基于径向基函数的结构可靠性分析算法研究
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摘要
在基于代理模型的可靠性分析中,计算效率是相关研究所关注的一项重要课题。为此,本文引入径向基函数(RBF)构造代理模型,围绕基于代理模型的可靠性分析开展了系列研究,力求在算法的效率和精度等方面做出一些有益的探索和改进。基于此思路,本论文开展和完成了以下四方面的研究内容：
(1)由于一次二阶矩法(FORM)具有方便高效的优点,其在结构可靠性分析中得到了广泛的应用。然而,在面对复杂结构,特别是高维耗时的计算问题时,传统的FORM方法便不再适用。为克服这一缺陷,本论文提出了一种基于响应面(RS)的FORM方法。该方法结合拉丁超立方采样(LHS)策略,采用RBF来近似描述隐式极限状态函数,在此基础上,运用HL-RF算法来求解可靠性指标,进而得到相应的结构失效概率。
(2)功能度量法(PMA)在可靠性分析和基于可靠性的优化设计(RBDO)中应用广泛,与可靠性指标法(RIA)相比,其具有较高的计算效率和稳健的鲁棒性等优点。但在分析过程中,PMA重复估算概率约束,这对大规模计算问题是难以接受的。为此,本论文引入RBF,提出了一种基于PMA的高效可靠性分析技术。该方法采取RBF与LHS结合的策略来近似描述极限状态方程,进而运用改进的中心点法(AMV)得到在预定目标可靠度和相应功能度量下的最可能失效点。在多次确认采样中心并重构RBF,直到满足一定的收敛准则后,才能得到最终有效的MPP点。
(3)在复杂结构的可靠性分析中,蒙特卡罗方法(MCM)和有限元分析技术(FEM)将带来巨大的计算量,基于RBF的RS方法是解决此类问题的一条有效途径。在计算过程中,RBF的参数和基函数将对最终可靠性分析的结果产生较大影响,然而,当今并没有相应的规范来指导如何选取合适的参数和基函数。本论文基于RIA可靠性模型,对由RBF结合LHS策略描述的极限状态方程,采用HL-RF算法求解得到可靠性指标和相应的失效概率,着重对RBF所涉及的参数和各常见基函数,如Gaussian基函数、Multi-Quadric基函数、Inverse Multi-Quadric基函数、Thin Plate Spline基函数、Cubic基函数和Linear基函数,进行了比较研究。
(4)在结构可靠性分析中,响应面法因能利用少量具体点的函数值来近似用多项式表述极限状态方程而应用广泛。由于用解析表达的多项式函数代替了精确的极限状态方程,通常计算所耗时间会得到显著的下降,这是响应面法的主要优点。然而,一些学者的研究表明采样点位置的选取和响应面的性能仍是一个值得关注研究方向。基于此认识,本论文提出一种改进的响应面方法对结构的可靠性及其对参数的敏感性进行了分析。该方法采用无交叉项的一阶多项式近似描述极限状态方程,进而得到极限状态方程的敏感性向量,在此基础上,提出了一个包含4n+1个采样点的实验设计方案,其中2n+1个采样点位于标准正态空间(U-space)坐标系的轴线上,另外2n个采样点根据前述极限状态方程的敏感性向量进行旋转。此外,另一个二次多项式也被用于近似描述极限状态方程,运行HL-RF算法,即可得到相应的MPP点。
An important problem in metamodel-based structural reliability analysis is how to reduce the computation time. The objective of this dissertation is to develop the efficient and accurate reliability analysis techniques to support metamodel-based reliability analysis. Therefore, there are basically four tasks to be carried out:
First, the first-order reliability method (FORM) is one of the most widely used structural reliability analysis techniques due to its simplicity and efficiency. However, direct using FORM seems disability to work well for complex problems, especially related to high-dimensional variables and computation intensive numerical models. To expand the applicability of the FORM for more practical engineering problems, a response surface approach based FORM is proposed for structural reliability analysis. The radial basis function (RBF) is employed to approximate the implicit limit-state functions combined with Latin Hypercube Sampling strategy. To guarantee the numerical stability, the improved HL-RF (zHL-RF) algorithm is used to assess the reliability index and corresponding probability of failure based on the constructed response surfac model.
