以问题为中心展开深度互动

问题是学习活动的核心,教学时设置不同层次的问题有助于有效推进思维深入,展开深度互动。教师在英语教学时可设置情景式问题引“话题”互动,设置梯度性问题引“思维”互动,设置开放性问题引“语用”互动,设置文化辨析性问题引“文化”互动,让学生在互动中感知、理解和应用英语语言,从而有效提升学生的英语综合运用能力,提升学生的英语素养。

“问题教学法”与“梯度性提升学生问题能力”——以“中国近现代史纲要”课程为例

'问题教学法'是以问题为核心,通过大量有内在联系的'问题'的不断'刺激',促使学生在思考中掌握知识的一种教学方法。'梯度性提升学生问题能力'是在'问题教学法'实施的前提下,将学生的问题能力依次分为产生问题、追问问题和创生问题,并遵循循序渐进的规律,最终达到'问题是接生婆,它能帮助新思想的诞生'的境界。

数学教学中学生解题能力的培养研究

在数学教学中培养学生解题能力符合新课改的要求,能够增强学生学习数学的信心和兴趣。教师可以将抽象的知识形象化,降低解题难度;让学生逐层递进、从易到难地解决梯度性问题;利用微课平台对学生实施分层教学;利用生活化教学模式促进学生对知识的理解。

Pure bending of simply supported circular plate of transversely isotropic functionally graded material

This paper considers the pure bending problem of simply supported transversely isotropic circular plates with elastic compliance coefficients being arbitrary functions of the thickness coordinate. First, the partial differential equation, which is satisfied by the stress functions for the axisymmetric deformation problem is derived. Then, stress functions are obtained by proper manipulation. The analytical expressions of axial force, bending moment and displacements are then deduced through integration. And then, stress functions are employed to solve problems of transversely isotropic functionally graded circular plate, with the integral constants completely determined from boundary conditions. An elasticity solution for pure bending problem, which coincides with the available solution when degenerated into the elasticity solutions for homogenous circular plate, is thus obtained. A numerical example is finally presented to show the effect of material inhomogeneity on the elastic field in a simply supported circular plate of transversely isotropic functionally graded material (FGM).

Application of a Derivative-Free Method with Projection Skill to Solve an Optimization Problem

Improving numerical forecasting skill in the atmospheric and oceanic sciences by solving optimization problems is an important issue. One such method is to compute the conditional nonlinear optimal perturbation(CNOP), which has been applied widely in predictability studies. In this study, the Differential Evolution(DE) algorithm, which is a derivative-free algorithm and has been applied to obtain CNOPs for exploring the uncertainty of terrestrial ecosystem processes, was employed to obtain the CNOPs for finite-dimensional optimization problems with ball constraint conditions using Burgers' equation. The aim was first to test if the CNOP calculated by the DE algorithm is similar to that computed by traditional optimization algorithms, such as the Spectral Projected Gradient(SPG2) algorithm. The second motive was to supply a possible route through which the CNOP approach can be applied in predictability studies in the atmospheric and oceanic sciences without obtaining a model adjoint system, or for optimization problems with non-differentiable cost functions. A projection skill was first explanted to the DE algorithm to calculate the CNOPs. To validate the algorithm, the SPG2 algorithm was also applied to obtain the CNOPs for the same optimization problems. The results showed that the CNOPs obtained by the DE algorithm were nearly the same as those obtained by the SPG2 algorithm in terms of their spatial distributions and nonlinear evolutions. The implication is that the DE algorithm could be employed to calculate the optimal values of optimization problems, especially for non-differentiable and nonlinear optimization problems associated with the atmospheric and oceanic sciences.

A Probe Method of Gradient Projection Type

In this paper,a probe method for nonlinear programming wiht equality and inequality is given. Its iterative directions at an arbitrary point x can be obtained through solving a liear system. The terminate conditions and choices of the parameters are given. The global convergence of the method is proved. Further more,some well known gradient projection type algorithms [1-15] and new gradient projection type algorithms from the linear system are given in this paper.

Constraints in Lean Optimization Method Tested on a Real Engineering Problem

Optimizing a vehicle includes testing millions of parameters with hundreds of constraints of the performance. In this article, 162 parameters are optimized with 5 constraints using Lean Optimization combined with Linear Programming. The method converges in this example in about 100 evaluations. This is less than the gradient method needs for its first step.

