Computation of Flows in a Channel with a Facing Step by Schwarz Alternating Method

This paper presents some preliminary work on the application of Schwarz alternating method, combined with a multigrid solver, for solving the steady 2D viscous incompressible Navier-Stokes equations. The difficulty caused by the violation of the compatibility condition imposed on the normal velocity at the boundary, which arises when subdomain problems are not solved exactly at each iteration, is surrounded by adding artificial compressiblity constants to the continuity equation each time when a subdomain problem is to be solved. These constants permit to maintain the compatibility condition of the subdomain problems and one can easily show that they tend to zero when the iteration number tends to infinity (in fact, they can be expressed explicitly as functions of the iteration number and some geometrical parameters of the computational domain). Numerical experiences have been made for flows in a channel with a facing step.     

Numerical modelling of longitudinal vibrations of a sucker rod string

A new technique for analyzing the dynamic behavior of a sucker rod string used in the oil well industry is presented. The main difficulty in the numerical calculation of the examined structure is a multivalued velocity—force relation determined by Coulomb's friction and by loads generated during operation of pump valves. Both the monotonic and nonmonotonic velocity—force relations are considered. A quasi-variational inequality formulation of the problem is proposed. The solution of the inequality amounts to finding the minimum of a convex nonsmooth functional at each time step by means of the Newmark difference time scheme, successive iterations and finite element discretization. The problem of functional minimization is reduced to construction of a sequence of smooth nonlinear programming problems by introducing the auxiliary variables and applying the augmented Lagrangian method. The proposed approach is used to study the longitudinal vibrations of sucker rod strings under near-real conditions. In such systems the most commonly occurring vibration modes are the stick-slip vibrations and the vibrations with natural force excited twice a cycle. The nonmonotonic character of the friction law leads to intensification of these vibrations. In the case of nonmonotonic friction law the stick-slip vibrations can occur even under the action of constant external forces.     

流动数值模拟中一种并行自适应有限元算法

给出了一种流动数值模拟中的基于误差估算的并行网格自适应有限元算法.首先,以初网格上获得的当地事后误差估算值为权,应用递归谱对剖分方法划分初网格,使各子域上总体误差近似相等,以解决负载平衡问题.然后以误差值为判据对各子域内网格进行独立的自适应处理.最后应用基于粘接元的区域分裂法在非匹配的网格上求解N-S方程.区域分裂情形下N-S方程有限元解的误差估算则是广义Stokes问题误差估算方法的推广.为验证方法的可靠性,给出了不可压流经典算例的数值结果.     

The Lattice Solid Model to Simulate the Physics of Rocks and Earthquakes: Incorporation of Friction

The particle-based lattice solid model developed to study the physics of rocks and the nonlinear dynamics of earthquakes is refined by incorporating intrinsic friction between particles. The model provides a means for studying the causes of seismic wave attenuation, as well as frictional heat generation, fault zone evolution, and localisation phenomena. A modified velocity–Verlat scheme that allows friction to be precisely modelled is developed. This is a difficult computational problem given that a discontinuity must be accurately simulated by the numerical approach (i.e., the transition from static to dynamical frictional behaviour). This is achieved using a half time step integration scheme. At each half time step, a nonlinear system is solved to compute the static frictional forces and states of touching particle-pairs. Improved efficiency is achieved by adaptively adjusting the time step increment, depending on the particle velocities in the system. The total energy is calculated and verified to remain constant to a high precision during simulations. Numerical experiments show that the model can be applied to the study of earthquake dynamics, the stick–slip instability, heat generation, and fault zone evolution. Such experiments may lead to a conclusive resolution of the heat flow paradox and improved understanding of earthquake precursory phenomena and dynamics.     

Scalable parallel methods for monolithic coupling in fluid–structure interaction with application to blood flow modeling

We introduce and study numerically a scalable parallel finite element solver for the simulation of blood flow in compliant arteries. The incompressible Navier–Stokes equations are used to model the fluid and coupled to an incompressible linear elastic model for the blood vessel walls. Our method features an unstructured dynamic mesh capable of modeling complicated geometries, an arbitrary Lagrangian–Eulerian framework that allows for large displacements of the moving fluid domain, monolithic coupling between the fluid and structure equations, and fully implicit time discretization. Simulations based on blood vessel geometries derived from patient-specific clinical data are performed on large supercomputers using scalable Newton–Krylov algorithms preconditioned with an overlapping restricted additive Schwarz method that preconditions the entire fluid–structure system together. The algorithm is shown to be robust and scalable for a variety of physical parameters, scaling to hundreds of processors and millions of unknowns.     

