Local solutions of the Hamilton-Jacobi-Bellman equation for some stochastic problems

Consideration is given to the control problem of motion of a linear oscillator, which is subject to the external Gaussian and Poisson random actions, with the aim to minimize the mean energy with the aid of the external bounded control force. The hybrid solution method is suggested for the solution of the stated problem. This method relies on the search in a portion of the phase space for the exact analytical solution of the appropriate Hamilton-Jacobi-Bellman (HJB) equation and the numerical solution of this equation in the remaining (bounded) portion of the space. It is proved that the found analytical solutions represent the asymptotics of solutions of the Hamilton-Jacobi-Bellman equation. With the aid of the decomposition method, the obtained results are applied to the problem for the suppression with the aid of the actuator of vibrations of an elastic rod (plate) that is found to be under the action of Gaussian random actions. Results of the numerical modeling are given.     

离散双输出反馈控制的最优协调性合成设计

本文以离散二次型最优控制理论为基础，研究了线性时不变系统在双输出反馈控制情况下控制作用的协调性合成设计问题，构造了一个数值迭代算法来求解相应的最侉 控制规律。通过产例计算和对比分析，既证明了该方法的收敛性和有效性又说明了这种控制问题的一些本质特征。     

Some new approaches to building and implementation of wall-functions for modeling of near-wall turbulent flows

To study and develop wall-functions for modeling of near-wall turbulent flows, a linear model equation is introduced. This equation simulates major mathematical peculiarities of the low-Reynolds-number model including a near wall sub-layer and transition region. Dirichlet and Newman boundary-value problems are considered. The standard and analytical wall-functions are investigated on different properties including the mesh sensitivity of a solution. A Robin-type interpretation of wall-functions as boundary conditions is suggested. It is shown that solution of a problem is mesh independent and more accurate in this case. General type analytical and numerical wall-functions are developed on the basis of a boundary condition transfer. An effective numerical method of decomposition is suggested. The method can be used in application to either high-Reynolds-number models with the numerical wall-functions or low-Reynolds-number models directly. Although a model equation is considered, the formulas, methods and conclusions are valid and can be directly used for the Reynolds Averaged Navier-Stokes (RANS) equations.     

A statistical minimax approach to optimizing linear models under a priori uncertainty conditions

A statistical minimax method for optimizing linear models with parameters, given up to the accuracy of belonging to some uncertainty sets, is proposed. Statistical methods for constructing uncertainty sets as confidence regions with a given reliability level are presented. A numerical method for finding a minimax strategy is proposed for arbitrary uncertainty sets that meet convexity and compactness conditions. A number of examples are considered that admit the analytical solution to optimization problem. Results of numerical simulation are given.     

Numerical solution of the time-dependent Dirac equation in coordinate space without fermion-doubling

The validation and parallel implementation of a numerical method for the solution of the time-dependent Dirac equation is presented. This numerical method is based on a split operator scheme where the space–time dependence is computed in coordinate space using the method of characteristics. Thus, most of the steps in the splitting are calculated exactly, making for a very efficient and unconditionally stable method. We show that it is free from spurious solutions related to the fermion-doubling problem and that it can be parallelized very efficiently. We consider a few simple physical systems such as the time evolution of Gaussian wave packets and the Klein paradox. The numerical results obtained are compared to analytical formulas for the validation of the method.     

Petrov-Galerkin finite element solution for the first passage probability and moments of first passage time of the randomly accelerated free particle

A numerical solution to an initial boundary value problem governing the probability of failure of a randomly accelerated free particle is obtained using a Petrov-Galerkin finite element method. This direct solution is the first successful one, and no others have been reported in the literature.A solution of the Pontriagin-Vitt equation for the time to first passage of the particle is obtained first: in this case an analytical solution is available and used to evaluate the numerical algorithm. Extensions to the solution of other stochastic differential equations, in particular those governing the probability of failure of the linear oscillator, and applications to structural dynamics are discussed.     

利用有限差分法计算真实头模型脑电正问题

脑电研究领域的两个关键问题是脑电正问题和脑电逆问题,脑电正问题是脑电逆问题的基础.由于复杂、非规则真实头模型中的脑电正问题不存在解析解,因此脑电源分析依赖于正问题数值算法的精度和效率.文章首先详细推导了有限差分算法求解三维各向同性脑电正问题的数学模型,然后在三层同心球模型上通过与解析解比较验证了该算法的精度和效率,最后将该算法应用于真实头模型.仿真结果表明,有限差分法可以有效地处理任意形状几何体的电位场分布问题,是模拟计算真实头模型中脑电正问题的有力工具.     

