Questioning numerical integration methods for microsphere (and microplane) constitutive equations

In the last few years, more and more complex microsphere models have been proposed to predict the mechanical response of various polymers. Similarly than for microplane models, they consist in deriving a one-dimensional force vs. stretch equation and to integrate it over the unit sphere to obtain a three-dimensional constitutive equation. In this context, the focus of authors is laid on the physics of the one-dimensional relationship, but in most of the case the influence of the integration method on the prediction is not investigated.Here we compare three numerical integration schemes: a classical Gaussian scheme, a method based on a regular geometric meshing of the sphere, and an approach based on spherical harmonics. Depending on the method, the number of integration points may vary from 4 to 983,040! Considering simple quantities, i.e. principal (large) strain invariants, it is shown that the integration method must be carefully chosen. Depending on the quantities retained to described the one-dimensional equation and the required error, the performances of the three methods are discussed. Consequences on stress–strain prediction are illustrated with a directional version of the classical Mooney–Rivlin hyperelastic model. Finally, the paper closes with some advices for the development of new microsphere constitutive equations.     

Exactly satisfying initial conditions neural network models for numerical treatment of first Painlevé equation

In this paper, novel computing approach using three different models of feed-forward artificial neural networks (ANNs) are presented for the solution of initial value problem (IVP) based on first Painlevé equation. These mathematical models of ANNs are developed in an unsupervised manner with capability to satisfy the initial conditions exactly using log-sigmoid, radial basis and tan-sigmoid transfer functions in hidden layers to approximate the solution of the problem. The training of design parameters in each model is performed with sequential quadratic programming technique. The accuracy, convergence and effectiveness of the proposed schemes are evaluated on the basis of the results of statistical analyses through sufficient large number of independent runs with different number of neurons in each model as well. The comparisons of these results of proposed schemes with standard numerical and analytical solutions validate the correctness of the design models.     

Systematic convergence of nonlinear stochastic estimators on the Special Orthogonal Group SO(3)

This paper introduces two novel nonlinear stochastic attitude estimators developed on the Special Orthogonal Group with the tracking error of the normalized Euclidean distance meeting predefined transient and steady‐state characteristics. The tracking error is confined to initially start within a predetermined large set such that the transient performance is guaranteed to obey dynamically reducing boundaries and decrease smoothly and asymptotically to the origin in probability from almost any initial condition. The proposed estimators produce accurate attitude estimates with remarkable convergence properties using measurements obtained from low‐cost inertial measurement units. The estimators proposed in continuous form are complemented by their discrete versions for the implementation purposes. The simulation results illustrate the effectiveness and robustness of the proposed estimators against uncertain measurements and large initialization error, whether in continuous or discrete form.     

Moving Least-Squares Differential Quadrature Method for Free Vibration of Antisymmetric Laminates

In this paper, the moving least-squares differential quadrature (MLSDQ) method is employed for free vibration of thick antisymmetric laminates based on the first-order shear deformation theory. The generalized displacements of the laminates are independently approximated with the centered moving least-squares (MLS) technique within each domain of influence. The MLS nodal shape functions and their partial derivatives are computed quickly through back-substitutions after only one LU decomposition. Subsequently, the weighting coefficients in the MLSDQ discretization are determined with the nodal partial derivatives of the MLS shape functions. The MLSDQ method combines the merits of both the differential quadrature and meshless methods which can be conveniently applied to complex domains and irregular discretizations without loss of implementation efficiency and numerical accuracy. The natural frequencies of the laminates with various edge conditions, ply angles, and shapes are calculated and compared with the existing solutions to study the numerical accuracy and stability of the MLSDQ method. Effects of support size, order of completeness of basis functions, and node irregularity on the numerical accuracy are investigated in detail.     

带参数的一阶脉冲微分方程边值问题

利用上下解和单调迭代法。讨论了带参数的一阶脉冲微分方程。获得了单边Lipschitz条件下这类方程边值问题的解的存在性。     

Estimation and Optimization of the Principal H-Maser Parameters

The scale along the coordinate axes in the equations of motion hitherto used for an H-maser (in relative units) is shown to depend on Q as the quality factor for an H-maser. This hinders both analysis of the operation of the H-maser and optimization of its parameters. The modified equations of motion proposed here do not have that shortcoming. We determine the atomic flux into the bulb that gives the H-maser the best metrological characteristics.     

Coupling of free surface and groundwater flows

In this paper we present some preliminary results about the coupling of shallow water equations for free surface flows and Darcy equation for groundwater flows. A suitable set of interface conditions is discussed: the Beavers and Joseph formula for the bottom stress is used. An iterative algorithm to solve the coupled problem is proposed and some numerical results are presented.     

Stabilization Methods for Spectral Element Computations of Incompressible Flows

A stabilization method for the spectral element computation of incompressible flow problems is investigated. It is based on a filtering procedure which consists in filtering the velocity field by a spectral vanishing Helmholtz-type operator at each time step. Relationship between this filtering procedure and SVV-stabilization method, introduced recently in [JCP, 2004, 196(2), p680], is established. A number of numerical examples are presented to show the accuracy and stabilization capability of the method.     

一种复合型数值方法的改进与算法实现

文中对原有的复合方法进行了改进,提出了一种新的复合型数值方法,这种方法解决了原方法求解含源项的问题时存在数值震荡的缺点。为了验证算法的有效性,对计算流体力学中一阶浅水波问题的两个典型的算例进行了数值模拟,在解的光滑性、锐利性等几个方面做了分析,结果表明改进后的方法既消除了数值震荡现象,又能准确地逼近真解,求出锐利的间断,是一种行之有效的数值方法。     

泥沙输移矢量式及其应用

本文基于Parker等人泥沙输移矢量式的分析和研究，增加了上举力的作用，并考虑了泥沙颗粒受力非平衡条件，采用牛顿力学的矢量分析法，推导了泥沙输移的通用矢量式。经对公式简化，且用Ikeda和笔者试验资料的验证，证明计算值与试验值吻合较好。