Power management in heterogeneous networks with energy harvesting base stations

In this paper, heterogeneous cellular networks (HCNs) with base stations (BSs) powered from both renewable energy sources and the grid power are considered. Based on a techno-economic analysis, we demonstrate that by controlling both transmit power and stored energy usage of BSs, energy costs can be effectively reduced. Specifically, we propose a two-stage BS operation scheme where an optimization and control subproblem is solved at each stage, respectively. For the first subproblem, transmit power of BSs is adjusted while quality of service (QoS) experienced by users is preserved. In the second subproblem, we consider the strategic scheduling of renewable energy used to power the BSs. That is, harvested energy may be reserved in the battery for future use to minimize the cost of on-grid power that varies in real-time. We propose: (1) an optimization approach built on a lattice model with a method to process outage rate constraint, and (2) a control algorithm based on nonlinear model predictive control (NMPC) theory to solve the two subproblems, respectively. Simulation results include a collection of case studies that demonstrate as to how operators may manage energy harvesting BSs to reduce their electricity costs.     

A meshless method for the nonlinear generalized regularized long wave equation

This paper presents a meshless method for the nonlinear generalized regularized long wave(GRLW) equation based on the moving least-squares approximation.The nonlinear discrete scheme of the GRLW equation is obtained and is solved using the iteration method.A theorem on the convergence of the iterative process is presented and proved using theorems of the infinity norm.Compared with numerical methods based on mesh,the meshless method for the GRLW equation only requires the scattered nodes instead of meshing the domain of the problem.Some examples,such as the propagation of single soliton and the interaction of two solitary waves,are given to show the effectiveness of the meshless method.     

Wireless Network Optimization for Federated Learning with Model Compression in Hybrid VLC/RF Systems

In this paper, the optimization of network performance to support the deployment of federated learning (FL) is investigated. In particular, in the considered model, each user owns a machine learning (ML) model by training through its own dataset, and then transmits its ML parameters to a base station (BS) which aggregates the ML parameters to obtain a global ML model and transmits it to each user. Due to limited radio frequency (RF) resources, the number of users that participate in FL is restricted. Meanwhile, each user uploading and downloading the FL parameters may increase communication costs thus reducing the number of participating users. To this end, we propose to introduce visible light communication (VLC) as a supplement to RF and use compression methods to reduce the resources needed to transmit FL parameters over wireless links so as to further improve the communication efficiency and simultaneously optimize wireless network through user selection and resource allocation. This user selection and bandwidth allocation problem is formulated as an optimization problem whose goal is to minimize the training loss of FL. We first use a model compression method to reduce the size of FL model parameters that are transmitted over wireless links. Then, the optimization problem is separated into two subproblems. The first subproblem is a user selection problem with a given bandwidth allocation, which is solved by a traversal algorithm. The second subproblem is a bandwidth allocation problem with a given user selection, which is solved by a numerical method. The ultimate user selection and bandwidth allocation are obtained by iteratively compressing the model and solving these two subproblems. Simulation results show that the proposed FL algorithm can improve the accuracy of object recognition by up to 16.7% and improve the number of selected users by up to 68.7%, compared to a conventional FL algorithm using only RF.     

A mixed implicit–explicit finite difference scheme for heat transport in magnetised plasmas

An explicit/implicit domain decomposition method is applied to the time-dependent heat-conduction problem in a 2-d, strongly anisotropic medium (a magnetised plasma), using a formulation of the spatial derivatives which avoids the pollution of perpendicular by parallel heat fluxes. The time-stepping at the sub-domain boundaries is done using a DuFort–Frankel scheme, which leads to a time step limit given not by instabilities, but by the damping rate of numerical oscillations driven by inconsistencies in the formulation of initial conditions or the temporal variations in the true physical solution. These limitations can be minimized, however, by aligning the subdomain boundaries as much as possible with magnetic flux surfaces. The time step limit depends on the ratio of the implicit grid spacing to the distance between subdomain boundaries (DuFort–Frankel lines in 2-d, surfaces in 3-d).     