模糊非脆弱动态输出反馈H∞控制器设计

本文考虑受有限字长影响的离散时间模糊系统的非脆弱H∞控制问题. 假定所设计的控制器具有加性区间型增益变量, 该增益变量反映了控制器数字执行过程中有限字长的影响. 区间型增益变量导致控制器设计具有数值计算问题, 而模糊性质的引入进一步增加了控制器设计的复杂性, 使问题变得更具有挑战性. 本文采用结构的顶点分离器方法来解决数值计算问题, 从而给出一个基于线性矩阵不等式的模糊非脆弱H∞控制器设计的两步算法. 该设计结果保证闭环系统渐进稳定并具有指定的H∞性能指标. 最后给出一个数值例子验证所提出方法的有效性.     

Estimation of dynamic buckling for composite columns with open cross-section

Local, global and interactive dynamic responses of thin-walled columns with open cross-sections subjected to pulse compressive loading of different shape are discussed.An analytical–numerical method has been proposed to analyse the dynamic buckling problem. The applied method is based on the asymptotic Koiter’s theory for conservative systems in the second-order approximation. In order to obtain equations of column walls, the non-linear theory of orthotropic thin-walled plates has been modified in such a way that additionally it accounts for all components of inertia forces.The results of calculations obtained with the analytical–numerical method have been compared with those attained from the finite element method. The results are in good agreement.     

A parameter uniform B-spline collocation method for solving singularly perturbed turning point problem having twin boundary layers

A numerical scheme is proposed to solve singularly perturbed two-point boundary value problems with a turning point exhibiting twin boundary layers. The scheme comprises B-spline collocation method on a non-uniform mesh of Shishkin type. Asymptotic bounds are established for the derivative of the analytical solution of a turning point problem. The present method is boundary layer resolving as well as second-order accurate in the maximum norm. A brief analysis has been carried out to prove the uniform convergence with respect to the singular perturbation parameter ? by decomposing the solution into smooth and singular components. Some relevant numerical examples are also illustrated to verify computationally the theoretical aspects.     

Solution of a one-dimensional phase change problem with non-uniform initial temperature

The variable time step methods, proposed earlier by the authors with regard to phase change problems, are applied with some improvements to a problem having a non-uniform initial temperature distribution. A comparison between the numerical results from the present methods and those due to authors, who solve the problem with different techniques, shows a very good agreement. Approximate analytical expressions are also obtained for the temperature distribution as well as for the movement of the interface using the integral method. The computed values from these expressions are found to be very close to those obtained from the numerical methods.     

Impulsive displacement of a quasi-linear viscoelastic material through accurate numerical inversion of the Laplace transform

An analytical model for the one-dimensional, impulsive displacement of a quasi-linear viscoelastic material has been developed. The quasi-linear model of Fung [1] has been used successfully for a wide range of soft biological tissues. Due to the integral definition of linear viscoelastic materials, solutions are conveniently performed in the Laplace transform plane. Complex kernels like the quasilinear model are challenging to invert back to the real plane. Here, the method of Gaver [2] and Stehfest [3] is used to numerically carry Laplace space solutions to the real plane. Parametric results for a basic impulsive disturbance problem are presented. Results indicate that stress wave propagation is weakly dependent on the fast time, slow time ratio and more strongly dependent on the logarithmic damping parameter. Limitations of the numerical inversion method in the face of discontinuities are discussed as well using asymptotic methods. As an alternative to the numerical/polynomial-based Gaver-Stehfest method, a semianalytical regularization function useful near large gradient regions method is developed. A composite method that utilizes both the fully numerical and semianalytical convolution-based method is also described. The composite model provides improved results in terms of reducing computational undershoot and overshoot (wiggles) which limit both the fully numerical and the semianalytical models alone.     

Numerical method of estimating the maximal likelihood of a smooth parametric manifold

Consideration was given to the numerical estimation of the maximum likelihood for the parameter vector describing a smooth manifold. Estimation is based on the results of observing motion of a dynamic plant whose trajectory belongs to this manifold and is measured with random errors having normal distribution with certain parameters. Application of the maximum likelihood method to such problems gives rise to the problem of nonlinear highdimensionality programming. Some constructive analytical results obtained enable significant reduction in the problem dimensionality. The problem of identifying the plane of motion of a dynamic plant was examined.     

