Smooth, second order, non‐negative meshfree approximants selected by maximum entropy
Smooth, second order, non‐negative meshfree approximants selected by maximum entropy
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DOI：
10.1002/nme.2597
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发表时间：
2009-09
影响因子：
2.9
通讯作者：
C. Cyron;M. Arroyo;Magdalena Ortiz
中科院分区：
文献类型：
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作者：
C. Cyron;M. Arroyo;Magdalena Ortiz
我们提出了一类近似格式，我们称之为二阶最大熵近似格式，它将一阶局部最大熵近似格式扩展到二阶一致性。该方法保留了一阶极大元格式的基本性质，即形状函数是光滑的，非负的，并且在边界处满足弱Kronecker - delta性质。这最后一个性质使得在偏微分方程的数值解中施加必要的边界条件变得微不足道。形状函数的求值不显式，但它是非常有效和鲁棒的。据我们所知，所提出的方法是第一个从任意维非负形状函数的非结构化数据中进行函数逼近的高阶格式。因此，数值实验表明，与拉格朗日有限元、基于MLS的无网格方法甚至B样条近似相比，该近似具有变化递减特性，并且在结构振动问题中具有优异的性能。与通常的基于MLS的二阶无网格方法相比，本文提出的形状函数更容易在伽辽金方法中集成，如标准基准问题所示。版权所有©2009 John Wiley & Sons， Ltd
We present a family of approximation schemes, which we refer to as second‐order maximum‐entropy (max‐ent) approximation schemes, that extends the first‐order local max‐ent approximation schemes to second‐order consistency. This method retains the fundamental properties of first‐order max‐ent schemes, namely the shape functions are smooth, non‐negative, and satisfy a weak Kronecker‐delta property at the boundary. This last property makes the imposition of essential boundary conditions in the numerical solution of partial differential equations trivial. The evaluation of the shape functions is not explicit, but it is very efficient and robust. To our knowledge, the proposed method is the first higher‐order scheme for function approximation from unstructured data in arbitrary dimensions with non‐negative shape functions. As a consequence, the approximants exhibit variation diminishing properties, as well as an excellent behavior in structural vibrations problems as compared with the Lagrange finite elements, MLS‐based meshfree methods and even B‐Spline approximations, as shown through numerical experiments. When compared with usual MLS‐based second‐order meshfree methods, the shape functions presented here are much easier to integrate in a Galerkin approach, as illustrated by the standard benchmark problems. Copyright © 2009 John Wiley & Sons, Ltd.