An adaptive approach for modeling a fiber-matrix composite with the FE2 method

In materials with a complicated microstructre [1], the macroscopic material behaviour is unknown. In this work a Fiber-Matrix composite is considered with elasto-plastic fibers. A homogenization of the microscale leads to the macroscopic material properties. In the present work, this is realized in the frame of a FE² formulation. It combines two nested finite element simulations. On the macroscale, the boundary value problem is modelled by finite elements, at each integration point a second finite element simulation on the microscale is employed to calculate the stress response and the material tangent modulus. One huge disadvantage of the approach is the high computational effort. Certainly, an accompanying homogenization is not necessary if the material behaves linear elastic. This motivates the present approach to deal with an adaptive scheme. An indicator, which makes use of the different boundary conditions (BC) of the BVP on microscale, is suggested. The homogenization with the Dirichlet BC overestimates the material tangent modulus whereas the Neumann BC underestimates the modulus [2]. The idea for an adaptive modeling is to use both of the BCs during the loading process of the macrostructure. Starting initially with the Neumann BC leads to an overestimation of the displacement response and thus the strain state of the boundary value problem on the macroscale. An accompanying homogenization is performed after the strain reaches a limit strain. Dirichlet BCs are employed for the accompanying homogenization. Some numerical examples demonstrate the capability of the presented method. (© 2017 Wiley-VCH Verlag GmbH & Co. KGaA, Weinheim)     

Elliptic equations with measurable coefficients in Reifenberg domains

We prove W1,p estimates for elliptic equations in divergence form under the assumption that for each point and for each sufficiently small scale there is a coordinate system so that the coefficients have small oscillation in (n−1) directions. We assume the boundary to be δ-Reifenberg flat and the coefficients having small oscillation in the flat direction of the boundary.     

A Variational Finite Element Model for Large-Eddy Simulations of Turbulent Flows

The authors introduce a new Large Eddy Simulation model in a channel,based on the projection on finite element spaces as filtering operation in its variational form,for a given triangulation{Th}h>0.The eddy viscosity is expressed in terms of the friction velocity in the boundary layer due to the wall,and is of a standard sub grid-model form outside the boundary layer.The mixing length scale is locally equal to the grid size.The computational domain is the channel without the linear sub-layer of the boundary layer.The no-slip boundary condition(or BC for short)is replaced by a Navier(BC)at the computational wall.Considering the steady state case,the authors show that the variational finite element model they have introduced,has a solution(vh,ph)h>0that converges to a solution of the steady state Navier-Stokes equation with Navier BC.     

Construction and Analysis of Lattice Boltzmann Methods Applied to a 1D Convection-Diffusion Equation

To solve the 1D (linear) convection-diffusion equation, we construct and we analyze two LBM schemes built on the D1Q2 lattice. We obtain these LBM schemes by showing that the 1D convection-diffusion equation is the fluid limit of a discrete velocity kinetic system. Then, we show in the periodic case that these LBM schemes are equivalent to a finite difference type scheme named LFCCDF scheme. This allows us, firstly, to prove the convergence in L ^∞ of these schemes, and to obtain discrete maximum principles for any time step in the case of the 1D diffusion equation with different boundary conditions. Secondly, this allows us to obtain most of these results for the Du Fort-Frankel scheme for a particular choice of the first iterate. We also underline that these LBM schemes can be applied to the (linear) advection equation and we obtain a stability result in L ^∞ under a classical CFL condition. Moreover, by proposing a probabilistic interpretation of these LBM schemes, we also obtain Monte-Carlo algorithms which approach the 1D (linear) diffusion equation. At last, we present numerical applications justifying these results.     

空间分数阶电报方程的格子Boltzmann方法

应用格子Boltzmann方法（LBM）对Riemann Liouville空间分数阶电报方程进行了数值模拟研究.首先,将分数阶算子中的积分项进行离散化处理,并进行了收敛阶分析.然后,构建了带修正函数项的一维三速度(D1Q3)的LBM演化模型.利用Chapman Enskog多尺度技术和Taylor展开技术,推导出各平衡态分布函数和修正函数的具体表达式,准确地从所建的演化模型恢复出宏观方程.最后,数值计算结果表明该模型是稳定、有效的.     

