报告题目：非线性随机系统的平稳响应近似闭合解
报告人：陈林聪 教授 华侨大学土木工程学院
照片：
邀请人：李伟
报告时间：2021年12月3日上午9：30-10:30
报告地点：腾讯会议:393 137 383
报告人简介：陈林聪，教授，博士生导师。福建省杰青获得者，入围2018年国家优青会评；2004年7月哈尔滨工程大学工程力学专业；2009年6月研究生毕业于浙江大学航空航天学院工程力学专业，获工学博士学位，导师朱位秋院士。2009年8月起入职华侨大学土木工程学院。期间于2014年8月至2015年8月与2018年12月至2020年1月，美国加州大学Merced分校访问学者，合作导师为ASME会士孙建桥教授。陈林聪教授主要从事工程结构随机振动、结构振动与控制、先进纤维材料结构等方面研究。先后主持多项目国家自然科学基金、福建省自然科学基金项目及教育部博士点创新基金。在国内主流期刊上发表SCI收录论文80余篇。
报告摘要：非线性随机动力学已引起工程界和科学界的广泛关注。非线性随机系统响应的统计特性完全由FPK方程所支配的概率密度函数来描述。然而，关于FPK方程的求解，仍然是一个巨大挑战。本报告介绍稳态FPK方程求解的新方法——一种修正加权残值法。区别于传统的加权残值法，修正加权残值法在试函数的构造技巧和待定系数的确定方法都富有创新性。针对试函数的构造方面，以详细平衡法为基础，提出以系统的概率势流与概率环流来构造稳态FPK方程的试函数；在确定待定系数的方面，引入迭代技术至传统的伽辽金加权残值法中，逐步优化权函数。诸多经典算例研究表明：引入迭代的伽辽金加权残值法，同时适用光滑系统和非光滑系统，计算结果具有较高的精度，甚至可收敛于精确解（精确解存在），是寻求非线性随机系统精确平稳解的潜在一种新方法。然而，引入迭代的伽辽金加权残值法，由于无法克服高维数值积分带来的庞大计算量，很难推广到多自由度高维系统。为此，提出了以数据科学理论为基础的最小二乘法来确定试函数中的待定系数。通过若干个经典算例研究表明，基于数据科学理论的最小二乘法具有较高的计算效率，计算结果也具有较高的精度，当系统存在精确解时，甚至可以收敛于精确解。
Abstract: Nonlinear stochastic dynamics has attracted extensive attention in engineering and science. The statistical characteristics of the response of nonlinear stochastic systems are completely described by the probability density function that dominated by the FPK equation. However, it is still a great challenge to solve the FPK equation. In this report, a new procedure of a modified weighted residual method for solving the reduced FPK equation is proposed. Different from the classical weighted residual method, the modified weighted residual method is innovative in the construction of trial function and the determination of undetermined coefficients. Aiming at the construction of trial function, the trial function is constructed with the probability potential flow and probability circulation of the system. In determining the undetermined coefficients, the iterative technique is introduced into the classical Galerkin weighted residual method to optimize the weighting function. Many examples have shown that the iterative Galerkin weighted residual method is suitable for both smooth and non-smooth systems and the computational results have high accuracy, even convergent to exact solutions ( if exact solutions exist). The iterative Galerkin weighted residual method can be a potential new method for seeking exact stationary solutions of nonlinear stochastic systems. On the other hand, the iterative Galerkin weighted residual method is still difficult to be extended to MDOF high-dimensional systems as it cannot overcome the huge amount of computation caused by high-dimensional numerical integration. Therefore, the least square method based on the theory of data science is proposed to determine the undetermined coefficients in the trial function. Some classical examples have shown that the least square method based on data science theory has high computational efficiency and high precision, and even converges to the exact solution when the system exists.
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