Phase Field Model of Thermo-Induced Marangoni Effects in the Mixtures and Its Numerical Simulations with Mixed Finite Element Method

In this paper, we study the Marangoni effects in the mixture of two Newtonian fluids due to the thermo-induced surface tension heterogeneity on the interface. We employ an energetic variational phase field model to describe its physical phenomena, and obtain the corresponding governing equations defined by a modified Navier-Stokes equations coupled with phase field and energy transport. A mixed Taylor-Hood finite element discretization together with full Newton's method are applied to this strongly nonlinear phase field model on a specific domain. Under different boundary conditions of temperature, the resulting numerical solutions illustrate that the thermal energy plays a fundamental role in the interfacial dynamics of two-phase flows. In particular, it gives rise to a dynamic interfacial tension that depends on the direction of temperature gradient, determining the movement of the interface along a sine/cosine-like curve.     

Effects of dynamic changes of tissue properties during laser-induced interstitial thermotherapy (LITT)

A two-dimensional model including the effects of dynamic changes in the physical properties on tissue temperature and damage was developed to describe laser energy transport, heat transfer, and damage accumulation during laser-induced interstitial thermotherapy (LITT). The Monte Carlo method was used to simulate photon transport in a tissue in the nonuniform optical property field, with the finite difference method used to solve the Pennes bioheat equation to calculate the temperature distribution and the Arrhenius equation used to predict the extent of thermal damage. The numerical results showed that the dynamic changes in the optical properties, thermal properties, and blood perfusion rate significantly affected damage volume accumulation and temperature history and should be included in numerical simulations of the LITT treatment.     

Weak Galerkin and Continuous Galerkin Coupled Finite Element Methods for the Stokes-Darcy Interface Problem

We consider a model of coupled free and porous media flow governed by Stokes equation and Darcy's law with the Beavers-Joseph-Saffman interface condition. In this paper, we propose a new numerical approach for the Stokes-Darcy system. The approach employs the classical finite element method for the Darcy region and the weak Galerkin finite element method for the Stokes region. We construct corresponding discrete scheme and prove its well-posedness. The estimates for the corresponding numerical approximation are derived. Finally, we present some numerical experiments to validate the efficiency of the approach for solving this problem.     

Numerical Discretization of Variational Phase Field Model for Phase Transitions in Ferroelectric Thin Films

Phase field methods have been widely used to study phase transitions and polarization switching in ferroelectric thin films. In this paper, we develop an efficient numerical scheme for the variational phase field model based on variational forms of the electrostatic energy and the relaxation dynamics of the polarization vector. The spatial discretization combines the Fourier spectral method with the finite difference method to handle three-dimensional mixed boundary conditions. It allows for an efficient semi-implicit discretization for the time integration of the relaxation dynamics. This method avoids explicitly solving the electrostatic equilibrium equation (a Poisson equation) and eliminates the use of associated Lagrange multipliers. We present several numerical examples including phase transitions and polarization switching processes to demonstrate the effectiveness of the proposed method.     

Permanent Charge Effects on Ionic Flow: A Numerical Study of Flux Ratios and Their Bifurcation

Ionic flow carries electrical signals for cells to communicate with each other. The permanent charge of an ion channel is a crucial protein structure for flow properties while boundary conditions play a role of the driving force. Their effects on flow properties have been analyzed via a quasi-one-dimensional Poisson-Nernst-Planck model for small and relatively large permanent charges. The analytical studies have led to the introduction of flux ratios that reflect permanent charge effects and have a universal property. The studies also show that the flux ratios have different behaviors for small and large permanent charges. However, the existing analytical techniques can reveal neither behaviors of flux ratios nor transitions between small and large permanent charges. In this work we present a numerical investigation on flux ratios to bridge between small and large permanent charges. Numerical results verify the analytical predictions for the two extremal regions. More significantly, emergence of non-trivial behaviors is detected as the permanent charge varies from small to large. In particular, saddle-node bifurcations of flux ratios are revealed, showing rich phenomena of permanent charge effects by the power of combining analytical and numerical techniques. An adaptive moving mesh finite element method is used in the numerical studies.     

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.     

An Efficient Finite Element Method with Exponential Mesh Refinement for the Solution of the Allen-Cahn Equation in Non-Convex Polygons

In this paper we consider the numerical solution of the Allen-Cahn type diffuse interface model in a polygonal domain. The intersection of the interface with the re-entrant corners of the polygon causes strong corner singularities in the solution. To overcome the effect of these singularities on the accuracy of the approximate solution, for the spatial discretization we develop an efficient finite element method with exponential mesh refinement in the vicinity of the singular corners, that is based on ($k$−1)-th order Lagrange elements, $k$≥2 an integer. The problem is fully discretized by employing a first-order, semi-implicit time stepping scheme with the Invariant Energy Quadratization approach in time, which is an unconditionally energy stable method. It is shown that for the error between the exact and the approximate solution, an accuracy of $\mathcal{O}$($h^k$+$τ$) is attained in the $L^2$-norm for the number of $\mathcal{O}$($h^{−2}$ln$h^{−1}$) spatial elements, where $h$ and $τ$ are the mesh and time steps, respectively. The numerical results obtained support the analysis made.     

