# Browsing by Author "Xu, S"

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Item Open Access A Level Set Method for the Simulation of Moving Contact Lines in Three Dimensions(Communications in Computational Physics, 2022-01-01) Zhao, Q; Xu, S; Ren, WWe propose an efficient numerical method for the simulation of the two-phase flows with moving contact lines in three dimensions. The mathematical model consists of the incompressible Navier-Stokes equations for the two immiscible fluids with the standard interface conditions, the Navier slip condition along the solid wall, and a contact angle condition (Ren et al. (2010) [28]). In the numerical method, the governing equations for the fluid dynamics are coupled with an advection equation for a level-set function. The latter models the dynamics of the fluid interface. Following the standard practice, the interface conditions are taken into account by introducing a singular force on the interface in the momentum equation. This results in a single set of governing equations in the whole fluid domain. Similarly, the contact angle condition is imposed by introducing a singular force, which acts in the normal direction of the contact line, into the Navier slip condition. The new boundary condition, which unifies the Navier slip condition and the contact angle condition, is imposed along the solid wall. The model is solved using the finite difference method. Numerical results are presented for the spreading of a droplet on both homogeneous and inhomogeneous solid walls, as well as the dynamics of a droplet on an inclined plate under gravity.Item Open Access A phase field model for compressible immiscible fluids with a new equation of state(International Journal of Multiphase Flow, 2022-04-01) Dai, H; Xu, S; Xu, Z; Zhao, N; Zhu, CX; Zhu, CIn this paper, we propose a new compressible phase field model of two different immiscible fluid components, in which the density of each phase is variable. In order to establish the compressible phase field model, a new P-V-T equation of state is introduced to solve the pressure. The new model is derived by the physics law of conservation, and conforms to the second law of thermodynamics. The model adopts an innovative expression of Helmholtz free energy, taking into account the new state equation of pressure and the varying material properties of each phase. A high-order accurate numerical scheme is introduced to solve the model equations. The convection terms of the governing equations are discretized by the fifth-order WENO scheme, and the residual terms are discretized by the Lax–Friedrichs method. Finally, the reliability and validity of the compressible two-phase model are verified by numerical simulations.Item Open Access A phase field model for mass transport with semi-permeable interfaces(Journal of Computational Physics, 2022-09-01) Qin, Y; Huang, H; Zhu, Y; Liu, C; Xu, SIn this paper, a thermaldynamical consistent model for mass transfer across permeable moving interfaces is proposed by using the energy variation method. We consider a restricted diffusion problem where the flux across the interface depends on its conductance and the difference of the concentration on each side. The diffusive interface phase-field framework used here has several advantages over the sharp interface method. First of all, explicit tracking of the interface is no longer necessary. Secondly, interfacial conditions can be incorporated with a variable diffusion coefficient. Finally, topological changes of interfaces can be handed easily. A detailed asymptotic analysis confirms the diffusive interface model converges to the existing sharp interface model as the interface thickness goes to zero. An energy stable numerical scheme is developed to solve this highly nonlinear coupled system.Numerical simulations first illustrate the consistency of theoretical results on the sharp interface limit. Then a convergence study and energy decay test are conducted to ensure the efficiency and stability of the numerical scheme. To illustrate the effectiveness of our phase-field approach, several examples are provided, including a study of a two-phase mass transfer problem where droplets with deformable interfaces are suspended in a moving fluid.Item Open Access An energetic variational approach for ION transport(Communications in Mathematical Sciences, 2014-03-06) Xu, S; Sheng, P; Liu, CThe transport and distribution of charged particles are crucial in the study of many physical and biological problems. In this paper, we employ an Energy Variational Approach to derive the coupled Poisson-Nernst-Planck-Navier-Stokes system. All of the physics is included in the choices of corresponding energy law and kinematic transport of particles. The variational derivations give the coupled force balance equations in a unique and deterministic fashion. We also discuss the situations with different types of boundary conditions. Finally, we show that the Onsager's relation holds for the electrokinetics, near the initial time of a step function applied field. © 2014 International Press.Item Open Access An Energy Stable $C^0$ Finite Element Scheme for A Phase-Field Model of Vesicle Motion and Deformation(SIAM Journal on Scientific Computing, 2022-01) Shen, L; Xu, Z; Lin, P; Huang, H; Xu, SItem Open Access An energy stable C^{0}finite element scheme for a quasi-incompressible phase-field model of moving contact line with variable density(Journal of Computational Physics, 2020-03-15) Shen, L; Huang, H; Lin, P; Song, Z; Xu, SIn this paper, we focus on modeling and simulation of two-phase flow problems with moving contact lines and variable density. A