Functional Differential Equations With Delay and Random Effects

The authors study the existence of mild solutions to a functional differential equation with delay and random effects. They use a random fixed point theorem with a stochastic domain. An example is included to illustrate their results.     

A cell functional minimization scheme for parabolic problem

We are interested in a robust and accurate finite volume scheme for 2-D parabolic problems derived from the cell functional minimization approach. The scheme has a local stencil, is locally conservative, treats discontinuity rigorously and leads to a symmetric positive definite linear system. Since the scheme has both cell centered unknowns and cell edge unknowns, the computational cost is an issue and a parallel algorithm is then suggested based on nonoverlapping domain decomposition approach. The interface condition is of the Dirichlet–Robin type and has a parameter λ. By choosing this parameter properly, the convergence of the iteration process could be sped up. Numerical results for linear and nonlinear problems demonstrate the good performance of the cell functional minimization scheme and its parallel version on distorted meshes.     

一类二阶多点边值问题正解的存在性

利用锥上不动点定理讨论了二阶常微分方程组多点边值问题正解的存在性，所得结论推广了最近的一些结果．     

Coupled Navier–Stokes—Molecular dynamics simulations using a multi‐physics flow simulation framework

Simulation of nano‐scale channel flows using a coupled Navier–Stokes/Molecular Dynamics (MD) method is presented. The flow cases serve as examples of the application of a multi‐physics computational framework put forward in this work. The framework employs a set of (partially) overlapping sub‐domains in which different levels of physical modelling are used to describe the flow. This way, numerical simulations based on the Navier–Stokes equations can be extended to flows in which the continuum and/or Newtonian flow assumptions break down in regions of the domain, by locally increasing the level of detail in the model. Then, the use of multiple levels of physical modelling can reduce the overall computational cost for a given level of fidelity. The present work describes the structure of a parallel computational framework for such simulations, including details of a Navier–Stokes/MD coupling, the convergence behaviour of coupled simulations as well as the parallel implementation. For the cases considered here, micro‐scale MD problems are constructed to provide viscous stresses for the Navier–Stokes equations. The first problem is the planar Poiseuille flow, for which the viscous fluxes on each cell face in the finite‐volume discretization are evaluated using MD. The second example deals with fully developed three‐dimensional channel flow, with molecular level modelling of the shear stresses in a group of cells in the domain corners. An important aspect in using shear stresses evaluated with MD in Navier–Stokes simulations is the scatter in the data due to the sampling of a finite ensemble over a limited interval. In the coupled simulations, this prevents the convergence of the system in terms of the reduction of the norm of the residual vector of the finite‐volume discretization of the macro‐domain. Solutions to this problem are discussed in the present work, along with an analysis of the effect of number of realizations and sample duration. The averaging of the apparent viscosity for each cell face, i.e. the ratio of the shear stress predicted from MD and the imposed velocity gradient, over a number of macro‐scale time steps is shown to be a simple but effective method to reach a good level of convergence of the coupled system. Finally, the parallel efficiency of the developed method is demonstrated. Copyright © 2009 John Wiley & Sons, Ltd.     

On the cut‐off phenomenon for the transitivity of randomly generated subgroups

Consider K ≥ 2 independent copies of the random walk on the symmetric group S_N starting from the identity and generated by the products of either independent uniform transpositions or independent uniform neighbor transpositions. At any time $n\in \mathbb{N}$, let G_n be the subgroup of S_N generated by the K positions of the chains. In the uniform transposition model, we prove that there is a cut‐off phenomenon at time N ln(N)/(2K) for the non‐existence of fixed point of G_n and for the transitivity of G_n, thus showing that these properties occur before the chains have reached equilibrium. In the uniform neighbor transposition model, a transition for the non‐existence of a fixed point of G_n appears at time of order $N^{1+\frac{2}{K}}$ (at least for K ≥ 3), but there is no cut‐off phenomenon. In the latter model, we recover a cut‐off phenomenon for the non‐existence of a fixed point at a time proportional to N by allowing the number K to be proportional to ln(N). The main tools of the proofs are spectral analysis and coupling techniques. © 2011 Wiley Periodicals, Inc. Random Struct. Alg., 2012     

Hybrid mesh generation with embedded surfaces using a multiple marching direction approach

