Oscillation properties of eigenfunctions for Sturm-Liouville problems with interface conditions via Prufer transformation

A class of Sturm-Liouville problems with discontinuity is studied in this paper.The oscillation properties of eigenfunctions for Sturm-Liouville problems with interface conditions are obtained.The main method used in this paper is based on Prufer transformation,which is different from the classical ones.Moreover,we give two examples to verify our main results.

LOCAL AND PARALLEL FINITE ELEMENT METHOD FOR THE MIXED NAVIER-STOKES/DARCY MODEL WITH BEAVERS-JOSEPH INTERFACE CONDITIONS

In this paper, we consider the mixed Navier-Stokes/Darcy model with BeaversJoseph interface conditions. Based on two-grid discretizations, a local and parallel finite element algorithm for this mixed model is proposed and analyzed. Optimal errors are obtained and numerical experiments are presented to show the efficiency and effectiveness of the local and parallel finite element algorithm.

BOUNDARY CONTROL OF TWO PDE'S SEPARATED BY INTERFACE CONDITIONS

This paper discusses the null boundary controllability of two PDE's,modeling a compositesolid with different physical properties in each layer.Interface conditions are imposed.

Hierarchical Absorbing Interface Conditions for Wave Propagation on Non-Uniform Meshes

In this paper,we propose hierarchical absorbing interface conditions to solve the problem of wave propagation in domains with a non-uniform space discretization or grid size inhomogeneity using Pad´e Via Lanczos(PVL)method.The proposed interface conditions add an auxiliary variable in the wave system to eliminate the spurious reflection at the interface between regions with different mesh sizes.The auxiliary variable with proper boundary condition can suppress the spurious reflection by cancelling the boundary source term produced by the space inhomogeneity in variational perspective.The new hierarchical interface conditions with the help of PVL implementation can effectively reduce the degree of freedom in solving the wave propagation problem.

A Finite Volume Method Preserving Maximum Principle for the Conjugate Heat Transfer Problems with General Interface Conditions

In this paper,we present a unified finite volume method preserving discrete maximum principle(DMP)for the conjugate heat transfer problems with general interface conditions.We prove the existence of the numerical solution and the DMP-preserving property.Numerical experiments show that the nonlinear iteration numbers of the scheme in[24]increase rapidly when the interfacial coefficients decrease to zero.In contrast,the nonlinear iteration numbers of the unified scheme do not increase when the interfacial coefficients decrease to zero,which reveals that the unified scheme is more robust than the scheme in[24].The accuracy and DMP-preserving property of the scheme are also veri ed in the numerical experiments.

Absorbing Interface Conditions for the Simulation of Wave Propagation on Non-Uniform Meshes

We proposed absorbing interface conditions for the simulation of linear wave propagation on non-uniform meshes.Based on the superposition principle of second-order linear wave equations,we decompose the interface condition problem into two subproblems around the interface:for the first one the conventional artificial absorbing boundary conditions is applied,while for the second one,the local analytic solutions can be derived.The proposed interface conditions permit a two-way transmission of low-frequency waves across mesh interfaces which can be supported by both coarse and fine meshes,and perform a one-way absorption of high-frequency waves which can only be supported by fine meshes when they travel from fine mesh regions to coarse ones.Numerical examples are presented to illustrate the efficiency of the proposed absorbing interface conditions.

A Normal Mode Stability Analysis of Numerical Interface Conditions for Fluid/Structure Interaction

In multi physics computations where a compressible fluid is coupled with a linearly elastic solid,it is standard to enforce continuity of the normal velocities and of the normal stresses at the interface between the fluid and the solid.In a numerical scheme,there are many ways that velocity-and stress-continuity can be enforced in the discrete approximation.This paper performs a normal mode stability analysis of the linearized problem to investigate the stability of different numerical interface conditions for a model problem approximated by upwind type finite difference schemes.The analysis shows that depending on the ratio of densities between the solid and the fluid,some numerical interface conditions are stable up to the maximal CFL-limit,while other numerical interface conditions suffer from a severe reduction of the stable CFL-limit.The paper also presents a new interface condition,obtained as a simplified characteristic boundary condition,that is proved to not suffer from any reduction of the stable CFL-limit.Numerical experiments in one space dimension show that the new interface condition is stable also for computations with the non-linear Euler equations of compressible fluid flow coupled with a linearly elastic solid.

