参数不确定问题的区间与仿射分析方法理论与应用分析
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摘要
不确定性问题，特别是不确定参数问题，在理论研究和工程应用中广泛存在。本论文以区间参数问题为研究对象，以含区间参数的函数界限计算的研究为基础，以区间和仿射算法为研究工具，探索性地研究了复杂函数的全局优化问题、结构参数和外部激励为区间变量时系统的静力响应问题、复杂非凸不连续可行域系统的可靠度计算问题以及基于仿射逆方法的结构分析问题。
主要内容如下：
1．含区间参数的函数界限计算
将函数中的不确定性参数用区间数表示，利用区间算法对函数界限进行分析。讨论了区间细分方法和不同型式的区间方法以提高计算精度，为含区间参数函数界限的计算提供了多种选择。并将区间方法应用于结构的非概率可靠度分析及静力分析中。
2．复杂函数的全局优化问题研究
分析了确定性和随机性优化、局部和全局优化之间的区别与联系，针对传统区间算法求解全局优化问题、耗时长、空间复杂度较高及收敛速度较慢的缺点，将仿射算法及局部优化算法引入了全局优化问题，给出了一种全局优化求解的仿射算法。由局部优化算法和各求解区间上待优化函数的仿射运算得到全局最优解的一个上界，再依据对各区间仿射运算的下界与全局最优解上界的比较来确定相应区间的去留，通过对不含全局最优解的子区间的删除来确定出最优解所在的子区间，并最终找到全局最优解。数值试验表明，该算法相对于传统的区间优化算法有较高的收敛速度，且占用了较少的系统资源。
3．含区间参数的系统响应界限分析
针对仿射运算时新符号噪声的引入必然造成误差放大的不足，在函数上下界计算中引入了矩阵形式的上下界的仿射计算公式，提出了一种计算上下界的改进仿射算法。该算法在仿射变量进行乘法运算时不会引入新的噪声，相对与传统的仿射算法能得到更紧凑的界限；并通过实例计算演示了该公式的计算过程及计算方法的有效性。将有界不确定性变量的仿射型及改进的仿射运算引入不确定系统响应上下界的计算。仿真结果表明，相对于区间算法及传统的仿射算法，该算法得到解的界限更为紧凑。
4．基于域分析的控制系统稳定域求解
针对不确定系统的区间表示不能描述变量间的相关性、相应的区间算法容易导致误差爆炸的问题，提出了不确定系统的仿射表示法及系统稳定性的仿射不等式判断方法。首先将系统中的不确定信息用仿射参数来表示，得到不确定控制系统传递函数的仿射形式，然后通过求解含仿射参数的不等式组求得了满足系统的稳定性条件时各噪声允许的范围。由于考虑了变量间的相关性，相对于区间算法，所提出的方法可以在更大的不确定范围内判断出系统的稳定性。
5．复杂非凸不连续可行域系统的可靠度计算
提出了一种复杂函数或电路系统的可行域计算的仿射区间方法，利用仿射区间方法对设计域内函数的函数界限进行分析，利用分支定界法将该区域分类为：可行、不可行域及不确定域；再将不确定区域进行细分，并对每个细分后的子区域进行进一步的函数界限分析及分类，直至子区域半径达到设计要求，然后进行可行域统计计算，将每个可行域的面积进行求和可得到函数的总可行域。该方法可以对非凸函数甚至可行域不连续函数的可行域进行估计。通过算例演示了该方法的计算过程并验证了该方法的有效性。
6．基于区间逆阵求解及仿射逆阵求解的结构分析
对于一些复杂问题，更常见的是多个变量组合起来，形成向量或者矩阵出现在方程中，当这些区间/仿射变量本身在某个区间内变动时，这些区间/仿射变量可以组合成区间/仿射向量或区间/仿射矩阵。以区间/仿射矩阵及其逆阵的解法为工具，对不确定工程结构的动力与静力分析进行研究。介绍了区间/仿射向量、区间/仿射矩阵及相关的一些概念，重点讨论了区间矩阵与仿射矩阵的逆阵求解方法，以实例演示了文中的方法并对其有效性进行了验证。
Uncertain problems, especially for parameter uncertain problems,is Widely exist inthe actual scientific research and engineering research.In this thesis,the parameteruncertain problems are taken as research objects,the function with uncertain parameterlimits problem are taken as research base,the interval and affine arithmetic are tacken asresearch tools. The exploratory research including:the global optimization problems;static and dynamic response problems of systems with uncertain parameters; reliabilitycauculation of non-convex and not simply connected fuction;structural analasis problemsbased on the solution of uncertain interval or affine matrix inverse problems.
The main research works can be described as follows:
1. The bounds analysis of fuction with uncertain parameter
By representing the uncertain parameters as interval numbers, the uncertaintymodels are obtained.The interval arithmetic and affine arithmetic are adopted to get thebounds of uncertainty models.In order to get more higher precision, subdivisions andrefinements method and several form of interval extensions are applied. An exampleswere provided to illustrate the validity and feasibility of the present method.Anextended beam and a three-truss example are provided to illustrate the validity andfeasibility of the presented procedures.
2. The study on the global optimization problems
The affine algorithm for global optimization was proposed by introducing theaffine algorithm and local optimization into the problem of global optimization, whichaim at solving disadvantage of the time consuming, higher space co mplexity and slowconvergence speed for the traditional interval algorithm in the solution of globaloptimization problem. The upper bound of global optimal solution was obtained bylocal optimization algorithm and affine arithmetic for objective function in eachsubinterval. And then, whether corresponding interval was discarded or not depends oncomparing the lower bound of affine arithmetic for objective function in eachsubinterval with the upper bound of global optimization solution. The subinterval whichcontains the optimal value was obtained by deleting the subinterval which did notcontain the optimal value. Lastly, the global optimal solution was found. NumericalSimulation results show that the proposed algorithm has higher convergence speedcompare with traditional interval optimization algorithm. At the same time, it alsooccupies less system resource.
3. Uncertain system response bounds analysis with interval arithmetic andaffinearithmetic
The introduction of new noise symbols causes error amplification in affinearithmetic inevitably. To avoid this disadvantage, this paper presents a modified affinearithmetic in matrix form for bounds computation of functions. The modified affinearithmetic does not introduce new noises during multiplication operation of affinevariables, and it can obtain compacter bounds compared to conventional affinearithmetic. The formulas computing processes and the validity of proposed method aredemonstrated by an example. The affine form of bounded uncertain variables andmodified affine arithmetic are brought in to calculate response bounds of uncertainsystem. The simulations show that, the proposed approach can obtain closer responsebounds than interval arithmetic and conventional affine arithmetic.
4. Stability region analysis using interval and affine method
As the interval form of uncertainty system can not express the pertinence ofuncertain variable, and interval algorithm may cause error explosion. The affine form ofuncertainty system and the affine inequality judge method for uncertainty system arepresented, at first, the certain parameter in certain system was substituted by affineparameter, we get the affine form transmit function of uncertain system, secondly, bysolving Matrix inequalities,we can get tolerable noise range with which the systemstability condition hold. An example shows that as take the pertinence of uncertainvariable into consideration, this method can judge the stability in a larger region, showsthe validity and the advantage of this method.
5. Reliability cauculation of non-convex and not simply connected fuction
In this chapter an affine-interval arithmetic-based method for the feasible regionevaluation of function or electronic circuits is presented. This method use affine-intervalarithmetic analyze the bounds of the function, and use branch and bound methoddivided these intervals into three kinds: accept regions, refuse regions and those ofuncertain regions; and the next, all the uncertain regions are re-divided and the boundscalculation and classification performed again until the subintervals small enough. Thestatistics on each of accept regions are performed next to get the sum of the acceptregions. The proposed technique guarantees an efficient, reliable and accurateevaluation of the yield, even for non-convex and not simply connected feasible region.The example presented shows the features of the approach.
6. The static analysis of interval structures based on interval matrix inversion andaffine matrix inversion
It is commonly used that many variables combined together, forms array or matixin engineering problems.The interval or affine array and matix was formed when one or more variable change in a certain range. The static analysis of interval structures basedon interval matrix inversion and affine matrix inversion.The basic conception wasdiscussed and the work focus on how to get interval matrix inversion and affine matrixinversion in a efficient method. The example presented shows the features and validityof this method.
