基于修正高精度广义胞元法的复合材料细观力学分析

高精度广义胞元法是多尺度分析复合材料模量和微观应力应变场的有效方法之一．然而，由于位移插值函数中缺少二次耦合项，很大程度上影响了复合材料局部应力、应变场，特别是剪切场的计算精度．本文通过引入二次方向耦合项，提出了一种修正的高精度广义胞元法插值函数．在施加周期性边界条件、平均应力和平均位移连续性条件后，可以确定位移插值函数中的系数．通过对多相复合材料弹性模量和局部场分析，并且与有限元分析和实验测量结果比较，验证了修正高精度广义胞元法的准确性．与高精度广义胞元相比，本文提出的修正高精度广义胞元法在不需要引入额外未知变量，不影响计算效率的前提下，对复合材料的局部应力场计算得更加准确．     

An improved local radial point interpolation method for transient heat conduction analysis

The smoothing thin plate spline （STPS） interpolation using the penalty function method according to the optimization theory is presented to deal with transient heat conduction problems. The smooth conditions of the shape functions and derivatives can be satisfied so that the distortions hardly occur. Local weak forms are developed using the weighted residual method locally from the partial differential equations of the transient heat conduction. Here the Heaviside step function is used as the test function in each sub-domain to avoid the need for a domain integral. Essential boundary conditions can be implemented like the finite element method （FEM） as the shape functions possess the Kronecker delta property. The traditional two-point difference method is selected for the time discretization scheme. Three selected numerical examples are presented in this paper to demonstrate the availability and accuracy of the present approach comparing with the traditional thin plate spline （TPS） radial basis functions.     

A model of students’ combinatorial thinking

Combinatorial topics have become increasingly prevalent in K-12 and undergraduate curricula, yet research on combinatorics education indicates that students face difficulties when solving counting problems. The research community has not yet addressed students’ ways of thinking at a level that facilitates deeper understanding of how students conceptualize counting problems. To this end, a model of students’ combinatorial thinking was empirically and theoretically developed; it represents a conceptual analysis of students’ thinking related to counting and has been refined through analyzing students’ counting activity. In this paper, the model is presented, and relationships between formulas/expressions, counting processes, and sets of outcomes are elaborated. Additionally, the usefulness and potential explanatory power of the model are demonstrated through examining data both from a study the author conducted, and from existing literature on combinatorics education.     

Uniform interpolation and the existence of sequent calculi

This paper presents a uniform and modular method to prove uniform interpolation for several intermediate and intuitionistic modal logics. The proof-theoretic method uses sequent calculi that are extensions of the terminating sequent calculus $\mathsf{G4ip}$ for intuitionistic propositional logic. It is shown that whenever the rules in a calculus satisfy certain structural properties, the corresponding logic has uniform interpolation. It follows that the intuitionistic versions of $\mathsf{K}$ and $\mathsf{KD}$ (without the diamond operator) have uniform interpolation. It also follows that no intermediate or intuitionistic modal logic without uniform interpolation has a sequent calculus satisfying those structural properties, thereby establishing that except for the seven intermediate logics that have uniform interpolation, no intermediate logic has such a sequent calculus.     

Piecewise Linear Interpolation With Nonequidistant Nodes

Pointwise and norm estimates for approximation of continuous function by piecewise linear interpolant with nonequidistant nodes are proved. Special attention is paid to the value of the absolute constants in front of the modulus.     

Orthogonal Polynomials Associated with a Δ-Sobolev Inner Product

We will study the family of polynomials which are orthogonal with respect to the discrete inner product involving difference operators $$(\,f,g)^{(K)}_\Delta=\sum^K_{m,k=0} \langle \lambda_{m,k}u,\Delta{^m}f\Delta^{k}g\rangle, \quad K \ge 0$$     

A C2‐continuous high‐resolution upwind convection scheme

A bounded upwinding scheme for numerical solution of hyperbolic conservation laws and Navier–Stokes equations is presented. The scheme is based on convection boundedness criterion and total variation diminishing stability criteria and developed by employing continuously differentiable functions. The accuracy of the scheme is verified by assessing the error and observed convergence rate on 1‐D benchmark test cases. A comparative study between the new scheme and conventional total variation diminishing/convection boundedness criterion‐based upwind schemes to solve standard nonlinear hyperbolic conservation laws is also accomplished. The scheme is then examined in the simulation of Newtonian and non‐Newtonian fluid flows of increasing complexity; a satisfactory agreement has been observed in terms of the overall behavior. Finally, the scheme is used to study the hydrodynamics of a gas‐solid flow in a bubbling fluidized bed. Copyright © 2013 John Wiley & Sons, Ltd.     

A Darboux-type Theory of Integrability for Discrete Dynamical Systems

There is a method for searching first integrals for polynomial ordinary differential equations (usually called Darboux method) based on the knowledge of several of their invariant algebraic hypersurfaces. We extend this method to discrete dynamical systems, providing a way of searching invariants for them and we give several examples of application.     

A new approach to numerical differentiation

In this paper we consider the numerical differentiation of functions specified by noisy data. A new approach, which is based on an integral equation of the first kind with a suitable compact operator, is presented and discussed. Since the singular system of the compact operator can be obtained easily, TSVD is chosen as the needed regularization technique and we show that the method calls for a discrete sine transform, so the method can be implemented easily and fast. Numerical examples are also given to show the efficiency of the method.