Stability of Finite Difference Discretizations of Multi-Physics Interface Conditions

We considermulti-physics computationswhere theNavier-Stokes equations of compressible fluid flow on some parts of the computational domain are coupled to the equations of elasticity on other parts of the computational domain.The different subdomains are separated by well-defined interfaces.We consider time accurate computations resolving all time scales.For such computations,explicit time stepping is very efficient.We address the issue of discrete interface conditions between the two domains of different physics that do not lead to instability,or to a significant reduction of the stable time step size.Finding such interface conditions is non-trivial.We discretize the problem with high order centered difference approximations with summation by parts boundary closure.We derive L2 stable interface conditions for the linearized one dimensional discretized problem.Furthermore,we generalize the interface conditions to the full non-linear equations and numerically demonstrate their stable and accurate performance on a simple model problem.The energy stable interface conditions derived here through symmetrization of the equations contain the interface conditions derived through normal mode analysis by Banks and Sj¨ogreen in[8]as a special case.

Predicting impact forces on pipelines from deep-sea fluidized slides:A comprehensive review of key factors

Deep-sea pipelines play a pivotal role in seabed mineral resource development,global energy and resource supply provision,network communication,and environmental protection.However,the placement of these pipelines on the seabed surface exposes them to potential risks arising from the complex deep-sea hydrodynamic and geological environment,particularly submarine slides.Historical incidents have highlighted the substantial damage to pipelines due to slides.Specifically,deep-sea fluidized slides(in a debris/mud flow or turbidity current physical state),characterized by high speed,pose a significant threat.Accurately assessing the impact forces exerted on pipelines by fluidized submarine slides is crucial for ensuring pipeline safety.This study aimed to provide a comprehensive overview of recent advancements in understanding pipeline impact forces caused by fluidized deep-sea slides,thereby identifying key factors and corresponding mechanisms that influence pipeline impact forces.These factors include the velocity,density,and shear behavior of deep-sea fluidized slides,as well as the geometry,stiffness,self-weight,and mechanical model of pipelines.Additionally,the interface contact conditions and spatial relations were examined within the context of deep-sea slides and their interactions with pipelines.Building upon a thorough review of these achievements,future directions were proposed for assessing and characterizing the key factors affecting slide impact loading on pipelines.A comprehensive understanding of these results is essential for the sustainable development of deep-sea pipeline projects associated with seabed resource development and the implementation of disaster prevention measures.

The coupled deep neural networks for coupling of the Stokes and Darcy–Forchheimer problems

We present an efficient deep learning method called coupled deep neural networks(CDNNs) for coupling of the Stokes and Darcy–Forchheimer problems. Our method compiles the interface conditions of the coupled problems into the networks properly and can be served as an efficient alternative to the complex coupled problems. To impose energy conservation constraints, the CDNNs utilize simple fully connected layers and a custom loss function to perform the model training process as well as the physical property of the exact solution. The approach can be beneficial for the following reasons: Firstly, we sample randomly and only input spatial coordinates without being restricted by the nature of samples.Secondly, our method is meshfree, which makes it more efficient than the traditional methods. Finally, the method is parallel and can solve multiple variables independently at the same time. We present the theoretical results to guarantee the convergence of the loss function and the convergence of the neural networks to the exact solution. Some numerical experiments are performed and discussed to demonstrate performance of the proposed method.

Central Schemes and Second Order Boundary Conditions for 1D Interface and Piston Problems in Lagrangian Coordinates

We study high-resolution central schemes in Lagrangian coordinates for the one-dimensional system of conservation laws describing the evolution of two gases in slab geometry separated by an interface.By using Lagrangian coordinates,the interface is transformed to a fixed coordinate in the computational domain and,as a consequence,the movement of the interface is obtained as a byproduct of the numerical solution.The main contribution is the derivation of a special equation of state to be imposed at the interface in order to avoid non-physical oscillations.Suitable boundary conditions at the piston that guarantee second order convergence are described.We compare the solution of the piston problem to other results available in the literature and to a reference solution obtained within the adiabatic approximation.A shock-interface interaction problem is also treated.The results on these tests are in good agreement with those obtained by other methods.