Second, the performance measure approach (PMA) is widely adopted for reliability analysis and reliability-based design optimization because of its robustness and efficiency compared to reliability analysis approach. However, it has been reported that PMA involves repeat evaluations of probabilistic constraints therefore it is prohibitively expensive for many large-scale applications. In order to overcome these disadvantages, this study proposes an efficient PMA-based reliability analysis technique using radial basis function. The RBF is adopted to approximate the implicit limit state functions in combination with Latin Hypercube Sampling strategy. The advanced mean value method is applied to obtain the most probable point with the prescribed target reliability and corresponding probabilistic performance measure to improve analysis accuracy. A sequential framework is proposed to relocate the sampling center to the obtained most probable point and reconstruct RBF until a criteria is satisfied.
Third, the Monte Carlo method and the finite element method for the structural reliability analysis lead often to a prohibitive computational cost. In the reliability estimation of complex structures, a response surface based on RBF has been suggested as a way to estimate the implicit limit state function. However, the parameters and basis functions of the RBF effects to the structural reliability analysis results but, there is no guidance how to select appropriate values for the parameters and basis functions. Therefore, this study researches effect of parameters and basis functions on RIA-based structural reliability estimates using the radial basis functions such as Gaussian, Multi-Quadric, Inverse Multi-Quadric, Thin Plate Spline, Cubic and Linear. The RBFs is adopted to approximate the limit state functions in combination with Latin Hypercube Sampling strategy. The HL-RF algorithm is applied to obtain the reliability index and probability of failure based on the constructed response surface model.
Fourth, the response surface method is a powerful structural reliability method using the values of the function at specific points that approximates the limit state function with a polynomial expression. The analytical function replaces the exact limit state function which the computational time required for the assessment of the reliability of structural systems can be reduced significantly. However, the location of the sample points has been investigated by several authors and the performance of the response surface method is still under discussion. Therefore, this study proposes a new response surface method for sensitivity estimation of parameters in structural reliability analysis. A first order polynomial without cross terms is adopted to approximate the limit-state function, and the sensitivity vector of the limit state function can be obtained. An experimental design with4n+1sampling points includes2n+l sampling points are chosen along the coordinate axes of the U-space of standard normal random variables and2n sampling points is rotated according to the sensitivity vector of the limit state function is built. A quadratic polynomial is adopted to approximate the limit-state function, and the most probable point can be obtained by conducting HL-RF algorithm based on the created response surface.
(1)由于一次二阶矩法(FORM)具有方便高效的优点,其在结构可靠性分析中得到了广泛的应用。然而,在面对复杂结构,特别是高维耗时的计算问题时,传统的FORM方法便不再适用。为克服这一缺陷,本论文提出了一种基于响应面(RS)的FORM方法。该方法结合拉丁超立方采样(LHS)策略,采用RBF来近似描述隐式极限状态函数,在此基础上,运用HL-RF算法来求解可靠性指标,进而得到相应的结构失效概率。
(2)功能度量法(PMA)在可靠性分析和基于可靠性的优化设计(RBDO)中应用广泛,与可靠性指标法(RIA)相比,其具有较高的计算效率和稳健的鲁棒性等优点。但在分析过程中,PMA重复估算概率约束,这对大规模计算问题是难以接受的。为此,本论文引入RBF,提出了一种基于PMA的高效可靠性分析技术。该方法采取RBF与LHS结合的策略来近似描述极限状态方程,进而运用改进的中心点法(AMV)得到在预定目标可靠度和相应功能度量下的最可能失效点。在多次确认采样中心并重构RBF,直到满足一定的收敛准则后,才能得到最终有效的MPP点。
(3)在复杂结构的可靠性分析中,蒙特卡罗方法(MCM)和有限元分析技术(FEM)将带来巨大的计算量,基于RBF的RS方法是解决此类问题的一条有效途径。在计算过程中,RBF的参数和基函数将对最终可靠性分析的结果产生较大影响,然而,当今并没有相应的规范来指导如何选取合适的参数和基函数。本论文基于RIA可靠性模型,对由RBF结合LHS策略描述的极限状态方程,采用HL-RF算法求解得到可靠性指标和相应的失效概率,着重对RBF所涉及的参数和各常见基函数,如Gaussian基函数、Multi-Quadric基函数、Inverse Multi-Quadric基函数、Thin Plate Spline基函数、Cubic基函数和Linear基函数,进行了比较研究。
(4)在结构可靠性分析中,响应面法因能利用少量具体点的函数值来近似用多项式表述极限状态方程而应用广泛。由于用解析表达的多项式函数代替了精确的极限状态方程,通常计算所耗时间会得到显著的下降,这是响应面法的主要优点。然而,一些学者的研究表明采样点位置的选取和响应面的性能仍是一个值得关注研究方向。基于此认识,本论文提出一种改进的响应面方法对结构的可靠性及其对参数的敏感性进行了分析。该方法采用无交叉项的一阶多项式近似描述极限状态方程,进而得到极限状态方程的敏感性向量,在此基础上,提出了一个包含4n+1个采样点的实验设计方案,其中2n+1个采样点位于标准正态空间(U-space)坐标系的轴线上,另外2n个采样点根据前述极限状态方程的敏感性向量进行旋转。此外,另一个二次多项式也被用于近似描述极限状态方程,运行HL-RF算法,即可得到相应的MPP点。
