In this paper, we introduce a quadrature rule for the numerical evaluation of certain hyper singular integrals. The rule is obtained by using Hermite interpolation polynomial. Error bound is also made.
A new family of numerical integration formula is presented, which uses the function evaluation at the midpoint of the interval and odd derivatives at the endpoints. Because the weights for the odd derivatives sum to z...A new family of numerical integration formula is presented, which uses the function evaluation at the midpoint of the interval and odd derivatives at the endpoints. Because the weights for the odd derivatives sum to zero, the derivative calculations cancel out for the interior points in the composite form, so that these derivatives must only be calculated at the endpoints of the overall interval of integration. When using N subintervals, the basic rule which uses the midpoint function evaluation and the first derivative at the endpoints achieves fourth order accuracy for the cost of N/2 function evaluations and 2 derivative evaluations, whereas the three point open Newton-Cotes method uses 3N/4 function evaluations to achieve the same order of accuracy. These derivative-based midpoint quadrature methods are shown to be more computationally efficient than both the open and closed Newton-Cotes quadrature rules of the same order. This family of derivative-based midpoint quadrature rules are derived using the concept of precision, along with the error term. A theorem concerning the order of accuracy of quadrature rule using the concept of precision is provided to justify its use to determine the leading order error term.展开更多
Similar to having done for the mid-point and trapezoid quadrature rules,we obtain alternative estimations of error bounds for the Simpson＇s quadrature rule involving n-time（1 ≤ n ≤ 4） differentiable mappings and ...Similar to having done for the mid-point and trapezoid quadrature rules,we obtain alternative estimations of error bounds for the Simpson＇s quadrature rule involving n-time（1 ≤ n ≤ 4） differentiable mappings and then to the estimations of error bounds for the adaptive Simpson＇s quadrature rule.展开更多
Finite part integrals introduced by Hadamard in connection with hyperbolic partial differential equations,have been useful in a number of engineering applications.In this paper we investigate some numerical methods fo...Finite part integrals introduced by Hadamard in connection with hyperbolic partial differential equations,have been useful in a number of engineering applications.In this paper we investigate some numerical methods for computing finite-part integrals.展开更多
We present a numerically stable one-point quadrature rule for the stiffness matrix and mass matrix of the three-dimensional numerical manifold method(3D NMM).The rule simplifies the integration over irregularly shaped...We present a numerically stable one-point quadrature rule for the stiffness matrix and mass matrix of the three-dimensional numerical manifold method(3D NMM).The rule simplifies the integration over irregularly shaped manifold elements and overcomes locking issues,and it does not cause spurious modes in modal analysis.The essential idea is to transfer the integral over a manifold element to a few moments to the element center,thereby deriving a one-point integration rule by the moments and making modifications to avoid locking issues.For the stiffness matrix,after the virtual work is decomposed into moments,higher-order moments are modified to overcome locking issues in nearly incompressible and bending-dominated conditions.For the mass matrix,the consistent and lumped types are derived by moments.In particular,the lumped type has the clear advantage of simplicity.The proposed method is naturally suitable for 3D NMM meshes automatically generated from a regular grid.Numerical tests justify the accuracy improvements and the stability of the proposed procedure.展开更多
A new Gaussian approximation nonlinear filter called generalized cubature quadrature Kalman filter (GCQKF) is introduced for nonlinear dynamic systems. Based on standard GCQKF, two extensions are developed, namely squ...A new Gaussian approximation nonlinear filter called generalized cubature quadrature Kalman filter (GCQKF) is introduced for nonlinear dynamic systems. Based on standard GCQKF, two extensions are developed, namely square root generalized cubature quadrature Kalman filter (SR-GCQKF) and iterated generalized cubature quadrature Kalman filter (I-GCQKF). In SR-GCQKF, the QR decomposition is exploited to alter the Cholesky decomposition and both predicted and filtered error covariances have been propagated in square root format to make sure the numerical stability. In I-GCQKF, the measurement update step is executed iteratively to make full use of the latest measurement and a new terminal criterion is adopted to guarantee the increase of likelihood. Detailed numerical experiments demonstrate the superior performance on both tracking stability and estimation accuracy of I-GCQKF and SR-GCQKF compared with GCQKF.展开更多
