An analogue of the four-point Newton-Cotes formula for a function with a boundary-layer component

The construction of the Newton-Cotes formulas is based on approximating the integrand by a Lagrange polynomial. The error of such quadrature formulas can be great for a function with a boundary-layer component. In this paper, an analog of the four-point Newton-Cotes rule is constructed. The construction is based on using a nonpolynomial interpolation that is exact for the boundary layer component. Error estimates of the quadrature rule independent of the boundary layer component gradients are obtained. Numerical experiments are performed.     

Cubature formulas for a two-variable function with boundary-layer components

Cubature formulas for evaluating the double integral of a two-variable function with boundary-layer components are constructed and studied. Because of the boundary-layer components, the cubature formulas based on Newton-Cotes formulas become considerably less accurate. Analogues of the trapezoidal and Simpson rules that are exact for the boundary-layer components are constructed. Error estimates for the constructed formulas are derived that are uniform in the gradients of the integrand in the boundary layers.     

Analogue of Newton–Cotes formulas for numerical integration of functions with a boundary-layer component

The numerical integration of functions with a boundary-layer component whose derivatives are not uniformly bounded is investigated. The Newton–Cotes formulas as applied to such functions can lead to significant errors. An analogue of Newton–Cotes formulas that is exact for the boundary-layer component is constructed. For the resulting formula, an error estimate that is uniform with respect to the boundary-layer component and its derivatives is obtained. Numerical results that agree with the error estimates are presented.     

Spline interpolation on a uniform grid for functions with a boundary-layer component

Spline interpolation of functions of one variable with a boundary-layer component is examined. Functions of this type can arise in the solution of a singularly perturbed boundary value problem on an interval. Spline interpolation formulas that are exact for the boundary-layer component are constructed, and their errors are estimated. Formulas for calculating the derivative based on the constructed interpolants are obtained. Numerical results are presented.     

Modification of the Euler quadrature formula for functions with a boundary-layer component

The Euler quadrature formula for the numerical integration of functions with a boundary-layer component on a uniform grid is investigated. If the function under study has a rapidly growing component, the error can be significant. A uniformly accurate quadrature formula is constructed by modifying the Hermite interpolation formula so that the resulting one is exact for the boundary-layer component. An analogue of the Euler formula that is exact for the boundary-layer component is constructed. It is proved that the resulting composite quadrature formula is third-order accurate in space uniformly with respect to the boundary-layer component and its derivatives.     

Analysis of Numerical Differentiation Formulas in a Boundary Layer on a Shishkin Grid

A problem of numerical differentiation of functions with large gradients in a boundary layer is investigated. The problem is that for functions with large gradients and a uniform grid the relative error of the classical difference formulas for derivatives may be considerable. It is proposed to use a Shishkin grid to obtain a relative error of the formulas that is independent of a small parameter. Error estimates that depend on the number of nodes of the difference formulas for a derivative of a given order are obtained. It is proved that the error estimate is uniform with respect to the small parameter. In the case of a uniform grid, a boundary layer region is indicated outside of which the numerical differentiation formulas have an error that is uniform with respect to the small parameter. The results of numerical experiments are presented.     

Inertia Effects in Squeeze Flows of Viscoplastic Fluids

A squeeze flow of a viscoplastic fluid through a narrow clearance between two coaxial surfaces of revolution is considered. The problem is described by boundary-layer equations. With the use of the method of integral approaches, formulas for the pressure distribution are obtained. Generally, the flow of viscoplastic fluids given by the nonlinear Shulman model is considered. The flows of viscoplastic fluids given by the Herschel, Bulkley, Bingham, Ostwald-de Waele, and Newton models are discussed in detail. Numerical examples of pressure distributions in the clearance between parallel disks are presented.     

