Efficient high-order immersed interface methods for heat equations with interfaces

An efficient high-order immersed interface method （IIM） is proposed to solve two-dimensional （2D） heat problems with fixed interfaces on Cartesian grids, which has the fourth-order accuracy in the maximum norm in both time and space directions. The space variable is discretized by a high-order compact （HOC） difference scheme with correction terms added at the irregular points. The time derivative is integrated by a Crank-Nicolson and alternative direction implicit （ADI） scheme. In this case, the time accuracy is just second-order. The Richardson extrapolation method is used to improve the time accuracy to fourth-order. The numerical results confirm the convergence order and the efficiency of the method.

Superconvergence of Direct Discontinuous Galerkin Methods:Eigen-structure Analysis Based on Fourier Approach

This paper investigates superconvergence properties of the direct discontinuous Galerkin(DDG)method with interface corrections and the symmetric DDG method for diffusion equations.We apply the Fourier analysis technique to symbolically compute eigenvalues and eigenvectors of the amplification matrices for both DDG methods with different coefficient settings in the numerical fluxes.Based on the eigen-structure analysis,we carry out error estimates of the DDG solutions,which can be decomposed into three parts:(i)dissipation errors of the physically relevant eigenvalue,which grow linearly with the time and are of order 2k for P^(k)(k=2,3)approximations;(ii)projection error from a special projection of the exact solution,which is decreasing over the time and is related to the eigenvector corresponding to the physically relevant eigenvalue;(iii)dissipative errors of non-physically relevant eigenvalues,which decay exponentially with respect to the spatial mesh sizeΔx.We observe that the errors are sensitive to the choice of the numerical flux coefficient for even degree P^(2)approximations,but are not for odd degree P^(3)approximations.Numerical experiments are provided to verify the theoretical results.

Immersed Interface Finite Element Methods for Elasticity Interface Problems with Non-Homogeneous Jump Conditions

In this paper,a class of new immersed interface finite element methods (IIFEM) is developed to solve elasticity interface problems with homogeneous and non-homogeneous jump conditions in two dimensions.Simple non-body-fitted meshes are used.For homogeneous jump conditions,both non-conforming and conforming basis functions are constructed in such a way that they satisfy the natural jump conditions. For non-homogeneous jump conditions,a pair of functions that satisfy the same non-homogeneous jump conditions are constructed using a level-set representation of the interface.With such a pair of functions,the discontinuities across the interface in the solution and flux are removed;and an equivalent elasticity interface problem with homogeneous jump conditions is formulated.Numerical examples are presented to demonstrate that such methods have second order convergence.

An Adaptive Mesh Refinement Strategy for Immersed Boundary/Interface Methods

An adaptive mesh refinement strategy is proposed in this paper for the Immersed Boundary and Immersed Interface methods for two-dimensional elliptic interface problems involving singular sources.The interface is represented by the zero level set of a Lipschitz functionϕ(x,y).Our adaptive mesh refinement is done within a small tube of|ϕ(x,y)|δwith finer Cartesian meshes.The discrete linear system of equations is solved by a multigrid solver.The AMR methods could obtain solutions with accuracy that is similar to those on a uniform fine grid by distributing the mesh more economically,therefore,reduce the size of the linear system of the equations.Numerical examples presented show the efficiency of the grid refinement strategy.

Numerical study of air cavity characteristics of bow wave breaking of KCS ship under different speeds

Ship bow wave breaking contains complex flow mechanism,which is very important for ship performance.In this study,a practical numerical simulation scheme for bow wave breaking is proposed and the scheme is applied to the simulation of bow wave breaking of KCS ship model with Fr=0.26,0.30,0.35,0.40,analyzing the impact of speed on the bow wave breaking.The results indicate that an increase in speed leads to a significant rise in viscous pressure resistance and more pronounced bow wave breaking.Moreover,it is found that the traditional wave height function in OpenFOAM is not suitable for detailed studies of bow wave breaking.This study extracts different free surfaces through top and bottom views to further analyze the free surface overturning,droplet splashing,and cavity entrainment in bow wave breaking.Additionally,the spatial and temporal distribution of cavities at Fr=0.40 is analyzed,revealing that cavity distribution is closely related to vortex structures and exhibits a periodic pulsation characteristic of approximately 12 s.