thermodynamically consistent phase-field model with general Navier boundary condition is developed based on the concept of quasi-incompressibility and the energy variational method. A mass conserving C0 finite element scheme is proposed to solve the PDE system. Energy stability is achieved at the fully discrete level. Various numerical results confirm that the proposed scheme for both P1 element and P2 element are energy stable.Item Open Access Analysis of main risk factors causing stroke in Shanxi Province based on machine learning models(Informatics in Medicine Unlocked, 2021-01-01) Liu, J; Sun, Y; Ma, J; Tu, J; Deng, Y; He, P; Li, R; Hu, F; Huang, H; Zhou, X; Xu, SBackground: In China, stroke has been the first leading cause of death in recent years. It is a major cause of long-term physical and cognitive impairment, which bring great pressure on the National Public Health System. On the other hand, China is a big country, evaluation of the risk of getting stroke is important for the prevention and treatment of stroke in China. Methods: A data set with 2000 hospitalized stroke patients in 2018 and 27583 residents during the year 2017 to 2020 is analyzed in this study. With the cleaned data, three models on stroke risk levels are built by using machine learning methods. The importance of “8+2” factors from China National Stroke Prevention Project (CSPP) is evaluated via decision tree and random forest models. The importance of more detailed features and their SHAP values are evaluated and ranked via random forest model. Furthermore, a logistic regression model is applied to evaluate the probability of getting stroke for different risk levels. Results: Among all “8+2” risk factors of getting stroke, the decision tree model reveals that top three factors are Hypertension (0.4995), Physical Inactivity (0.08486) and Diabetes Mellitus (0.07889), and the random forest model shows that top three factors are Hypertension (0.3966), Hyperlipidemia (0.1229) and Physical Inactivity (0.1146). In addition to “8+2” factors the importance of features for lifestyle information, demographic information and medical measurement are evaluated via random forest model. It shows that top five features are Systolic Blood Pressure (SBP) (0.3670), Diastolic Blood Pressure (DBP) (0.1541), Physical Inactivity (0.0904), Body Mass Index (BMI) (0.0721) and Fasting Blood Glucose (FBG)(0.0531). SHAP values show that DBP, Physical Inactivity, SBP, BMI, Smoking, FBG, and Triglyceride(TG) are positively correlated to the risk of getting stroke. High-density Lipoprotein (HDL) is negatively correlated to the risk of getting stroke. Combining with the data of 2000 hospitalized stroke patients, the logistic regression model shows that the average probabilities of getting stroke are 7.20%±0.55% for the low-risk level patients, 19.02%±0.94% for the medium-risk level patients and 83.89%±0.97% for the high-risk level patients. Conclusion: Based on the census data from Shanxi Province, we investigate stroke risk factors and their ranking. It shows that Hypertension, Physical Inactivity, and Overweight are ranked as the top three high stroke risk factors in Shanxi. The probability of getting a stroke is also estimated through our interpretable machine learning methods.Item Open Access Behavior of different numerical schemes for random genetic drift(BIT Numerical Mathematics, 2019-09-01) Xu, S; Chen, M; Liu, C; Zhang, R; Yue, XIn the problem of random genetic drift, the probability density of one gene is governed by a degenerated convection-dominated diffusion equation. Dirac singularities will always be developed at boundary points as time evolves, which is known as the fixation phenomenon in genetic evolution. Three finite volume methods: FVM1-3, one central difference method: FDM1 and three finite element methods: FEM1-3 are considered. These methods lead to different equilibrium states after a long time. It is shown that only schemes FVM3 and FEM3, which are the same, preserve probability, expectation and positiveness and predict the correct probability of fixation. FVM1-2 wrongly predict the probability of fixation due to their intrinsic viscosity, even though they are unconditionally stable. Contrarily, FDM1 and FEM1-2 introduce different anti-diffusion terms, which make them unstable and fail to preserve positiveness.Item Open Access Behavior of Raman modes in InPBi alloys under hydrostatic pressure(AIP Advances, 2019-03) Zheng, C; Wang, X; Ning, J; Ding, K; Sun, B; Wang, S; Xu, SItem Open Access Derivation of a continuum model and the energy law for moving contact lines with insoluble surfactants(Physics of Fluids, 2014-06-05) Zhang, Z; Xu, S; Ren, WA continuous model is derived for the dynamics of two immiscible fluids with moving contact lines and insoluble surfactants based on thermodynamic principles. The continuum model consists of the Navier-Stokes equations for the dynamics of the two fluids and a convection-diffusion equation for the evolution of the surfactant on the fluid interface. The interface condition, the boundary condition for the slip velocity, and the condition for the dynamic contact angle are derived from the consideration of energy dissipations. Different types of energy dissipations, including the viscous dissipation, the dissipations on the solid wall and at the contact line, as well as the dissipation due to the diffusion of surfactant, are identified from the analysis. A finite element method is developed for the continuum model. Numerical experiments are performed to demonstrate the influence of surfactant on the contact line dynamics. The different types of energy dissipations are compared