| 齐栋梁,刘建秀.超收敛等几何无网格配点法[J].计算力学学报,2025,42(3):485~492 |
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| 超收敛等几何无网格配点法 |
| A superconvergent isogeometric meshfree collocation method |
| 投稿时间：2023-10-25 修订日期：2024-03-12 |
| DOI：10.7511/jslx20231025001 |
| 中文关键词: 无网格法 等几何配点法 再生梯度 递推梯度 超收敛 |
| 英文关键词:meshfree method isogeometric collocation analysis reproduced gradient recursive gradient superconvergence |
| 基金项目:河北省自然科学基金(A2022412001);河北水利电力学院基本科研业务费研究项目(SYKY2201);河北水利电力学院博士启动资金(SYBJ2201)资助项目. |
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| 摘要点击次数: 7 |
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| 中文摘要: |
| 无网格法和等几何分析采用的形函数或基函数均具有高阶光滑特性,但应用于配点法分析时表现出奇数次基函数精度掉阶问题。本文以等几何基函数的一致性条件和无网格再生梯度理论为基础,提出了一种超收敛等几何无网格配点法。首先基于等几何基函数的无网格表示理论,构建了由等几何基函数再生点定义的混合梯度基向量,发展了一种等几何基函数梯度的无网格衍化形式,数值实现非常简捷。然后,将无网格法中形函数变换技术和递推梯度算法引入到配点法分析中,构建了变换等几何无网格形函数的二阶递推梯度。该梯度构造形式与传统等几何基函数梯度相比,满足额外高一阶再生条件,进而为实现超收敛配点法分析提供了保障。最后,文中通过一系列数值算例系统验证了超收敛等几何无网格配点法的精度和收敛性。数值结果表明,所提方法比传统等几何配点法具有更高计算精度,且在奇数次基函数下误差收敛阶次比传统方法高两阶,呈现超收敛特性。 |
| 英文摘要: |
| Both meshfree methods and isogeometric analysis enjoy the highly smooth shape functions or basis functions.However the basis degree discrepancy issue is observed for the meshfree and isogeometric collocation analysis.This study presents a superconvergent isogeometric meshfree collocation method,which is based upon the consistent conditions of isogeometric basis functions and meshfree gradient reproducing theory.In the proposed method,a mixed gradient reproducing basis vector is defined in accordance with the reproducing kernel meshfree formulation of isogeometric basis functions.Subsequently,a meshfree framework is developed to construct the gradients of isogeometric basis functions in a meshfree fashion,which avoids the costly gradient computation of isogeometric basis functions,which is trivial for numerical implementation.Meanwhile,the second order recursive gradients of the transformed isogeometric meshfree shape functions are established by introducing the basis transformation technique and the recursive gradient formulation.It is noted that the proposed recursive transformed gradients meet extra one-order higher reproducing conditions,which provides an important safeguard for superconvergent collocation analysis.Finally,a series of typical numerical examples are used to systematically verify the convergence and accuracy of the proposed method.Numerical results well demonstrate that the proposed methodology exhibits superior computational accuracy in contrast to the conventional isogeometric collocation,and a superconvergence with two additional orders of accuracy is realized by the proposed approach. |
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