主要内容如下：
1．含区间参数的函数界限计算
将函数中的不确定性参数用区间数表示，利用区间算法对函数界限进行分析。讨论了区间细分方法和不同型式的区间方法以提高计算精度，为含区间参数函数界限的计算提供了多种选择。并将区间方法应用于结构的非概率可靠度分析及静力分析中。
2．复杂函数的全局优化问题研究
分析了确定性和随机性优化、局部和全局优化之间的区别与联系，针对传统区间算法求解全局优化问题、耗时长、空间复杂度较高及收敛速度较慢的缺点，将仿射算法及局部优化算法引入了全局优化问题，给出了一种全局优化求解的仿射算法。由局部优化算法和各求解区间上待优化函数的仿射运算得到全局最优解的一个上界，再依据对各区间仿射运算的下界与全局最优解上界的比较来确定相应区间的去留，通过对不含全局最优解的子区间的删除来确定出最优解所在的子区间，并最终找到全局最优解。数值试验表明，该算法相对于传统的区间优化算法有较高的收敛速度，且占用了较少的系统资源。
3．含区间参数的系统响应界限分析
针对仿射运算时新符号噪声的引入必然造成误差放大的不足，在函数上下界计算中引入了矩阵形式的上下界的仿射计算公式，提出了一种计算上下界的改进仿射算法。该算法在仿射变量进行乘法运算时不会引入新的噪声，相对与传统的仿射算法能得到更紧凑的界限；并通过实例计算演示了该公式的计算过程及计算方法的有效性。将有界不确定性变量的仿射型及改进的仿射运算引入不确定系统响应上下界的计算。仿真结果表明，相对于区间算法及传统的仿射算法，该算法得到解的界限更为紧凑。
4．基于域分析的控制系统稳定域求解
针对不确定系统的区间表示不能描述变量间的相关性、相应的区间算法容易导致误差爆炸的问题，提出了不确定系统的仿射表示法及系统稳定性的仿射不等式判断方法。首先将系统中的不确定信息用仿射参数来表示，得到不确定控制系统传递函数的仿射形式，然后通过求解含仿射参数的不等式组求得了满足系统的稳定性条件时各噪声允许的范围。由于考虑了变量间的相关性，相对于区间算法，所提出的方法可以在更大的不确定范围内判断出系统的稳定性。
5．复杂非凸不连续可行域系统的可靠度计算
提出了一种复杂函数或电路系统的可行域计算的仿射区间方法，利用仿射区间方法对设计域内函数的函数界限进行分析，利用分支定界法将该区域分类为：可行、不可行域及不确定域；再将不确定区域进行细分，并对每个细分后的子区域进行进一步的函数界限分析及分类，直至子区域半径达到设计要求，然后进行可行域统计计算，将每个可行域的面积进行求和可得到函数的总可行域。该方法可以对非凸函数甚至可行域不连续函数的可行域进行估计。通过算例演示了该方法的计算过程并验证了该方法的有效性。
6．基于区间逆阵求解及仿射逆阵求解的结构分析
对于一些复杂问题，更常见的是多个变量组合起来，形成向量或者矩阵出现在方程中，当这些区间/仿射变量本身在某个区间内变动时，这些区间/仿射变量可以组合成区间/仿射向量或区间/仿射矩阵。以区间/仿射矩阵及其逆阵的解法为工具，对不确定工程结构的动力与静力分析进行研究。介绍了区间/仿射向量、区间/仿射矩阵及相关的一些概念，重点讨论了区间矩阵与仿射矩阵的逆阵求解方法，以实例演示了文中的方法并对其有效性进行了验证。
Uncertain problems, especially for parameter uncertain problems,is Widely exist inthe actual scientific research and engineering research.In this thesis,the parameteruncertain problems are taken as research objects,the function with uncertain parameterlimits problem are taken as research base,the interval and affine arithmetic are tacken asresearch tools. The exploratory research including:the global optimization problems;static and dynamic response problems of systems with uncertain parameters; reliabilitycauculation of non-convex and not simply connected fuction;structural analasis problemsbased on the solution of uncertain interval or affine matrix inverse problems.
The main research works can be described as follows:
1. The bounds analysis of fuction with uncertain parameter
By representing the uncertain parameters as interval numbers, the uncertaintymodels are obtained.The interval arithmetic and affine arithmetic are adopted to get thebounds of uncertainty models.In order to get more higher precision, subdivisions andrefinements method and several form of interval extensions are applied. An exampleswere provided to illustrate the validity and feasibility of the present method.Anextended beam and a three-truss example are provided to illustrate the validity andfeasibility of the presented procedures.
2. The study on the global optimization problems
The affine algorithm for global optimization was proposed by introducing theaffine algorithm and local optimization into the problem of global optimization, whichaim at solving disadvantage of the time consuming, higher space co mplexity and slowconvergence speed for the traditional interval algorithm in the solution of globaloptimization problem. The upper bound of global optimal solution was obtained bylocal optimization algorithm and affine arithmetic for objective function in eachsubinterval. And then, whether corresponding interval was discarded or not depends oncomparing the lower bound of affine arithmetic for objective function in eachsubinterval with the upper bound of global optimization solution. The subinterval whichcontains the optimal value was obtained by deleting the subinterval which did notcontain the optimal value. Lastly, the global optimal solution was found. NumericalSimulation results show that the proposed algorithm has higher convergence speedcompare with traditional interval optimization algorithm. At the same time, it alsooccupies less system resource.
3. Uncertain system response bounds analysis with interval arithmetic andaffinearithmetic
The introduction of new noise symbols causes error amplification in affinearithmetic inevitably. To avoid this disadvantage, this paper presents a modified affinearithmetic in matrix form for bounds computation of functions. The modified affinearithmetic does not introduce new noises during multiplication operation of affinevariables, and it can obtain compacter bounds compared to conventional affinearithmetic. The formulas computing processes and the validity of proposed method aredemonstrated by an example. The affine form of bounded uncertain variables andmodified affine arithmetic are brought in to calculate response bounds of uncertainsystem. The simulations show that, the proposed approach can obtain closer responsebounds than interval arithmetic and conventional affine arithmetic.
4. Stability region analysis using interval and affine method
As the interval form of uncertainty system can not express the pertinence ofuncertain variable, and interval algorithm may cause error explosion. The affine form ofuncertainty system and the affine inequality judge method for uncertainty system arepresented, at first, the certain parameter in certain system was substituted by affineparameter, we get the affine form transmit function of uncertain system, secondly, bysolving Matrix inequalities,we can get tolerable noise range with which the systemstability condition hold. An example shows that as take the pertinence of uncertainvariable into consideration, this method can judge the stability in a larger region, showsthe validity and the advantage of this method.
5. Reliability cauculation of non-convex and not simply connected fuction
In this chapter an affine-interval arithmetic-based method for the feasible regionevaluation of function or electronic circuits is presented. This method use affine-intervalarithmetic analyze the bounds of the function, and use branch and bound methoddivided these intervals into three kinds: accept regions, refuse regions and those ofuncertain regions; and the next, all the uncertain regions are re-divided and the boundscalculation and classification performed again until the subintervals small enough. Thestatistics on each of accept regions are performed next to get the sum of the acceptregions. The proposed technique guarantees an efficient, reliable and accurateevaluation of the yield, even for non-convex and not simply connected feasible region.The example presented shows the features of the approach.
6. The static analysis of interval structures based on interval matrix inversion andaffine matrix inversion
It is commonly used that many variables combined together, forms array or matixin engineering problems.The interval or affine array and matix was formed when one or more variable change in a certain range. The static analysis of interval structures basedon interval matrix inversion and affine matrix inversion.The basic conception wasdiscussed and the work focus on how to get interval matrix inversion and affine matrixinversion in a efficient method. The example presented shows the features and validityof this method.
引文
[1]钟万勰.计算结构力学与最优控制[M].大连,大连理工大学出版社,1993.
[2]程耿东.工程结构优化设计基础[M].北京,水利电力出版社,1984.
[3]王占权,云庆夏,杨东援.改进的遗传规划研究[J].系统工程与理论实践,2000,5:66-69.
[4]庆灵,冬梅.不确定广义系统的分析与综合[M].沈阳:东北大学出版社,2003.
[5]Chen J J, Che J W, Sun H A, et al. Probabilistic dynamic analys is of truss structures[J]. StructuralEngineering and Mechanics,2002,13(2):231-239.
[6]王登刚,李杰.计算不确定结构系统静态响应的一种可靠方法[J].计算力学学报,2003,20(6):662-669.
[7]吕震宙,冯蕴雯.结构可靠性问题研究的若干进展[J].力学进展,2000,30(1):21-28.
[8]俞立,陈国定,杨马英.不确定系统具有圆盘区域极点约束的鲁棒控制[J].自动化学报,2000,26(1):116-120.
[9]谢少锋,陈晓怀.测量系统不确定度分析及其动态性研究[J].计量学报,2002,23(3):237-240.
[10]史进渊.汽轮机动叶片的可靠性设计方法[J].应用力学学报,2008,24(2):331-334.
[11]徐可君,江龙平.叶片振动的非概率可靠性研究[J].机械工程学报,2002,38(10):17-19.
[12]李润方，王建军.齿轮系统动力学[M].北京:科学出版社,1997.
[13]张建云,丘大谋.齿轮刚度及制造误差对多平行轴转子系统动力学性能的影响[J].机械科学与技术,1996,15(5):749-754.
[14]李永祥.数控机床热误差建模新方法及其应用研究[D].上海交通大学,2007.
[15]丁文政,黄筱调.不确定度评定法预测再制造机床定位精度及其工程应用[J].现代制造工程,2006,10:59-62.
[16]陈力.参数不确定空间机械臂系统的鲁棒自适应混合控制[J].控制理论与应用,2004,21(4):512-516.
[17]代颖,郑南宁.一类关于不确定性机器人的鲁棒控制策略[J].自动化学报,1999,25(2):204-209.
[18]谢箭,刘国良,颜世佐,等.基于神经网络的不确定性空间机器人自适应控制方法研究[J].宇航学报,2010,31(1):123-129.
[19]李忠献,李忠诚,梁万顺.考虑地基岩土参数不确定性的核电厂结构随机地震反应分析[J].核动力工程,2006,27(2):30-35.
[20]江近仁,孙景江.考虑参数不确定性的结构动力可靠度分析[J].世界地震工程,1992,1:30-36.
[21]张建宁,于建国.地震属性应用中的不确定性分析[J].石油物探,2006,45(4):373-379.
[22]刘玉彬,王光远.工程结构广义可靠性理论[M].北京:科学出版社,2005.
[23]Gao W, Chen J J, Ma H B, et al. Dynamic response analysis of closed loop control system forintelligent truss structures based on probability[J]. Structural Engineering and Mechanics,2003,15(2):239-248.
[24]Ewins D J, Han Z S. Resonant vibration levels of a mistuned bladed disk[J]. Journal of Vibration,Acoustics Stress and Reliability in Design,1984,106(2):211-217..