numerically. © 2014 AIP Publishing LLC.Item Open Access Diffuse interface model for cell interaction and aggregation with Lennard-Jones type potential(Computer Methods in Applied Mechanics and Engineering, 2023-10-01) Shen, L; Lin, P; Xu, Z; Xu, SThis study introduces a phase-field model designed to simulate the interaction and aggregation of multicellular systems under flow conditions within a bounded spatial domain. The model incorporates a multi-dimensional Lennard-Jones potential to account for short-range repulsion and adhesive bonding between cells. To solve the governing equations while preserving energy law, a second-order accurate C0 finite element method is employed. The validity of the model is established through numerical tests, and experimental data from cell stretch tests is utilized for model calibration and validation. Additionally, the study investigates the impact of varying adhesion properties in red blood cells. Overall, this work presents a thermodynamically consistent and computationally efficient framework for simulating cell–cell and cell–wall interactions under flow conditions.Item Open Access Droplet dynamics: A phase-field model of mobile charges, polarization, and its leaky dielectric approximation(Physics of Fluids, 2023-08-01) Qin, Y; Huang, H; Song, Z; Xu, SThis paper presents a Poisson–Nernst–Planck–Navier–Stokes–Cahn–Hillard (PNP–NS–CH) model for an electrically charged droplet suspended in a viscous fluid under an external electric field. Our model incorporates spatial variations in electric permittivity and diffusion constants, as well as interfacial capacitance. Based on a time scale analysis, we derive two approximations of the original model: a dynamic model for the net charge (assuming unchanged conductance) and a leaky-dielectric model (assuming unchanged conductance and net charge). For the leaky-dielectric model, we perform a detailed asymptotic analysis to demonstrate the convergence of the diffusive-interface leaky-dielectric model to the sharp interface model as the interface thickness approaches zero. Numerical computations are conducted to validate the asymptotic analysis and demonstrate the model's effectiveness in handling topology changes, such as electro-coalescence. Our numerical results from these two approximation models reveal that the polarization force, induced by the spatial variation in electric permittivity perpendicular to the external electric field, consistently dominates the Lorentz force arising from the net charge. The equilibrium shape of droplets is determined by the interplay between these two forces along the direction of the electric field. Moreover, in the presence of interfacial capacitance, a local variation in effective permittivity results in the accumulation of counter-ions near the interface, leading to a reduction in droplet deformation. Our numerical solutions also confirm that the leaky-dielectric model is a reasonable approximation of the original PNP–NS–CH model when the electric relaxation time is sufficiently short. Both the Lorentz force and droplet deformation decrease significantly when the diffusion of net charge increases.Item Open Access Homogenization for chemical vapor infiltration process(Communications in Mathematical Sciences, 2017-01-01) Zhang, C; Bai, Y; Xu, S; Yue, XMulti-scale modeling and numerical simulations of the isothermal chemical vapor infiltration (CVI) process for the fabrication of carbon fiber reinforced silicon carbide (C/SiC) composites were presented in [Bai, Yue and Zeng, Commun. Comput. Phys., 7(3):597-612, 2010]. The homogenization theory, which played a fundamental role in the multi-scale algorithm, will be rigorously established in this paper. The governing system, which is a multi-scale reaction-diffusion equation, is different in the two stages of CVI process, so we will consider the homogenization for the two stages respectively. One of the main features is that the reaction only occurs on the surface of fiber, so it behaves as a singular surface source. The other feature is that in the second stage of the process when the micro pores inside the fiber bundles are all closed, the diffusion only occurs in the macro pores between fiber bundles and we face up with a problem in a locally periodic perforated domain.Item Open Access HOMOGENIZATION OF A DISCRETE NETWORK MODEL FOR CHEMICAL VAPOR INFILTRATION PROCESS(COMMUNICATIONS IN MATHEMATICAL SCIENCES, 2021) Xiao, C; Xu, S; Yuex, X; Zhang, C; Zhang, CItem Open Access Homogenization of thermal-hydro-mass transfer processes(Discrete and Continuous Dynamical Systems - Series S, 2015-02-01) Xu, S; Yue, XIn the repository, multi-physics processes are induced due to the long-time heat-emitting from the nuclear waste, which is modeled as a nonlinear system with oscillating coefficients. In this paper we first derive the homogenized system for the thermal-hydro-mass transfer processes by the technique of two-scale convergence, then present some error estimates for the first order expansions.Item Open Access Homogenization theory of ion transportation in multicellular tissue(Discrete and Continuous Dynamical Systems - B, 2023) Xiao, C; Huang, H; Xu, S; Yu, T; Yue, XItem Open Access Homogenization: In mathematics or physics?