A G space theory and a weakened weak (W2) form for a unified formulation of compatible and incompatible methods: Part I theory
A G space theory and a weakened weak (W2) form for a unified formulation of compatible and incompatible methods: Part I theory
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DOI：
10.1002/nme.2719
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发表时间：
2010-02
影响因子：
2.9
通讯作者：
Guirong Liu
中科院分区：
文献类型：
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作者：
Guirong Liu
本文利用广义梯度光顺技术引入了G空间理论和弱弱形式(W2)，统一了一大类相容方法和非相容方法。W2公式既适用于有限元方法设置，也适用于无网格设置，并且W2模型可以具有特殊的属性，包括软化行为、上限和超精度。本文的第一部分重点介绍了W2配方的理论和基础。赋范G空间首先被定义为包括连续函数和不连续函数两者，从而允许使用更多类型的方法/技术来创建用于数值模型的形函数。然后从理论上证明了G空间的重要性质和一组有用的不等式，并对其进行了详细的分析。这些性质保证了基于W2公式发展的数值方法在空间上是稳定的并且收敛于精确解，只要物理问题是适定的。该理论适用于标准弱公式适用的任何问题，并能提供具有“接近精确”刚度、上界和超精度等特殊性质的数值解。版权所有©2009 John Wiley&Sons，Ltd.
This paper introduces a G space theory and a weakened weak form (W2) using the generalized gradient smoothing technique for a unified formulation of a wide class of compatible and incompatible methods. The W2 formulation works for both finite element method settings and mesh‐free settings, and W2 models can have special properties including softened behavior, upper bounds and ultra accuracy. Part I of this paper focuses on the theory and fundamentals for W2 formulations. A normed G space is first defined to include both continuous and discontinuous functions allowing the use of much more types of methods/techniques to create shape functions for numerical models. Important properties and a set of useful inequalities for G spaces are then proven in the theory and analyzed in detail. These properties ensure that a numerical method developed based on the W2 formulation will be spatially stable and convergent to the exact solutions, as long as the physical problem is well posed. The theory is applicable to any problems to which the standard weak formulation is applicable, and can offer numerical solutions with special properties including ‘close‐to‐exact’ stiffness, upper bounds and ultra accuracy. Copyright © 2009 John Wiley & Sons, Ltd.