conservative variables near the material interface and use this information to modify the numerical solution for an interfacing cell when the interface has passed the cell. The modification procedure can be used on the GFM with any base schemes. In this paper we use the fifth order finite difference WENO scheme for the spatial discretization and the third order TVD Runge-Kutta method for the time discretization. The level set method is used to capture the interface. Numerical experiments show that the method is at least mass and momentum conservative and is in general comparable in numerical resolution with the original GFM.     

Deep Unfitted Nitsche Method for Elliptic Interface Problems

This paper proposes a deep unfitted Nitsche method for solving elliptic interface problems with high contrasts in high dimensions. To capture discontinuities of the solution caused by interfaces, we reformulate the problem as an energy minimization problem involving two weakly coupled components. This enables us to train two deep neural networks to represent two components of the solution in high-dimensional space. The curse of dimensionality is alleviated by using the Monte-Carlo method to discretize the unfitted Nitsche energy functional. We present several numerical examples to show the performance of the proposed method.     

A Phase-Field Model Coupled with Lattice Kinetics Solver for Modeling Crystal Growth in Furnaces

In this study, we present a new numerical model for crystal growth in a vertical solidification system. This model takes into account the buoyancy induced convective flow and its effect on the crystal growth process. The evolution of the crystal growth interface is simulated using the phase-field method. A semi-implicit lattice kinetics solver based on the Boltzmann equation is employed to model the unsteady incompressible flow. This model is used to investigate the effect of furnace operational conditions on crystal growth interface profiles and growth velocities. For a simple case of macroscopic radial growth, the phase-field model is validated against an analytical solution. The numerical simulations reveal that for a certain set of temperature boundary conditions, the heat transport in the melt near the phase interface is diffusion dominant and advection is suppressed.     

A Multi-Domain Hybrid DG and WENO Method for Hyperbolic Conservation Laws on Hybrid Meshes

In [SIAM J. Sci. Comput., 35(2)(2013), A1049–A1072], a class of multi-domain hybrid DG and WENO methods for conservation laws was introduced. Recent applications of this method showed that numerical instability may encounter if the DG flux with Lagrangian interpolation is applied as the interface flux during the moment of conservative coupling. In this continuation paper, we present a more robust approach in the construction of DG flux at the coupling interface by using WENO procedures of reconstruction. Based on this approach, such numerical instability is overcome very well. In addition, the procedure of coupling a DG method with a WENO-FD scheme on hybrid meshes is disclosed in detail. Typical testing cases are employed to demonstrate the accuracy of this approach and the stability under the flexibility of using either WENO-FD flux or DG flux at the moment of requiring conservative coupling.     

A Simple 3D Immersed Interface Method for Stokes Flow with Singular Forces on Staggered Grids

In this paper, a fairly simple 3D immersed interface method based on the CG-Uzawa type method and the level set representation of the interface is employed for solving three-dimensional Stokes flow with singular forces along the interface. The method is to apply the Taylor's expansions only along the normal direction and incorporate the jump conditions up to the second normal derivatives into the finite difference schemes. A second order geometric iteration algorithm is employed for computing orthogonal projections on the surface with third-order accuracy. The Stokes equations are discretized involving the correction terms on staggered grids and then solved by the conjugate gradient Uzawa type method. The major advantages of the present method are the special simplicity, the ability in handling the Dirichlet boundary conditions, and no need of the pressure boundary condition. The method can also preserve the volume conservation and the discrete divergence free condition very well. The numerical results show that the proposed method is second order accurate and efficient.     

A Nominally Second-Order Cell-Centered Finite Volume Scheme for Simulating Three-Dimensional Anisotropic Diffusion Equations on Unstructured Grids

We present a finite volume based cell-centered method for solving diffusion equations on three-dimensional unstructured grids with general tensor conduction. Our main motivation concerns the numerical simulation of the coupling between fluid flows and heat transfers. The corresponding numerical scheme is characterized by cell-centered unknowns and a local stencil. Namely, the scheme results in a global sparse diffusion matrix, which couples only the cell-centered unknowns. The space discretization relies on the partition of polyhedral cells into sub-cells and on the partition of cell faces into sub-faces. It is characterized by the introduction of sub-face normal fluxes and sub-face temperatures, which are auxiliary unknowns. A sub-cell-based variational formulation of the constitutive Fourier law allows to construct an explicit approximation of the sub-face normal heat fluxes in terms of the cell-centered temperature and the adjacent sub-face temperatures. The elimination of the sub-face temperatures with respect to the cell-centered temperatures is achieved locally at each node by solving a small and sparse linear system. This system is obtained by enforcing the continuity condition of the normal heat flux across each sub-cell interface impinging at the node under consideration. The parallel implementation of the numerical algorithm and its efficiency are described and analyzed. The accuracy and the robustness of the proposed finite volume method are assessed by means of various numerical test cases.     