Full Discretization Scheme for the Dynamics of Elliptic Membrane Shell Model

In this study, the dynamics of elliptic membrane shell model has been proposed and discussed numerically for the first time. Firstly, we show that the solution of this model exists and is unique. Secondly, we consider spatial and time discretizations of the time-dependent elliptic membrane shell by finite element method and Newmark scheme, respectively. Then, the corresponding existence, uniqueness, stability, convergence and a priori error estimate are given. Finally, we present numerical results involving a portion of an ellipsoidal shell and a portion of a spherical shell to verify the efficiency and convergence of the numerical scheme.     

Optimal Error Estimates of Compact Finite Difference Discretizations for the Schrödinger-Poisson System

We study compact finite difference methods for the Schrödinger-Poisson equation in a bounded domain and establish their optimal error estimates under proper regularity assumptions on wave function $ψ$ and external potential $V(x)$. The Crank-Nicolson compact finite difference method and the semi-implicit compact finite difference method are both of order $\mathcal{O}$($h^4$+$τ^2$) in discrete $l^2$, $H^1$ and $l^∞$ norms with mesh size $h$ and time step $τ$. For the errors of compact finite difference approximation to the second derivative and Poisson potential are nonlocal, thus besides the standard energy method and mathematical induction method, the key technique in analysis is to estimate the nonlocal approximation errors in discrete $l^∞$ and $H^1$ norm by discrete maximum principle of elliptic equation and properties of some related matrix. Also some useful inequalities are established in this paper. Finally, extensive numerical results are reported to support our error estimates of the numerical methods.     

One-Dimensional Blood Flow with Discontinuous Properties and Transport: Mathematical Analysis and Numerical Schemes

In this paper we consider the one-dimensional blood flow model with discontinuous mechanical and geometrical properties, as well as passive scalar transport, proposed in [E.F. Toro and A. Siviglia. Flow in collapsible tubes with discontinuous mechanical properties: mathematical model and exact solutions. Communications in Computational Physics. 13(2), 361-385, 2013], completing the mathematical analysis by providing new propositions and new proofs of relations valid across different waves. Next we consider a first order DOT Riemann solver, proposing an integration path that incorporates the passive scalar and proving the well-balanced properties of the resulting numerical scheme for stationary solutions. Finally we describe a novel and simple well-balanced, second order, non-linear numerical scheme to solve the equations under study; by using suitable test problems for which exact solutions are available, we assess the well-balanced properties of the scheme, its capacity to provide accurate solutions in challenging flow conditions and its accuracy.     

Medpor修复眶底缺损三维有限元模型的建立与生物力学分析

目的 对Medpor修复眶底缺损进行三维有限元的建模,并分析其生物力学性质,研究Medpor材料植入眶底后的位移及应力情况,为临床提供参考。方法 将患者术前头颅螺旋CT数据导入Mimics软件后,进行左侧眶下壁三维重建,应用有限元软件对模型进行网格划分,通过CT扫描灰度值的转换,对各部分材质进行赋值,再对整体模型施加重力载荷,计算并分析Medpor材料植入眶底后在不固定、1枚螺钉固定及2枚螺钉固定的不同状态下的位移与应力情况。结果 构建了Medpor修复眶底缺损的三维有限元模型。无固定螺钉情况下,Medpor材料位移较大,几乎无应力;在有固定螺钉情况下,1枚固定螺钉的应力值与2枚固定螺钉应力值没有明显差异。结论 首次建立了关于Medpor修复眶底缺损的三维有限元模型。利用有限元方法发现,螺钉的植入能够对Medpor材料起到良好的固定作用,其中1枚螺钉与2枚螺钉的固定作用相似。     

Stability of Soft Quasicrystals in a Coupled-Mode Swift-Hohenberg Model for Three-Component Systems

In this article, we discuss the stability of soft quasicrystalline phases in a coupled-mode Swift-Hohenberg model for three-component systems, where the characteristic length scales are governed by the positive-definite gradient terms. Classic two-mode approximation method and direct numerical minimization are applied to the model. In the latter approach, we apply the projection method to deal with the potentially quasiperiodic ground states. A variable cell method of optimizing the shape and size of higher-dimensional periodic cell is developed to minimize the free energy with respect to the order parameters. Based on the developed numerical methods, we rediscover decagonal and dodecagonal quasicrystalline phases, and find diverse periodic phases and complex modulated phases. Furthermore, phase diagrams are obtained in various phase spaces by comparing the free energies of different candidate structures. It does show not only the important roles of system parameters, but also the effect of optimizing computational domain. In particular, the optimization of computational cell allows us to capture the ground states and phase behavior with higher fidelity. We also make some discussions on our results and show the potential of applying our numerical methods to a larger class of mean-field free energy functionals.     

