Uncertainty propagation, one of the structural engineering problems, is receiving increasing attention owing to the fact that most significant loads are random in nature and structural parameters are typically subject...Uncertainty propagation, one of the structural engineering problems, is receiving increasing attention owing to the fact that most significant loads are random in nature and structural parameters are typically subject to variation. In the study, the collocation interval analysis method based on the first class Chebyshev polynomial approximation is presented to investigate the least favorable responses and the most favorable responses of interval-parameter structures under random excitations. Compared with the interval analysis method based on the first order Taylor expansion, in which only information including the function value and derivative at midpoint is used, the collocation interval analysis method is a non-gradient algorithm using several collocation points which improve the precision of results owing to better approximation of a response function. The pseudo excitation method is introduced to the solving procedure to transform the random problem into a deterministic problem. To validate the procedure, we present numerical results concerning a building under seismic ground motion and aerofoil under continuous atmosphere turbulence to show the effectiveness of the collocation interval analysis method.展开更多
Structural reliability is an important method to measure the safety performance of structures under the influence of uncertain factors.Traditional structural reliability analysis methods often convert the limit state ...Structural reliability is an important method to measure the safety performance of structures under the influence of uncertain factors.Traditional structural reliability analysis methods often convert the limit state function to the polynomial form to measure whether the structure is invalid.The uncertain parameters mainly exist in the form of intervals.This method requires a lot of calculation and is often difficult to achieve efficiently.In order to solve this problem,this paper proposes an interval variable multivariate polynomial algorithm based on Bernstein polynomials and evidence theory to solve the structural reliability problem with cognitive uncertainty.Based on the non-probabilistic reliability index method,the extreme value of the limit state function is obtained using the properties of Bernstein polynomials,thus avoiding the need for a lot of sampling to solve the reliability analysis problem.The method is applied to numerical examples and engineering applications such as experiments,and the results show that the method has higher computational efficiency and accuracy than the traditional linear approximation method,especially for some reliability problems with higher nonlinearity.Moreover,this method can effectively improve the reliability of results and reduce the cost of calculation in practical engineering problems.展开更多
This paper proposes a new non-intrusive trigonometric polynomial approximation interval method for the dynamic response analysis of nonlinear systems with uncertain-but-bounded parameters and/or initial conditions.Thi...This paper proposes a new non-intrusive trigonometric polynomial approximation interval method for the dynamic response analysis of nonlinear systems with uncertain-but-bounded parameters and/or initial conditions.This method provides tighter solution ranges compared to the existing approximation interval methods.We consider trigonometric approximation polynomials of three types:both cosine and sine functions,the sine function,and the cosine function.Thus,special interval arithmetic for trigonometric function without overestimation can be used to obtain interval results.The interval method using trigonometric approximation polynomials with a cosine functional form exhibits better performance than the existing Taylor interval method and Chebyshev interval method.Finally,two typical numerical examples with nonlinearity are applied to demonstrate the effectiveness of the proposed method.展开更多
In electrical circuit analysis, it is often necessary to find the set of all direct current （d.c.） operating points （either voltages or currents） of nonlinear circuits. In general, these nonlinear equations are of...In electrical circuit analysis, it is often necessary to find the set of all direct current （d.c.） operating points （either voltages or currents） of nonlinear circuits. In general, these nonlinear equations are often represented as polynomial systems. In this paper, we address the problem of finding the solutions of nonlinear electrical circuits, which are modeled as systems of n polynomial equations contained in an n-dimensional box. Branch and Bound algorithms based on interval methods can give guaranteed enclosures for the solution. However, because of repeated evaluations of the function values, these methods tend to become slower. Branch and Bound algorithm based on Bernstein coefficients can be used to solve the systems of polynomial equations. This avoids the repeated evaluation of function values, but maintains more or less the same number of iterations as that of interval branch and bound methods. We propose an algorithm for obtaining the solution of polynomial systems, which includes a pruning step using Bernstein Krawczyk operator and a Bernstein Coefficient Contraction algorithm to obtain Bernstein coefficients of the new domain. We solved three circuit analysis problems using our proposed algorithm. We compared the performance of our proposed algorithm with INTLAB based solver and found that our proposed algorithm is more efficient and fast.展开更多
Motivated by the idea of Shen, et al.'s work, which proposed a hybrid procedure for real root isolation of polynomial equations based on homotopy continuation methods and interval analysis,this paper presents a hy...Motivated by the idea of Shen, et al.'s work, which proposed a hybrid procedure for real root isolation of polynomial equations based on homotopy continuation methods and interval analysis,this paper presents a hybrid procedure to compute sample points on each connected component of a real algebraic set by combining a special homotopy method and interval analysis with a better estimate on initial intervals. For a real algebraic set given by a polynomial system, the new method ?rst constructs a square polynomial system which represents the sample points, and then solve this system by a special homotopy continuation method introduced recently by Wang, et al.(2017). For each root returned by the homotopy continuation method, which is a complex approximation of some(complex/real) root of the polynomial system, interval analysis is used to verify whether it is an approximation of a real root and ?nally get real points on the given real algebraic set. A new estimate on initial intervals is presented which helps compute smaller initial intervals before performing interval iteration and thus saves computation. Experiments show that the new method works pretty well on tested examples.展开更多
