Eigendecomposition‐based convergence analysis of the Neumann series for laminated composites and discretization error estimation

Bellis Cédric; Moulinec Hervé; Suquet Pierre

摘要

Summary >In computational homogenization for periodic composites, the Lippmann‐Schwinger integral equation constitutes a convenient formulation to devise numerical methods to compute local fields and their macroscopic responses. Among them, the iterative scheme based on the Neumann series is simple and efficient. For such schemes, a priori global error estimates on local fields and effective property are not available, and this is the concern of this article, which focuses on the simple, but illustrative, conductivity problem in laminated composites. The global error is split into an iteration error, associated with the Neumann series expansion, and a discretization error. The featured nonlocal Green's operator is expressed in terms of the averaging operator, which circumvents the use of the Fourier transform. The Neumann series is formulated in a discrete setting, and the eigendecomposition of the iterated matrix is performed. The ensuing analysis shows that the local fields are computed using a particular subset of eigenvectors, the iteration error being governed by the associated eigenvalues. Quadratic error bounds on the effective property are also discussed. The discretization error is shown to be related to the accuracy of the trapezoidal quadrature scheme. These results are illustrated numerically, and their extension to other configurations is discussed. </abstract> </span> <span class="z_kbtn z_kbtnclass hoverxs" style="display: none;">展开▼</span> </div> <div class="translation abstracttxt"> <span class="zhankaihshouqi fivelineshidden" id="abstract"> <span>机译：</span><Abstract Type =“Main”XML：Lang =“EN”XML：ID =“NME6206-ABS-ABS-0001”> <标题类型=“main”>摘要</ title> 在对周期复合材料的计算均匀化中，Lippmann-Schwinger整体方程构成了方便的制剂，以设计计算局部场和宏观反应的数字方法。其中，基于Neumann系列的迭代方案简单富有高效。对于此类方案，不可用的先验全局错误估算和有效属性，这是本文的关注，它侧重于层压复合材料中简单但说明的导电性问题。全局错误分为与Neumann系列扩展相关的迭代错误和离散错误。特色的非局部绿色的运算符在平均操作员方面表达，这避免了使用傅里叶变换。 NeuMann系列在离散设置中配制，并且执行迭代矩阵的实际复容。随后的分析表明，使用特定的特定eIgenvectors子集来计算本地字段，迭代错误由相关的特征值控制。还讨论了有效属性上的二次误差界限。离散化误差显示与梯形正交方案的准确性有关。这些结果在数值上进行了说明，讨论了它们对其他配置的扩展。</ p> </摘要> </span> <span class="z_kbtn z_kbtnclass hoverxs" style="display: none;">展开▼</span> </div> </div> <div class="record"> <h2 class="all_title" id="enpatent33" > 著录项</h2> <ul> <li> <span class="lefttit">来源</span> <div style="width: 86%;vertical-align: text-top;display: inline-block;"> <a href='/journal-foreign-20801/'>《International Journal for Numerical Methods in Engineering 》</a> <b style="margin: 0 2px;">|</b><span>2020年第2期</span><b style="margin: 0 2px;">|</b> <span>共33页</span> </div> </li> <li> <div class="author"> <span class="lefttit">作者</span> <p id="fAuthorthree" class="threelineshidden zhankaihshouqi"> <a href="/search.html?doctypes=4_5_6_1-0_4-0_1_2_3_7_9&sertext=Bellis Cédric&option=202" target="_blank" rel="nofollow">Bellis Cédric;</a> <a href="/search.html?doctypes=4_5_6_1-0_4-0_1_2_3_7_9&sertext=Moulinec Hervé&option=202" target="_blank" rel="nofollow">Moulinec Hervé;</a> <a href="/search.html?doctypes=4_5_6_1-0_4-0_1_2_3_7_9&sertext=Suquet Pierre&option=202" target="_blank" rel="nofollow">Suquet Pierre;</a> </p> <span class="z_kbtnclass z_kbtnclassall hoverxs" id="zkzz" style="display: none;">展开▼</span> </div> </li> <li> <div style="display: flex;"> <span class="lefttit">作者单位</span> <div style="position: relative;margin-left: 3px;max-width: 639px;"> <div class="threelineshidden zhankaihshouqi" id="fOrgthree"> <p>Aix Marseille Univ CNRS Centrale Marseille LMAMarseille France;</p> <p>Aix Marseille Univ CNRS Centrale Marseille LMAMarseille France;</p> <p>Aix Marseille Univ CNRS Centrale Marseille LMAMarseille France;</p> </div> <span class="z_kbtnclass z_kbtnclassall hoverxs" id="zhdw" style="display: none;">展开▼</span> </div> </div> </li> <li> <span class="lefttit">收录信息</span> <span style="width: 86%;vertical-align: text-top;display: inline-block;"></span> </li> <li> <span class="lefttit">原文格式</span> <span>PDF</span> </li> <li> <span class="lefttit">正文语种</span> <span>eng</span> </li> <li> <span class="lefttit">中图分类</span> <span> <a href="https://www.zhangqiaokeyan.com/clc/6940.html" title="工程数学">工程数学 ;</a></span> </li> <li class="antistop"> <span class="lefttit">关键词</span> <p style="width: 86%;vertical-align: text-top;"> <a style="color: #3E7FEB;" href="/search.html?doctypes=4_5_6_1-0_4-0_1_2_3_7_9&sertext=computational homogenization&option=203" rel="nofollow">computational homogenization;</a> <a style="color: #3E7FEB;" href="/search.html?doctypes=4_5_6_1-0_4-0_1_2_3_7_9&sertext=error estimates&option=203" rel="nofollow">error estimates;</a> <a style="color: #3E7FEB;" href="/search.html?doctypes=4_5_6_1-0_4-0_1_2_3_7_9&sertext=Green's operator&option=203" rel="nofollow">Green's operator;</a> <a style="color: #3E7FEB;" href="/search.html?doctypes=4_5_6_1-0_4-0_1_2_3_7_9&sertext=Lippmann‐Schwinger equation&option=203" rel="nofollow">Lippmann‐Schwinger equation;</a> </p> <div class="translation"> 机译：计算均匀化;错误估计;绿色的运算符;Lippmann-Schwinger方程式; </div> </li> </ul> </div> </div> <div class="literature cardcommon"> <div class="similarity "> <h3 class="all_title" id="enpatent66">相似文献</h3> <div class="similaritytab clearfix"> <ul> <li class="active" >外文文献</li> <li >中文文献</li> <li >专利</li> </ul> </div> <div class="similarity_details"> <ul > 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