高斯型多维积分公式公开课一等奖课件省赛课获奖课件

高斯型多维积分公式报告人：肖青导师：周少武第1页HunanUniversityofScienceandTechnology内容4、总结3、多变量函数1、研究背景2、单变量函数CollegeofMechanicalandElectricalEngineering2.1数值积分

研3.1张量积研3.2稀疏网格法2.2多项式混沌展开研3.3容积量法3.4算例第2页HunanUniversityofScienceandTechnology1.研究背景

CollegeofMechanicalandElectricalEngineering输入X非线性传递系统输出y(1)第3页HunanUniversityofScienceandTechnology1.研究背景

CollegeofMechanicalandElectricalEngineering(2)第4页HunanUniversityofScienceandTechnology1.研究背景

CollegeofMechanicalandElectricalEngineering(3)第5页HunanUniversityofScienceandTechnology2.单变量函数

CollegeofMechanicalandElectricalEngineering(4)(5)第6页HunanUniversityofScienceandTechnology2.单变量函数

CollegeofMechanicalandElectricalEngineering(6)(7)(8)第7页HunanUniversityofScienceandTechnology2.1数值积分

CollegeofMechanicalandElectricalEngineering(9)(10)第8页HunanUniversityofScienceandTechnology2.1数值积分CollegeofMechanicalandElectricalEngineering图1：基于泰勒展开式对sin(x)逼近第9页HunanUniversityofScienceandTechnology2.1数值积分CollegeofMechanicalandElectricalEngineering图2：基于Hermite多项式对sin(x)逼近第10页HunanUniversityofScienceandTechnology2.1数值积分

CollegeofMechanicalandElectricalEngineering第11页HunanUniversityofScienceandTechnology2.1数值积分

CollegeofMechanicalandElectricalEngineering第12页HunanUniversityofScienceandTechnology2.1数值积分CollegeofMechanicalandElectricalEngineering(11)第13页HunanUniversityofScienceandTechnology2.1数值积分

CollegeofMechanicalandElectricalEngineering第14页HunanUniversityofScienceandTechnology2.1数值积分

CollegeofMechanicalandElectricalEngineering(12)第15页HunanUniversityofScienceandTechnology2.1数值积分CollegeofMechanicalandElectricalEngineering(13)

第16页HunanUniversityofScienceandTechnology2.1数值积分

CollegeofMechanicalandElectricalEngineering第17页HunanUniversityofScienceandTechnology2.2多项式混沌展开

CollegeofMechanicalandElectricalEngineering(14)(15)第18页HunanUniversityofScienceandTechnology2.2多项式混沌展开CollegeofMechanicalandElectricalEngineering第19页HunanUniversityofScienceandTechnology2.2多项式混沌展开CollegeofMechanicalandElectricalEngineering第20页HunanUniversityofScienceandTechnology2.2多项式混沌展开CollegeofMechanicalandElectricalEngineering第21页HunanUniversityofScienceandTechnology2.2多项式混沌展开

CollegeofMechanicalandElectricalEngineering第22页HunanUniversityofScienceandTechnology2.2多项式混沌展开

CollegeofMechanicalandElectricalEngineering第23页HunanUniversityofScienceandTechnology2.2多项式混沌展开

CollegeofMechanicalandElectricalEngineering(16)第24页HunanUniversityofScienceandTechnology2.2多项式混沌展开

CollegeofMechanicalandElectricalEngineering(17)(18)(19)第25页HunanUniversityofScienceandTechnology2.2多项式混沌展开基于式（16）计算数学盼望，有：

CollegeofMechanicalandElectricalEngineering(20)第26页HunanUniversityofScienceandTechnology2.2多项式混沌展开

CollegeofMechanicalandElectricalEngineering(21)第27页HunanUniversityofScienceandTechnology2.2多项式混沌展开若计算二阶原点矩，数值积分算法为：对比可知：在求取二阶原点矩时，数值积分更方便。另外，延用这种思想能够方便求取更高阶原点矩；若采取多项式混沌展开法，用系数表达输出量高阶矩，计算很繁琐。或许说，在计算高阶矩时，数值积分对信息处理更有效率。CollegeofMechanicalandElectricalEngineering(22)第28页HunanUniversityofScienceandTechnology3.

多变量函数

CollegeofMechanicalandElectricalEngineering(23)(24)(25)(26)第29页HunanUniversityofScienceandTechnology3.1张量积

CollegeofMechanicalandElectricalEngineering(27)(28)第30页HunanUniversityofScienceandTechnology3.1张量积CollegeofMechanicalandElectricalEngineering表1：Gauss-Hermite积分节点和权重n权重节点123第31页HunanUniversityofScienceandTechnology3.1张量积

CollegeofMechanicalandElectricalEngineering第32页HunanUniversityofScienceandTechnology3.1张量积

CollegeofMechanicalandElectricalEngineering第33页HunanUniversityofScienceandTechnology3.2稀疏网格法

CollegeofMechanicalandElectricalEngineering(29)(30)第34页HunanUniversityofScienceandTechnology3.2稀疏网格法CollegeofMechanicalandElectricalEngineering表2：稀疏网格法计算量计算量优点：代数精度较高、计算量较低。缺点：仍不适用于较多变量。第35页HunanUniversityofScienceandTechnology3.3容积量法

CollegeofMechanicalandElectricalEngineering(31)第36页HunanUniversityofScienceandTechnology3.3容积量法为满足所有方程，有：计算量：优点：若要求代数精度较低时，计算量较低。缺点：不适用于高阶精度，且目前尚无较好配

点办法。CollegeofMechanicalandElectricalEngineering(32)第37页HunanUniversityofScienceandTechnology3.4算例

CollegeofMechanicalandElectricalEngineering(33)方法计算量结果张量积（3节点）10.00稀疏网格法（L=2）10.00稀疏网格法（L=3）10.00表3：不一样积分法计算成果第38页HunanUniversityofScienceandTechnology3.4算例

CollegeofMechanicalandElectricalEngineering(34)方法计算量结果张量积（3节点）9.00稀疏网格法（L=2）0.00稀疏网格法（L=3）9.00表4：不一样积分法计算成果第39页HunanUniversityofScienceandTechnology3.4算例

CollegeofMechanicalandElectricalEngineering(35)表5：不一样积分法计算成果方法计算量结果张量积（3节点）8.00稀疏网格法（L=2）0.00稀疏网格法（L=3）0.00稀疏网格法（L=4）8.00第40页HunanUniversityofScienceandTechnology3.4算例

CollegeofMechanicalandElectricalEngineering(36)表6：不一样积分法计算成果方法计算量结果张量积（3节点）139.20稀疏网格法（L=2）6.43稀疏网格法（L=3）20.65稀疏网格法（L=4）44