Modeling and simulation of plasma jet by lattice Boltzmann method

In order to find a simple and efficient simulation for plasma spray process, an attempt of modeling was made to calculate velocity and temperature field of the plasma jet by hexagonal 7-bit lattice Boltzmann method (LBM) in this paper. Utilizing the methods of Chapman–Enskog expansion and multi-scale expansion, the authors derived the macro equations of the plasma jet from the lattice Boltzmann evolution equations on the basis of selecting two opportune equilibrium distribution functions. The present model proved to be valid when the predictions of the current model were compared with both experimental and previous model results. It is found that the LBM is simpler and more efficient than the finite difference method (FDM). There is no big variation of the flow characteristics, and the isotherm distribution of the turbulent plasma jet is compared with the changed quantity of the inlet velocity. Compared with the velocity at the inlet, the temperature at the inlet has a less influence on the characteristics of plasma jet.     

Weighted and regularity estimates for nonlinear equations on Reifenberg flat domains

Global weighted Lp estimates are obtained for the gradient of solutions to nonlinear elliptic Dirichlet boundary value problems over a bounded nonsmooth domain. Morrey and Hölder regularity of solutions are also established, as a consequence. These results generalize various existing estimates for nonlinear equations. The nonlinearities are of at most linear growth and assumed to have a uniform small mean oscillation. The boundary of the domain, on the other hand, may exhibit roughness but assumed to be sufficiently flat in the sense of Reifenberg. Our approach uses maximal function estimates and Vitali covering lemma, and also known regularity results of solutions to nonlinear homogeneous equations.     

Boundary Layers Associated with Incompressible Navier-Stokes Equations: The Noncharacteristic Boundary Case

The goal of this article is to study the boundary layer of wall bounded flows in a channel at small viscosity when the boundaries are uniformly noncharacteristic, i.e., there is injection and/or suction everywhere at the boundary. Following earlier work on the boundary layer for linearized Navier-Stokes equations in the case where the boundaries are characteristic (no-slip at the boundary and non-permeable), we consider here the case where the boundary is permeable and thus noncharacteristic. The form of the boundary layer and convergence results are derived in two cases: linearized equation and full nonlinear equations. We prove that there exists a boundary layer at the outlet (downwind) of the form e^−Uz/ε where U is the speed of injection/suction at the boundary, z is the distance to the outlet of the channel, and ε is the kinematic viscosity. We improve an earlier result of S. N. Alekseenko (1994, Siberian Math. J.35, No. 2, 209-230) where the convergence in L² of the solutions of the Navier-Stokes equations to that of the Euler equations at vanishing viscosity was established. In the two dimensional case we are able to derive the physically relevant uniform in space (L^∞ norm) estimates of the boundary layer. The uniform in space estimate is derived by properly developing our previous idea of better control on the tangential derivative and the use of an anisotropic Sobolev imbedding. To the best of our knowledge this is the first rigorously proved result concerning boundary layers for the full (nonlinear) Navier-Stokes equations for incompressible fluids.     

The effects of bed form roughness on total suspended load via the Lattice Boltzmann Method

Bed forms in natural rivers and man-made channels provide the dominant contribution to overall flow resistance and hence significantly affect sediment transport rate. Many laboratory experiments and field observations have been conducted on bed forms, and it was found that theoretical flat-bed assumptions do not give the correct estimation for the total suspended load (TSL). In this study, we present a systematic numerical investigation of turbulent open-channel flows over bed forms using the Lattice Boltzmann Method (LBM). A static Smagorinsky model is incorporated into LBM to account for turbulence, and the dynamic interface between fluid and air is captured by a free-surface model. The time-averaged flow velocity, turbulence intensity and Reynolds shear stress in LBM simulations show an excellent agreement with the available experimental data. In addition, the coherent flow structures induced by the bed forms qualitatively agree with previous numerical results from Large Eddy Simulations based the Navier–Stokes equations. We then proceed to investigate the effects of bed form roughness, quantified by the total friction factor f_T, on sediment transport. It is found that the prediction of the TSL based on the theoretical flat-bed assumptions may lead to an overestimation of up to 30%, depending on the bed form roughness. In addition, the normalized TSL is linearly proportional to f_T and nearly inversely proportional to the ratio of downward settling velocity and upward turbulence induced diffusion. Our work proposes a general law linking these quantities to estimate the TSL, which has the potential for a more efficient and accurate engineering design of man-made channels and improved river management.     