An important problem in metamodel-based structural reliability analysis is how to reduce the computation time. The objective of this dissertation is to develop the efficient and accurate reliability analysis techniques to support metamodel-based reliability analysis. Therefore, there are basically four tasks to be carried out:
First, the first-order reliability method (FORM) is one of the most widely used structural reliability analysis techniques due to its simplicity and efficiency. However, direct using FORM seems disability to work well for complex problems, especially related to high-dimensional variables and computation intensive numerical models. To expand the applicability of the FORM for more practical engineering problems, a response surface approach based FORM is proposed for structural reliability analysis. The radial basis function (RBF) is employed to approximate the implicit limit-state functions combined with Latin Hypercube Sampling strategy. To guarantee the numerical stability, the improved HL-RF (zHL-RF) algorithm is used to assess the reliability index and corresponding probability of failure based on the constructed response surfac model.
Second, the performance measure approach (PMA) is widely adopted for reliability analysis and reliability-based design optimization because of its robustness and efficiency compared to reliability analysis approach. However, it has been reported that PMA involves repeat evaluations of probabilistic constraints therefore it is prohibitively expensive for many large-scale applications. In order to overcome these disadvantages, this study proposes an efficient PMA-based reliability analysis technique using radial basis function. The RBF is adopted to approximate the implicit limit state functions in combination with Latin Hypercube Sampling strategy. The advanced mean value method is applied to obtain the most probable point with the prescribed target reliability and corresponding probabilistic performance measure to improve analysis accuracy. A sequential framework is proposed to relocate the sampling center to the obtained most probable point and reconstruct RBF until a criteria is satisfied.
Third, the Monte Carlo method and the finite element method for the structural reliability analysis lead often to a prohibitive computational cost. In the reliability estimation of complex structures, a response surface based on RBF has been suggested as a way to estimate the implicit limit state function. However, the parameters and basis functions of the RBF effects to the structural reliability analysis results but, there is no guidance how to select appropriate values for the parameters and basis functions. Therefore, this study researches effect of parameters and basis functions on RIA-based structural reliability estimates using the radial basis functions such as Gaussian, Multi-Quadric, Inverse Multi-Quadric, Thin Plate Spline, Cubic and Linear. The RBFs is adopted to approximate the limit state functions in combination with Latin Hypercube Sampling strategy. The HL-RF algorithm is applied to obtain the reliability index and probability of failure based on the constructed response surface model.
Fourth, the response surface method is a powerful structural reliability method using the values of the function at specific points that approximates the limit state function with a polynomial expression. The analytical function replaces the exact limit state function which the computational time required for the assessment of the reliability of structural systems can be reduced significantly. However, the location of the sample points has been investigated by several authors and the performance of the response surface method is still under discussion. Therefore, this study proposes a new response surface method for sensitivity estimation of parameters in structural reliability analysis. A first order polynomial without cross terms is adopted to approximate the limit-state function, and the sensitivity vector of the limit state function can be obtained. An experimental design with4n+1sampling points includes2n+l sampling points are chosen along the coordinate axes of the U-space of standard normal random variables and2n sampling points is rotated according to the sensitivity vector of the limit state function is built. A quadratic polynomial is adopted to approximate the limit-state function, and the most probable point can be obtained by conducting HL-RF algorithm based on the created response surface.
引文
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