同时稳定两单输入系统控制器的解析解

研究两个单输入控制系统的同时稳定问题。证明了同时稳定两个单输入系统控制器解存在的充分条件，给出了设计同时稳定两单输入系统控制器的具体表达式。该方法计算简单，便于实际控制系统设计使用。通过算例给出了该方法的具体计算步骤。     

On the solution of the three forces problem and its application in optimal designing of a class of symmetric plane frameworks of least weight

Two problems of minimum weight design of plane trusses are dealt with. The first problem concerns construction of the lightest fully stressed truss subject to three self-equilibrated forces applied at three given points. This problem has been solved analytically by H.S.Y. Chan in 1966. This analytical solution is re-derived in the present paper. It compares favourably with new numerical solutions found here by the method developed recently by the first author. The solution to the three forces problem paves the way to half-analytical as well as numerical solutions to the problem of minimum weight design of plane symmetric frameworks transmitting two symmetrically located vertical forces to two fixed supports lying along the line linking the points of application of the forces.     

Collocation method using artificial viscosity for solving stiff singularly perturbed turning point problem having twin boundary layers

A numerical scheme is proposed to solve singularly perturbed two-point boundary value problems with a turning point exhibiting twin boundary layers. The scheme comprises a B-spline collocation method on a uniform mesh, which leads to a tridiagonal linear system. Asymptotic bounds are established for the derivative of the analytical solution of a turning point problem. The analysis is done on a uniform mesh, which permits its extension to the case of adaptive meshes which may be used to improve the solution. The design of an artificial viscosity parameter is confirmed to be a crucial ingredient for simulating the solution of the problem. Some relevant numerical examples are also illustrated to verify computationally the theoretical aspects.     

The effects of inertia on the structure of viscous corner eddies

Viscous eddies in the region close to a sharp corner are examined. The asymmetry in their structure that is apparent in numerical solutions for moderate values of the Reynolds number is derived analytically. A comparison is given with previous numerical studies and the agreement is found to be good. Some numerical verification of the analytical results is obtained from a study of the driven cavity flow problem for Reynolds numbers in the range 0–1000.     

Analytical solution of the optimal slew problem for an axisymmetric spacecraft in the class of conical motions

The traditional problem is discussed of an optimal spacecraft slew in terms of minimum energy costs. The spacecraft is considered as a rigid body with one symmetry axis under arbitrary boundary conditions for the angular position and angular velocity of the spacecraft in the quaternion formulation. Using substitutions of variables, the original problem is simplified (in terms of dynamic Euler equations) to the optimal slew problem for a rigid body with a spherical mass distribution. The simplified problem contains one additional scalar differential equation. A new analytical solution is presented for this problem in the class of conical motions, leading to constraints on the initial and final values of the angular velocity vector. In addition, the optimal slew problem is modified in the class of conical motions to derive an analytical solution under arbitrary boundary conditions for the angular position and angular velocity of the spacecraft. A numerical example is given for the conical motion of the spacecraft, as well as examples showing the closeness of the solutions of the traditional and modified optimal slew problems for an axisymmetric spacecraft.     

Capacitance of microstrip lines on anisotropic substrate using the method of lines and finite elements

A finite elements-based Method of Lines is used to calculate static capacitances for microstrip line structures with layered substrates of arbitrary permittivity tensor. The problem is solved by discretizing the associated static equations in one coordinate variable using finite elements, and obtaining an analytical solution with respect to the remaining variable. Proper boundary and transition conditions are imposed. To validate the solution, comparison is made with results derived from other methods. A significant increase in numerical efficiency afforded by the presented method, relative to the conventional, fully discrete, finite element solution, is demonstrated quantitatively.     

辛几何形态下不同边界条件的薄板解析解

利用平面弹性与板弯曲的相似性理论,用直接法研究辛几何形态下的薄板弯曲问题。当薄板对边边界条件形式不同时,将其进行降阶形成对偶方程组,再利用分离变量法把问题转化为本征值问题求解。通过本征函数、辛正交关系、展开求解等手段得到了薄板的解析解。算例表明辛求解的有效性与快速收敛性。