A Bilinear Immersed Finite Volume Element Method for the Diffusion Equation with Discontinuous Coefficient

This paper is to present a finite volume element (FVE) method based on the bilinear immersed finite element (IFE) for solving the boundary value problems of the diffusion equation with a discontinuous coefficient (interface problem). This method possesses the usual FVE method's local conservation property and can use a structured mesh or even the Cartesian mesh to solve a boundary value problem whose coefficient has discontinuity along piecewise smooth nontrivial curves. Numerical examples are provided to demonstrate features of this method. In particular, this method can produce a numerical solution to an interface problem with the usual O(h²) (in L² norm) and O(h) (in H¹ norm) convergence rates.     

An Efficient Neural-Network and Finite-Difference Hybrid Method for Elliptic Interface Problems with Applications

A new and efficient neural-network and finite-difference hybrid method is developed for solving Poisson equation in a regular domain with jump discontinuities on embedded irregular interfaces. Since the solution has low regularity across the interface, when applying finite difference discretization to this problem, an additional treatment accounting for the jump discontinuities must be employed. Here, we aim to elevate such an extra effort to ease our implementation by machine learning methodology. The key idea is to decompose the solution into singular and regular parts. The neural network learning machinery incorporating the given jump conditions finds the singular solution, while the standard five-point Laplacian discretization is used to obtain the regular solution with associated boundary conditions. Regardless of the interface geometry, these two tasks only require supervised learning for function approximation and a fast direct solver for Poisson equation, making the hybrid method easy to implement and efficient. The two- and three-dimensional numerical results show that the present hybrid method preserves second-order accuracy for the solution and its derivatives, and it is comparable with the traditional immersed interface method in the literature. As an application, we solve the Stokes equations with singular forces to demonstrate the robustness of the present method.     

A Coupled Immersed Interface and Level Set Method for Three-Dimensional Interfacial Flows with Insoluble Surfactant

In this paper, a numerical method is presented for simulating the 3D interfacial flows with insoluble surfactant. The numerical scheme consists of a 3D immersed interface method (IIM) for solving Stokes equations with jumps across the interface and a 3D level-set method for solving the surfactant convection-diffusion equation along a moving and deforming interface. The 3D IIM Poisson solver modifies the one in the literature by assuming that the jump conditions of the solution and the flux are implicitly given at the grid points in a small neighborhood of the interface. This assumption is convenient in conjunction with the level-set techniques. It allows standard Lagrangian interpolation for quantities at the projection points on the interface. The interface jump relations are re-derived accordingly. A novel rotational procedure is given to generate smooth local coordinate systems and make effective interpolation. Numerical examples demonstrate that the IIM Poisson solver and the Stokes solver achieve second-order accuracy. A 3D drop with insoluble surfactant under shear flow is investigated numerically by studying the influences of different physical parameters on the drop deformation.     

ADI-FDTD Method for Two-Dimensional Transient Electromagnetic Problems

An efficient and accurate numerical scheme is proposed for solving the transverse electric (TE) mode electromagnetic (EM) propagation problem in two-dimensional earth. The scheme is based on the alternating direction finite-difference time-domain (ADI-FDTD) method. Unlike the conventional upward continuation approach for the earth-air interface, an integral formulation for the interface boundary is developed and it is effectively incorporated to the ADI solver. Stability and convergence analysis together with an error estimate are presented. Numerical simulations are carried out to validate the proposed method, and the advantage of the present method over the popular Du-Fort-Frankel scheme is clearly demonstrated. Examples of the electromagnetic field propagation in the ground with anomaly further verify the effectiveness of the proposed scheme.     

An Approximate Riemann Solver for Fluid-Solid Interaction Problems with Mie-Grüneisen Equations of State

We propose an approximate solver for compressible fluid-elastoplastic solid Riemann problems. The fluid and hydrostatic components of the solid are described by a family of general Mie-Grüneisen equations of state, and the hypo-elastoplastic constitutive law we studied includes the perfect plasticity and linearly hardened plasticity. The approximate solver provides the interface stress and normal velocity by an iterative method. The well-posedness and convergence of our solver are verified with mild assumptions on the equations of state. The proposed solver is applied in computing the numerical flux at the phase interface for our compressible multi-medium flow simulation on Eulerian girds. Several numerical examples, including Riemann problems, underground explosion and high speed impact applications, are presented to validate the approximate solver.     