[25]Dai J, Chen J J, Li Y G. Dynamic response optimization design for engineering structures basedon reliability[J]. Applied Mathematics and Mechanics,2003,24(1):43-52.
[26]Srinivasan A V. Vibrations of Bladed-Disk Assemblies—A Selected Survey (Survey Paper)[J].Journal of Vibration, Acoustics Stress and Reliability in Des ign,1984,106(2):165-168.
[27]刘强,尔联洁,刘金琨.参数不确定机械伺服系统的鲁棒非线性摩擦补偿控制[J].自动化学报,2008,29(4):628-632.
[28]张继周,缪林昌,王华敬.土性参数不确定性描述方法的探讨[J].岩土工程学报,2009,31(12):1936-1940.
[29]胡南辉,金朝永,陈德银.不确定时滞广义系统的H∞保性能控制[J].电机与控制学报,2008,12(3):331-336.
[30]王攀,吕震宙,李贵杰.参数不确定情况下结构系统重要性分析[J].中国科学:技术科学,2011,41(11):1512-1518.
[31]陈洁,周绍磊,宋召青.基于不确定性的高超声速飞行器动态面自适应反演控制系统设计[J].宇航学报,2010,31(11):2550-2556.
[32]郑剑飞,冯勇,郑雪梅,等.不确定非线性系统的自适应反演终端滑模控制[J].控制理论与应用,2009,26(4):410-414.
[33]王飞跃.基于不确定性理论的尾矿坝稳定性分析及综合评价研究[D].中南大学博士学位论文,2009.
[34]张福民,曲兴华,叶声华.面向对象的大尺寸测量不确定度分析[J].光学精密工程,2008,16(11):2239-2243.
[35]陈群,唐岷,朱分清.强度参数的不确定性对土石坝坝坡失稳概率的影响[J].岩土工程学报,2008,30(11):1595-1598.
[36]陈怀艳,曹芸,韩洁.基于蒙特卡罗法的测量不确定度评定[J].电子测量与仪器学报,2011,25(4):301-308.
[37]郭书祥,吕震宙,冯立富.模糊运算和模糊有限元静力控制方程的求解[J].应用数学和力学,2002,23(9):936-942.
[38]张玲华,郑宝玉.随机信号处理[M].北京:清华大学出版社,2003.
[39]谢东.模糊信息处理及应用[M].北京:科学出版社,2003.
[40]王光远.未确知信息及其数学处理[J].哈尔滨建筑大学学报,1990,23(4):1-9.
[41]王清印.泛灰集与泛灰数的代数运算[J].华中理工大学学报,1992,4:151-156.
[42]王宏伟,马广富.基于模糊模型的混沌时间序列预测[J].物理学报,2005,53(10):3293-3297.
[43]谢宏,牛东晓,张国立,等.一种模糊模型的混合建模方法及在短期负荷预测中的应用[J].中国电机工程学报,2005,25(8):17-22.
[44]李炜,蒋栋年.基于TS模糊模型的非线性网络化控制系统的H∞鲁棒容错控制[J].控制与决策,2010,25(4):598-604.
[45]唐国元,宾鸿赞.基于模糊模型的车辆稳定性控制方法研究[J].中国机械工程,2005,15(22):2064-2067.
[46]王书斌,胡品慧,林立.基于TS模糊模型的状态反馈预测控制[J].控制理论与应用,2007,24(5):819-824.
[47]廖龙涛,李少远,黄广斌.规则可生长与修剪的非线性系统TS模糊模型辨识[J].自动化学报,2007,33(10):1097-1100.
[48]张捷,薄煜明,吕明.基于模糊模型的无线网络控制系统故障检测[J].系统工程与电子技术,2010,32(4):842-845.
[49]郑英,王彦伟,方华京.基于T—S模型的网络化控制系统的鲁棒容错控制[J].华中科技大学学报(自然科学版),2008,36(3):111-113.
[50]屈福政,费烨,王欣.复杂机械方案多属性灰色模糊优选模型及应用[J].大连理工大学学报,2005,45(2):201-205.
[51]宋刚,胡德金.基于Sugeno模糊模型的数控机床故障诊断法[J].上海交通大学学报,2005,39(1):91-94.
[52]李晓梅,丁宁,朱喜林.表面粗糙度模糊神经网络在线辨识模型[J].机械工程学报,2007,43(3):212-217.
[53]姚成玉,赵静一.基于T—S模型的液压系统模糊故障树分析方法研究[J].中国机械工程,2009,20(16):1913-1917.
[54]崔根群,李春书,王丽敏,等.偏曲轴少齿差行星传动机构优化数学模型的建立[J].机械设计,2005,22(2):37-39.
[55]吕震宙,孙颉,徐友良.机械结构系统模糊可靠性分析的数字计算方法[J].机械工程学报,2005,41(9):19-23.
[56]胡芳,陈无畏.基于非线性模型的汽车空气悬架系统模糊控制研究[J].合肥工业大学学报,2005,28(7):772-777.
[57]余跃庆,周刚,方道星.基于模糊PID融合的柔性机械臂振动压电主动控制研究[J].中国机械工程,2008,19(15):1836-1841.
[58]胡桥,何正嘉,张周锁,等.经验模式分解模糊特征提取的支持向量机混合诊断模型[J].西安交通大学学报,2005,39(3):290-294.
[59]刘开第,赵奇,周少玲,等.机械产品方案设计模糊综合评价中隶属度转换的新方法[J].机械工程学报,2009,45(12):162-166.
[60]冯柯,崔永固,李静,等.基于模糊逻辑和遗传算法的工程机械故障诊断[J].解放军理工大学学报,2006,7(4):385-389.
[61]潘晓东,秦从律,钱磊.与建筑抗震设计规范相对应的地面地震动随机模型参数研究[J].地震研究,2005,28(1):82-85.
[62]张秋虎,方圣恩,任伟新.基于随机响应面模型的随机模型修正方法[C].乌鲁木齐,第22届全国结构工程学术会议论文集第Ⅲ册.2013.
[63]鲍春梅.随机参数下齿轮传动系统的响应分析与模拟[D].兰州交通大学,2013.
[64]索建臣.基于高阶泰勒展开式的随机结构的随机振动响应分析[D].武汉理工大学,2007.
[65]李美兰.非线性随机时滞系统的稳定与控制问题研究[D].中南大学,2008.
[66]梅勇兵.随机信号处理在工程中的应用[D].四川大学,2005.
[67]王新刚,张义民,王宝艳.机械零部件的动态可靠性灵敏度分析[J].机械工程学报,2010,46(10):188-193.
[68]王启瑞,刘立强,陈无畏.基于随机次优控制的汽车电动助力转向与主动悬架集成控制[J].中国机械工程,2005,16(8):743-747.
[69]刘春,陈宇东,陈塑寰.随机参数结构振动控制闭环特征值的标准差[J].机械强度,2005,26(6):600-604.
[70]王正,谢里阳,李兵.考虑载荷作用次数的零部件可靠性模型[J].机械强度,2008,30(1):68-71.
[71]吕震宙,孙颉,徐友良.机械结构系统模糊可靠性分析的数字计算方法[J].机械工程学报,2005,41(9):19-23.
[72]戴君.基于四分之一车辆模型的具有随机结构参数车辆的随机动力分析[J].振动与冲击,2010,29(6):211-215.
[73]董霞,王恪典.一种间隙副连杆模型及其在复杂机构精度分析中的应用[J].机械科学与技术,2005,24(4):479-483.
[74]赵永翔,杨冰,张卫华.应变疲劳可靠性理论与方法的新进展[J].机械强度,2005,27(5):604-611.
[75]王磊,刘文珽.基于随机模糊参数的结构模糊可靠性分析模型[J].北京航空航天大学学报,2005,31(4):412-415.
[76]于旭东,龙兴武,汤建勋.机械抖动激光陀螺的随机振动响应分析[J].光学精密工程,2007,15(11):1760-1766.
[77]赵永翔,杨冰,梁红琴.一维机械强度参数概率模型的合理重构[J].工程力学,2008,25(1):42-48.
[78]王世宇,宋轶民,张策,等.行星齿轮传动的共振失效概率[J].天津大学学报,2006,38(12):1122-1128.
[79]Elishakoff I. Three versions of the finite element method based on concept of stochasticty,fuzziness or anti-optimization[J]. Applied Mechanics Review,1998,51(3):209-218.
[80]Chen S H, Yang X W. Interval finite element method for beam structures[J]. Finite Elements inanalys is and Design,2000,34(1):75-88.
[81]Ellishakoff I. Essay on uncertainties in elastic and viscoelastic structures: from A MFreudenthal's criticisms to modern convex modeling[J]. Computers&Structures,1995,56(6):871-895.
[82]Elishakoff I. Possible limitations of probabilistic methods in engineering[J]. Applied MechanicsReview,2000,53(2):19-36.
[83]刘开第，未确知数学[M].武汉,华中理工大学出版社,1997.
[84]刘开第,曹庆奎,庞彦军.基于未确知集合的故障诊断方法[J].自动化学报,2005,30(5):747-756.
[85]Ge Q, Wu H Q. Several unascertained model on developing strategies in the Yangte riverbasin[C]. ISDSRC,1991.
[86]王光远.工程软科学理论[M].北京:科学出版社,1992.
[87]Zhu Z Q, Liang Z T, Chen J J. Unascertained factor method of dynamic characteristic analysisfor antenna structures[J]. Journal of China Ordnance,2008,4(3):167-172.
[88]梁震涛,陈建军,胡太彬.未确知桁架结构有限元分析的未确知因子法[J].机械强度,2005,27(4):498-503.