Quantitative analysis on influencing factors for interface propagation-based thermal conductivity measurement method during solid-liquid transition

The recently proposed interface propagation-based method has shown its advantages in obtaining the thermal conductivity of phase change materials during solid-liquid transition over conventional techniques. However, in previous investigation, the analysis on the measurement error was qualitative and only focused on the total effects on the measurement without decoupling the influencing factors. This paper discusses the effects of influencing factors on the measurement results for the interface propagation-based method. Numerical simulations were performed to explore the influencing factors, namely model simplification, subcooling and natural convection, along with their impact on the measurement process and corresponding measurement results. The numerical solutions were provided in terms of moving curves of the solid-liquid interface and the predicted values of thermal conductivity. Results indicated that the impact of simplified model was strongly dependent on Stefan number of the melting process. The degree of subcooling would lead to underestimated values for thermal conductivity prediction. The natural convection would intensify the heat transfer rate in the liquid region, thereby overestimating the obtained results of thermal conductivity. Correlations and experimental guidelines are provided. The relative errors are limited in ±1.5%,±3%and ±2% corresponding to the impact of simplified model, subcooling and natural convection, respectively.

The Algebraic Immersed Interface and Boundary Method for Elliptic Equations with Jump Conditions

A new simple fictitious domain method, the algebraic immersed interface and boundary (AIIB) method, is presented for elliptic equations with immersed interface conditions. This method allows jump conditions on immersed interfaces to be discretized with a good accuracy on a compact stencil. Auxiliary unknowns are created at existing grid locations to increase the degrees of freedom of the initial problem. These auxiliary unknowns allow imposing various constraints to the system on interfaces of complex shapes. For instance, the method is able to deal with immersed interfaces for elliptic equations with jump conditions on the solution or discontinuous coefficients with a second order of spatial accuracy. As the AIIB method acts on an algebraic level and only changes the problem matrix, no particular attention to the initial discretization is required. The method can be easily implemented in any structured grid code and can deal with immersed boundary problems too. Several validation problems are presented to demonstrate the interest and accuracy of the method.

Sharp interface direct forcing immersed boundary methods： A summary of some algorithms and applications

Body-fitted mesh generation has long been the bottleneck of simulating fluid flows involving complex geometries. Immersed boundary methods are non-boundary-conforming methods that have gained great popularity in the last two decades for their simplicity and flexibility, as well as their non-compromised accuracy. This paper presents a summary of some numerical algori- thms along the line of sharp interface direct forcing approaches and their applications in some practical problems. The algorithms include basic Navier-Stokes solvers, immersed boundary setup procedures, treatments of stationary and moving immersed bounda- ries, and fluid-structure coupling schemes. Applications of these algorithms in particulate flows, flow-induced vibrations, biofluid dynamics, and free-surface hydrodynamics are demonstrated. Some concluding remarks are made, including several future research directions that can further expand the application regime of immersed boundary methods.

A sharp interface approach for cavitation modeling using volume-of-fluid and ghost-fluid methods

This paper describes a novel sharp interface approach for modeling the cavitation phenomena in incompressible viscous flows. A one-field formulation is adopted for the vapor-liquid two-phase flow and the interface is tracked using a volume of fluid（VOF） method. Phase change at the interface is modeled using a simplification of the Rayleigh-Plesset equation. Interface jump conditions in velocity and pressure field are treated using a level set based ghost fluid method. The level set function is constructed from the volume fraction function. A marching cubes method is used to compute the interface area at the interface grid cells. A parallel fast marching method is employed to propagate interface information into the field. A description of the equations and numerical methods is presented. Results for a cavitating hydrofoil are compared with experimental data.

