再生光滑梯度无网格法动力特性研究  

Characteristics of dynamic meshfree analysis with reproducing kernel smoothed gradients

作　　者：付赛赛 邓立克 吴俊超 王东东[1] 张灿辉[1] FU Saisai;DENG Like;WU Junchao;WANG Dongdong;ZHANG Canhui(Fujian Key Laboratory of Digital Simulations for Coastal Civil Engineering,Department of Civil Engineering,Xiamen University,361005 Xiamen,China;Key Laboratory for Intelligent Infrastructure and Monitoring of Fujian Province,College of Civil Engineering,Huaqiao University,361021 Xiamen,China)

机构地区：[1]厦门大学土木工程系,福建省滨海土木工程数字仿真重点实验室,厦门361005 [2]华侨大学土木工程学院,福建省智慧基础设施与监测重点实验室,厦门361021

出　　处：《应用力学学报》2022年第6期1065-1075,共11页Chinese Journal of Applied Mechanics

基　　金：国家自然科学基金资助项目(No.12072302,12102138);福建省自然科学基金资助项目(No.2021J02003)。

摘　　要：无网格形函数的非多项式特性导致梯度计算复杂耗时,同时高斯积分方法不满足积分约束条件,因此传统伽辽金无网格法难以达到理论收敛率。再生光滑梯度的构造特点使其自然满足积分约束条件,并有效地避免计算无网格形函数的直接梯度,因而具有高效和精确的特点。为了探究再生光滑梯度无网格法的动力特性,本研究构造了基于再生光滑梯度理论的伽辽金无网格法的动力分析方法,详细研究了再生光滑梯度无网格法的动力计算精度,包括频散特性、自由振动和时程动力分析。再生光滑梯度无网格法采用再生光滑梯度替代传统的无网格形函数梯度,由于其本身与积分约束条件的内在一致性,直接采用基函数对应阶次的低阶高斯积分方法对质量和刚度矩阵进行数值积分,即可保证最优收敛率和精度。理论分析与数值计算结果均表明,再生光滑梯度无网格法的频散特性、频率收敛率和时程动力计算精度,都明显优于采用高阶高斯积分方法的传统无网格法。The gradient valuation of non-polynomial type meshfree shape functions is usually very complex and costly,which leads to an undesirable fact,i.e.,even very high order Gauss quadrature rules cannot ensure an optimal convergence of Galerkin meshfree formulation.The reproducing kernel smoothed gradients of meshfree shape functions naturally meet the Galerkin integration constraint,which completely avoid the time consuming computation of meshfree shape function gradients,and thus is highly efficient and accurate.In order to assess the dynamic performance of reproducing kernel smoothed gradient meshfree algorithm,this work presents a dynamic meshfree analysis with reproducing kernel smoothed gradients and the convergence and accuracy are particularly investigated in detail,including the dispersion analysis,free vibration analysis and transient analysis.Since the present algorithm inherits the integration consistency of Galerkin formalism,the relatively low order Gauss quadrature rules corresponding to the basis order in meshfree approximation are capable of producing the optimal convergence with superior accuracy.Both theoretical and computational results consistently demonstrate that,regarding the dispersion analysis,frequency convergence as well as transient response,the reproducing kernel smoothed gradient Galerkin meshfree formulation gains much more favorable solution accuracy for dynamic meshfree analysis,compared with the conventional meshfree scheme using high order Gauss integration.

关 键 词：无网格法 动力分析 再生光滑梯度 收敛性 精度 

分 类 号：O242.2[理学—计算数学]