MULTISCALE HOMOGENIZATION OF NONLINEAR HYPERBOLIC EQUATIONS WITH SEVERAL TIME SCALES

We study the multiscale homogenization of a nonlinear hyperbolic equation in a periodic setting.We obtain an accurate homogenization result.We also show that as the nonlinear term depends on the microscopic time variable,the global homogenized problem thus obtained is a system consisting of two hyperbolic equations.It is also shown that in spite of the presence of several time scales,the global homogenized problem is not a reiterated one.     

NEW METHOD FOR IMPROVED CALCULATIONS OF UNSTEADY COMPLEX FLOWS IN LARGE ARTERIES

Using an improved computational fluid dynamics (CFD) method developed for highly unsteady three-dimensional flows, numerical simulations for oscillating flow cycles and detailed unsteady simulations of the flow and forces on the aortic vessels at the iliac bifurcation, for both healthy and diseased patients, are analyzed. Improvements in computational efficiency and acceleration in convergence are achieved by calculating both an unsteady pressure gradient which is due to fluid acceleration and a good global pressure field correction based on mass flow for the pressure Poisson equation. Applications of the enhanced method to oscillatory flow in curved pipes yield an order of magnitude increase in speed and efficiency, thus allowing the study of more complex flow problems such as flow through the mammalian abdominal aorta at the iliac arteries bifurcation. To analyze the large forces which can exist on stent graft of patients with abdominal aortic aneurysm (AAA) disease, a complete derivation of the force equations is presented. The accelerated numerical algorithm and the force equations derived are used to calculate flow and forces for two individuals whose geometry is obtained from CT data and whose respective blood pressure measurements are obtained experimentally. Although the use of endovascular stent grafts in diseased patients can alter vessel geometries, the physical characteristics of stents are still very different when compared to native blood vessels of healthy subjects. The geometry for the AAA stent graph patient studied in this investigation induced flows that resulted in large forces that are primarily caused by the blood pressure. These forces are also directly related to the flow cross-sectional area and the angle of the iliac arteries relative to the main descending aorta. Furthermore, the fluid flow is significantly disturbed in the diseased patient with large flow recirculation and stagnant regions which are not present for healthy subjects.     

IMPLEMENTATION OF MIXED METHODS AS FINITE DIFFERENCE METHODS AND APPLICATIONS TO NONISOTHERMAL MULTIPHASE FLOW IN POROUS MEDIA

In this paper we consider mixed finite element methods for second order elliptic prob-lems. In the case of the lowest order Brezzi-Douglas-Marini elements (if d=2) or Brezzi-Douglas-Durán-Fortin elements (if d=3) on rectangular parallelepipeds, we show thatthe mixed method system, by incorporating certain quadrature rules, can be written as asimple, cell-centered finite difference method. This leads to the solution of a sparse, pos-itive semidefinite linear system for the scalar unknown. For a diagonal tensor coefficient,the sparsity pattern for the scalar unknown is a five point stencil if d=2, and seven ifd=3. For a general tensor coefficient, it is a nine point stencil, and nineteen, respectively.Applications of the mixed method implementation as finite differences to nonisothermalmultiphase, multicomponent flow in porous media are presented.     

DNS and scaling law analysis of compressible turbulent channel flow

Fully developed compressible turbulent channel flow (Ma = 0.8, Re = 3300) is numerically simulated, and the data base of turbulence is established. The statistics such as density-weighted mean velocity and RMS velocity fluctuations in semi-local coordinates agree well with those from other DNS data. High order statistics (skewness and flatness factors) of velocity fluctuations of compressible turbulence are reported for the first time. Compressibility effects are also discussed. Pressure-dilatation absorbs part of the kinetic energy and makes the streaks of compressible channel flow more smooth. The scaling laws of compressible channel flow are also discussed. The conclusions are: (a) Scaling law is found in the center area of the channel, (b) In this area, ESS is also found, (c) When Mach number is not very high, compressibility has little effect on scaling exponents.     