(Discrete and Continuous Dynamical Systems - Series S, 2016-10-01) Xu, S; Yue, X; Zhang, CIn mathematics, homogenization theory considers the limitations of the sequences of the problems and their solutions when a parameter tends to zero. This parameter is regarded as the ratio of the characteristic size between the micro scale and macro scale. So what is considered is a sequence of problems in axed domain while the characteristic size in micro scale tends to zero. But in the real physics or engineering situations, the micro scale of a medium isxed and can not be changed. In the process of homogenization, it is the size in macro scale which becomes larger and larger and tends to innity. We observe that the homogenization in physics is not equivalent to the homogenization in mathematics up to some simple rescaling. With some direct error estimates, we explain in what sense we can accept the homogenized problem as the limitation of the original real physical problems. As a byproduct, we present some results on the mathematical homogenization of some problems with source term being only weakly compacted in H1, while in standard homogenization theory, the source term is assumed to be at least compacted in H1. A real example is also given to show the validation of our observation and results.Item Open Access Modeling and Simulating Asymmetrical Conductance Changes in Gramicidin Pores(Computational and Mathematical Biophysics, 2014-01-01) Xu, S; Chen, M; Majd, S; Yue, X; Liu, CGramicidin A is a small and well characterized peptide that forms an ion channel in lipid membranes. An important feature of gramicidin A (gA) pore is that its conductance is affected by the electric charges near the its entrance. This property has led to the application of gramicidin A as a biochemical sensor for monitoring and quantifying a number of chemical and enzymatic reactions. Here, a mathematical model of conductance changes of gramicidin A pores in response to the presence of electrical charges near its entrance, either on membrane surface or attached to gramicidin A itself, is presented. In this numerical simulation, a two dimensional computational domain is set to mimic the structure of a gramicidin A channel in the bilayer surrounded by electrolyte. The transport of ions through the channel is modeled by the Poisson-Nernst-Planck (PNP) equations that are solved by Finite Element Method (FEM). Preliminary numerical simulations of this mathematical model are in qualitative agreement with the experimental results in the literature. In addition to the model and simulations, we also present the analysis of the stability of the solution to the boundary conditions and the convergence of FEM method for the two dimensional PNP equations in our model.Item Open Access Numerical method for multi-alleles genetic drift problem(SIAM Journal on Numerical Analysis, 2019-01-01) Xu, S; Chen, X; Liu, C; Yue, XGenetic drift describes random fluctuations in the number of genes variants in a population. One of the most popular models is the Wright-Fisher model. The diffusion limit of this model is a degenerate diffusion-convection equation. Due to the degeneration and convection, Dirac singularities will always develop at the boundaries as time evolves, i.e., the fixation phenomenon occurs. Theoretical analysis has proven that the weak solution of this equation, regarded as measure, conserves total probability and expectations. In the current work, we propose a scheme for 3-alleles model with absolute stability and generalize it to N-alleles case (N > 3). Our method can conserve not only total probability and expectations, but also positivity. We also prove that the discrete solution converges to a measure as the mesh size tends to zero, which is the exact measure solution of the original problem. The simulations illustrate that the probability density decays to zero first on the inner nodes, then also on the edge nodes except at the three vertex nodes, on which the density finally concentrates. The results correctly predict the fixation probability and are consistent with theoretical ones and with direct Monte Carlo simulations.Item Open Access Optic nerve microcirculation: Fluid flow and electrodiffusion(Physics of Fluids, 2021-04-01) Zhu, Y; Xu, S; Eisenberg, RS; Huang, HComplex fluids flow in complex ways in complex structures. Transport of water and various organic and inorganic molecules in the central nervous system (CNS) are important in a wide range of biological and medical processes [C. Nicholson and S. Hrabětová, "Brain extracellular space: The final frontier of neuroscience,"Biophys. J. 113(10), 2133 (2017)]. However, the exact driving mechanisms are often not known. In this paper, we investigate flows induced by action potentials in an optic nerve as a prototype of the CNS. Different from traditional fluid dynamics problems, flows in biological tissues such as the CNS are coupled with ion transport. It is driven by osmosis created by the concentration gradient of ionic solutions, which in turn influence the transport of ions. Our mathematical model is based on the known structural and biophysical properties of the experimental system used by the Harvard group [R. K. Orkand, J. G. Nicholls, and S. W. Kuffler, "Effect of nerve impulses on the membrane potential of glial cells in the central nervous system of amphibia,"J. Neurophysiol. 29(4), 788 (1966)]. Asymptotic analysis and numerical computation show the significant role of water in convective ion transport. The full model (including water) and the electrodiffusion model (excluding water) are compared in detail to reveal an interesting interplay between water and ion transport. In the full model, convection due to water flow dominates inside the glial domain. This water flow in the glia contributes significantly to the spatial buffering of potassium in the extracellular space. Convection in the extracellular domain does not contribute significantly to spatial buffering. Electrodiffusion is the dominant mechanism for flows confined to the extracellular domain.