An accurate and efficient differential quadrature time element method （DQTEM） is proposed for solving ordi- nary differential equations （ODEs）, the numerical dissipation and dispersion of DQTEM is much smaller tha...An accurate and efficient differential quadrature time element method （DQTEM） is proposed for solving ordi- nary differential equations （ODEs）, the numerical dissipation and dispersion of DQTEM is much smaller than that of the direct integration method of single/multi steps. Two methods of imposing initial conditions are given, which avoids the tediousness when derivative initial conditions are imposed, and the numerical comparisons indicate that the first method, in which the analog equations of initial displacements and velocities are used to directly replace the differential quadra- ture （DQ） analog equations of ODEs at the first and the last sampling points, respectively, is much more accurate than the second method, in which the DQ analog equations of initial conditions are used to directly replace the DQ analog equations of ODEs at the first two sampling points. On the contrary to the conventional step-by-step direct integration schemes, the solutions at all sampling points can be obtained simultaneously by DQTEM, and generally, one differential quadrature time element may be enough for the whole time domain. Extensive numerical comparisons validate the effi- ciency and accuracy of the proposed method.展开更多
Optimization of mapping rule of bit-interleaved Turbo coded modulation with 16 quadrature amplitude modulation (QAM) is investigated based on different impacts of various encoded bits sequence on Turbo decoding perfor...Optimization of mapping rule of bit-interleaved Turbo coded modulation with 16 quadrature amplitude modulation (QAM) is investigated based on different impacts of various encoded bits sequence on Turbo decoding performance. Furthermore, bit-interleaved in-phase and quadrature phase (I-Q) Turbo coded modulation scheme are designed similarly with I-Q trellis coded modulation (TCM). Through performance evaluation and analysis, it can be seen that the novel mapping rule outperforms traditional one and the I-Q Turbo coded modulation can not achieve good performance as expected. Therefore, there is not obvious advantage in using I-Q method in bit-interleaved Turbo coded modulation.展开更多
In this paper we consider polynomials orthogonal with respect to the linear functional L:P→C,defined on the space of all algebraic polynomials P by L[p]=∫_(-1)^(1)p(x)(1−x)^(α−1/2)(1+x)^(β−1/2)exp(iζx)dx,whereα,...In this paper we consider polynomials orthogonal with respect to the linear functional L:P→C,defined on the space of all algebraic polynomials P by L[p]=∫_(-1)^(1)p(x)(1−x)^(α−1/2)(1+x)^(β−1/2)exp(iζx)dx,whereα,β>−1/2 are real numbers such thatℓ=|β−α|is a positive integer,andζ∈R\{0}.We prove the existence of such orthogonal polynomials for some pairs ofαandζand for all nonnegative integersℓ.For such orthogonal polynomials we derive three-term recurrence relations and also some differential-difference relations.For such orthogonal polynomials the corresponding quadrature rules of Gaussian type are considered.Also,some numerical examples are included.展开更多
单变量维数缩减法可以高效、准确地进行结构响应矩的分析。与传统的一阶可靠度算法FORM(First Order Reliability Method),二阶可靠度算法SORM(Second Order Reliability Method)相比,该方法不需要响应的导数,也不需要迭代搜索最可能失...单变量维数缩减法可以高效、准确地进行结构响应矩的分析。与传统的一阶可靠度算法FORM(First Order Reliability Method),二阶可靠度算法SORM(Second Order Reliability Method)相比,该方法不需要响应的导数,也不需要迭代搜索最可能失效点。然而近期的研究发现,该方法中基于矩的积分方法MBQR(Moment Based Quadrature Rule)在积分点增加之后求解线性方程组时,会出现系数矩阵的奇异性并导致数值结果不稳定,从而影响了该方法的效率和精度。提出了归一化的基于矩的积分方法,有效地解决了数值求解过程中的不稳定问题。利用降维法求解结构响应统计矩,并通过Pearson系统计算响应的概率密度函数,从而获得失效概率。算例表明了本文方法的计算效率和精度。展开更多
文摘In this paper, we introduce a quadrature rule for the numerical evaluation of certain hyper singular integrals. The rule is obtained by using Hermite interpolation polynomial. Error bound is also made.
文摘A new family of numerical integration formula is presented, which uses the function evaluation at the midpoint of the interval and odd derivatives at the endpoints. Because the weights for the odd derivatives sum to zero, the derivative calculations cancel out for the interior points in the composite form, so that these derivatives must only be calculated at the endpoints of the overall interval of integration. When using N subintervals, the basic rule which uses the midpoint function evaluation and the first derivative at the endpoints achieves fourth order accuracy for the cost of N/2 function evaluations and 2 derivative evaluations, whereas the three point open Newton-Cotes method uses 3N/4 function evaluations to achieve the same order of accuracy. These derivative-based midpoint quadrature methods are shown to be more computationally efficient than both the open and closed Newton-Cotes quadrature rules of the same order. This family of derivative-based midpoint quadrature rules are derived using the concept of precision, along with the error term. A theorem concerning the order of accuracy of quadrature rule using the concept of precision is provided to justify its use to determine the leading order error term.
基金Supported by the Natural Science Foundation of Zhejiang Province（Y6090361）
文摘Similar to having done for the mid-point and trapezoid quadrature rules,we obtain alternative estimations of error bounds for the Simpson＇s quadrature rule involving n-time（1 ≤ n ≤ 4） differentiable mappings and then to the estimations of error bounds for the adaptive Simpson＇s quadrature rule.
文摘Finite part integrals introduced by Hadamard in connection with hyperbolic partial differential equations,have been useful in a number of engineering applications.In this paper we investigate some numerical methods for computing finite-part integrals.