Refined-mesh interpolation method for functions with a boundary-layer component

Linear and quadratic spline interpolation methods for a one-variable function with a boundary-layer component are examined. It is shown that the interpolation method for such a function leads to considerable errors when applied on a uniform mesh. The error of linear and quadratic spline interpolations on meshes that are refined in the boundary layer is estimated. Numerical results are presented.     

桁架结构的灵敏度分析及其在满应力设计中的应用

就桁架结构而言,本文在结构变化定理的基础上提出了计算杆内力、杆应力和节点位移关于杆面积偏导数(梯度)的一组公式.与已有的计算结构响应的梯度公式比较,在一般情形下,用本文公式进行计算所需的附加载荷个数最少,因而计算量也小.这对于广泛使用响应梯度的许多优化方法有减少机时的实用价值.另外,我们还将导出的梯度公式用于满应力设计,得到一个改进的满应力迭代公式.算例表明与简单应力比法相比,改进的方法大大地减少了收敛于满应力设计所需的结构重分析次数.     

Bochner-Weitzenböck formulas and curvature actions on Riemannian manifolds

Gradients are natural first order differential operators depending on Riemannian metrics. The principal symbols of them are related to the enveloping algebra and higher Casimir elements. We give formulas in the enveloping algebra that induce not only identities for higher Casimir elements but also all Bochner-Weitzenböck formulas for gradients. As applications, we give some vanishing theorems.

     

Turbulent boundary layer on a plate. Reynolds analogy and new formulations of the temperature defect law

A consistent asymptotic theory describing hydrodynamic and thermal turbulent boundary layers on a flat plate in zero pressure gradient is developed. The fact that the flow depends on a limited number of governing parameters allows us to formulate algebraic closure conditions that relate the turbulent shear stress and turbulent heat flux to mean velocity and temperature gradients. As a result of an exact asymptotic solution of the boundary-layer equations, the known laws of the wall for the velocity and temperature and the velocity and temperature defect laws as well as the expressions for the skin-friction coefficient, Stanton number, and Reynolds-analogy factor are obtained. The latter implies two new formulations for the temperature defect law one of which is completely similar to the velocity defect law and does not contain the Stanton number and the turbulent Prandtl number, and the other does not contain the skin-friction coefficient. A heat-transfer law is obtained that relates only thermal quantities. The theoretical conclusions agree well with experimental data. (© 2015 Wiley-VCH Verlag GmbH & Co. KGaA, Weinheim)     

Generalized gradients of Lipschitz functionals

The purpose of this article is threefold: (i) to present in a unified fashion the theory of generalized gradients, whose elements are at present scattered in various sources; (ii) to give an account of the ways in which the theory has been applied; (iii) to prove new results concerning generalized gradients of summation functionals, pointwise maxima, and integral functionals on subspaces of L^∞. These last-mentioned formulas are obtained with an eye to future applications in the calculus of variations and optimal control. Their proofs can be regarded as applications of the existing theory of subgradients of convex functionals as developed by Rockafellar, Ioffe and Levin, Valadier, and others.     

Semi-Linear Difference Schemes

A class of semi-linear numerical differentiation formulas is designed for functions with steep gradients. A semi-linear second-order difference scheme is constructed to solve the two-point singular perturbation problem. It is shown that this semi-linear scheme has one more order of approximation precision than the central difference scheme for small $\epsilon$ and saves computation time for required accuracy. Numerical results agreeing with the above analysis are included.     

A general approach to get series solution of non-similarity boundary-layer flows

An analytic method for strongly non-linear problems, namely the homotopy analysis method (HAM), is applied to give convergent series solution of non-similarity boundary-layer flows. As an example, the non-similarity boundary-layer flows over a stretching flat sheet are used to show the validity of this general analytic approach. Without any assumptions of small/large quantities, the corresponding non-linear partial differential equation with variable coefficients is transferred into an infinite number of linear ordinary differential equations with constant coefficients. More importantly, an auxiliary artificial parameter is used to ensure the convergence of the series solution. Different from previous analytic results, our series solutions are convergent and valid for all physical variables in the whole domain of flows. This work illustrates that, by means of the homotopy analysis method, the non-similarity boundary-layer flows can be solved in a similar way like similarity boundary-layer flows. Mathematically, this analytic approach is rather general in principle and can be applied to solve different types of non-linear partial differential equations with variable coefficients in science and engineering.     