Interfacial phase formation of Al-Cu bimetal by solid-liquid casting method

The solid-liquid method was used to prepare the continuous casting of copper cladding aluminium by liquid aluminum alloy and solid copper, and the interfacial phase formation of Al-Cu bimetal at different pouring temperatures(700, 750, 800 oC) was investigated by means of metallograph, scanning electron microscopy(SEM) and energy dispersive spectrometry(EDS) methods. The results showed that the pouring temperature of aluminum melt had an important influence on the element diffusion of Cu from the solid Cu to Al alloy melt and the reactions between Al and Cu, as well as the morphology of the Al-Cu interface. When the pouring temperature was 800 oC, there were abundant Al-Cu intermetallic compounds(IMCs) near the interface. However, a lower pouring temperature(700 oC) resulted in the formation of cavities which was detrimental to the bonding and mechanical properties. Under the conditions in this study, the good metallurgical bonding of Al-Cu was achieved at a pouring temperature of 750 oC.

Deformation tests and failure process analysis of an anchorage structure

In order to study the failure process of an anchorage structure and the evolution law of the body＇s defor- mation field, anchor push-out tests were carried out based on digital speckle correlation methods （DSCM）. The stress distribution of the anchorage interface was investigated using the particle flow numerical simulation method. The results indicate that there are three stages in the deformation and fail- ure process of an anchorage structure： elastic bonding stage, a de-bonding stage and a failure stage. The stress distribution in the interface controls the stability of the structure. In the elastic bonding stage, the shear stress peak point of the interface is close to the loading end, and the displacement field gradually develops into a ＂V＂ shape, in the de-bonding stage, there is a shear stress plateau in the center of the anchorage section, and shear strain localization begins to form in the deformation field. In the failure stage, the bonding of the interface fails rapidly and the shear stress peak point moves to the anchorage free end. The anchorage structure moves integrally along the macro-cracl~ The de-bonding stage is a research focus in the deformation and failure process of an anchorage structure, and plays an important guiding role in roadway support design and prediction of the stability of the surrounding rock.

Analytical methods for nano-bio interface interactions

Knowledge on the interactions between engineered nanomaterials(ENMs) and biological systems is critical both for the assessment of biological effects of ENMs and for the rational design of ENM-based products. However, probing the events that occur at the nano-bio interface remains extremely challenging due to their complex and dynamic nature. So far, the understanding of mechanisms underlying nano-bio interactions has been mainly limited by the lack of proper analytical techniques with sufficient sensitivity, selectivity and resolution for characterization of nano-bio interface events. Moreover, many classic bioanalytical methods are not suitable for direct measurement of nano-bio interface interactions. These have made establishing analytical methodologies for systematic and comprehensive study of nano-bio interface one of the most focused areas in nanobiology. In this review we have discussed some representative developments regarding analytical techniques for nano-bio interface characterization, including the improvements of traditional methods and the emergence of powerful new technologies. These developments have allowed ultrasensitive, real-time analysis of interactions between ENMs and biomolecules, transformations of ENMs in biological environment, and impacts of ENMs on living systems on molecular or cellular level.

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.

Interface Finite Volume Method with Low Calculating Amount for Step Size Varying Electro-Quasistatic Field Problem

In this paper,a novel numerical method called interface finite volume method(I-FVM)for calculation of step-varying Electro-quasistatic(EQS)field is proposed.First,the principle of I-FVM is derived.Then,with numerical example of double layers parallel plate structure under step voltage which has an analytical solution,effectiveness and correctness of the I-FVM are verified.It can be found the calculating time of I-FVM is only 30%of normal FVM without decreasing accuracy during the whole calculating process.Furthermore,an engineering example about the electric field of DBC(Direct Bonding Copper)structure in a high voltage IGBT device is given.It can be found that accuracy of the I-FVM is the same as normal FVM,while time cost of I-FVM is only 20.8%of normal FVM.At last,the I-FVM is extended to one dimension based on the two-direction tri-diagonal matrix algorithm(TDMA)method given in this paper which can save processing of LU decomposition compared to one-dimensional traditional TDMA.In conclusion,the novel method called I-FVM proposed in this paper can decrease calculating amount for a step size varying electro-quasistatic field calculation problem.It may be a good method for large-scale EQS field calculation.

