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 Discontinuity and Cusp Capturing PINN for Stokes Interface Problems with Discontinuous Viscosity and Singular Forces

In this paper,we present a discontinuity and cusp capturing physicsinformed neural network(PINN)to solve Stokes equations with a piecewiseconstant viscosity and singular force along an interface.We first reformulate the governing equations in each fluid domain separately and replace the singular force effect with the traction balance equation between solutions in two sides along the interface.Since the pressure is discontinuous and the velocity has discontinuous derivatives across the interface,we hereby use a network consisting of two fully-connected sub-networks that approximate the pressure and velocity,respectively.The two sub-networks share the same primary coordinate input arguments but with different augmented feature inputs.These two augmented inputs provide the interface information,so we assume that a level set function is given and its zero level set indicates the position of the interface.The pressure sub-network uses an indicator function as an augmented input to capture the function discontinuity,while the velocity sub-network uses a cusp-enforced level set function to capture the derivative discontinuities via the traction balance equation.We perform a series of numerical experiments to solve two-and three-dimensional Stokes interface problems and perform an accuracy comparison with the augmented immersed interface methods in literature.Our results indicate that even a shallow network with a moderate number of neurons and sufficient training data points can achieve prediction accuracy comparable to that of immersed interface methods.

Numerical Simulation of Moving Contact Lines with Surfactant by Immersed Boundary Method

In this paper,we present an immersed boundary method for simulating moving contact lines with surfactant.The governing equations are the incompressible Navier-Stokes equations with the usual mixture of Eulerian fluid variables and Lagrangian interfacial markers.The immersed boundary force has two components:one from the nonhomogeneous surface tension determined by the distribution of surfactant along the fluid interface,and the other from unbalanced Young’s force at the moving contact lines.An artificial tangential velocity has been added to the Lagrangian markers to ensure that the markers are uniformly distributed at all times.The corresponding modified surfactant equation is solved in a way such that the total surfactant mass is conserved.Numerical experiments including convergence analysis are carefully conducted.The effect of the surfactant on the motion of hydrophilic and hydrophobic drops are investigated in detail.

The Immersed Boundary Method for Two-Dimensional Foam with Topological Changes

We extend the immersed boundary(IB)method to simulate the dynamics of a 2D dry foam by including the topological changes of the bubble network.In the article[Y.Kim,M.-C.Lai,and C.S.Peskin,J.Comput.Phys.229:5194-5207,2010],we implemented an IB method for the foam problem in the two-dimensional case,and tested it by verifying the von Neumann relation which governs the coarsening of a two-dimensional dry foam.However,the method implemented in that article had an important limitation;we did not allow for the resolution of quadruple or higher order junctions into triple junctions.A total shrinkage of a bubble with more than four edges generates a quadruple or higher order junction.In reality,a higher order junction is unstable and resolves itself into triple junctions.We here extend the methodology previously introduced by allowing topological changes,and we illustrate the significance of such topological changes by comparing the behaviors of foams in which topological changes are allowed to those in which they are not.

A Front-Tracking Method for Motion by Mean Curvature with Surfactant

In this paper,we present a finite difference method to track a network of curves whose motion is determined by mean curvature.To study the effect of inhomogeneous surface tension on the evolution of the network of curves,we include surfactant which can diffuse along the curves.The governing equations consist of one parabolic equation for the curve motion coupled with a convection-diffusion equation for the surfactant concentration along each curve.Our numerical method is based on a direct discretization of the governing equations which conserves the total surfactant mass in the curve network.Numerical experiments are carried out to examine the effects of inhomogeneous surface tension on the motion of the network,including the von Neumann law for cell growth in two space dimensions.

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.

Preface Special Issue for SCPDE08: Numerical Methods and Analysis for PDEs and Inverse Problems

The Third International Conference on Scientific Computing and Partial Differential Equations(SCPDE)was held from December 8 to December 12,2008 at China Hong Kong Baptist University.It was a sequel to similar conferences held in Hong Kong region(2002 and 2005).The conference aims to promote research interests in scientific computation.In SCPDE 2008,there were 118 participants from seventeen countries and regions participated in the conference.The Programme included seventeen plenary addresses,thirty invited talks,twenty five contributed talks and seven poster presentations.