An improved nonlinear conjugate gradient method with an optimal property

Conjugate gradient methods have played a special role in solving large scale nonlinear problems. Recently, the author and Dai proposed an efficient nonlinear conjugate gradient method called CGOPT, through seeking the conjugate gradient direction closest to the direction of the scaled memoryless BFGS method. In this paper, we make use of two types of modified secant equations to improve CGOPT method. Under some assumptions, the improved methods are showed to be globally convergent. Numerical results are also reported.

A Barzilai-Borwein conjugate gradient method

The linear conjugate gradient method is an optimal method for convex quadratic minimization due to the Krylov subspace minimization property. The proposition of limited-memory BFGS method and Barzilai-Borwein gradient method, however, heavily restricted the use of conjugate gradient method for largescale nonlinear optimization. This is, to the great extent, due to the requirement of a relatively exact line search at each iteration and the loss of conjugacy property of the search directions in various occasions. On the contrary, the limited-memory BFGS method and the Barzilai-Bowein gradient method share the so-called asymptotical one stepsize per line-search property, namely, the trial stepsize in the method will asymptotically be accepted by the line search when the iteration is close to the solution. This paper will focus on the analysis of the subspace minimization conjugate gradient method by Yuan and Stoer(1995). Specifically, if choosing the parameter in the method by combining the Barzilai-Borwein idea, we will be able to provide some efficient Barzilai-Borwein conjugate gradient(BBCG) methods. The initial numerical experiments show that one of the variants, BBCG3, is specially efficient among many others without line searches. This variant of the BBCG method might enjoy the asymptotical one stepsize per line-search property and become a strong candidate for large-scale nonlinear optimization.

Inverse Heat Transfer Problem of Thermal Contact Conductance Estimation in Periodically Contacting Surfaces

The problems involving periodic contacting surfaces have different practical applications. An inverse heat conductionproblem for estimating the periodic Thermal Contact Conductance (TCC) between one-dimensional, constantproperty contacting solids has been investigated with conjugate gradient method (CGM) of function estimation.This method converges very rapidly and is not so sensitive to the measurement errors. The advantage of thepresent method is that no a priori information is needed on the variation of the unknown quantities, since the solutionautomatically determines the functional form over the specified domain. A simple, straight forward techniqueis utilized to solve the direct, sensitivity and adjoint problems, in order to overcome the difficulties associatedwith numerical methods. Two general classes of results, the results obtained by applying inexact simulatedmeasured data and the results obtained by using data taken from an actual experiment are presented. In addition,extrapolation method is applied to obtain actual results. Generally, the present method effectively improves theexact TCC when exact and inexact simulated measurements input to the analysis. Furthermore, the results obtainedwith CGM and the extrapolation results are in agreement and the little deviations can be negligible.

Inverse Coefficient Problems for Elliptic Hemivariational Inequalities

This paper is devoted to a class of inverse coefficient problems for nonlinear elliptic hemivariational inequalities. The unknown coefficient of elliptic hemivariational inequalities depends on the gradient of the solution and belongs to a set of admissible coefficients. It is shown that the nonlinear elliptic hemivariational inequalities are uniquely solvable for the given class of coefficients. The result of existence of quasisolutions of the inverse problems is obtained.

Linear response eigenvalue problem solved by extended locally optimal preconditioned conjugate gradient methods

The locally optimal block preconditioned 4-d conjugate gradient method(LOBP4dC G) for the linear response eigenvalue problem was proposed by Bai and Li(2013) and later was extended to the generalized linear response eigenvalue problem by Bai and Li(2014). We put forward two improvements to the method: A shifting deflation technique and an idea of extending the search subspace. The deflation technique is able to deflate away converged eigenpairs from future computation, and the idea of extending the search subspace increases convergence rate per iterative step. The resulting algorithm is called the extended LOBP4 dC G(ELOBP4dC G).Numerical results of the ELOBP4 dC G strongly demonstrate the capability of deflation technique and effectiveness the search space extension for solving linear response eigenvalue problems arising from linear response analysis of two molecule systems.