BDF Schemes in Stable Generalized Finite Element Methods for Parabolic Interface Problems with Moving Interfaces

There are several difficulties in generalized/extended finite element methods(GFEM/XFEM)for moving interface problems.First,the GFEM/XFEM may be unstable in a sense that condition numbers of system matrices could be much bigger than those of standard FEM.Second,they may not be robust in that the condition numbers increase rapidly as interface curves approach edges of meshes.Furthermore,time stepping schemes need carrying out carefully since both enrichment functions and enriched nodes in the GFEM/XFEM vary in time.This paper is devoted to proposing the stable and robust GFEM/XFEM with efficient time stepping schemes for the parabolic interface problems with moving interfaces.A so-called stable GFEM(SGFEM)developed for elliptical interface problems is extended to the parabolic interface problems for spatial discretizations;while backward difference formulae(BDF)are used for the time stepping.Numerical studies demonstrate that the SGFEM with the first and second order BDF(also known as backward Euler method and BDF2)is stable,robust,and achieves optimal convergence rates.Comparisons of the proposed SGFEM with various commonly-used GFEM/XFEM are made,which show advantages of the SGFEM over the other GFEM/XFEM in aspects of stability,robustness,and convergence.

Stability analysis of the moving interface in piston- and non-piston-like displacements

From the macroscopic point of view, expressions involving reservoir and operational parameters are established for investigating the stability of moving interface in piston- and non-piston-like displacements. In the case of axisymmetrical piston-like displacement, the stability is related to the moving interface position and water to oil mobility ratio. The capillary effect on the stability of moving interface depends on whether or not the moving interface is already stable and correlates with the wettability of the reservoir rock. In the case of non-piston-like displacement, the stability of the front is governed by both the relative permeability and the mobility ratio.

Simulation of flows with moving contact lines on a dual-resolution Cartesian grid using a diffuse-interface immersed-boundary method

In this paper, we investigate flows with moving contact lines on curved solid walls on a dual-resolution grid using a diffuse-interface immersed-boundary（DIIB） method. The dual-resolution grid, on which the flows are solved on a coarse mesh while the interface is resolved on a fine mesh, was shown to significantly improve the computational efficiency when simulating multiphase flows. On the other hand, the DIIB method is able to resolve dynamic wetting on curved substrates on a Cartesian grid, but it usually requires a mesh of high resolution in the vicinity of a moving contact line to resolve the local flow. In the present study, we couple the DIIB method with the dual-resolution grid, to improve the interface resolution for flows with moving contact lines on curved solid walls at an affordable cost. The dynamic behavior of moving contact lines is validated by studying drop spreading, and the numerical results suggest that the effective slip length λ_n can be approximated by 1.9Cn, where Cn is a dimensionless measure of the thickness of the diffuse interface. We also apply the method to drop impact onto a convex substrate, and the results on the dual-resolution grid are in good agreement with those on a single-resolution grid. It shows that the axisymmetric simulations using the DIIB method on the dual-resolution grid saves nearly 60% of the computational time compared with that on a single-resolution grid.

A Method of Lines Based on Immersed Finite Elements for Parabolic Moving Interface Problems

This article extends the finite element method of lines to a parabolic initial boundary value problem whose diffusion coefficient is discontinuous across an interface that changes with respect to time.The method presented here uses immersed finite element(IFE)functions for the discretization in spatial variables that can be carried out over a fixedmesh(such as a Cartesianmesh if desired),and this featuremakes it possible to reduce the parabolic equation to a system of ordinary differential equations(ODE)through the usual semi-discretization procedure.Therefore,with a suitable choice of the ODE solver,this method can reliably and efficiently solve a parabolic moving interface problem over a fixed structured(Cartesian)mesh.Numerical examples are presented to demonstrate features of this new method.

SPECTRAL METHOD FOR CALCULATING MOVING INTERFACE BETWEEN FLUIDS IN POROUS MEDIUM

In the application of spectral method to the calculation of moving interface between fluids in porous medium there are two difficulties: the spectral calcula- tion of function defined by a singular integral and the numerical quadrature of highly oscillating function. This paper proposes a spectral method for calculating the problem and finds the way to overcome the two difficulties. Example calcula- tions show that the method can describe successfully interfacial motion and, with almost the same order of computational amount, is more accurate and stabler than the corresponding finite difference method.

A review of research progress in air-to-water sound transmission

International and domestic research progress in theory and experiment and applications of the air-to-water sound transmission are presented in this paper. Four classical numerical methods of calculating the underwater sound field gener- ated by an airborne source, i.e., the ray theory, the wave solution, the normal-mode theory and the wavenumber integration approach, are introduced. Effects of two special conditions, i.e., the moving airborne source or medium and the rough air-water interface, on the air-to-water sound transmission are reviewed. In experimental studies, the depth and range distributions of the underwater sound field created by different kinds of airborne sources in near-field and far-field, the longitudinal horizontal correlation of underwater sound field and application methods for inverse problems are reviewed.

2008 Stability Analysis of the Immersed Boundary Method for a Two-Dimensional Membrane with Bending Rigidity

In this paper,we analyze the stability of the Immersed Boundary Methodapplied to a membrane-fluid system with a plasma membrane immersed in an incompressibleviscous fluid.We show that for small deformations,the planar rest state isstable for a membrane with bending rigidity.The smoothed version,using a standardregularization technique for the singular force,is also shown to be stable.Furthermore,we show that the coupled fluid-membrane system is stiff and smoothing helpsto reduce the stiffness.Compared to the system of elastic fibers immersed in an incompressiblefluid,membrane with bending rigidity consist of a wider range of decayrates.Therefore numerical instability could occur more easily for an explicit methodwhen the time step size is not sufficiently small,even though the continuous problemis stable.

A Constrained Level Set Method for Simulating the Formation of Liquid Bridges

In this paper,we investigate the dynamic process of liquid bridge formation between two parallel hydrophobic plates with hydrophilic patches,previously studied in[1].We propose a dynamic Hele-Shaw model to take advantage of the small aspect ratio between the gap width and the plate size.A constrained level set method is applied to solve the model equations numerically,where a global constraint is imposed in the evolution[2]stage together with local constraints in the reinitialization[3]stage of level set function in order to limit numerical mass loss.In contrast to the finite element method used in[2],we use a finite difference method with a 5th order HJWENO scheme for spatial discretization.To illustrate the effectiveness of the constrained method,we have compared the results obtained by the standard level set method with those from the constrained version.Our results show that the constrained level set method produces physically reasonable results while that of the standard method is less reliable.Our numerical results also show that the dynamic nature of the flow plays an important role in the process of liquid bridge formation and criteria based on static energy minimization approach has limited applicability.

Numerical Simulation for a Droplet Fission Process of Electrowetting on Dielectric Device

Electrowetting has been proposed as a technique for manipulating dropletssurrounded by air or oil. In this paper, we discuss the modeling and simulation of thedroplet fission process between two parallel plates inside an electrowetting on dielectric (EWOD) device. Since the gap between the plates is small, we use the two-phaseHele-Shaw flow as a model. While there are several high order methods around, suchas the immersed interface methods [1, 2], we decide to use two first-order methods forsimplicity. A ghost-fluid (GF) method is employed to solve the governing equationsand a local level set method is used to track the drop interface. For comparison purposes, the same set of two-phase Hele-Shaw equations are also solved directly usingthe immersed boundary (IB) method. Numerical results are consistent with experimental observations reported in the literature.