Numerical Analysis of Asphalt Pavements under Moving Wheel Loads

The responses of the pavement in service are the basis for the design of the semi-rigid base course asphalt pavement. Due to the dynamic characteristics of wheel loacis and the temperature loads, the dynamic response analysis is very significant. In this article, the dynamic analysis of asphalt pavement under moving wheel loads is carried out using finite dement method canpled with non-reflective boundary method. The influences of the base modulus, thickness, the vehicle velocity, the tire pressure, and the contact condition at the interface are studied using parametric analysis. The results of numerical analysis show that it is not appropriate to simply increase the base modulus or thickness in the design. It would be beneficial if the base design is optimized synthetically. The increase of damping is also beneficial to the pavements because of the surface deflection and the stresses declination. Furthermore, the good contact condition at the interface results in good performance because it combines every layer of the pavement to work together. As overload aggravates the working condition of the pavement, it is not allowed.

A, ■-Ω METHOD FOR 3-D EDDY CURRENT ANALYSIS

After the field equations and the snonumuoo conditions between the interfaces for 3D eddy current problems Under various gauges were discussed, it was pointed cut in this paper that using the magnetic vector potential A. the electric scalar potential and Coulomb gauge △ .A = 0 in eddy current regions and using the magntetic scalar potential Ω in the non-conducting regions are more suitable. All field equations, the boundary conditions, the interface continuity conditions and the corresponding variational principle of this method are also given

Multi-domain physics-informed neural network for solving heat conduction and conjugate natural convection with discontinuity of temperature gradient on interface

Deep learning has been increasingly recognized as a promising tool in solving kinds of physical problems beyond powerful approximations. A multi-domain physics-informed neural network(mPINN) is proposed to solve the non-uniform heat conduction and conjugate natural convection with the discontinuity of temperature gradient on the interface. Local radial basis function method(LRBF) is applied to compute the case without the analytical solution and is regarded as the benchmark solver.Each physical domain matches a private neural network and all neural networks are connected by the shared information of temperature and heat flux on the interface. Joint training and separate training are utilized to minimize the loss function, which usually consists of the residual of boundary conditions, interface conditions and governing equations. Joint training minimizes the sum of all losses from neural networks with one shared optimizer, while separate training owns its private optimizer. Local adaptive activation function(LAAF) is used to accelerate the convergence and acquire a lower loss value when compared with its fixed counterpart. The numerical experiments on three types of residual points, uniform, Gauss-Lobatto and random, are conducted and it can be concluded that the uniform residual points can obtain the most accurate solution than the random and Gauss-Lobatto. Joint training is more accurate than the separate training when the number of residual points is relatively small,while the separate training performs better than the joint training for the large number of residual points. Numerous test cases on multi-domain heat transfer and fluid flow show the accuracy of the proposed m PINN. Local and global heat transfer rates show good agreements with the results from LRBF. Excepting the forward problems, the thermal conductivity ratio, the constant source and the characteristic parameters of natural convection are accurately learned from sparsely distributed data points.

Effects of sulfate-reducing bacteria on methylmercury at the sediment–water interface

Sediment cores（containing sediment and overlying water） from Baihua Reservoir（SW China）were cultured under different redox conditions with different microbial activities, to understand the effects of sulfate-reducing bacteria（SRB） on mercury（Hg） methylation at sediment–water interfaces. Concentrations of dissolved methyl mercury（DMe Hg） in the overlying water of the control cores with bioactivity maintained（BAC） and cores with only sulfate-reducing bacteria inhibited（SRBI） and bacteria fully inhibited（BACI） were measured at the anaerobic stage followed by the aerobic stage. For the BAC and SRBI cores, DMe Hg concentrations in waters were much higher at the anaerobic stage than those at the aerobic stage, and they were negatively correlated to the dissolved oxygen concentrations（r =- 0.5311 and r =- 0.4977 for BAC and SRBI, respectively）. The water DMe Hg concentrations of the SRBI cores were 50% lower than those of the BAC cores, indicating that the SRB is of great importance in Hg methylation in sediment–water systems, but there should be other microbes such as iron-reducing bacteria and those containing specific gene cluster（hgc AB）, besides SRB,causing Hg methylation in the sediment–water system.