In the model of geometric programming, values of parameters cannot be gotten owing to data fluctuation and incompletion. But reasonable bounds of these parameters can be attained. This is to say, parameters of this mo...In the model of geometric programming, values of parameters cannot be gotten owing to data fluctuation and incompletion. But reasonable bounds of these parameters can be attained. This is to say, parameters of this model can be regarded as interval grey numbers. When the model contains grey numbers, it is hard for common programming method to solve them. By combining the common programming model with the grey system theory, and using some analysis strategies, a model of grey polynomial geometric programming, a model of θ positioned geometric programming and their quasi-optimum solution or optimum solution are put forward. At the same time, we also developed an algorithm for the problem. This approach brings a new way for the application research of geometric programming. An example at the end of this paper shows the rationality and feasibility of the algorithm.展开更多
This paper presents a symbolic algorithm to compute the topology of a plane curve.This is a full version of the authors’CASC15 paper.The algorithm mainly involves resultant computations and real root isolation for un...This paper presents a symbolic algorithm to compute the topology of a plane curve.This is a full version of the authors’CASC15 paper.The algorithm mainly involves resultant computations and real root isolation for univariate polynomials.Compared to other symbolic methods based on elimination techniques,the novelty of the proposed method is that the authors use a technique of interval polynomials to solve the system f(α,y),?f/?y(α,y)and simultaneously obtain numerous simple roots of f(α,y)=0 on theαfiber.This significantly improves the efficiency of the lifting step because the authors are no longer required to compute the simple roots of f(α,y)=0.After the topology is computed,a revised Newton’s method is presented to compute an isotopic meshing of the plane algebraic curve.Though the approximation method is numerical,the authors can ensure that the proposed method is a certified one,and the meshing is topologically correct.Several nontrivial examples confirm that the proposed algorithm performs well.展开更多
Wu's elimination method is an important method for solving multivariate polynomial equations. In this paper, we apply interval arithmetic to Wu's method and convert the problem of solving polynomial equations ...Wu's elimination method is an important method for solving multivariate polynomial equations. In this paper, we apply interval arithmetic to Wu's method and convert the problem of solving polynomial equations into that of solving interval polynomial equations. Parallel results such as zero-decomposition theorem are obtained for interval polynomial equations. The advantages of the new approach are two-folds: First, the problem of the numerical instability arisen from floating-point arithmetic is largely overcome. Second,the low efficiency of the algorithm caused by large intermediate coefficients introduced by exact compaction is dramatically improved. Some examples are provided to illustrate the effectiveness of the proposed algorithm.展开更多
基金supported by the National Natural Science Foundation of China (Grant Nos. 10872017, 90816024 and 10876100)111 Project (Grant No. B07009)
文摘Uncertainty propagation, one of the structural engineering problems, is receiving increasing attention owing to the fact that most significant loads are random in nature and structural parameters are typically subject to variation. In the study, the collocation interval analysis method based on the first class Chebyshev polynomial approximation is presented to investigate the least favorable responses and the most favorable responses of interval-parameter structures under random excitations. Compared with the interval analysis method based on the first order Taylor expansion, in which only information including the function value and derivative at midpoint is used, the collocation interval analysis method is a non-gradient algorithm using several collocation points which improve the precision of results owing to better approximation of a response function. The pseudo excitation method is introduced to the solving procedure to transform the random problem into a deterministic problem. To validate the procedure, we present numerical results concerning a building under seismic ground motion and aerofoil under continuous atmosphere turbulence to show the effectiveness of the collocation interval analysis method.
文摘Structural reliability is an important method to measure the safety performance of structures under the influence of uncertain factors.Traditional structural reliability analysis methods often convert the limit state function to the polynomial form to measure whether the structure is invalid.The uncertain parameters mainly exist in the form of intervals.This method requires a lot of calculation and is often difficult to achieve efficiently.In order to solve this problem,this paper proposes an interval variable multivariate polynomial algorithm based on Bernstein polynomials and evidence theory to solve the structural reliability problem with cognitive uncertainty.Based on the non-probabilistic reliability index method,the extreme value of the limit state function is obtained using the properties of Bernstein polynomials,thus avoiding the need for a lot of sampling to solve the reliability analysis problem.The method is applied to numerical examples and engineering applications such as experiments,and the results show that the method has higher computational efficiency and accuracy than the traditional linear approximation method,especially for some reliability problems with higher nonlinearity.Moreover,this method can effectively improve the reliability of results and reduce the cost of calculation in practical engineering problems.
