无网格局部Petrov-Galerkin法及程序设计（1）——研究背景

研究背景

无单元Galerkin(EFG)方法需要背景单元网格，以便在计算系统矩阵时进行积分。需要背景单元进行积分的原因是使用Galerkin弱形式来生成离散系统方程。是否可以不使用弱形式？答案是肯定的：已经开发出了可用于强形式的无网格方法，例如不规则有限差分法 ^{[1] [2]} ，有限点法 ^[3] 和局部点配置法 ^{[4] [5] [6] [7] [8]} 。但是，这些方法通常对于节点不规则性不是很稳定，并且获得的结果可能不太准确。人们仍在致力于使这些方法变得更稳定，特别是在使用具有适当设计的正则化技术的局部径向函数的方向上 ^{[7] [8]} 。

在使用加权残量法时，如果我们像在点配置方法中那样，尝试使用点的局部域中的信息逐点满足方程，则可以在局部域上通过进行数值积分来局部实现积分形式。由Atluri和Zhu ^[9] 提出的无网格局部Petrov-Galerkin(MLPG)方法使用了Petrov-Galerkin残量公式的所谓局部弱形式。多年来，对MLPG进行了微调、改进和扩展 ^{[10] [11] [12] [13] [14] [15] [16] [17]} 。我们详细介绍了用于二维（2D）固体力学问题的MLPG方法。

在MLPG实现中，采用移动最小二乘（MLS）近似来构造形状函数。因此，类似于EFG方法，存在强加基本边界条件的问题。在 ^{[10] [11]} 中提出的原始MLPG使用惩罚方法。在 ^[13] 中的公式中，使用了一种称为直接插值的方法。这里介绍了这两种方法，此外还有用于自由振动问题的正交变换方法。

我们给出了许多基准示例，以说明MLPG方法的过程和有效性。通过这些示例还研究了不同参数（包括MLPG不同域的尺寸）对结果准确性的影响。

尽管MLPG中的逐节点过程与配点法非常相似，但由于使用了局部积分加权残量的局部弱形式，因此MLPG对于节点不规则性更为稳定。由于在局部Petrov-Galerkin公式中使用了MLS形状函数，因此MLPG可以重现包含在MLS形状函数基础上的多项式。补丁程序测试示例将证明这一事实。

封面图： ^[18]

参考

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18. ^ https://timgsa.baidu.com/timg?image&quality=80&size=b9999_10000&sec=1596901511635&di=cbf344e39fe097656590b0eecc1573b9&imgtype=0&src=http%3A%2F%2Fa.hiphotos.baidu.com%2Fbaike%2Fpic%2Fitem%2F3b87e950352ac65c58a14bd0fbf2b21192138ae8.jpg