差商算子的多尺度分解与B样条级数的非均匀细分算法

This paper presents a decomposition scheme of the m-th order divided difference operator on non-uniform knots. Through it we introduce a subdivision algorithm of arbitrary B-spline series. Two numerical experiments on function approximation and curve fitting are given to show the efficiency of our algorithm.     

复合材料弹性结构的高精度多尺度算法与数值模拟

１．引言 由于复合材料结构物理参数的非均匀各向异性以及细部几何构形的复杂性,在计算它的位移场、应力、应变场时,传统的有限元法因网格生成困难和计算规模太大而难以实现．７０年代初,Ｉ．Ｂａｂｕｓｋａ,Ｊ．Ｌ．Ｌｉｏｎｓ等人针对复合材料弹性结构提出了均匀化方法,见文［１］,数值实验表明,均匀化方法对描述复合材料弹性结构的有效材料常数及刚度性质是有效的,但它不能刻画应力和应变场的局部变化,因而作为复合材料强度理论的判断依据,均匀化理论显然是不够的．为此,Ｊ．Ｌ．Ｌｉｏｎｓ,Ｏ．Ａ．Ｏｌｅｉｎｉｋ等分别就…     

复合材料特征值的高阶多尺度Rayleigh商校正

本文讨论了周期结构复合材料特征值的多尺度计算,提出了高阶多尺度Rayleigh商校正算法,并给出了收敛性分析. 最后,通过大量数值实验结果表明,新算法是有效且必要的.     

CONVEXITY AND SYMMETRY OF TRANSLATING SOLITONS IN MEAN CURVATURE FLOWS

This paper proves that any rotationally symmetric translating soliton of mean curvature flow in R3 is strictly convex if it is not a plane and it intersects its symmetric axis at one point. The authors also study the symmetry of any translating soliton of mean curvature flow in Rn.     

三维复合介质波动方程多尺度辛几何算法

本文针对三维复合介质波动方程,提出了一类多尺度辛几何算法.其主要内容有:1.快速振荡系数三维波动方程的多尺度渐近分析与收敛性估计;2.均匀化波动方程的辛几何算法;3.多尺度辛几何算法与数值实验结果.     

CORRECTION METHODS FOR STEADY INCOMPRESSIBLE FLOWS

1.IntroductionFOrsemi-periodicincompressiblefluidflows,S.C.R.Dennisandco-workers[1--4]solvethevorticity-streamfunctionformulationofthegoverningequationsbytheseriestruncationandfinitedifferencemethod.Sincenoboundaryconditionforthevorticity,theypropose...     

矩阵形式二次修正Maxwell-Dirac系统的多尺度算法

带二次修正项的Dirac方程在拓扑绝缘体、石墨烯、超导等新材料电磁光特性分析中有着十分广泛的应用.本文工作的创新点有：一是首次提出了矩阵形式带有二次修正项的Dirac方程,它是比较一般的数学框架,涵盖了上述材料体系很多重要的物理模型,具体见附录A;二是针对上述材料体系的电磁响应问题,提出了有界区域Weyl规范下具有周期间断系数矩阵形式带二次修正项Maxwell-Dirac系统的多尺度渐近方法,结合Crank-Nicolson有限差分方法和自适应棱单元方法,发展了一类多尺度算法.数值试验结果验证了多尺度渐近方法的正确性和算法的有效性.     

网络流在清理三角债问题中的应用

本文把清理三角债中两种优化数学模型问题，化成求解相应网络上最小费用流的问题，从而得到（强）多项式算法，并把另外的一种优化数学模型问题。化成线性规划问题.于是解答了文[3]中提出的清理三角债的三个基本问题．     

一类多孔介质气流模型的正解

多孔介质气体流动模型是一类特殊的边值问题，一方面它具有奇异性，另一方面，其微分算子不具有比较性质．本文通过证明这类问题使某种极值原理成立，从而得出了正解的存在性．     

多孔介质中非Fick流的校正方法

本文首先给出H~1-模意义下多孔介质中非Fick流的矩形双线性元的渐进误差展开,进而通过插值后处理方法得到一种插值校正格式来提高有限元近似解的精度.     

