Affine approximation of isoparametric elements

The use of rectangular isoparametric elements in finite element analysis of second-order boundary-value problems requires evaluating integrals of rational polynomial functions. Gaussian quadrature formulas are currently the most popular method of obtaining approximations to the exact integrals. A new method is described in which the isoparametric finite element function spaces are approximated. The resulting integrals can be evaluated exactly, avoiding the computational expense of the Gaussian quadrature schemes, particularly the 27 point formula used in three-dimensional elements.     

Robust design with arbitrary distributions using Gauss-type quadrature formula

In this paper, we present the Gauss-type quadrature formula as a rigorous method for statistical moment estimation involving arbitrary input distributions and further extend its use to robust design optimization. The mathematical background of the Gauss-type quadrature formula is introduced and its relation with other methods such as design of experiments (DOE) and point estimate method (PEM) is discussed. Methods for constructing one dimensional Gauss-type quadrature formula are summarized and the insights are provided. To improve the efficiency of using it for robust design optimization, a semi-analytic design sensitivity analysis with respect to the statistical moments is proposed for two different multi-dimensional integration methods, the tensor product quadrature (TPQ) formula and the univariate dimension reduction (UDR) method. Through several examples, it is shown that the Gauss-type quadrature formula can be effectively used in robust design involving various non-normal distributions. The proposed design sensitivity analysis significantly reduces the number of function calls of robust optimization using the TPQ formulae, while using the UDR method, the savings of function calls are observed only in limited situations.     

On Hamiltonian realization of time-varying nonlinear systems

This paper investigates Hamiltonian realization of time-varying nonlinear (TVN) systems, and proposes a number of new methods for the problem. It is shown that every smooth TVN system can be expressed as a generalized Hamiltonian system if the origin is the equilibrium of the system. If the Jacopian matrix of a TVN system is nonsingu-lar, the system has a generalized Hamiltonian realization whose structural matrix and Hamiltonian function are given explicitly. For the case that the Jacobian matrix is singular, this paper provides a constructive decomposition method, and then proves that a TVN system has a generalized Hamiltonian realization if its Jacobian matrix has a non-singular main diagonal block. Furthermore, some sufficient (necessary and sufficient) conditions for dissipative Hamiltonian realization of TVN systems are also presented in this paper.     

Generalized neurofuzzy network modeling algorithms usingBezier-Bernstein polynomial functions and additive decomposition

This paper introduces a new neurofuzzy model construction algorithm for nonlinear dynamic systems based upon basis functions that are Bezier-Bernstein polynomial functions. This paper is generalized in that it copes with n-dimensional inputs by utilising an additive decomposition construction to overcome the curse of dimensionality associated with high n. This new construction algorithm also introduces univariate Bezier-Bernstein polynomial functions for the completeness of the generalized procedure. Like the B-spline expansion based neurofuzzy systems, Bezier-Bernstein polynomial function based neurofuzzy networks hold desirable properties such as nonnegativity of the basis functions, unity of support, and interpretability of basis function as fuzzy membership functions, moreover with the additional advantages of structural parsimony and Delaunay input space partition, essentially overcoming the curse of dimensionality associated with conventional fuzzy and RBF networks. This new modeling network is based on additive decomposition approach together with two separate basis function formation approaches for both univariate and bivariate Bezier-Bernstein polynomial functions used in model construction. The overall network weights are then learnt using conventional least squares methods. Numerical examples are included to demonstrate the effectiveness of this new data based modeling approach.     

Normal forms and the multiple-times expansion method for finite-dimensional integrable Hamiltonian systems

The authors apply the method of multiple-times expansion to finite-dimensional integrable Hamiltonian systems of polynomial type in order to determine integrable Hamiltonian systems and to derive new integrable systems from previously known ones. Recursion operators for the derived integrable systems are obtained. Normal forms for finite-dimensional integrable Hamiltonian systems are also constructed. It is demonstrated that the Hamiltonians found by the multiple-times expansion method are indeed the normal form expansions.     

Quadrature imposition of compatibility conditions in Chebyshev methods

Often, in solving an elliptic equation with Neumann boundary conditions, a compatibility condition has to be imposed for well-posedness. This condition involves integrals of the forcing function. When pseudospectral Chebyshev methods are used to discretize the partial differential equation, these integrals have to be approximated by an appropriate quadrature formula. The Gauss-Chebyshev (or any variant of it, like the Gauss-Lobatto) formula cannot be used here since the integrals under consideration do not include the weight function. A natural candidate to be used in approximating the integrals is the Clenshaw-Curtis formula; however, we show in this article that this is the wrong choice and it may lead to divergence if time-dependent methods are used to march the solution to steady state. We develop, in this paper, the correct quadrature formula for these problems. This formula takes into account the degree of the polynomials involved. We show that this formula leads to a well-conditioned Chebyshev approximation to the differential equations and that the compatibility condition is automatically satisfied.     