文摘This paper proposes a new non-intrusive trigonometric polynomial approximation interval method for the dynamic response analysis of nonlinear systems with uncertain-but-bounded parameters and/or initial conditions.This method provides tighter solution ranges compared to the existing approximation interval methods.We consider trigonometric approximation polynomials of three types:both cosine and sine functions,the sine function,and the cosine function.Thus,special interval arithmetic for trigonometric function without overestimation can be used to obtain interval results.The interval method using trigonometric approximation polynomials with a cosine functional form exhibits better performance than the existing Taylor interval method and Chebyshev interval method.Finally,two typical numerical examples with nonlinearity are applied to demonstrate the effectiveness of the proposed method.
文摘In electrical circuit analysis, it is often necessary to find the set of all direct current （d.c.） operating points （either voltages or currents） of nonlinear circuits. In general, these nonlinear equations are often represented as polynomial systems. In this paper, we address the problem of finding the solutions of nonlinear electrical circuits, which are modeled as systems of n polynomial equations contained in an n-dimensional box. Branch and Bound algorithms based on interval methods can give guaranteed enclosures for the solution. However, because of repeated evaluations of the function values, these methods tend to become slower. Branch and Bound algorithm based on Bernstein coefficients can be used to solve the systems of polynomial equations. This avoids the repeated evaluation of function values, but maintains more or less the same number of iterations as that of interval branch and bound methods. We propose an algorithm for obtaining the solution of polynomial systems, which includes a pruning step using Bernstein Krawczyk operator and a Bernstein Coefficient Contraction algorithm to obtain Bernstein coefficients of the new domain. We solved three circuit analysis problems using our proposed algorithm. We compared the performance of our proposed algorithm with INTLAB based solver and found that our proposed algorithm is more efficient and fast.
基金supported by the National Natural Science Foundation of China under Grant Nos.61732001 and 61532019
文摘Motivated by the idea of Shen, et al.'s work, which proposed a hybrid procedure for real root isolation of polynomial equations based on homotopy continuation methods and interval analysis,this paper presents a hybrid procedure to compute sample points on each connected component of a real algebraic set by combining a special homotopy method and interval analysis with a better estimate on initial intervals. For a real algebraic set given by a polynomial system, the new method ?rst constructs a square polynomial system which represents the sample points, and then solve this system by a special homotopy continuation method introduced recently by Wang, et al.(2017). For each root returned by the homotopy continuation method, which is a complex approximation of some(complex/real) root of the polynomial system, interval analysis is used to verify whether it is an approximation of a real root and ?nally get real points on the given real algebraic set. A new estimate on initial intervals is presented which helps compute smaller initial intervals before performing interval iteration and thus saves computation. Experiments show that the new method works pretty well on tested examples.
基金Supported by the NSF Jiangsu Province(BK2003211)Supported by the NSF of Henan Province(2003120001)
文摘In the model of geometric programming, values of parameters cannot be gotten owing to data fluctuation and incompletion. But reasonable bounds of these parameters can be attained. This is to say, parameters of this model can be regarded as interval grey numbers. When the model contains grey numbers, it is hard for common programming method to solve them. By combining the common programming model with the grey system theory, and using some analysis strategies, a model of grey polynomial geometric programming, a model of θ positioned geometric programming and their quasi-optimum solution or optimum solution are put forward. At the same time, we also developed an algorithm for the problem. This approach brings a new way for the application research of geometric programming. An example at the end of this paper shows the rationality and feasibility of the algorithm.
基金supported by the National Natural Science Foundation of China under Grant No.11471327“The Research Funds for Beijing Universities”under Grant No.KM201910009001“The Research and Development Funds of Hubei University of Science and Technology”under Grant No.BK202024.
文摘This paper presents a symbolic algorithm to compute the topology of a plane curve.This is a full version of the authors’CASC15 paper.The algorithm mainly involves resultant computations and real root isolation for univariate polynomials.Compared to other symbolic methods based on elimination techniques,the novelty of the proposed method is that the authors use a technique of interval polynomials to solve the system f(α,y),?f/?y(α,y)and simultaneously obtain numerous simple roots of f(α,y)=0 on theαfiber.This significantly improves the efficiency of the lifting step because the authors are no longer required to compute the simple roots of f(α,y)=0.After the topology is computed,a revised Newton’s method is presented to compute an isotopic meshing of the plane algebraic curve.Though the approximation method is numerical,the authors can ensure that the proposed method is a certified one,and the meshing is topologically correct.Several nontrivial examples confirm that the proposed algorithm performs well.
基金supported by the Outstanding Youth Grant of NSF of China(Grant No.60225002)the National Key Basic Research Project of China(Grnat No.2004CB318000)the TRAPOYT in Higher Education Institute of MOE of China.
文摘Wu's elimination method is an important method for solving multivariate polynomial equations. In this paper, we apply interval arithmetic to Wu's method and convert the problem of solving polynomial equations into that of solving interval polynomial equations. Parallel results such as zero-decomposition theorem are obtained for interval polynomial equations. The advantages of the new approach are two-folds: First, the problem of the numerical instability arisen from floating-point arithmetic is largely overcome. Second,the low efficiency of the algorithm caused by large intermediate coefficients introduced by exact compaction is dramatically improved. Some examples are provided to illustrate the effectiveness of the proposed algorithm.
