A G space theory and a weakened weak (W2) form for a unified formulation of compatible and incompatible methods: Part II applications to solid mechanics problems
A G space theory and a weakened weak (W2) form for a unified formulation of compatible and incompatible methods: Part II applications to solid mechanics problems
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DOI：
10.1002/nme.2720
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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
在本文的第一部分中，我们建立了G空间理论和W2公式化的基本原理。第二部分着重于G空间理论在固体力学问题W2模型中的应用。我们首先定义了一个双线性形式，证明了一些重要的性质，并证明了W2配方将是空间稳定的，并收敛到精确解。然后，我们给出了一些可能的W2模型的例子，包括SFEM，NS‐FEM，ES‐FEM，NS‐PIM，ES‐PIM和CS‐PIM。我们证明了这些模型的主要性质：（1）如果在适当的H空间中寻求解，它们在常规意义下是变分相容的（2）当在具有间断函数的真G空间中求解时，它们通过了标准的分片检验（不相容的情况）;（3）与有限元方法（FEM）模型相比并且可能与精确模型相比，离散化模型的刚度减小，允许我们获得关于FEM和精确解两者的上限解，以及（4）W2模型对网格的质量不太敏感，并且可以使用三角形网格而没有任何精度问题。这些性质和理论已经通过使用一些W2模型，包括兼容和不兼容的情况下解决的例子，数值证实。我们将看到，G空间理论和W2形式可以制定各种稳定和收敛的数值方法，有限元作为一个特殊情况。版权所有© 2009约翰威利父子有限公司。
In part I of this paper, we have established the G space theory and fundamentals for W2 formulation. Part II focuses on the applications of the G space theory to formulate W2 models for solid mechanics problems. We first define a bilinear form, prove some of the important properties, and prove that the W2 formulation will be spatially stable, and convergent to exact solutions. We then present examples of some of the possible W2 models including the SFEM, NS‐FEM, ES‐FEM, NS‐PIM, ES‐PIM, and CS‐PIM. We show the major properties of these models: (1) they are variationally consistent in a conventional sense, if the solution is sought in a proper H space (compatible cases); (2) They pass the standard patch test when the solution is sought in a proper G space with discontinuous functions (incompatible cases); (3) the stiffness of the discretized model is reduced compared with the finite element method (FEM) model and possibly to the exact model, allowing us to obtain upper bound solutions with respect to both the FEM and the exact solutions and (4) the W2 models are less sensitive to the quality of the mesh, and triangular meshes can be used without any accuracy problems. These properties and theories have been confirmed numerically via examples solved using a number of W2 models including compatible and incompatible cases. We shall see that the G space theory and the W2 forms can formulate a variety of stable and convergent numerical methods with the FEM as one special case. Copyright © 2009 John Wiley & Sons, Ltd.