Optimal quadrature formula nonlinear estimators

This paper presents a method for the realization of nonlinear estimators based on an optimal quadrature approximation. The optimal quadrature formula is obtained by solving a set of nonlinear algebraic equations induced from a monospline subject to a set of interpolatory conditions. All the weights of the optimal quadrature formulas derived from monosplines which do not involve the derivatives of the integrand at the end points are positive. This guarantees the positiveness of the quantized density functions in numerical approximation of Bayesian recursive computations. The numerical errors associated with the optimal quadrature approximation to Bayesian recursive computations are discussed. Finally methods of quantizing and updating the prediction and filtering densities are derived.     

Extension of numerical quadrature formulae to cater for end point singular behaviours over finite intervals

The N-point Gaussian quadrature method is generalised to cater for various possible singular behaviours at the end points of the interval of integration at the expense of being algebraically exact for a polynomial of lower order than usual. Weights and abscissae are chosen to exactly integrate an integrand which is the sum of the singular functions and an arbitrary polynomial. This allows us to cater for several different end-point singularities in the same quadrature formula and in this way differs from published quadratures where a singular behaviour is incorporated in a weight function that multiplies an arbitrary polynomial. We present tables of weights and abscissae that cater for (i) logarithmic end point singularities and (ii) logarithmic plus inverse square root singular behaviours. Also a 10-point quadrature is presented that exactly caters for log(x), x^-1/4,x^-1/2,x^-3/4 singular behaviours and is recommended for programmable calculator use. Finally a brief comparison study of the various (10-point) quadratures herein considered is made.     

求解Cauchy型奇异积分方程的数值方法

1.引 言 断裂力学中许多裂纹问题的数学模型都可归结为奇异积分方程(SIE)[1,2].由于这些奇异积分方程的封闭解一般情况下都难以得到,因而数值方法受到广泛的注意.Muskhelishvili[3]对奇异积分方程的一般理论进行了深入的研究.这些研究成果为奇异积分方程的求解,不论     

分维自适应稀疏网格积分非线性滤波器

针对含加性高斯噪声的非线性离散系统,提出了可分别根据各维状态及量测方程的非线性函数特性来确定采样点及其权重的积分滤波器.设计了基于嵌入式高斯采样积分和稀疏网格法则的自适应多变量采样积分方法,可在匹配函数高阶泰勒展开项时,利用低阶采样点,提出了高效的数据结构和遍历算法,便于采用该积分方法分别估计系统状态/量测的预测均值和协方差矩阵.该滤波器既能根据各维非线性函数的特性确定采样点,又实现了对采样值和权重的完全复用,保证了算法效率.理论分析和仿真表明,该滤波算法中自适应调整的运算量小于计算非线性函数采样值.该滤波器与无迹卡尔曼滤波相比,提高了滤波精度,与固定形式的稀疏网格滤波器相比,提高了采样效率,且该方法为两者的广义形式.仿真实验也验证了状态估计的精确性和函数采样的高效性.     

Asymptotic stability of port‑hamiltonian systems with constant inputs

In this paper, the asymptotic stability of Port-Hamiltonian (PH) systems with constant inputs is studied. Constant inputs are useful for stabilizing systems at their nonzero equilibria and can be realized by step signals. To achieve this goal, two methods based on integral action and comparison principle are presented in this paper. These methods change the convex Hamiltonian function and the restricted damping matrix of the previous results into a Hamiltonian function with a local minimum and a positive semidefinite matrix, respectively. Due to common conditions of Hamiltonian function and damping matrix, the proposed method asymptotically stabilizes more classes of PH systems with constant inputs than the existing methods. Finally, the validity and advantages of the presented methods are shown in an example.     

Asymptotically sharp error estimates for modified compound quadrature formulae for Cauchy principal value integrals

One of the standard methods for computing Cauchy principal value integrals is to subtract the singularity, and then to apply a given quadrature formula. This results in a quadrature formula for the Cauchy principal value integral which is called a modified quadrature formula. Here, we consider the case that this given quadrature formula is a compound quadrature formula, and derive error estimates of the form |R[f]| ≤κ _i ∥f ⁽ⁱ⁾ ∥_∞ (whereR[f] is the error of the modified quadrature formula). In contrast to previous estimates, the behaviour ofκ _i when the number of quadrature nodes tends to infinity is determined exactly.     