An Improved Peridynamic Model with Energy-Based Micromodulus Correction Method for Fracture in Particle Reinforced Composites

We introduce an improved bond-based peridynamic (BPD) model for simulating brittle fracture in particle reinforced composites based on a micromodulus correction approach. In the peridynamic (PD) constitutive model of particle reinforced composites, three kinds of interactive bond forces are considered, and precise definition of mechanical properties for PD bonds is essential for the fracture analysis in particle reinforced composites. A new micromodulus model of PD bonds for particle reinforced composites is proposed based on the equivalence between the elastic strain energy density of classical continuum mechanics and peridynamic model and the harmonic average approach. The damage of particle reinforced composites is defined locally at the level of pairwise bond, and the critical stretch criterion is described as a function of fracture energy based on the composite failure theory. The algorithm procedure for the improved BPD model based on the finite element/discontinuous Galerkin finite element method is brought forward in detail. Several numerical examples are performed to test the feasibility and effectiveness of the proposed model and algorithm in analysis of elastic deformation, crack nucleation and propagation in particle reinforced composites. Additionally, the impact of distribution, shape and size of particles on the fractures of composite materials are also investigated. Numerical results demonstrate that the improved BPD model can effectively be used to analyze the fracture in particle reinforced composites.     

A Stabilized Finite Element Method for Modified Poisson-Nernst-Planck Equations to Determine Ion Flow Through a Nanopore

The conventional Poisson-Nernst-Planck equations do not account for the finite size of ions explicitly. This leads to solutions featuring unrealistically high ionic concentrations in the regions subject to external potentials, in particular, near highly charged surfaces. A modified form of the Poisson-Nernst-Planck equations accounts for steric effects and results in solutions with finite ion concentrations. Here, we evaluate numerical methods for solving the modified Poisson-Nernst-Planck equations by modeling electric field-driven transport of ions through a nanopore. We describe a novel, robust finite element solver that combines the applications of the Newton's method to the nonlinear Galerkin form of the equations, augmented with stabilization terms to appropriately handle the drift-diffusion processes.To make direct comparison with particle-based simulations possible, our method is specifically designed to produce solutions under periodic boundary conditions and to conserve the number of ions in the solution domain. We test our finite element solver on a set of challenging numerical experiments that include calculations of the ion distribution in a volume confined between two charged plates, calculations of the ionic current though a nanopore subject to an external electric field, and modeling the effect of a DNA molecule on the ion concentration and nanopore current.     

A Lowest-Order Mixed Finite Element Method for the Elastic Transmission Eigenvalue Problem

The goal of this paper is to develop numerical methods computing a few smallest elastic interior transmission eigenvalues, which are of practical importance in inverse elastic scattering theory. The problem is challenging since it is nonlinear, non-self-adjoint, and of fourth order. In this paper, we construct a lowest-order mixed finite element method which is close to the Ciarlet-Raviart mixed finite element method. The scheme is based on Lagrange finite element and is one of the less expensive methods in terms of the amount of degrees of freedom. Due to the non-self-adjointness, the discretization of elastic transmission eigenvalue problem leads to a non-classical mixed problem which does not fit into the framework of classical theoretical analysis. Instead, we obtain the convergence analysis based on the spectral approximation theory of compact operator. Numerical examples are presented to verify the theory. Both real and complex eigenvalues can be obtained.     

A Positivity-Preserving Second-Order BDF Scheme for the Cahn-Hilliard Equation with Variable Interfacial Parameters

We present and analyze a new second-order finite difference scheme for the Macromolecular Microsphere Composite hydrogel, Time-Dependent Ginzburg-Landau (MMC-TDGL) equation, a Cahn-Hilliard equation with Flory-Huggins-deGennes energy potential. This numerical scheme with unconditional energy stability is based on the Backward Differentiation Formula (BDF) method in time derivation combining with Douglas-Dupont regularization term. In addition, we present a pointwise bound of the numerical solution for the proposed scheme in the theoretical level. For the convergent analysis, we treat three nonlinear logarithmic terms as a whole and deal with all logarithmic terms directly by using the property that the nonlinear error inner product is always non-negative. Moreover, we present the detailed convergent analysis in $ℓ^∞$(0,$T$;$H_h^{-1}$)∩$ℓ^2$(0,$T$;$H_h^1$) norm. At last, we use the local Newton approximation and multigrid method to solve the nonlinear numerical scheme, and various numerical results are presented, including the numerical convergence test, positivity-preserving property test, spinodal decomposition, energy dissipation and mass conservation properties.     