[89]Liang Z T, Chen J J, Gao W, et al. Reliability allocation of large spaceborne antenna deploymentmechanism system using unascertained method[C].1st International Symposium on Systemsand Control in Aerospace and Astronautics (ISSCAA), Harbin,2006,2006:1098-1103.
[90]Elishakoff I, Eliseef P, Glegg S. Convex modeling of material uncertainty in vibrations of aviscoelastic structure[J]. AIAA Journal,1994,32:843-849.
[91]Ben-Haim Y, Elishakoff I. Convex models of uncertainty in applied mechanics[M]. Amsterdam:Elsevier science Publishers,1990.
[92]Qiu Z P, Gu Y X. Extension of convex models and its improvement on the approximatesolution[J]. ACTA Mechanic A SINICA (English series),1996,12(4):349-357.
[93]Ganzerli S, Pantelides C P. Optimum structural design via convex model superposition[J].Computers&Structures,2000,74(6):639-647.
[94]Elishako I. Convex Modeling-A Generalization Of Interval Analysis For NonprobabilisticTreatment Of Uncertainty[C]//These Proceedings.1995.
[95]Ganzerli S, Pantelides C P. Load and resistance convex models for optimum design[J]. Structuraloptimization,1999,17(4):259-268.
[96]Pantelides C P, Booth B C. Computer-aided des ign of optimal structures with uncertainty[J].Computers&Structures,2000,74(3):293-307.
[97]曹鸿钧,段宝岩.基于凸集合模型的非概率可靠性研究[J].计算力学学报,2006,22(5):546-549.
[98]Chen S H, Qiu Z, Liu Z. Perturbation method for computing eigenvalue bounds in structuralvibration systems with interval parameters[J]. Communications in numerical methods inengineering,1994,10(2):121-134.
[99]Chen S, Qiu Z, Song D. A new method for computing the upper and lower bounds onfrequencies of structures with interval parameters[J]. Mechanics research communications,1994,21(6):583-592.
[100]Qiu Z P, Chen S H, Elishakoff I. Bounds of eigenvalues for structures with an intervaldescription of uncertain-but-non-random parameters[J]. Chaos, Soliton, and Fractral,1996,7(3):425-434.
[101]Qiu Z P, Chen S H, Elishakoff. Natural frequencies of structures withuncertain-but-non-random parameters[J]. Journal of Optimization Theory and Applications,1995,86(3):669-683.
[102]Burkill J C. Functions of intervals[J]. Proceedings of the London Mathematical Society,1924,2(1):275-310.
[103]Young R C. The algebra of many-valued quantities[J]. Mathematische Annalen,1931,104(1):260-290.
[104]Moore R E. Interval arithmetic and automatic error analysis in digital computing[M]. Stanford:Stanford University,1962.
[105]Moore R E. Interval analysis[M]. New Jersey: Prentice-Hall,1966.
[106]Moore R E, Moore R E. Methods and applications of interval analysis[M]. Philadelphia: Siam,1979.
[107]Alefeld G, Herzberger J. Introduction to interval computation[M]. Academic press,1984.
[108]Moore R E, Kearfott R B, Cloud M J. Introduction to interval analysis[M]. Siam,2009.
[109]Stolfi J, De Figueiredo L H. An introduction to affine arithmetic[J].2003.
[110]Bühler K. Linear interval estimations for parametric objects theory and application[C]//Computer Graphics Forum. Blackwell Publishers Ltd,2001,20(3):522-531.
[111]Bühler K. Taylor models and affine arithmetics-towards a more sophisticated use of reliablearithmetics in computer graphics[C]. In: Proceedings of the17th Spring Conference inComputer Graphics(SCCG’01), Budmerice, Slovakia,2001:40-48.
[112]Bühler K, Barth W. A new intersection algorithm for parametric surfaces based on linearinterval estimations[C]. In: Scientific Computing, Validated Numerics, Interval Methods,Boston/Dordrecht/London: Kluwer Academic Publishers,2001:179-190.
[113]Bühler K. A new subdivision algorithm for the intersection of parametric surfaces [M]. Vienna:Vienna University of Technology,2001.
[114]Buehler K. Fast and reliable plotting of implicit curves[M]. Springer US,2002.
[115]De Figueiredo L H. Surface intersection using affine arithmetic [C]. In: Proceedings of GraphicsInterface, Toronto, Ontario, Canada,1996:168-175.
[116]Figueiredo L H, Stolfi J. Adaptive enumeration of implicit surfaces with affinearithmetic[C]//Computer Graphics Forum. Blackwell Science Ltd,1996,15(5):287-296.
[117]HENRIQUE A D E C J L, GATTASS F M. Interval methods for ray casting implicit surfaceswith affine arithmetic[J]. Proceedings of XII SIBGRAPI,1999,1:65-71
[118]Heidrich W, Seidel H P. Ray tracing procedural displacement shaders[C]. In: Proceedings ofGraphics Interface, Vancouver, British Columbia, Canada,1998:8-16.
[119]Heidrich W, Slusallek P, Seidel H P. Sampling of procedural shaders using affine arithmetic [J].ACM Transactions on Graphics,1998,17(3):158-176.
[120]Voiculescu I, Berchtold J, Bowyer A, et al. Interval and Affine Arithmetic for Surface Locationof Power—and Bernstein—Form Polynomials[M]//The Mathematics of Surfaces IX. SpringerLondon,2000:410-423.
[121]Henrique de Figueiredo L, Stolfi J, Velho L. Approximating parametric curves with strip treesusing affine arithmetic[C]//Computer Graphics Forum. Blackwell Publishing, Inc,2003,22(2):171-179.
[122]Bowyer A, Martin R, Shou H, et al. Affine intervals in a CSG geometricmodeller[M]//Uncertainty in Geometric Computations. Springer US,2002:1-14.
[123]郭书祥,吕震宙.区间运算和静力区间有限元[J].应用数学和力学,2001,22(12):1249-1254.
[124]杨晓伟,陈塑寰,滕绍勇.基于单元的静力区间有限元法[J].计算力学学报,2002,19(2):179-183.
[125]王晓军,李云龙,王磊,等.基于区间扩阶系统方法的结构静力分析[J].工程力学,2013,30(1):22-30.
[126]马娟,陈建军,张建国,等.不确定性桁架结构区间有限元分析的区间因子法[J].机械设计与研究,2006,21(6):6-9.
[127]张建国,陈建军,马孝松.具有区间参数的不确定结构静力区间分析的一种算法[J].机械科学与技术,2006,24(10):1158-1162.
[128]李金平,陈建军,朱增青,等.结构区间有限元方程组的一种解法[J].工程力学,2010,27(4):79-83.
[129]郭书祥,吕震宙.线性区间有限元静力控制方程的组合解法[J].计算力学学报,2003,20(1):34-38.
[130]孙靖.用于区间参数多目标优化问题的遗传算法[D].中国矿业大学,2012.
[131]陈怀海.非确定结构系统区间分析的直接优化法[J].南京航空航天大学学报,1999,31(2):146-150.
[132]Dessombz O, Thouverez F, La néJ P, et al. Analysis of mechanical systems using intervalcomputations applied to finite element methods[J]. Journal of Sound and Vibration,2001,239(5):949-968.
[133]Markov S. An iterative method for algebraic solution to interval equations[J]. AppliedNumerical Mathematics,1999,30(2):225-239.
[134]郭书祥,吕震宙.区间有限元静力控制方程的一种迭代解法[J].西北工业大学学报,2002,20(1):20-23.
[135]佘远国,沈成武.改进的区间有限元静力控制方程迭代解法[J].武汉理工大学学报:交通科学与工程版,2005,29(2):248-251.
[136]姜浩.基于区间算法求解非线性方程组和全局最优化问题[D].国防科学技术大学,2008.
[137]Ben-Haim Y. A non-probabilistic concept of reliability[J]. Structural Safety,1994,14(4):227-245.
[138]ELISHAKOFF I. Discussion on：A non-probabilistic concept of reliability[J]. Structural Safety,1995,17(3)：195-199
[139]Ben-Haim Y. A non-probabilistic measure of reliability of linear systems based on expansion ofconvex models[J]. Structural Safety,1995,17(2):91-109.
[140]郭书祥,吕震宙.结构的非概率可靠性方法和概率可靠性方法的比较[J].应用力学学报,2003,20(3):107-110.
[141]郭书祥,张陵,李颖.结构非概率可靠性指标的求解方法[J].计算力学学报,2005,22(2):227-231.
[142]亢战,罗阳军.基于凸模型的结构非概率可靠性优化[J].力学学报,2006,38(6):807-815.
[143]张新锋,赵彦,施浒立.基于凸集的结构非概率可靠性度量研究[J].机械强度,2007,29(4):589-592.
[144]宋利锋,邱志平.含模糊区间变量的结构非概率可靠性优化设计[J].工程力学,2013,30(6):36-40.
[145]Elishakoff I, Haftka R T, Fang J. Structural design under bounded uncertainty—optimizationwith anti-optimization[J]. Computers&structures,1994,53(6):1401-1405.
[146]Lombardi M. Optimization of uncertain structures using non-probabilistic models[J].Computers&structures,1998,67(1):99-103.
[147]Lombardi M, Haftka R T. Anti-optimization technique for structural design under loaduncertainties[J]. Computer methods in applied mechanics and engineering,1998,157(1):19-31.
[148]Ganzerli S, Pantelides C P. Load and resistance convex models for optimum design[J].Structural optimization,1999,17(4):259-268.
[149]姜潮,刘丽新,龙湘云.一种概率-区间混合结构可靠性的高效计算方法[J].计算力学学报,2013,30(5):605-609.
[150]吕春梅,张义民,李鹤,等.动态结构系统的频率可靠性稳健设计研究[J].计算力学学报,2013,30(5):653-656.