Optimal weighted chebyshev-type quadrature formulas

A weighted quadrature formula is called of Chebyshev type if it has equal coefficients and real (but not necessarily distinct) nodes. Among such quadrature rules we construct an optimal one, i. e., one which has maximum algebraic degree of accuracy and minimum error when applied to the first power not exactly integrated. Optimal quadrature rules, typically, have multiple nodes. Their construction requires the complete solution of systems of algebraic equations involving generalized power sums. Numerical results are presented for the case of constant weight function on a finite interval, as well as for weight functions of the Hermite and Laguerre type on infinite intervals. The work of the second author was supported in part by the National Science Foundation under grant GP-36557     

An analytical approach to one-parameter MIMO systems passivity enforcement

Stable linear time-invariant systems can be made passive by a feedforward action. In this article, an analytical approach to obtain the matrix which allows to enforce passivity in the system is proposed. This matrix depends only on one parameter, namely α. The introduced method is based on the calculation of the characteristic polynomial of the Hamiltonian matrix associated to the Positive Real problem. This polynomial is then used to derive a finite set of values of the parameter α, in which the value assuring passivity enforcement with minimum dissipation can be selected. Numerical examples are reported.     

Band formulas for calculating the numerator’s coefficients of the transfer function of a single-input system

Based on band matrices, a new formula for finding polynomial coefficients of the numerator of the transfer function for a linear single-input single-output system is derived. Its properties are studied for analyzing and synthesizing linear systems with many outputs, as a result of which a method is developed for the synthesis of the specified polynomial of zero dynamics (specified output) for a linear system with many outputs.     

Algebraically rectifiable parametric curves

Sufficient and necessary conditions for the arc length of a polynomial parametric curve to be an algebraic function of the parameter are formulated. It is shown that if the arc length is algebraic, it is no more complicated than the square root of a polynomial. Polynomial curves that have this property encompass the Pythagorean-hodograph curves—for which the arc length is just a polynomial in the parameter—as a proper subset. The algebraically rectifiable cubics, other than Pythagorean-hodograph curves, constitute a single-parameter family of cuspidal curves. The implications of the general algebraic rectifiability criterion are also completely enumerated in the case of quartics, in terms of their cusps and intrinsic shape freedoms. Finally, the characterization and construction of algebraically rectifiable quintics is briefly sketched. These forms offer a rich repertoire of curvilinear profiles, whose lengths are readily determined without numerical quadrature, for practical design problems.     

A Unified Approach to Linear and Nonlinear Normal Forms for Hamiltonian Systems

We prove that in all but one case the normal form of a real or complex Hamiltonian matrix which is irreducible and appropriately normalized can be computed by Lie series methods in formally the same manner as one computes the normal form of a nonlinear Hamiltonian function. Calculations are emphasized; the methods are illustrated with detailed examples, and for the sake of completeness the exceptional case is also reviewed and illustrated. Alternate methods are also discussed, along with detailed examples.     

GPS高程拟合在水利测量中的应用研究

常见高程系统包括大地高、正高及正常高系统,由于 GPS 测量的大地高在实际测量中的应用十分有限,须采用高程拟合技术,将大地高转换成为正常高,但转换过程中会出现高程异常,因此采用拟合方程对已知高程进行异常处理。高程拟合方法包括多项式与多面函数拟合法,在 GPS 拟合原理基础上,对二次多项式、 三次多项式、多面函数 3 种拟合方法的高程异常及拟合差值进行分析计算,并将拟合方法应用于地形起伏差异不同的 2 个测区,对 3 种拟合方法的精度评定进行比较分析。结果表明：三次多项式拟合方法的精度要优于二次多项式和多面函数的拟合精度,说明三次多项式拟合更适用于水利测绘工程,且通过三次多项式拟合方法可减少水利测量工作,提高测量精度和工作效率。     

Characterizing the measures on the unit circle with exact quadrature formulas in the space of polynomials

In the present paper we characterize the measures on the unit circle for which there exists a quadrature formula with a fixed number of nodes and weights and such that it exactly integrates all the polynomials with complex coefficients. As an application we obtain quadrature rules for polynomial modifications of the Bernstein measures on [−1,1], having a fixed number of nodes and quadrature coefficients and such that they exactly integrate all the polynomials with real coefficients.