This paper proposes a method for the creation of hybrid meshes with embedded surfaces for viscous flow simulations as an extension of the multiple marching direction approach (AIAA J. 2007; 45 (1):162–167). The multiple marching direction approach enables to place semi‐structured elements around singular points, where valid semi‐structured elements cannot be placed using conventional hybrid mesh generation methods. This feature is discussed first with a couple of examples. Elements sometimes need to be clustered inside a computational domain to obtain more accurate results. For example, solution features, such as shocks, vortex cores and wake regions, can be extracted during the process of adaptive mesh generation. These features can be represented as surface meshes embedded in a computational domain. Semi‐structured elements can be placed around the embedded surface meshes using the multiple marching direction approach with a pretreatment method. Tetrahedral elements can be placed easily instead. A couple of results are presented to demonstrate the capability of the mesh generation method. Copyright © 2008 John Wiley & Sons, Ltd.     

APPROXIMATING COMMON FIXED POINTS OF NEARLY ASYMPTOTICALLY NONEXPANSIVE MAPPINGS

We use an iteration scheme to approximate common fixed points of nearly asymptotically nonexpansive mappings.We generalize corresponding theorems of [1] to the case of two nearly asymptotically nonexpansive mappings and those of [9] not only to a larger class of mappings but also with better rate of convergence.     

基于摩擦磨损的KDP晶体固结磨料抛光垫优化

本文通过固结磨料球与KDP晶体对磨的单因素试验探究固结磨料球中反应物种类、磨粒浓度、反应物浓度、基体硬度对摩擦系数、磨痕截面积和磨痕处粗糙度的影响,试验结果表明:KHCO₃固结磨料球对磨后磨痕对称性好,磨痕处的粗糙度值低;磨痕截面积随磨粒和反应物浓度的增加而增大,随基体硬度的增大而降低;磨痕处粗糙度随磨粒和反应物浓度的增加先降低后上升,随基体硬度的增大先上升后降低;摩擦系数受磨粒和反应物浓度影响不明显,随基体硬度的增大而降低。选择KHCO₃作为反应物,Ⅰ基体,磨粒浓度为基体质量的100%,反应物浓度为15%制备固结磨料球与KDP晶体对磨后的磨痕轮廓对称度好且磨痕处粗糙度值低,以该组分制备固结磨料垫干式抛光KDP晶体,可实现晶体表面粗糙度Sa值为18.50 nm,材料去除率为130 nm/min的高效精密加工。     

A 3D mesh deformation technique for irregular in‐flight ice accretion

Aircraft holding around busy airports may be requested to sustain as much as 45 min of icing before landing or being diverted to another airport. In this paper, a three‐dimensional mesh deformation scheme, based on a structural frame analogy, is proposed for the numerical simulation of ice accretion during extended exposure to adverse weather conditions. The goal is to provide an approach that is robust and efficient enough to delay or altogether avoid re‐meshing while preserving (enforcing) nearly orthogonal elements at the highly distorted ice surface. Robustness is achieved by suitably modifying the axial and torsional stiffness components of the frame elements in order to handle large and irregular grid displacements typical of in‐flight icing. Computational efficiency is obtained by applying the mesh displacement to an automatically selected small subset of the entire computational domain. The methodology is validated first in the case of deformations typical of fluid‐structure interaction problems, including wing bending, a helicopter rotor in forward flight, and the twisting of a high‐lift wing configuration. The approach is then assessed for aero‐icing on two swept wings and compared against experimental measurements where available. Copyright © 2015 John Wiley & Sons, Ltd.     

A weakly compressible MPS method for modeling of open‐boundary free‐surface flow

A mesh‐free particle method, based on the moving particle semi‐implicit (MPS) interaction model, has been developed for the simulation of two‐dimensional open‐boundary free‐surface flows. The incompressibility model in the original MPS has been replaced with a weakly incompressible model. The effect of this replacement on the efficiency and accuracy of the model has been investigated. The new inflow–outflow boundary conditions along with the particle recycling strategy proposed in this study extend the application of the model to open‐boundary problems. The final model is able to simulate open‐boundary free surface flow in cases of large deformation and fragmentation of free surface. The models and proposed algorithms have been validated and applied to sample problems. The results confirm the model's efficiency and accuracy. Copyright © 2009 John Wiley & Sons, Ltd.