On the elliptic mesh generation in domains containing multiple inclusions and undergoing large deformations

We present an improved method to generate a sequence of structured meshes even when the physical domain contains deforming inclusions. This method belongs to the class of Arbitrary Lagrangian–Eulerian (ALE) methods for solving moving boundary problems. Its tools are either (a) separate mappings of the domain boundaries and enforcing the node distribution on lines emanating from singular points or (b) domain decomposition and separate mappings of each subdomain using suitable coordinate systems. The latter is shown to be more versatile and general. In both cases a set of elliptic equations is used to generate the grid extending in this way the method advanced by Dimakopoulos and Tsamopoulos [Y. Dimakopoulos, J.A. Tsamopoulos, A quasi-elliptic transformation for moving boundary problems with large anisotropic deformations, J. Comput. Phys. 192 (2003) 494–522]. We shall present examples where this earlier method and all other mesh generating methods which are based on a conformal mapping or solving a quasi-elliptic set of PDEs fail to produce an acceptable mesh and accurate solutions in such geometries. Furthermore, in contrast to other methods, appropriate boundary conditions and constraints such as, orthogonality of specific mesh lines and prespecified node distributions on them, can be easily implemented along a specific part of the domain or its boundary. Hence, no attractive terms at specific corners or singular points are needed. To increase the mesh resolution around the moving interfaces while keeping low the memory requirements and the computational time, a local mesh refinement technique has been incorporated as well. The method is demonstrated in two challenging examples where no remeshing is required in spite of the large domain deformations. In the first one, the transient growth of two bubbles embedded in a viscoelastic filament undergoing stretching in the axial direction is examined, while in the second one the linear and non-linear dynamics of two bubbles in a viscous medium are determined in an acoustic field. The large elasticity of the filament in the first case or the large inertia in the second case coupled with the externally induced large deformations of the liquid domain requires the accurate calculation which is achieved by the method we propose herein. The governing equations are solved using the finite element/Galerkin method with appropriate modifications to solve the hyperbolic constitutive equation of a viscoelastic fluid. These are coupled with an implicit Euler method for time integration or with Arnoldi’s algorithm for normal mode analysis.     

Probabilistically induced domain decomposition methods for elliptic boundary-value problems

Monte Carlo as well as quasi-Monte Carlo methods are used to generate only few interfacial values in two-dimensional domains where boundary-value elliptic problems are formulated. This allows for a domain decomposition of the domain. A continuous approximation of the solution is obtained interpolating on such interfaces, and then used as boundary data to split the original problem into fully decoupled subproblems. The numerical treatment can then be continued, implementing any deterministic algorithm on each subdomain. Both, Monte Carlo (or quasi-Monte Carlo) simulations and the domain decomposition strategy allow for exploiting parallel architectures. Scalability and natural fault tolerance are peculiarities of the present algorithm. Examples concern Helmholtz and Poisson equations, whose probabilistic treatment presents additional complications with respect to the case of homogeneous elliptic problems without any potential term and source.     

Explicit Multi-Symplectic Splitting Methods for the Nonlinear Dirac Equation

In this paper, we propose two new explicit multi-symplectic splitting methods for the nonlinear Dirac (NLD) equation. Based on its multi-symplectic formulation, the NLD equation is split into one linear multi-symplectic system and one nonlinear infinite Hamiltonian system. Then multi-symplectic Fourier pseudospectral method and multi-symplectic Preissmann scheme are employed to discretize the linear subproblem, respectively. And the nonlinear subsystem is solved by a symplectic scheme. Finally, a composition method is applied to obtain the final schemes for the NLD equation. We find that the two proposed schemes preserve the total symplecticity and can be solved explicitly. Numerical experiments are presented to show the effectiveness of the proposed methods.     

多尺度模拟中网格守恒重映算法

针对耦合微观分子动力学(MD)和宏观有限元方法(FE)的多尺度模拟,提出一类新的基于贡献单元法的网格守恒重映算法.由于物理量是由有限元节点以及相应区域的原子信息通过积分重构得到的,对结构和非结构网格都能适用.对于未知量定义在顶点的情形,引入辅助网格.数值例子验证了算法的准确性和有效性.     

Two-Level Defect-Correction Method for Steady Navier-Stokes Problem with Friction Boundary Conditions

In this paper, we present two-level defect-correction finite element method for steady Navier-Stokes equations at high Reynolds number with the friction boundary conditions, which results in a variational inequality problem of the second kind. Based on Taylor-Hood element, we solve a variational inequality problem of Navier-Stokes type on the coarse mesh and solve a variational inequality problem of Navier-Stokes type corresponding to Newton linearization on the fine mesh. The error estimates for the velocity in the $H^1$ norm and the pressure in the $L^2$ norm are derived. Finally, the numerical results are provided to confirm our theoretical analysis.     