An analytical technique for solving a class of strongly nonlinear conservative systems

Recently, an analytical technique has been developed to determine approximate solutions of strongly nonlinear differential equations containing higher order harmonic terms. Usually, a set of nonlinear algebraic equations is solved in this method. However, analytical solutions of these algebraic equations are not always possible, especially in the case of a large oscillation. Previously such algebraic equations for the Duffing equation were solved in powers of a small parameter; but the solutions measure desired results when the amplitude is an order of 1. In this article different parameters of the same nonlinear problems are found, for which the power series produces desired results even for the large oscillation. Moreover, two or three terms of this power series solution measure a good result when the amplitude is an order of 1. Besides these, a suitable truncation formula is found in which the solution measures better results than existing solutions. The method is mainly illustrated by the Duffing oscillator but it is also useful for many other nonlinear problems.     

Oscillations of higher order differential equations of neutral type

In this paper, sufficient conditions have been obtained for oscillation of solutions of a class of nth order linear neutral delay-differential equations. Some of these results have been used to study oscillatory behaviour of solutions of a class of boundary value problems for neutral hyperbolic partial differential equations.     

Chaos: A bridge from microscopic uncertainty to macroscopic randomness

It is traditionally believed that the macroscopic randomness has nothing to do with the micro-level uncertainty. Besides, the sensitive dependence on initial condition (SDIC) of Lorenz chaos has never been considered together with the so-called continuum-assumption of fluid (on which Lorenz equations are based), from physical and statistic viewpoints. A very fine numerical technique [6] with negligible truncation and round-off errors, called here the “clean numerical simulation” (CNS), is applied to investigate the propagation of the micro-level unavoidable uncertain fluctuation (caused by the continuum-assumption of fluid) of initial conditions for Lorenz equation with chaotic solutions. Our statistic analysis based on CNS computation of 10,000 samples shows that, due to the SDIC, the uncertainty of the micro-level statistic fluctuation of initial conditions transfers into the macroscopic randomness of chaos. This suggests that chaos might be a bridge from micro-level uncertainty to macroscopic randomness, and thus would be an origin of macroscopic randomness. We reveal in this article that, due to the SDIC of chaos and the inherent uncertainty of initial data, accurate long-term prediction of chaotic solution is not only impossible in mathematics but also has no physical meanings. This might provide us a new, different viewpoint to deepen and enrich our understandings about the SDIC of chaos.     

一种计算超声速流中由湍流剪切层脉动引起的激波振荡频率的方法

在超声速或高超声速绕流中,一种很严重的脉动压力环境是由激波边界层相互作用引起的激波振荡.这种高强度的振荡激波可能诱发结构共振.因这一现象非常复杂,已发表的文章都采用经验或半经验方法.本文首次从基本流体动力学方程出发,给出了由湍流剪切层引起的激波振荡频率的理论解,得到了振荡频率随气流Mach数M_∞和压缩折转角θ的变化规律,计算结果与实验值是相符的.本文为激波振荡导致的气动弹性问题提供了一种有价值的理论方法.     

Boundary Integral Equation Method in the Steady State Oscillation Problems for Anisotropic Bodies

The three-dimensional steady state oscillation problems of the elasticity theory for homogeneous anisotropic bodies are studied. By means of the limiting absortion principle the fundamental matrices maximally decaying at infinity are constructed and the generalized Sommerfeld–Kupradze type radiation conditions are formulated. Special functional spaces are introduced in which the basic and mixed exterior boundary value problems of the steady state oscillation theory have unique solutions for arbitrary values of the oscillation parameter. Existence theorems are proved by reduction of the original boundary value problems to equivalent boundary integral (pseudodifferential) equations. © 1997 by B.G. Teubner Stuttgart-John Wiley & Sons, Ltd.     

