Heterogeneous Smoothed Finite Element Method: Convergence/Superconvergence Proof and Its Performance in High-Contrast Composite Materials

IF 2.7 3区 工程技术 Q1 ENGINEERING, MULTIDISCIPLINARY
Yun Chen, Guirong Liu, Junzhi Cui, Qiaofu Zhang, Ziqiang Wang
{"title":"Heterogeneous Smoothed Finite Element Method: Convergence/Superconvergence Proof and Its Performance in High-Contrast Composite Materials","authors":"Yun Chen,&nbsp;Guirong Liu,&nbsp;Junzhi Cui,&nbsp;Qiaofu Zhang,&nbsp;Ziqiang Wang","doi":"10.1002/nme.7636","DOIUrl":null,"url":null,"abstract":"<div>\n \n <p>Smoothed Finite Element Method (S-FEM) has been widely used in many engineering simulations. However, there are still a lot of theoretical problems to be solved, especially with regard to composite materials and convergence proof. Based on a novel least-squares approximation and conservation property, we extend S-FEM to heterogeneous materials. Firstly, the orthogonality, Softening Effect, and energy function are checked. Secondly, the interpolation error in the maximum norm is estimated in any dimension. Consequently, we get a theoretical convergence rate which has been sought since the year 2010. When restricted to one-dimensional problems, we construct a special test function to prove the superconvergence: (1) S-FEM flux is exact in the meaning of element-wise integral; (2) numerical flux is exact at some point in each element; (3) physical flux can be quadratically approximated at the center of each element. At last, we present two numerical experiments: (1) conventional S-FEM fails in high-contrast composite materials while our new scheme performs well; (2) our flux converges quadratically.</p>\n </div>","PeriodicalId":13699,"journal":{"name":"International Journal for Numerical Methods in Engineering","volume":"126 3","pages":""},"PeriodicalIF":2.7000,"publicationDate":"2024-12-11","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":"0","resultStr":null,"platform":"Semanticscholar","paperid":null,"PeriodicalName":"International Journal for Numerical Methods in Engineering","FirstCategoryId":"5","ListUrlMain":"https://onlinelibrary.wiley.com/doi/10.1002/nme.7636","RegionNum":3,"RegionCategory":"工程技术","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":null,"EPubDate":"","PubModel":"","JCR":"Q1","JCRName":"ENGINEERING, MULTIDISCIPLINARY","Score":null,"Total":0}
引用次数: 0

Abstract

Smoothed Finite Element Method (S-FEM) has been widely used in many engineering simulations. However, there are still a lot of theoretical problems to be solved, especially with regard to composite materials and convergence proof. Based on a novel least-squares approximation and conservation property, we extend S-FEM to heterogeneous materials. Firstly, the orthogonality, Softening Effect, and energy function are checked. Secondly, the interpolation error in the maximum norm is estimated in any dimension. Consequently, we get a theoretical convergence rate which has been sought since the year 2010. When restricted to one-dimensional problems, we construct a special test function to prove the superconvergence: (1) S-FEM flux is exact in the meaning of element-wise integral; (2) numerical flux is exact at some point in each element; (3) physical flux can be quadratically approximated at the center of each element. At last, we present two numerical experiments: (1) conventional S-FEM fails in high-contrast composite materials while our new scheme performs well; (2) our flux converges quadratically.

平滑有限元法(S-FEM)已被广泛应用于许多工程模拟中。然而,仍有许多理论问题有待解决,尤其是在复合材料和收敛性证明方面。基于新颖的最小二乘近似和守恒特性,我们将 S-FEM 扩展到异质材料。首先,检验了正交性、软化效应和能量函数。其次,在任意维度上估算最大法的插值误差。因此,我们得到了自 2010 年以来一直在寻求的理论收敛速率。当局限于一维问题时,我们构建了一个特殊的测试函数来证明超收敛性:(1)S-FEM 通量在元素全积分的意义上是精确的;(2)数值通量在每个元素的某一点上是精确的;(3)物理通量可以在每个元素的中心进行二次逼近。最后,我们介绍两个数值实验:(1) 传统的 S-FEM 在高对比度复合材料中失效,而我们的新方案性能良好;(2) 我们的通量可二次收敛。
本文章由计算机程序翻译,如有差异,请以英文原文为准。
求助全文
约1分钟内获得全文 求助全文
来源期刊
CiteScore
5.70
自引率
6.90%
发文量
276
审稿时长
5.3 months
期刊介绍: The International Journal for Numerical Methods in Engineering publishes original papers describing significant, novel developments in numerical methods that are applicable to engineering problems. The Journal is known for welcoming contributions in a wide range of areas in computational engineering, including computational issues in model reduction, uncertainty quantification, verification and validation, inverse analysis and stochastic methods, optimisation, element technology, solution techniques and parallel computing, damage and fracture, mechanics at micro and nano-scales, low-speed fluid dynamics, fluid-structure interaction, electromagnetics, coupled diffusion phenomena, and error estimation and mesh generation. It is emphasized that this is by no means an exhaustive list, and particularly papers on multi-scale, multi-physics or multi-disciplinary problems, and on new, emerging topics are welcome.
×
引用
GB/T 7714-2015
复制
MLA
复制
APA
复制
导出至
BibTeX EndNote RefMan NoteFirst NoteExpress
×
提示
您的信息不完整,为了账户安全,请先补充。
现在去补充
×
提示
您因"违规操作"
具体请查看互助需知
我知道了
×
提示
确定
请完成安全验证×
copy
已复制链接
快去分享给好友吧!
我知道了
右上角分享
点击右上角分享
0
联系我们:info@booksci.cn Book学术提供免费学术资源搜索服务,方便国内外学者检索中英文文献。致力于提供最便捷和优质的服务体验。 Copyright © 2023 布克学术 All rights reserved.
京ICP备2023020795号-1
ghs 京公网安备 11010802042870号
Book学术文献互助
Book学术文献互助群
群 号:481959085
Book学术官方微信