Conservative space-time mesh refinement methods for the FDTD solution of Maxwell’s equations

A new variational space-time mesh refinement method is proposed for the FDTD solution of Maxwell’s equations. The main advantage of this method is to guarantee the conservation of a discrete energy that implies that the scheme remains L² stable under the usual CFL condition. The only additional cost induced by the mesh refinement is the inversion, at each time step, of a sparse symmetric positive definite linear system restricted to the unknowns located on the interface between coarse and fine grid. The method is presented in a rather general way and its stability is analyzed. An implementation is proposed for the Yee scheme. In this case, various numerical results in 3-D are presented in order to validate the approach and illustrate the practical interest of space-time mesh refinement methods.     

基于区域分裂的非均匀Lattice Boltzmann方法

基于区域分裂技术,设计了一种使用非均匀网格的LatticeBoltzmann（LB）模型,其基本思想是：将流场分解为若干个规划的子区域,在每个子区域使用均匀网格上的LB模型求解,子区域的边界条件利用插值方法决定,用该模型对二维圆柱绕流问题进行了模拟,并与其它方法的结果作了比较。     

Minimum entropy deconvolution of pulse-echo signals acquired from attenuative layered media.

In this article deconvolution of ultrasonic pulse-echo data acquired from attenuative layered media is considered. The problem is divided in two subproblems: treating the sparse reflection sequence caused by the layered structure of the media and treating the frequency-dependent attenuation. The first subproblem is solved by means of joint maximum a posteriori estimation of the assumed zero mean, white, nonstationary reflection sequence and its corresponding sequence of unknown standard deviations. This approach leads to an algorithm that seeks minimum entropy solutions for the reflection sequence and therefore the algorithm serves as a novel link between the classical Wiener filter and methods for sparse or minimum entropy deconvolution. The second subproblem is solved by introducing a new signal processing-oriented, linear discrete-time model for frequency-dependent attenuation in isotropic and homogeneous media. The deconvolution algorithm is tested using simulated data and its performance for real normal incidence pulse-echo data from a composite material is also demonstrated. The results show that the algorithm, in combination with the attenuation model, yields estimates that reveal the internal structure of the composite and, thus, simplify the interpretation of the ultrasonic data.     

多流管网格上九点格式的节点插值方法

多流管方法是二维多介质辐射流体力学数值模拟中一类常用的求解方法,它采用Lagrange-Euler混合型四边形网格,称为多流管网格。通常其网格品质高于一般的四边形网格。在这类网格上,可以利用网格特性对九点扩散格式中的节点插值方法进行改进。本文利用调和平均点和梯度离散构造的方法提出几种节点插值方法。并给出数值实验,说明现有应用程序中的节点插值方法损失精度,而新的节点插值方法能够使得九点格式在多流管网格上具有二阶精度。     

Thermal explosion in a droplet-gas cloud

The effect of the presence of a spray of liquid fuel on thermal explosion in a combustible droplet-gas cloud is investigated. By ‘thermal explosion’ we refer exclusively to the initial stages of the behaviour of the combustible medium as its temperature begins to rise and various competing physical and chemical processes are called into play. A qualitative analysis of the system of governing equations is carried out using an advanced geometrical asymptotic technique (the integral manifold method). Possible types of dynamical behaviour of the system are classified and parametric regions of their existence are determined analytically. It is demonstrated that the original problem can be decomposed into two subproblems, due to the underlying hierarchical time scale structure. The first subproblem relates to the droplet heat up period, for which a relatively rapid time scale is applicable. The second subproblem begins at the saturation point. For the latter, more significant second stage, it is found that there are five main dynamical regimes: slow regimes, conventional fast explosive regimes, thermal explosion with freeze delay and two different types of thermal explosion with delay (the concentration of the combustible gas decreases or increases). Upper and lower bounds for the delay time are derived analytically and compared with results of numerical simulations, with rather satisfactory agreement.     

Sparse spectral-tau method for the three-dimensional helically reduced wave equation on two-center domains

We describe a multidomain spectral-tau method for solving the three-dimensional helically reduced wave equation on the type of two-center domain that arises when modeling compact binary objects in astrophysical applications. A global two-center domain may arise as the union of Cartesian blocks, cylindrical shells, and inner and outer spherical shells. For each such subdomain, our key objective is to realize certain (differential and multiplication) physical-space operators as matrices acting on the corresponding set of modal coefficients. We then achieve sparse realizations through the integration “preconditioning” of Coutsias, Hagstrom, Hesthaven, and Torres. Since ours is the first three-dimensional multidomain implementation of the technique, we focus on the issue of convergence for the global solver, here the alternating Schwarz method accelerated by GMRES. Our methods may prove relevant for numerical solution of other mixed-type or elliptic problems, and in particular for the generation of initial data in general relativity.