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.

A Hybrid Immersed Interface Method for Driven Stokes Flow in an Elastic Tube

We present a hybrid numerical method for simulating fluid flow through a compliant,closed tube,driven by an internal source and sink.Fluid is assumed to be highly viscous with its motion described by Stokes flow.Model geometry is assumed to be axisymmetric,and the governing equations are implemented in axisymmetric cylindrical coordinates,which capture 3D flow dynamics with only 2D computations.We solve the model equations using a hybrid approach:we decompose the pressure and velocity fields into parts due to the surface forcings and due to the source and sink,with each part handled separately by means of an appropriate method.Because the singularly-supported surface forcings yield an unsmooth solution,that part of the solution is computed using the immersed interface method.Jump conditions are derived for the axisymmetric cylindrical coordinates.The velocity due to the source and sink is calculated along the tubular surface using boundary integrals.Numerical results are presented that indicate second-order accuracy of the method.

Phase Field Models Versus Parametric Front Tracking Methods: Are They Accurate and Computationally Efficient?

We critically compare the practicality and accuracy of numerical approximations of phase field models and sharp interface models of solidification.Here we focus on Stefan problems,and their quasi-static variants,with applications to crystal growth.New approaches with a high mesh quality for the parametric approximations of the resulting free boundary problems and new stable discretizations of the anisotropic phase field system are taken into account in a comparison involving benchmark problems based on exact solutions of the free boundary problem.

Interface stress element method and its application in analysis of anti-sliding stability of gravity dam

The rigid body limit equilibrium method （LEM） and the nonlinear finite element method （NFEM） are often used in the analysis of anti-sliding stability of gravity dam. But LEM cannot reflect the process of progressive instability and mechanical mecha- nism on failure for rock mass while NFEM is difficult to use to solve the displacement discontinuity of weak structural plane. Combining the research with Xiangjiaba Hydropower Station project, the analysis of anti-sliding stability for segment 12# of the dam has been carried out using interface stress element method （ISEM）. The results can reflect the most dangerous location, the scope and distribution of failure zone in weak structural plane, and present the process of progressive failure in dam foun- dation as well as the safety coefficient of possible sliding body. These achievements provide an important technical reference for dam foundation treatment measures. The computational results show that ISEM can naturally describe discontinuous de- formation of rock mass such as dislocation, openness and sliding. Besides, this method is characterized by good adaptability, convenient calculation and high compatibility, thus it is regarded as an effective way to make an analysis of anti-sliding stabil- ity of gravity dam

Multi-domain Spectral Immersed Interface Method for Solving Elliptic Equation with a Global Description of Discontinuous Functions

This paper presents the extension of the global description approach of a discontinuous function, which is proposed in the previous paper, to a spectral domain decomposition method. This multi-domain spectral immersed interlace method（IIM） divides the whole computation domain into the smooth and discontinuous parts. Fewer points on the smooth domains are used via taking advantage of the high accuracy property of the spectral method, but more points on the discontinuous domains are employed to enhance the resolution of the calculation. Two that the domain decomposition technique can placed around the discontinuity. The present reached, in spite of the enlarged computational discontinuous problems are tested to verify the present method. The results show reduce the error of the spectral IIM, especially when more collocation points are method is t：avorable for the reason that the same level of the accuracy can be domain.

A Simple 3D Immersed InterfaceMethod 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.