[151]方鹏亚,常新龙,胡宽,等.基于区间不确定性的涡轮盘强度可靠性优化设计[J].推进技术,2013,34(007):962-967.
[152]朱增青.区间和未确知参数结构(机构)分析方法研究及应用[D].西安电子科技大学,2009.
[153]Shou H, Martin R. Voiculescu I. Bowyer A, Wang G. Affine arithmetic in matrix form forpolynomial evaluation and algebraic curve drawing[J]. Progress in Natural Science,2002,12(1):77–80.
[154]Stol J, De Figueiredo L H. Self-validated numerical methods and applications[C]//Monographfor21st Brazilian Mathematics Colloquium, IMPA, Rio de Janeiro.1997.
[155]Christoph Grimm, Wilhelm Heupke, Klaus Waldschmidt. Analysis of Mixed-Signal Systemswith Affine Arithmetic[J]. IEEE Transactions on Computer-Aided Design of Integrated Circuitsand Systems,2005,24(1):118-123.
[156]M. Vitelli. Range Analysis in Electroquasistatic Field Linear Problems[J], IEEE Transactionson Dielectrics and Electrical Insulation,2003,10(1):155-167.
[157]Siome Klein Goldenstein, Christian Vogler, Dimitris Metaxas. Statistical Cue Integration inDAG Deformable Models[J]. IEEE Transactions on Pattern Analysis and Machine Intelligence,2003,25(7):801-813.
[158]Nicola Femia, Giovanni Spagnuolo. True Worst-Case Circuit Tolerance Analysis Using GeneticAlgorithms and Affine Arithmetic[J]. IEEE Transactions on Circuits and Systems,2000,47(9):1285-1296.
[159]Luiz Henrique de Figueiredo, Jorge Stolfi. Affine arithmetic: concepts and applications[J].Numerical Algorithms,2004,37:147–158.
[160]Ralph Martin, Huahao Shou, Irina Voiculescu, Adrian Bowyer, Guojin Wang. Comparison ofinterval methods for plotting algebraic curves[J]. Computer Aided Geometric Design,2002,19:553–587.
[161]Shou H, Lin H, Martin R, et al. Modified affine arithmetic is more accurate than centeredinterval arithmetic or affine arithmetic[M]//Mathematics of Surfaces. Springer BerlinHeidelberg,2003:355-365.
[162]邱志平，顾笑冬，李登峰.单自由度不确定滞回系统振动响应的区间分析方法[J]，动力学与控制学报,2007,(5):173-177.
[163]陈塑寰，裴春艳.不确定二阶振动控制系统动力响应的区间方法[J]，吉林大学学报(工学版),2008,38(1):94-98.
[164]邱志平，马丽红，王晓军.不确定非线性结构动力响应的区间分析方法[J],力学学报,2006,38(5):645-651.
[165]林立广,陈建军,马娟.基于区间因子法的不确定性桁架结构动力响应分析[J],应用力学学报,2008,25(4):612-617.
[166]郑松,侯迪波,周泽魁.动态调整选择策略的改进蚁群算法[J].控制与决策,2008,23(2):225-228.
[167]杨雪榕,梁加红,陈凌,等.多邻域改进粒子群算法[J].系统工程与电子技术,2010,32(11):2453-2458.
[168]赵文红,王宇平,王巍.快速寻优的全局优化进化算法[J].计算机工程,2008,34(8):208-212.
[169]吴慧卓,张可村.基于拉格朗日对偶的一类全局优化算法[J].西安交通大学学报,2008,42(8):1031-1034.
[170]Hong Zhou. Solving Fractional Problems Management Based on A Deterministic Algorithm[C].2009IITA International Conference on Control, Automation and Systems Engineering,2009:163-166.
[171]Lin Youdong, Stadtherr, Mark A. Deterministic Global Optimization for Dynamic SystemsUsing Interval Analysis[C].200612th GAMM-IMACS International Symposium on ScientificComputing, Computer Arithmetic and Validated Numerics,2007:323-31.
[172]Xiaowei Zhang and Sanyang Liu. Interval algorithm for global numerical optimization[J].Engineering Optimization,2008,40(9):849-868.
[173]M. sun, A. W. Johnson. Interval branch and bound with local sampling for constrained globaloptimization[J]. Journal of Global Optimization,2005,33:62-82.
[174]Luiz Henrique de Figueiredo, Jorge Stolfi. Affine arithmetic: concepts and applications[J].Numerical Algorithms,2004,37:147-158.
[175]Hansen E. Global optimization using interval analysis-the multidimensional case[J].Numerische Mathematik,1980,34(1):247-270.
[176]Berner S. New results on verified global optimization[J]. Computing,1996,57(2):323-343.
[177]裴春艳,陈塑寰,基于区间方法的不确定二阶控制系统稳定性的鲁棒性分析[J],吉林大学学报(工学版),2006,36(增刊):20-25.
[178]申涛,王孝红,袁铸钢,一类不确定系统的鲁棒稳定性分析[J],自动化学报,2007,33(4):426-427.
[179]井元伟,姜囡,郝彬彬,一类不确定系统的最优极小极大鲁棒控制[J],控制与决策,2008,23(2):208-212.
[180]Bertoni F C, da Silva I N, Pires M G. A Neurogenetic Approach and its Application toConstrained Nonlinear Convex Optimization Problems with Joint and Dis joint FeasibleRegions[C]//Hybrid Intelligent Systems,2008. HIS'08. Eighth International Conference on.IEEE,2008:90-95.
[181] Wei Bian, Xiaoping Xue.Subgradient-Based Neural Networks for Nonsmooth NonconvexOptimization Problems[J]. IEEE Transactions on Neural Networks.2009,20(6):1024-1038.
[182] Hsc c-c,Chang, S.-C. Yu, C. Y. Tolerance design of robust controllers for uncertain intervalsystems based on evolutionary algorithms[J]. Control Theory&Applications.2006,1(1):244-252.
[183] Didier Henrion, Jean-Bernard Lasserre. Inner Approximations for Polynomial MatrixInequalities and Robust Stability Regions[J]. IEEE Transactions on Automatic Control.2012,57(6):1456-1467.
[184] M. Soliman. H. Emara, A.Elshafei, A. Bahgat Robust Output Feedback Power SystemStabilizer Design: an LMI approach[C]. Power and Energy Society General Meeting-Conversion and Delivery of Electrical Energy in the21st Century,2008IEEE,pages:1-8.
[185] Spagnuolo, G. An interval arithmetic-based yield evaluation in circuit tolerance design[C].ISCAS2002. IEEE International Symposium on Circuits and system,2002:743-746.
[186] Jorge E. Hurtado, Diego A. Alvarez. The encounter of interval and probabilistic approaches tostructural reliability at the design point[J]. Computer Methods in Applied Mechanics andEngineering.2012,225-228(15):74-94.
[187] Comba J L D, Stolfi J. Affine arithmetic and its applications to computer graphics[C]. In:Proceedings of Anais do V II S B GRAPI, Recife, Brazil,1993:9-18.
[188]朱增青,陈建军,宋宗凤等.区间参数杆系结构非概率可靠性指标的改进仿射算法[J],工程力学,2010,27(2):49-58.
[189]谢永强,陈建军,徐亚兰.基于仿射算法的确定性全局优化算法[J],华南理工大学学报(自然科学版),2012,40(5):35-40.
[190] D. Degrauwe, G. Lombaert, G. De Roeck, Improving interval analysis in finite elementcalculations by means of affine arithmetic[J]. Computers and Structures,2010,88(3-4):247–254.
[191] Peter W. M. Tsang, Terry Y. F YUEN, W.C. Situ. Enhanced affine invariant matching ofbroken boundaries based on particle swarm optimization and the dynamic migrant principle[J].Applied Soft Computing Journal,2010,10(2):432-438.
[192]孙继涛,张银萍.关于时变区间矩阵的稳定性研究[J].控制理论与应用,1995,12(1):108-113.
[193]Degrauwe D, Lombaert G, De Roeck G. Improving interval analysis in finite elementcalculations by means of affine arithmetic[J]. Computers&structures,2010,88(3):247-254.
[2]程耿东.工程结构优化设计基础[M].北京,水利电力出版社,1984.
[3]王占权,云庆夏,杨东援.改进的遗传规划研究[J].系统工程与理论实践,2000,5:66-69.
[4]庆灵,冬梅.不确定广义系统的分析与综合[M].沈阳:东北大学出版社,2003.
[5]Chen J J, Che J W, Sun H A, et al. Probabilistic dynamic analys is of truss structures[J]. StructuralEngineering and Mechanics,2002,13(2):231-239.
[6]王登刚,李杰.计算不确定结构系统静态响应的一种可靠方法[J].计算力学学报,2003,20(6):662-669.
[7]吕震宙,冯蕴雯.结构可靠性问题研究的若干进展[J].力学进展,2000,30(1):21-28.
[8]俞立,陈国定,杨马英.不确定系统具有圆盘区域极点约束的鲁棒控制[J].自动化学报,2000,26(1):116-120.
[9]谢少锋,陈晓怀.测量系统不确定度分析及其动态性研究[J].计量学报,2002,23(3):237-240.
[10]史进渊.汽轮机动叶片的可靠性设计方法[J].应用力学学报,2008,24(2):331-334.
[11]徐可君,江龙平.叶片振动的非概率可靠性研究[J].机械工程学报,2002,38(10):17-19.
[12]李润方，王建军.齿轮系统动力学[M].北京:科学出版社,1997.
[13]张建云,丘大谋.齿轮刚度及制造误差对多平行轴转子系统动力学性能的影响[J].机械科学与技术,1996,15(5):749-754.
[14]李永祥.数控机床热误差建模新方法及其应用研究[D].上海交通大学,2007.