High-Order and High Accurate CFD Methods and Their Applications for Complex Grid Problems

The purpose of this article is to summarize our recent progress in high-order and high accurate CFD methods for flow problems with complex grids as well as to discuss the engineering prospects in using these methods.Despite the rapid development of high-order algorithms in CFD,the applications of high-order and high accurate methods on complex configurations are still limited.One of the main reasons which hinder the widely applications of thesemethods is the complexity of grids.Many aspects which can be neglected for low-order schemes must be treated carefully for high-order ones when the configurations are complex.In order to implement highorder finite difference schemes on complex multi-block grids,the geometric conservation lawand block-interface conditions are discussed.A conservativemetricmethod is applied to calculate the grid derivatives,and a characteristic-based interface condition is employed to fulfil high-order multi-block computing.The fifth-order WCNS-E-5 proposed by Deng[9,10]is applied to simulate flows with complex grids,including a double-delta wing,a transonic airplane configuration,and a hypersonic X-38 configuration.The results in this paper and the references show pleasant prospects in engineering-oriented applications of high-order schemes.

A Robin-Robin Domain Decomposition Method for a Stokes-Darcy Structure Interaction with a Locally Modified Mesh

A new numerical method based on locally modified Cartesian meshes is proposed for solving a coupled system of a fluid flow and a porous media flow.The fluid flow is modeled by the Stokes equations while the porous media flow is modeled by Darcy’s law.The method is based on a Robin-Robin domain decomposition method with a Cartesian mesh with local modifications near the interface.Some computational examples are presented and discussed.

Boundary Homogenization in the Spontaneous Potential Well-Logging

Motivated by the study on the spontaneous potential well-logging, this paper deals with the homogenization of boundary conditions for a class of elliptic problems with jump interface conditions.

A Weighted Multi-Domain Spectral Penalty Method with Inhomogeneous Grid for Supersonic Injective Cavity Flows

In[J.Comput.Phys.192(1),pp.325-354(2003)],we have developed a multidomain spectral method with stable and conservative penalty interface conditions for the numerical simulation of supersonic reactive recessed cavity flows with homogeneous grid.In this work,the previously developed methodology is generalized to inhomogeneous grid to simulate the two dimensional supersonic injector-cavity system.Non-physical modes in the solution generated at the domain interfaces due to the spatial grid inhomogeneity are minimized using the new weighted multi-domain spectral penalty method.The proposed method yields accurate and stable solutions of the injector-cavity system which agree well with experiments qualitatively.Through the direct numerical simulation of the injector-cavity system using the weighted method,the geometric effect of the cavity wall on pressure fluctuations is investigated.It is shown that the recessed slanted cavity attenuates pressure fluctuations inside cavity enabling the cavity to act potentially as a stable flameholder for scramjet engine.

Extension of Near-Wall Domain Decomposition to Modeling Flows with Laminar-Turbulent Transition

The near-wall domain decomposition method(NDD)has proved to be very efficient for modeling near-wall fully turbulent flows.In this paper the NDD is extended to non-equilibrium regimeswith laminar-turbulent transition(LTT)for the first time.The LTT is identified with the use of the e^(N)-method which is applied to both incompressible and compressible flows.TheNDD ismodified to take into account LTT in an efficientway.In addition,implementation of the intermittency expands the capabilities of NDD to model non-equilibrium turbulent flows with transition.Performance of the modified NDD approach is demonstrated on various test problems of subsonic and supersonic flows past a flat plate,a supersonic flow over a compression corner and a planar shock wave impinging on a turbulent boundary layer.The results of modeling with and without decomposition are compared in terms of wall friction and show good agreement with each other while NDD significantly reducing computational resources needed.It turns out that the NDD can reduce the computational time as much as three times while retaining practically the same accuracy of prediction.