SYMMETRY OF TRANSLATING SOLUTIONS TO MEAN CURVATURE FLOWS

First, we review the authors’ recent results on translating solutions to mean curvature flows in Euclidean space as well as in Minkowski space, emphasizing on the asymptotic expansion of rotationally symmetric solutions. Then we study the sufficient condition for which the translating solution is rotationally symmetric. We will use a moving plane method to show that this condition is optimal for the symmetry of solutions to fully nonlinear elliptic equations without ground state condition.     

GLOBAL SOLUTIONS FOR SOME OLDROYD MODELS OF NON-NEWTONIAN FLOWS

The authors consider here some Oldroyd models of non-Newtonian flows consisting of a strong coupling between incompressible Navier-Stokes equations and some transport equations. It is proved that there exist global weak solutiosn for general initial conditions. The existence proof relies upon showing the propagation in time of the compactness of solutions.     

ANALYSIS OF A DEGENERATED DIFFUSION SVEQIRV EPIDEMIC MODEL WITH GENERAL INCIDENCE IN A SPACE HETEROGENEOUS ENVIRONMENT

Considering the comprehensive impact of vaccination, quarantine and spatial heterogeneity on diseases dynamics, this paper formulates an SVEQIRV model with degenerate diffusion. Firstly, we discuss the well-posedness of the model solution. Then, we analyze the dynamic properties of model by using the semigroup theory and the global exponential attractor theory. We use the threshold feature λ? which is the principal eigenvalue of the eigenvalue problem associated with the linearized system at the disease free equilibrium, to describe the transmission dynamics of epidemics. The results show that the disease-free equilibrium is globally asymptotically stable when λ? < 0 and the system is uniformly persistent when λ? > 0. Finally, some numerical simulations and the sensitivity analysis are conducted to visualize the theoretical results and the effect of vaccination rate on disease dynamics.     

Coupling strategies for the numerical simulation of blood flow in deformable arteries by 3D and 1D models

The fluid structure interaction mechanism in vascular dynamics can be described by either 3D or 1D models, depending on the level of detail of the flow and pressure patterns needed for analysis. A successful strategy that has been proposed in the past years is the so-called geometrical multiscale approach, which consists of coupling both 3D and 1D models so as to use the former only in those regions where details of the fluid flow are needed and describe the remaining part of the vascular network by the simplified 1D model.In this paper we review recently proposed strategies to couple the 3D and 1D models, and within the 3D model, to couple the fluid and structure sub-problems. The 3D/1D coupling strategy relies on the imposition of the continuity of flow rate and total normal stress at the interface. On the other hand, the fluid–structure coupling strategy employs Robin transmission conditions. We present some numerical results and show the effectiveness of the new approaches.     

REDUCTIONS OF ADJOINT REPRESENTATIONS TO LAX REPRESENTATIONS FOR CONSTRAINED FLOWS

Within framwork of zero-curvature representation theory,the Lax reprsentations for x- and tn-constrained flows of soliton hierarchy are obtained from reductions of adjoint representations of the auxiliary linear problems. This method is applied to the third order spectral problem by taking modified Boussinesq hierarchy as an illustrative example.     

曲率流的参数化有限元逼近

许多物理现象可以在数学上描述为受曲率驱动的自由界面运动,例如薄膜和泡沫的演变、晶体生长,等等.这些薄膜和界面的运动常依赖于其表面曲率,从而可以用相应的曲率流来描述,其相关自由界面问题的数值计算和误差分析一直是计算数学领域中的难点.参数化有限元法是曲率流的一类有效计算方法,已经能够成功模拟一些曲面在几类基本的曲率流下的演化过程.本文重点讨论曲率流的参数化有限元逼近,它的产生、发展和当前的一些挑战.