[15]丁文政,黄筱调.不确定度评定法预测再制造机床定位精度及其工程应用[J].现代制造工程,2006,10:59-62.
[16]陈力.参数不确定空间机械臂系统的鲁棒自适应混合控制[J].控制理论与应用,2004,21(4):512-516.
[17]代颖,郑南宁.一类关于不确定性机器人的鲁棒控制策略[J].自动化学报,1999,25(2):204-209.
[18]谢箭,刘国良,颜世佐,等.基于神经网络的不确定性空间机器人自适应控制方法研究[J].宇航学报,2010,31(1):123-129.
[19]李忠献,李忠诚,梁万顺.考虑地基岩土参数不确定性的核电厂结构随机地震反应分析[J].核动力工程,2006,27(2):30-35.
[20]江近仁,孙景江.考虑参数不确定性的结构动力可靠度分析[J].世界地震工程,1992,1:30-36.
[21]张建宁,于建国.地震属性应用中的不确定性分析[J].石油物探,2006,45(4):373-379.
[22]刘玉彬,王光远.工程结构广义可靠性理论[M].北京:科学出版社,2005.
[23]Gao W, Chen J J, Ma H B, et al. Dynamic response analysis of closed loop control system forintelligent truss structures based on probability[J]. Structural Engineering and Mechanics,2003,15(2):239-248.
[24]Ewins D J, Han Z S. Resonant vibration levels of a mistuned bladed disk[J]. Journal of Vibration,Acoustics Stress and Reliability in Design,1984,106(2):211-217..
[25]Dai J, Chen J J, Li Y G. Dynamic response optimization design for engineering structures basedon reliability[J]. Applied Mathematics and Mechanics,2003,24(1):43-52.
[26]Srinivasan A V. Vibrations of Bladed-Disk Assemblies—A Selected Survey (Survey Paper)[J].Journal of Vibration, Acoustics Stress and Reliability in Des ign,1984,106(2):165-168.
[27]刘强,尔联洁,刘金琨.参数不确定机械伺服系统的鲁棒非线性摩擦补偿控制[J].自动化学报,2008,29(4):628-632.
[28]张继周,缪林昌,王华敬.土性参数不确定性描述方法的探讨[J].岩土工程学报,2009,31(12):1936-1940.
[29]胡南辉,金朝永,陈德银.不确定时滞广义系统的H∞保性能控制[J].电机与控制学报,2008,12(3):331-336.
[30]王攀,吕震宙,李贵杰.参数不确定情况下结构系统重要性分析[J].中国科学:技术科学,2011,41(11):1512-1518.
[31]陈洁,周绍磊,宋召青.基于不确定性的高超声速飞行器动态面自适应反演控制系统设计[J].宇航学报,2010,31(11):2550-2556.
[32]郑剑飞,冯勇,郑雪梅,等.不确定非线性系统的自适应反演终端滑模控制[J].控制理论与应用,2009,26(4):410-414.
[33]王飞跃.基于不确定性理论的尾矿坝稳定性分析及综合评价研究[D].中南大学博士学位论文,2009.
[34]张福民,曲兴华,叶声华.面向对象的大尺寸测量不确定度分析[J].光学精密工程,2008,16(11):2239-2243.
[35]陈群,唐岷,朱分清.强度参数的不确定性对土石坝坝坡失稳概率的影响[J].岩土工程学报,2008,30(11):1595-1598.
[36]陈怀艳,曹芸,韩洁.基于蒙特卡罗法的测量不确定度评定[J].电子测量与仪器学报,2011,25(4):301-308.
[37]郭书祥,吕震宙,冯立富.模糊运算和模糊有限元静力控制方程的求解[J].应用数学和力学,2002,23(9):936-942.
[38]张玲华,郑宝玉.随机信号处理[M].北京:清华大学出版社,2003.
[39]谢东.模糊信息处理及应用[M].北京:科学出版社,2003.
[40]王光远.未确知信息及其数学处理[J].哈尔滨建筑大学学报,1990,23(4):1-9.
[41]王清印.泛灰集与泛灰数的代数运算[J].华中理工大学学报,1992,4:151-156.
[42]王宏伟,马广富.基于模糊模型的混沌时间序列预测[J].物理学报,2005,53(10):3293-3297.
[43]谢宏,牛东晓,张国立,等.一种模糊模型的混合建模方法及在短期负荷预测中的应用[J].中国电机工程学报,2005,25(8):17-22.
[44]李炜,蒋栋年.基于TS模糊模型的非线性网络化控制系统的H∞鲁棒容错控制[J].控制与决策,2010,25(4):598-604.
[45]唐国元,宾鸿赞.基于模糊模型的车辆稳定性控制方法研究[J].中国机械工程,2005,15(22):2064-2067.
[46]王书斌,胡品慧,林立.基于TS模糊模型的状态反馈预测控制[J].控制理论与应用,2007,24(5):819-824.
[47]廖龙涛,李少远,黄广斌.规则可生长与修剪的非线性系统TS模糊模型辨识[J].自动化学报,2007,33(10):1097-1100.
[48]张捷,薄煜明,吕明.基于模糊模型的无线网络控制系统故障检测[J].系统工程与电子技术,2010,32(4):842-845.
[49]郑英,王彦伟,方华京.基于T—S模型的网络化控制系统的鲁棒容错控制[J].华中科技大学学报(自然科学版),2008,36(3):111-113.
[50]屈福政,费烨,王欣.复杂机械方案多属性灰色模糊优选模型及应用[J].大连理工大学学报,2005,45(2):201-205.
[51]宋刚,胡德金.基于Sugeno模糊模型的数控机床故障诊断法[J].上海交通大学学报,2005,39(1):91-94.
[52]李晓梅,丁宁,朱喜林.表面粗糙度模糊神经网络在线辨识模型[J].机械工程学报,2007,43(3):212-217.
[53]姚成玉,赵静一.基于T—S模型的液压系统模糊故障树分析方法研究[J].中国机械工程,2009,20(16):1913-1917.
[54]崔根群,李春书,王丽敏,等.偏曲轴少齿差行星传动机构优化数学模型的建立[J].机械设计,2005,22(2):37-39.
[55]吕震宙,孙颉,徐友良.机械结构系统模糊可靠性分析的数字计算方法[J].机械工程学报,2005,41(9):19-23.
[56]胡芳,陈无畏.基于非线性模型的汽车空气悬架系统模糊控制研究[J].合肥工业大学学报,2005,28(7):772-777.
[57]余跃庆,周刚,方道星.基于模糊PID融合的柔性机械臂振动压电主动控制研究[J].中国机械工程,2008,19(15):1836-1841.
[58]胡桥,何正嘉,张周锁,等.经验模式分解模糊特征提取的支持向量机混合诊断模型[J].西安交通大学学报,2005,39(3):290-294.
[59]刘开第,赵奇,周少玲,等.机械产品方案设计模糊综合评价中隶属度转换的新方法[J].机械工程学报,2009,45(12):162-166.
[60]冯柯,崔永固,李静,等.基于模糊逻辑和遗传算法的工程机械故障诊断[J].解放军理工大学学报,2006,7(4):385-389.
[61]潘晓东,秦从律,钱磊.与建筑抗震设计规范相对应的地面地震动随机模型参数研究[J].地震研究,2005,28(1):82-85.
[62]张秋虎,方圣恩,任伟新.基于随机响应面模型的随机模型修正方法[C].乌鲁木齐,第22届全国结构工程学术会议论文集第Ⅲ册.2013.
[63]鲍春梅.随机参数下齿轮传动系统的响应分析与模拟[D].兰州交通大学,2013.
[64]索建臣.基于高阶泰勒展开式的随机结构的随机振动响应分析[D].武汉理工大学,2007.
[65]李美兰.非线性随机时滞系统的稳定与控制问题研究[D].中南大学,2008.
[66]梅勇兵.随机信号处理在工程中的应用[D].四川大学,2005.
[67]王新刚,张义民,王宝艳.机械零部件的动态可靠性灵敏度分析[J].机械工程学报,2010,46(10):188-193.
[68]王启瑞,刘立强,陈无畏.基于随机次优控制的汽车电动助力转向与主动悬架集成控制[J].中国机械工程,2005,16(8):743-747.
[69]刘春,陈宇东,陈塑寰.随机参数结构振动控制闭环特征值的标准差[J].机械强度,2005,26(6):600-604.
[70]王正,谢里阳,李兵.考虑载荷作用次数的零部件可靠性模型[J].机械强度,2008,30(1):68-71.
[71]吕震宙,孙颉,徐友良.机械结构系统模糊可靠性分析的数字计算方法[J].机械工程学报,2005,41(9):19-23.
[72]戴君.基于四分之一车辆模型的具有随机结构参数车辆的随机动力分析[J].振动与冲击,2010,29(6):211-215.
[73]董霞,王恪典.一种间隙副连杆模型及其在复杂机构精度分析中的应用[J].机械科学与技术,2005,24(4):479-483.
[74]赵永翔,杨冰,张卫华.应变疲劳可靠性理论与方法的新进展[J].机械强度,2005,27(5):604-611.
[75]王磊,刘文珽.基于随机模糊参数的结构模糊可靠性分析模型[J].北京航空航天大学学报,2005,31(4):412-415.
[76]于旭东,龙兴武,汤建勋.机械抖动激光陀螺的随机振动响应分析[J].光学精密工程,2007,15(11):1760-1766.
[77]赵永翔,杨冰,梁红琴.一维机械强度参数概率模型的合理重构[J].工程力学,2008,25(1):42-48.
[78]王世宇,宋轶民,张策,等.行星齿轮传动的共振失效概率[J].天津大学学报,2006,38(12):1122-1128.