基金supported by the National Natural Science Foundation of China(Grant Nos.42302331,52130905 and 52079002)。
文摘We present a numerically stable one-point quadrature rule for the stiffness matrix and mass matrix of the three-dimensional numerical manifold method(3D NMM).The rule simplifies the integration over irregularly shaped manifold elements and overcomes locking issues,and it does not cause spurious modes in modal analysis.The essential idea is to transfer the integral over a manifold element to a few moments to the element center,thereby deriving a one-point integration rule by the moments and making modifications to avoid locking issues.For the stiffness matrix,after the virtual work is decomposed into moments,higher-order moments are modified to overcome locking issues in nearly incompressible and bending-dominated conditions.For the mass matrix,the consistent and lumped types are derived by moments.In particular,the lumped type has the clear advantage of simplicity.The proposed method is naturally suitable for 3D NMM meshes automatically generated from a regular grid.Numerical tests justify the accuracy improvements and the stability of the proposed procedure.
基金supported by the National Natural Science Foundation of China(6147322711472222)+2 种基金the Aerospace Technology Support Fund of China(2014-HT-XGD)the Natural Science Foundation of Shaanxi Province(2015JM6304)the Aeronautical Science Foundation of China(20151353018)
文摘A new Gaussian approximation nonlinear filter called generalized cubature quadrature Kalman filter (GCQKF) is introduced for nonlinear dynamic systems. Based on standard GCQKF, two extensions are developed, namely square root generalized cubature quadrature Kalman filter (SR-GCQKF) and iterated generalized cubature quadrature Kalman filter (I-GCQKF). In SR-GCQKF, the QR decomposition is exploited to alter the Cholesky decomposition and both predicted and filtered error covariances have been propagated in square root format to make sure the numerical stability. In I-GCQKF, the measurement update step is executed iteratively to make full use of the latest measurement and a new terminal criterion is adopted to guarantee the increase of likelihood. Detailed numerical experiments demonstrate the superior performance on both tracking stability and estimation accuracy of I-GCQKF and SR-GCQKF compared with GCQKF.
基金supported by the National Natural Science Foundation of China (11172028,10772014)
文摘An accurate and efficient differential quadrature time element method （DQTEM） is proposed for solving ordi- nary differential equations （ODEs）, the numerical dissipation and dispersion of DQTEM is much smaller than that of the direct integration method of single/multi steps. Two methods of imposing initial conditions are given, which avoids the tediousness when derivative initial conditions are imposed, and the numerical comparisons indicate that the first method, in which the analog equations of initial displacements and velocities are used to directly replace the differential quadra- ture （DQ） analog equations of ODEs at the first and the last sampling points, respectively, is much more accurate than the second method, in which the DQ analog equations of initial conditions are used to directly replace the DQ analog equations of ODEs at the first two sampling points. On the contrary to the conventional step-by-step direct integration schemes, the solutions at all sampling points can be obtained simultaneously by DQTEM, and generally, one differential quadrature time element may be enough for the whole time domain. Extensive numerical comparisons validate the effi- ciency and accuracy of the proposed method.
文摘Optimization of mapping rule of bit-interleaved Turbo coded modulation with 16 quadrature amplitude modulation (QAM) is investigated based on different impacts of various encoded bits sequence on Turbo decoding performance. Furthermore, bit-interleaved in-phase and quadrature phase (I-Q) Turbo coded modulation scheme are designed similarly with I-Q trellis coded modulation (TCM). Through performance evaluation and analysis, it can be seen that the novel mapping rule outperforms traditional one and the I-Q Turbo coded modulation can not achieve good performance as expected. Therefore, there is not obvious advantage in using I-Q method in bit-interleaved Turbo coded modulation.
基金supported in part by Serbian Ministry of Education and Science(Projects#174015 and Ⅲ44006).
文摘In this paper we consider polynomials orthogonal with respect to the linear functional L:P→C,defined on the space of all algebraic polynomials P by L[p]=∫_(-1)^(1)p(x)(1−x)^(α−1/2)(1+x)^(β−1/2)exp(iζx)dx,whereα,β>−1/2 are real numbers such thatℓ=|β−α|is a positive integer,andζ∈R\{0}.We prove the existence of such orthogonal polynomials for some pairs ofαandζand for all nonnegative integersℓ.For such orthogonal polynomials we derive three-term recurrence relations and also some differential-difference relations.For such orthogonal polynomials the corresponding quadrature rules of Gaussian type are considered.Also,some numerical examples are included.
文摘单变量维数缩减法可以高效、准确地进行结构响应矩的分析。与传统的一阶可靠度算法FORM(First Order Reliability Method),二阶可靠度算法SORM(Second Order Reliability Method)相比,该方法不需要响应的导数,也不需要迭代搜索最可能失效点。然而近期的研究发现,该方法中基于矩的积分方法MBQR(Moment Based Quadrature Rule)在积分点增加之后求解线性方程组时,会出现系数矩阵的奇异性并导致数值结果不稳定,从而影响了该方法的效率和精度。提出了归一化的基于矩的积分方法,有效地解决了数值求解过程中的不稳定问题。利用降维法求解结构响应统计矩,并通过Pearson系统计算响应的概率密度函数,从而获得失效概率。算例表明了本文方法的计算效率和精度。
