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A G space theory and a weakened weak (W 2 ) form for a unified formulation of compatible and incompatible methods: Part II applications to solid mechanics problems
Author(s) -
Liu G. R.
Publication year - 2009
Publication title -
international journal for numerical methods in engineering
Language(s) - English
Resource type - Journals
SCImago Journal Rank - 1.421
H-Index - 168
eISSN - 1097-0207
pISSN - 0029-5981
DOI - 10.1002/nme.2720
Subject(s) - finite element method , discretization , mathematics , space (punctuation) , bilinear form , weak formulation , polygon mesh , mathematical analysis , computer science , physics , geometry , boundary value problem , thermodynamics , operating system
In part I of this paper, we have established the G space theory and fundamentals for W 2 formulation. Part II focuses on the applications of the G space theory to formulate W 2 models for solid mechanics problems. We first define a bilinear form, prove some of the important properties, and prove that the W 2 formulation will be spatially stable, and convergent to exact solutions. We then present examples of some of the possible W 2 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 W 2 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 W 2 models including compatible and incompatible cases. We shall see that the G space theory and the W 2 forms can formulate a variety of stable and convergent numerical methods with the FEM as one special case. Copyright © 2009 John Wiley & Sons, Ltd.