An Implementation of MAC Grid-Based IIM-Stokes Solver for Incompressible Two-Phase Flows

In this paper, a novel implementation of immersed interface method combined with Stokes solver on a MAC staggered grid for solving the steady two-fluid Stokes equations with interfaces. The velocity components along the interface are introduced as two augmented variables and the resulting augmented equation is then solved by the GMRES method. The augmented variables and/or the forces are related to the jumps in pressure and the jumps in the derivatives of both pressure and velocity, and are interpolated using cubic splines and are then applied to the fluid through the jump conditions. The Stokes equations are discretized on a staggered Cartesian grid via a second order finite difference method and solved by the conjugate gradient Uzawa-type method. The numerical results show that the overall scheme is second order accuracy. The major advantages of the present IIM-Stokes solver are the efficiency and flexibility in terms of types of fluid flow and different boundary conditions. The proposed method avoids solution of the pressure Poisson equation, and comparisons are made to show the advantages of time savings by the present method. The generalized two-phase Stokes solver with correction terms has also been applied to incompressible two-phase Navier-Stokes flow.     

A Dynamical Concurrent Multiscale Method Employing a Transmitting Boundary to Minimize Wave Reflections at the Domain Interface

In this work, we present a reformulation of a concurrent multi-scale computation method, previously developed and demonstrated in a static setting at zero temperature, to extend the method for dynamic cases. Additionally, within the context of dynamics, a well-known problem faced in multi-scale simulations pertains to the spurious wave reflections that occur at the artificially introduced interface between the atomistic domain and the coarse scale domain. To address this computational issue, we derive and demonstrate from first principles a simple yet effective solution to mitigate the manifestation of such nonphysical wave reflections at the surface of an artificially truncated domain. We obtain an explicit relationship between the displacement and the velocity terms of the terminal atom and the additional force that needs to be applied on this atom in order to absorb any incident wave. We present how the proposed transmitting boundary can be implemented within the multi-scale formulation for dynamic cases and include numerical examples to verify the efficacy of the methods.     

Boundary Integral Modelling of Elastic Wave Propagation in Multi-Layered 2D Media with Irregular Interfaces

We present a semi-analytic method based on the propagation matrix formulation of indirect boundary element method to compute response of elastic (and acoustic) waves in multi-layered media with irregular interfaces. The method works recursively starting from the top-most free surface at which a stress-free boundary condition is applied, and the displacement-stress boundary conditions are then subsequently applied at each interface. The basic idea behind this method is the matrix formulation of the propagation matrix (PM) or more recently the reflectivity method as wide used in the geophysics community for the computation of synthetic seismograms in stratified media. The reflected and transmitted wave-fields between arbitrary shapes of layers can be computed using the indirect boundary element method (BEM, sometimes called IBEM). Like any standard BEM, the primary task of the BEM-based propagation matrix method (thereafter called PM-BEM) is the evaluation of element boundary integral of the Green's function, for which there are standard method that can be adapted. In addition, effective absorbing boundary conditions as used in the finite difference numerical method is adapted in our implementation to suppress the spurious arrivals from the artificial boundaries due to limited model space. To our knowledge, such implementation has not appeared in the literature. We present several examples in this paper to demonstrate the effectiveness of this proposed PM-BEM for modelling elastic waves in media with complex structure.     

A Diffuse-Interface Lattice Boltzmann Method for the Dendritic Growth with Thermosolutal Convection

In this work, we proposed a diffuse-interface model for the dendritic growth with thermosolutal convection. In this model, the sharp boundary between the fluid and solid dendrite is firstly replaced by a thin but nonzero thickness diffuse interface, which is described by the order parameter, and the diffuse-interface based governing equations for the dendritic growth are presented. To solve the model for the dendritic growth with thermosolutal convection, we also developed a diffuse-interface multi-relaxation-time lattice Boltzmann (LB) method. In this method, the order parameter in the phase-field equation is combined into the force caused by the fluid-solid interaction, and the treatment on the complex fluid-solid interface can be avoided. In addition, four LB models are designed for the phase-field equation, concentration equation, temperature equation and the Navier-Stokes equations in a unified framework. Finally, we performed some simulations of the dendritic growth to test the present diffuse-interface LB method, and found that the numerical results are in good agreements with some previous works.     

An Immersed Interface Method for Axisymmetric Electrohydrodynamic Simulations in Stokes Flow

A numerical scheme based on the immersed interface method (IIM) is developed to simulate the dynamics of an axisymmetric viscous drop under an electric field. In this work, the IIM is used to solve both the fluid velocity field and the electric potential field. Detailed numerical studies on the numerical scheme show a second-order convergence. Moreover, our numerical scheme is validated by the good agreement with previous analytical models [1, 31, 39], and numerical results from the boundary integral simulations [17]. Our method can be extended to Navier-Stokes fluid flow with nonlinear inertia effects.