基于卷积神经网络的无网格形函数影响域优化研究  

A CNN-based Approach for Optimizing Support Selection of Meshfree Methods

作　　者：刘宇翔 王东东[1,2] 樊礼恒 陈健 侯松阳 Yuxiang Liu;Dongdong Wang;Liheng Fan;Jian Chen;Songyang Hou(Department of Civil Engineering,Xiamen University,Xiamen 361005;Xiamen Engineering Technology Center for Intelligent Maintenance of Infrastructures,Xiamen 361005)

机构地区：[1]厦门大学土木工程系,厦门361005 [2]厦门市交通基础设施智能管养工程技术研究中心,厦门361005

出　　处：《固体力学学报》2021年第3期302-319,共18页Chinese Journal of Solid Mechanics

基　　金：国家自然科学基金项目(11772280,12072302)资助。

摘　　要：在无网格法中,离散节点之间的相互联系由节点形函数影响域的大小确定,因此形函数影响域的大小对无网格法的计算精度有着直接和重要的影响.但由于无网格形函数的形式较为复杂,目前形函数影响域大小的选择仍然缺乏系统的理论依据,通常在实际计算中仍凭借经验进行选取,难以保证计算精度.卷积神经网络是一类机器学习方法,其感受野与无网格形函数的影响域具有内在相似性,因此在形函数影响域选择方面有很好的适用性.基于该特性,论文通过引入卷积神经网络对无网格形函数的影响域进行了优化选择.首先,针对感受野和影响域的匹配关系,分析了卷积神经网络的结构设计和超参数选择,提出了一种无网格法内禀卷积神经网络结构的设计方法;然后依托该网络结构设计方法,建立了对无网格形函数影响域和数值解分别优化或同时优化的卷积神经网络.文中通过算例系统验证了所提无网格法内禀卷积神经网络对形函数影响域选择和计算结果的优化效应.In meshfree methods,the discretized particles or nodes are inter-related by the supports of meshfree shape functions,and thus the support sizes of meshfree shape functions have a significant impact on the meshfree solution accuracy.However,due to the complexity of meshfree shape functions,for instance,the reproducing kernel meshfree shape functions,currently there are still no general guidelines for meshfree support selection,and in practice the meshfree support sizes are often picked up by trial and error and a uniform support size is set for all particles which cannot always ensure the meshfree solution accuracy.Therefore,it is highly desirable if the meshfree support size can be rationally adjusted to improve the accuracy of meshfree analysis.In this work,a machine learning approach,namely,the convolutional neural network(CNN),is introduced to optimize the meshfree support selection.It is noted that in the CNN,each neuron processes data that are located within its receptive field.Thus it turns out that there is a strong similarity and linkage between the CNN receptive fields and meshfree supports,and accordingly a meshfree intrinsic CNN framework is proposed for the Galerkin meshfree analysis.Based upon this meshfree intrinsic CNN framework,the network architecture and the corresponding determination of hyperparameters,such as the convolution kernel size,receptive field size,and their relationships with the number of meshfree nodes,the monomial basis function order and the support size,are particularly detailed for the meshfree methods.Subsequently,in the context of the proposed meshfree intrinsic CNN framework,several CNN architectures are designed to facilitate the meshfree support selection and solution prediction,in which a single objective like meshfree support selection,or multi-objectives like both meshfree support selection and solution prediction can be efficiently realized.Numerical results systematically demonstrate that the proposed meshfree intrinsic CNN framework is very effective to optimize the

关 键 词：无网格法 卷积神经网络 形函数 影响域 感受野 计算精度 

分 类 号：TP183[自动化与计算机技术—控制理论与控制工程]