Towards optimal mixed finite elements for general shell structures

Abstract

Optimal finite elements are available for many engineering applications, such as incompressible media, beam and plate bending problems. However, no existing general shell elements have been shown to satisfy the mathematical conditions of optimality in the analysis of shell structures. The objective in this thesis is to work towards optimal shell elements that possess properties of uniform convergence independent of the problem solved. We develop a thorough numerical convergence study methodology for bending dominated and membrane-dominated shells and apply it to the family of quadrilateral MITC (Mixed-Interpolated Tensorial Components) shell elements proposed by Bathe, Dvorkin and Bucalem. The study demonstrates the superior performance of these elements. We also develop a new simple and effective triangular mixed-interpolated shell element and demonstrate the element reliability. Deriving the relation between the MITC approach and the classical constrained minimization problem leads to the conclusion that the optimal shell elements must satisfy the relevant inf-sup condition in the analysis of bending-dominated shells. In practice, this condition is virtually impossible to check analytically, especially when general geometries and/or distorted meshes are considered. Using appropriate bounds, we develop a new numerical inf-sup test aimed to indicate if a particular finite element discretization satisfies this fundamental condition. While the displacement based elements, of course, fail the test, the quadrilateral and the new triangular MITC elements pass the test.