Fluctuations for ∇φ interface model on a wall

We consider ∇φ interface model on a hard wall. The hydrodynamic large-scale space-time limit for this model is discussed with periodic boundary by Funaki et al. (2000, preprint). This paper studies fluctuations of the height variables around the hydrodynamic limit in equilibrium in one dimension imposing Dirichlet boundary conditions. The fluctuation is non-Gaussian when the macroscopic interface is attached to the wall, while it is asymptotically Gaussian when the macroscopic interface stays away from the wall. Our basic method is the penalization. Namely, we substitute in the dynamics the reflection at the wall by strong drift for the interface when it goes down beyond the wall and show the fluctuation result for such massive ∇φ interface model. Then, this is applied to prove the fluctuation for the ∇φ interface model on the wall.     

Steady laminar boundary layer of a viscous incompressible fluid in a convergent channel with distributed suction at the wall

In this paper an attempt has been made to find the solution of the boundary layer equations for two-dimensional laminar steady motion of a viscous incompressible fluid in a convergent channel (sink flow) with suction at the wall. Suction velocity v₀ (x) ～ 1/x has been imposed at the wall and an approximate solution has been obtained with the help of similarity transformation. A solution valid at a large distance from the wall and a series solution valid near the wall have been obtained and the two solutions have been joined at a suitable point. It is seen that the boundary layer thickness diminishes as the value of the suction parameter\(\lambda ( = v_0 x/\sqrt {u_1 v} )\) increases. The velocity profile and the boundary layer parameters for solid wall (λ = 0) obtained from this solution are found to be in close agreement with the profile and the parameters calculated from the known exact solution for the solid wall problem.     

Quasineutral limit of a time-dependent drift-diffusion-Poisson model for p-n junction semiconductor devices

In this paper the vanishing Debye length limit of the bipolar time-dependent drift-diffusion-Poisson equations modelling insulated semiconductor devices with p-n junctions (i.e., with a fixed bipolar background charge) is studied. For sign-changing and smooth doping profile with ‘good’ boundary conditions the quasineutral limit (zero-Debye-length limit) is performed rigorously by using the multiple scaling asymptotic expansions of a singular perturbation analysis and the carefully performed classical energy methods. The key point in the proof is to introduce a ‘density’ transform and two λ-weighted Liapunov-type functionals first, and then to establish the entropy production integration inequality, which yields the uniform estimate with respect to the scaled Debye length. The basic point of the idea involved here is to control strong nonlinear oscillation by the interaction between the entropy and the entropy dissipation.     

L p solvability of divergence type parabolic and elliptic systems with partially BMO coefficients

We prove the ${{\mathcal{H}}^{1}_{p,q}}$ solvability of second order systems in divergence form with leading coefficients A ^???? only measurable in (t, x ¹) and having small BMO (bounded mean oscillation) semi-norms in the other variables. In addition, we assume one of the following conditions is satisfied: (i) A ¹¹ is measurable in t and has a small BMO semi-norm in the other variables; (ii) A ¹¹ is measurable in x ¹ and has a small BMO semi-norm in the other variables. The corresponding results for the Cauchy problem and elliptic systems are also established. Some of our results are new even for scalar equations. Using the results for systems in the whole space, we obtain the solvability of systems on a half space and Lipschitz domain with either the Dirichlet boundary condition or the conormal derivative boundary condition.     

Oscillation results for higher even order nonlinear partial functional differential equations of neutral type

For some higher even order nonlinear partial functional differential equations of neutral type with continuous distributed delay, we obtain new oscillation criteria of solutions for boundary value problem of the equations, in our paper we omit the assumption which has been required for related results given before. The main tool used is the standard Philos integral averaging technique.