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

• Published in: International journal for numerical methods in engineering Vol. 81; no. 9; pp. 1127 - 1156
• Main Author: Liu, G. R.
• Format: Journal Article
• Language: English
• Published: Chichester, UK John Wiley & Sons, Ltd 26-02-2010; Wiley
• Subjects: Compatibility; Computational techniques; Exact sciences and technology; Exact solutions; Finite element method; Fundamental areas of phenomenology (including applications); G space; Galerkin weak form; Mathematical analysis; Mathematical methods in physics; Mathematical models; meshfree method; Numerical analysis; numerical method; Physics; point interpolation method; Solid mechanics; solution bound; Structural and continuum mechanics; variational principle; weakened weakform; Solid mechanics; meshfree method; Test standard; Automatic mesh generation; Modeling; Galerkin weak form; Geometrical model; Exact solution; weakened weakform; Finite element method; variational principle; Upper bound; solution bound; Weak solution; G space; Standard test; Polynomial interpolation; Meshless method; numerical method; point interpolation method; Reduced order systems; compatibility; Numerical convergence
• ISSN: 0029-5981; 1097-0207
• DOI: 10.1002/nme.2720

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.