Optimal operation of a grid-connected battery energy storage system over its lifetime

This paper deals with the optimal control of grid-connected Battery Energy Storage Systems (BESSs) operating for energy arbitrage. An important issue is that BESSs degrade over time, according to their use, and thus they are usable only for a limited number of cycles. Therefore, the time horizon of the optimization depends on the actual operation of the BESS. We focus on Li-ion batteries and use an empirical model to describe battery degradation. The BESS model includes an equivalent circuit for the battery and a simplified model for the power converter. In order to model the energy price variations, we use a linear stochastic model that includes the effect of the time-of-the-day. The problem of maximizing the revenues obtained over the BESS lifetime is formulated as a stochastic optimal control problem with a long, operation-dependent time horizon. First, we divide this problem into a finite set of subproblems, such that for each one of them, the State of Health (SoH) of the battery is approximately constant. Next, we reformulate approximately every subproblem into the minimization of the ratio of two long-time average-cost criteria and use a value-iteration-type algorithm to derive the optimal policy. Finally, we present some numerical results and investigate the effects of the energy loss parameters, degradation parameters, and price dynamics on the optimal policy.     

Time-Lapse 3-D Seismic Wave Simulation via the Generalized Multiscale Finite Element Method

Numerical solution of time-lapse seismic monitoring problems can be challenging due to the presence of finely layered reservoirs. Repetitive wave modeling using fine layered meshes also adds more computational cost. Conventional approaches such as finite difference and finite element methods may be prohibitively expensive if the whole domain is discretized with the cells corresponding to the grid in the reservoir subdomain. A common approach in this case is to use homogenization techniques to upscale properties of subsurface media and assign the background properties to coarser grid; however, inappropriate application of upscaling might result in a distortion of the model, which hinders accurate monitoring of the fluid change in subsurface. In this work, we instead investigate capabilities of a multiscale method that can deal with fine scale heterogeneities of the reservoir layer and more coarsely meshed rock properties in the surrounding domains in the same fashion. To address the 3-D wave problems, we also demonstrate how the multiscale wave modeling technique can detect the changes caused by fluid movement while the hydrocarbon production activity proceeds.     

Numerical Simulations of Rarefied Gases in Curved Channels: Thermal Creep,Circulating Flow,and Pumping Effect

We present numerical simulations of a new system of micro-pump based on the thermal creep effect described by the kinetic theory of gases. This device is made of a simple smooth and curved channel with a periodic temperature distribution. Using the Boltzmann-BGK model as the governing equation for the gas flow, we develop a numerical method based on a deterministic finite volume scheme, implicit in time, with an implicit treatment of the boundary conditions. This method is comparatively less sensitive to the slow flow velocity than the usual Direct Simulation Monte Carlo method in case of long devices, and turns out to be accurate enough to compute macroscopic quantities like the pressure field in the channel. Our simulations show the ability of the device to produce a one-way flow that has a pumping effect.     

Finite Neuron Method and Convergence Analysis

We study a family of $H^m$-conforming piecewise polynomials based on the artificial neural network, referred to as the finite neuron method (FNM), for numerical solution of $2m$-th-order partial differential equations in$\mathbb{R}^d$ for any $m,d≥1$ and then provide convergence analysis for this method. Given a general domain Ω$⊂\mathbb{R}^d$ and a partition$\mathcal{T}_h$ of Ω, it is still an open problem in general how to construct a conforming finite element subspace of $H^m$(Ω) that has adequate approximation properties. By using techniques from artificial neural networks, we construct a family of $H^m$-conforming functions consisting of piecewise polynomials of degree $k$ for any $k≥m$ and we further obtain the error estimate when they are applied to solve the elliptic boundary value problem of any order in any dimension. For example, the error estimates that $‖u−u_N‖_{H^m(\rm{Ω})}=\mathcal{O}(N^{−\frac{1}{2}−\frac{1}{d}})$ is obtained for the error between the exact solution $u$ and the finite neuron approximation $u_N$. A discussion is also provided on the difference and relationship between the finite neuron method and finite element methods (FEM). For example, for the finite neuron method, the underlying finite element grids are not given a priori and the discrete solution can be obtained by only solving a non-linear and non-convex optimization problem. Despite the many desirable theoretical properties of the finite neuron method analyzed in the paper, its practical value requires further investigation as the aforementioned underlying non-linear and non-convex optimization problem can be expensive and challenging to solve. For completeness and the convenience of the reader, some basic known results and their proofs are introduced.