[79]Elishakoff I. Three versions of the finite element method based on concept of stochasticty,fuzziness or anti-optimization[J]. Applied Mechanics Review,1998,51(3):209-218.
[80]Chen S H, Yang X W. Interval finite element method for beam structures[J]. Finite Elements inanalys is and Design,2000,34(1):75-88.
[81]Ellishakoff I. Essay on uncertainties in elastic and viscoelastic structures: from A MFreudenthal's criticisms to modern convex modeling[J]. Computers&Structures,1995,56(6):871-895.
[82]Elishakoff I. Possible limitations of probabilistic methods in engineering[J]. Applied MechanicsReview,2000,53(2):19-36.
[83]刘开第，未确知数学[M].武汉,华中理工大学出版社,1997.
[84]刘开第,曹庆奎,庞彦军.基于未确知集合的故障诊断方法[J].自动化学报,2005,30(5):747-756.
[85]Ge Q, Wu H Q. Several unascertained model on developing strategies in the Yangte riverbasin[C]. ISDSRC,1991.
[86]王光远.工程软科学理论[M].北京:科学出版社,1992.
[87]Zhu Z Q, Liang Z T, Chen J J. Unascertained factor method of dynamic characteristic analysisfor antenna structures[J]. Journal of China Ordnance,2008,4(3):167-172.
[88]梁震涛,陈建军,胡太彬.未确知桁架结构有限元分析的未确知因子法[J].机械强度,2005,27(4):498-503.
[89]Liang Z T, Chen J J, Gao W, et al. Reliability allocation of large spaceborne antenna deploymentmechanism system using unascertained method[C].1st International Symposium on Systemsand Control in Aerospace and Astronautics (ISSCAA), Harbin,2006,2006:1098-1103.
[90]Elishakoff I, Eliseef P, Glegg S. Convex modeling of material uncertainty in vibrations of aviscoelastic structure[J]. AIAA Journal,1994,32:843-849.
[91]Ben-Haim Y, Elishakoff I. Convex models of uncertainty in applied mechanics[M]. Amsterdam:Elsevier science Publishers,1990.
[92]Qiu Z P, Gu Y X. Extension of convex models and its improvement on the approximatesolution[J]. ACTA Mechanic A SINICA (English series),1996,12(4):349-357.
[93]Ganzerli S, Pantelides C P. Optimum structural design via convex model superposition[J].Computers&Structures,2000,74(6):639-647.
[94]Elishako I. Convex Modeling-A Generalization Of Interval Analysis For NonprobabilisticTreatment Of Uncertainty[C]//These Proceedings.1995.
[95]Ganzerli S, Pantelides C P. Load and resistance convex models for optimum design[J]. Structuraloptimization,1999,17(4):259-268.
[96]Pantelides C P, Booth B C. Computer-aided des ign of optimal structures with uncertainty[J].Computers&Structures,2000,74(3):293-307.
[97]曹鸿钧,段宝岩.基于凸集合模型的非概率可靠性研究[J].计算力学学报,2006,22(5):546-549.
[98]Chen S H, Qiu Z, Liu Z. Perturbation method for computing eigenvalue bounds in structuralvibration systems with interval parameters[J]. Communications in numerical methods inengineering,1994,10(2):121-134.
[99]Chen S, Qiu Z, Song D. A new method for computing the upper and lower bounds onfrequencies of structures with interval parameters[J]. Mechanics research communications,1994,21(6):583-592.
[100]Qiu Z P, Chen S H, Elishakoff I. Bounds of eigenvalues for structures with an intervaldescription of uncertain-but-non-random parameters[J]. Chaos, Soliton, and Fractral,1996,7(3):425-434.
[101]Qiu Z P, Chen S H, Elishakoff. Natural frequencies of structures withuncertain-but-non-random parameters[J]. Journal of Optimization Theory and Applications,1995,86(3):669-683.
[102]Burkill J C. Functions of intervals[J]. Proceedings of the London Mathematical Society,1924,2(1):275-310.
[103]Young R C. The algebra of many-valued quantities[J]. Mathematische Annalen,1931,104(1):260-290.
[104]Moore R E. Interval arithmetic and automatic error analysis in digital computing[M]. Stanford:Stanford University,1962.
[105]Moore R E. Interval analysis[M]. New Jersey: Prentice-Hall,1966.
[106]Moore R E, Moore R E. Methods and applications of interval analysis[M]. Philadelphia: Siam,1979.
[107]Alefeld G, Herzberger J. Introduction to interval computation[M]. Academic press,1984.
[108]Moore R E, Kearfott R B, Cloud M J. Introduction to interval analysis[M]. Siam,2009.
[109]Stolfi J, De Figueiredo L H. An introduction to affine arithmetic[J].2003.
[110]Bühler K. Linear interval estimations for parametric objects theory and application[C]//Computer Graphics Forum. Blackwell Publishers Ltd,2001,20(3):522-531.
[111]Bühler K. Taylor models and affine arithmetics-towards a more sophisticated use of reliablearithmetics in computer graphics[C]. In: Proceedings of the17th Spring Conference inComputer Graphics(SCCG’01), Budmerice, Slovakia,2001:40-48.
[112]Bühler K, Barth W. A new intersection algorithm for parametric surfaces based on linearinterval estimations[C]. In: Scientific Computing, Validated Numerics, Interval Methods,Boston/Dordrecht/London: Kluwer Academic Publishers,2001:179-190.
[113]Bühler K. A new subdivision algorithm for the intersection of parametric surfaces [M]. Vienna:Vienna University of Technology,2001.
[114]Buehler K. Fast and reliable plotting of implicit curves[M]. Springer US,2002.
[115]De Figueiredo L H. Surface intersection using affine arithmetic [C]. In: Proceedings of GraphicsInterface, Toronto, Ontario, Canada,1996:168-175.
[116]Figueiredo L H, Stolfi J. Adaptive enumeration of implicit surfaces with affinearithmetic[C]//Computer Graphics Forum. Blackwell Science Ltd,1996,15(5):287-296.
[117]HENRIQUE A D E C J L, GATTASS F M. Interval methods for ray casting implicit surfaceswith affine arithmetic[J]. Proceedings of XII SIBGRAPI,1999,1:65-71
[118]Heidrich W, Seidel H P. Ray tracing procedural displacement shaders[C]. In: Proceedings ofGraphics Interface, Vancouver, British Columbia, Canada,1998:8-16.
[119]Heidrich W, Slusallek P, Seidel H P. Sampling of procedural shaders using affine arithmetic [J].ACM Transactions on Graphics,1998,17(3):158-176.
[120]Voiculescu I, Berchtold J, Bowyer A, et al. Interval and Affine Arithmetic for Surface Locationof Power—and Bernstein—Form Polynomials[M]//The Mathematics of Surfaces IX. SpringerLondon,2000:410-423.
[121]Henrique de Figueiredo L, Stolfi J, Velho L. Approximating parametric curves with strip treesusing affine arithmetic[C]//Computer Graphics Forum. Blackwell Publishing, Inc,2003,22(2):171-179.
[122]Bowyer A, Martin R, Shou H, et al. Affine intervals in a CSG geometricmodeller[M]//Uncertainty in Geometric Computations. Springer US,2002:1-14.
[123]郭书祥,吕震宙.区间运算和静力区间有限元[J].应用数学和力学,2001,22(12):1249-1254.
[124]杨晓伟,陈塑寰,滕绍勇.基于单元的静力区间有限元法[J].计算力学学报,2002,19(2):179-183.
[125]王晓军,李云龙,王磊,等.基于区间扩阶系统方法的结构静力分析[J].工程力学,2013,30(1):22-30.
[126]马娟,陈建军,张建国,等.不确定性桁架结构区间有限元分析的区间因子法[J].机械设计与研究,2006,21(6):6-9.
[127]张建国,陈建军,马孝松.具有区间参数的不确定结构静力区间分析的一种算法[J].机械科学与技术,2006,24(10):1158-1162.
[128]李金平,陈建军,朱增青,等.结构区间有限元方程组的一种解法[J].工程力学,2010,27(4):79-83.
[129]郭书祥,吕震宙.线性区间有限元静力控制方程的组合解法[J].计算力学学报,2003,20(1):34-38.
[130]孙靖.用于区间参数多目标优化问题的遗传算法[D].中国矿业大学,2012.
[131]陈怀海.非确定结构系统区间分析的直接优化法[J].南京航空航天大学学报,1999,31(2):146-150.
[132]Dessombz O, Thouverez F, La néJ P, et al. Analysis of mechanical systems using intervalcomputations applied to finite element methods[J]. Journal of Sound and Vibration,2001,239(5):949-968.
[133]Markov S. An iterative method for algebraic solution to interval equations[J]. AppliedNumerical Mathematics,1999,30(2):225-239.
[134]郭书祥,吕震宙.区间有限元静力控制方程的一种迭代解法[J].西北工业大学学报,2002,20(1):20-23.
[135]佘远国,沈成武.改进的区间有限元静力控制方程迭代解法[J].武汉理工大学学报:交通科学与工程版,2005,29(2):248-251.
[136]姜浩.基于区间算法求解非线性方程组和全局最优化问题[D].国防科学技术大学,2008.
[137]Ben-Haim Y. A non-probabilistic concept of reliability[J]. Structural Safety,1994,14(4):227-245.
[138]ELISHAKOFF I. Discussion on：A non-probabilistic concept of reliability[J]. Structural Safety,1995,17(3)：195-199
[139]Ben-Haim Y. A non-probabilistic measure of reliability of linear systems based on expansion ofconvex models[J]. Structural Safety,1995,17(2):91-109.