Thrust nozzle optimization including boundary-layer effects

An optimization analysis employing the calculus of variations and a numerical relaxation scheme for applying the results are presented for the design of axisymmetric, maximumthrust nozzle contours including boundary-layer effects. The method is illustrated by comparison with a known solution without boundary-layer effects. A parametric study of boundary-layer effects for a range of parameters typical of scramjet thrust nozzles is presented.This work was sponsored by the Air Force Aero-Propulsion Laboratory, Wright-Patterson Air Force Base, Ohio, under Contract No. F33615-67-C-1068.     

Boundary Layers and Boundary Hinge-Support Conditions for Thin Plates

The boundary-layer phenomenon is investigated for a thin three-dimensional plate. An arbitrary anisotropy of elastic properties and nonhomogeneity in longitudinal and transversal directions are assumed. Several initial asymptotic terms are constructed and a way of continuing the asymptotic procedure is described. Precise estimates of the difference between the exact and approximate solutions are obtained for the energy norm. An amalgamated problem is formed that gives the two-term asymptotics of the fields of displacements and stresses at a distance from the lateral side of the plate, whereas the edge effects are simulated by conditions of elastic fastening containing integral characteristics of the boundary-layer problem in the semistrip. Hinge-support conditions due to a sufficiently narrow clamped zone at the edge of the plate are derived and justified as well. Bibliography: 48 titles.     

有吸除的层流边界层问题

本文对于有吸除的普遍层流边界层方程求得其渐近解,然后推导了位移厚度、动量厚度和表面摩擦的计算公式.此外还处理了确定分离点位置的问题.最后以具有恒定吸除的平板均匀绕流情形为例,就某些边界层特征参数作了数值计算,我们所得的结果与Iglisch获得的结果很好地符合.     

On the computability of the Steenrod squares

We give explicitely the formulas of a sequence of morphisms which measure the failure of commutativity of the cup product on the cochain level, provided that we work with simplicial sets; these formulas are established in terms of the component morphisms of a given Eilenberg-Zilber contraction. As a consequence, in the case in which the simplicial set is finite in each dimension, we obtain an algorithm for calculating Steenrod squares.     

On Generic Existence and Uniqueness in Nonconvex Optimal Control Problems

We derive conditions for generic existence and uniqueness of optimal control and trajectories for some class of finite-dimensional optimal control problems in the absence of traditional convexity assumptions. It is shown that for these problems existence and uniqueness of optimal control for a given initial point x is equivalent to the differentiability of optimal value functions at x. These results are obtained in the general framework of representation formulas for (sub-)gradients of inf-envelope functions which appear in nonsmooth analysis.     

Stabilization of Laminar Boundary-Layer Flow using Dielectric Barrier Discharge Plasma Actuators

A numerical method is developed to study the stabilizing effect of dielectric barrier discharge plasma actuators on laminar boundary-layer flow. A finite difference approach based on a Keller box discretization is chosen to solve the Falkner-Skan transformed boundary-layer equations. The fluid dynamic effect of the flow-control device is implemented as a body-force field, derived quantitatively from previous measurements using particle image velocimetry. The resulting laminar boundary-layer flow is compared to experimental wind tunnel measurements and the effect on hydrodynamic stability is investigated in the framework of linear stability theory. A good agreement between experimentally acquired and numerically predicted transition locations based on an empirical function is obtained, rendering the numerical scheme valuable as a design tool for DBD based flow control applications. (© 2012 Wiley-VCH Verlag GmbH & Co. KGaA, Weinheim)