[140]郭书祥,吕震宙.结构的非概率可靠性方法和概率可靠性方法的比较[J].应用力学学报,2003,20(3):107-110.
[141]郭书祥,张陵,李颖.结构非概率可靠性指标的求解方法[J].计算力学学报,2005,22(2):227-231.
[142]亢战,罗阳军.基于凸模型的结构非概率可靠性优化[J].力学学报,2006,38(6):807-815.
[143]张新锋,赵彦,施浒立.基于凸集的结构非概率可靠性度量研究[J].机械强度,2007,29(4):589-592.
[144]宋利锋,邱志平.含模糊区间变量的结构非概率可靠性优化设计[J].工程力学,2013,30(6):36-40.
[145]Elishakoff I, Haftka R T, Fang J. Structural design under bounded uncertainty—optimizationwith anti-optimization[J]. Computers&structures,1994,53(6):1401-1405.
[146]Lombardi M. Optimization of uncertain structures using non-probabilistic models[J].Computers&structures,1998,67(1):99-103.
[147]Lombardi M, Haftka R T. Anti-optimization technique for structural design under loaduncertainties[J]. Computer methods in applied mechanics and engineering,1998,157(1):19-31.
[148]Ganzerli S, Pantelides C P. Load and resistance convex models for optimum design[J].Structural optimization,1999,17(4):259-268.
[149]姜潮,刘丽新,龙湘云.一种概率-区间混合结构可靠性的高效计算方法[J].计算力学学报,2013,30(5):605-609.
[150]吕春梅,张义民,李鹤,等.动态结构系统的频率可靠性稳健设计研究[J].计算力学学报,2013,30(5):653-656.
[151]方鹏亚,常新龙,胡宽,等.基于区间不确定性的涡轮盘强度可靠性优化设计[J].推进技术,2013,34(007):962-967.
[152]朱增青.区间和未确知参数结构(机构)分析方法研究及应用[D].西安电子科技大学,2009.
[153]Shou H, Martin R. Voiculescu I. Bowyer A, Wang G. Affine arithmetic in matrix form forpolynomial evaluation and algebraic curve drawing[J]. Progress in Natural Science,2002,12(1):77–80.
[154]Stol J, De Figueiredo L H. Self-validated numerical methods and applications[C]//Monographfor21st Brazilian Mathematics Colloquium, IMPA, Rio de Janeiro.1997.
[155]Christoph Grimm, Wilhelm Heupke, Klaus Waldschmidt. Analysis of Mixed-Signal Systemswith Affine Arithmetic[J]. IEEE Transactions on Computer-Aided Design of Integrated Circuitsand Systems,2005,24(1):118-123.
[156]M. Vitelli. Range Analysis in Electroquasistatic Field Linear Problems[J], IEEE Transactionson Dielectrics and Electrical Insulation,2003,10(1):155-167.
[157]Siome Klein Goldenstein, Christian Vogler, Dimitris Metaxas. Statistical Cue Integration inDAG Deformable Models[J]. IEEE Transactions on Pattern Analysis and Machine Intelligence,2003,25(7):801-813.
[158]Nicola Femia, Giovanni Spagnuolo. True Worst-Case Circuit Tolerance Analysis Using GeneticAlgorithms and Affine Arithmetic[J]. IEEE Transactions on Circuits and Systems,2000,47(9):1285-1296.
[159]Luiz Henrique de Figueiredo, Jorge Stolfi. Affine arithmetic: concepts and applications[J].Numerical Algorithms,2004,37:147–158.
[160]Ralph Martin, Huahao Shou, Irina Voiculescu, Adrian Bowyer, Guojin Wang. Comparison ofinterval methods for plotting algebraic curves[J]. Computer Aided Geometric Design,2002,19:553–587.
[161]Shou H, Lin H, Martin R, et al. Modified affine arithmetic is more accurate than centeredinterval arithmetic or affine arithmetic[M]//Mathematics of Surfaces. Springer BerlinHeidelberg,2003:355-365.
[162]邱志平，顾笑冬，李登峰.单自由度不确定滞回系统振动响应的区间分析方法[J]，动力学与控制学报,2007,(5):173-177.
[163]陈塑寰，裴春艳.不确定二阶振动控制系统动力响应的区间方法[J]，吉林大学学报(工学版),2008,38(1):94-98.
[164]邱志平，马丽红，王晓军.不确定非线性结构动力响应的区间分析方法[J],力学学报,2006,38(5):645-651.
[165]林立广,陈建军,马娟.基于区间因子法的不确定性桁架结构动力响应分析[J],应用力学学报,2008,25(4):612-617.
[166]郑松,侯迪波,周泽魁.动态调整选择策略的改进蚁群算法[J].控制与决策,2008,23(2):225-228.
[167]杨雪榕,梁加红,陈凌,等.多邻域改进粒子群算法[J].系统工程与电子技术,2010,32(11):2453-2458.
[168]赵文红,王宇平,王巍.快速寻优的全局优化进化算法[J].计算机工程,2008,34(8):208-212.
[169]吴慧卓,张可村.基于拉格朗日对偶的一类全局优化算法[J].西安交通大学学报,2008,42(8):1031-1034.
[170]Hong Zhou. Solving Fractional Problems Management Based on A Deterministic Algorithm[C].2009IITA International Conference on Control, Automation and Systems Engineering,2009:163-166.
[171]Lin Youdong, Stadtherr, Mark A. Deterministic Global Optimization for Dynamic SystemsUsing Interval Analysis[C].200612th GAMM-IMACS International Symposium on ScientificComputing, Computer Arithmetic and Validated Numerics,2007:323-31.
[172]Xiaowei Zhang and Sanyang Liu. Interval algorithm for global numerical optimization[J].Engineering Optimization,2008,40(9):849-868.
[173]M. sun, A. W. Johnson. Interval branch and bound with local sampling for constrained globaloptimization[J]. Journal of Global Optimization,2005,33:62-82.
[174]Luiz Henrique de Figueiredo, Jorge Stolfi. Affine arithmetic: concepts and applications[J].Numerical Algorithms,2004,37:147-158.
[175]Hansen E. Global optimization using interval analysis-the multidimensional case[J].Numerische Mathematik,1980,34(1):247-270.
[176]Berner S. New results on verified global optimization[J]. Computing,1996,57(2):323-343.
[177]裴春艳,陈塑寰,基于区间方法的不确定二阶控制系统稳定性的鲁棒性分析[J],吉林大学学报(工学版),2006,36(增刊):20-25.
[178]申涛,王孝红,袁铸钢,一类不确定系统的鲁棒稳定性分析[J],自动化学报,2007,33(4):426-427.
[179]井元伟,姜囡,郝彬彬,一类不确定系统的最优极小极大鲁棒控制[J],控制与决策,2008,23(2):208-212.
[180]Bertoni F C, da Silva I N, Pires M G. A Neurogenetic Approach and its Application toConstrained Nonlinear Convex Optimization Problems with Joint and Dis joint FeasibleRegions[C]//Hybrid Intelligent Systems,2008. HIS'08. Eighth International Conference on.IEEE,2008:90-95.
[181] Wei Bian, Xiaoping Xue.Subgradient-Based Neural Networks for Nonsmooth NonconvexOptimization Problems[J]. IEEE Transactions on Neural Networks.2009,20(6):1024-1038.
[182] Hsc c-c,Chang, S.-C. Yu, C. Y. Tolerance design of robust controllers for uncertain intervalsystems based on evolutionary algorithms[J]. Control Theory&Applications.2006,1(1):244-252.
[183] Didier Henrion, Jean-Bernard Lasserre. Inner Approximations for Polynomial MatrixInequalities and Robust Stability Regions[J]. IEEE Transactions on Automatic Control.2012,57(6):1456-1467.
[184] M. Soliman. H. Emara, A.Elshafei, A. Bahgat Robust Output Feedback Power SystemStabilizer Design: an LMI approach[C]. Power and Energy Society General Meeting-Conversion and Delivery of Electrical Energy in the21st Century,2008IEEE,pages:1-8.
[185] Spagnuolo, G. An interval arithmetic-based yield evaluation in circuit tolerance design[C].ISCAS2002. IEEE International Symposium on Circuits and system,2002:743-746.
[186] Jorge E. Hurtado, Diego A. Alvarez. The encounter of interval and probabilistic approaches tostructural reliability at the design point[J]. Computer Methods in Applied Mechanics andEngineering.2012,225-228(15):74-94.
[187] Comba J L D, Stolfi J. Affine arithmetic and its applications to computer graphics[C]. In:Proceedings of Anais do V II S B GRAPI, Recife, Brazil,1993:9-18.
[188]朱增青,陈建军,宋宗凤等.区间参数杆系结构非概率可靠性指标的改进仿射算法[J],工程力学,2010,27(2):49-58.
[189]谢永强,陈建军,徐亚兰.基于仿射算法的确定性全局优化算法[J],华南理工大学学报(自然科学版),2012,40(5):35-40.
[190] D. Degrauwe, G. Lombaert, G. De Roeck, Improving interval analysis in finite elementcalculations by means of affine arithmetic[J]. Computers and Structures,2010,88(3-4):247–254.
[191] Peter W. M. Tsang, Terry Y. F YUEN, W.C. Situ. Enhanced affine invariant matching ofbroken boundaries based on particle swarm optimization and the dynamic migrant principle[J].Applied Soft Computing Journal,2010,10(2):432-438.
[192]孙继涛,张银萍.关于时变区间矩阵的稳定性研究[J].控制理论与应用,1995,12(1):108-113.
[193]Degrauwe D, Lombaert G, De Roeck G. Improving interval analysis in finite elementcalculations by means of affine arithmetic[J]. Computers&structures,2010,88(3):247-254.
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