A Mathematical Discussion of Corotational Finite Element Modeling

Abstract

This thesis discusses the mathematics of the Element Independent Corotational (EICR) Method and the more general Unified Small-Strain Corotational Formulation. The former was developed by Rankin, Brogan and Nour-Omid [106]. The latter, created by Felippa and Haugen [49], provides a theoretical frame work for the EICR and similar methods and its own enhanced methods. The EICR and similar corotational methods analyse non-linear deformation of a body by its discretization into finite elements, each with an orthogonal frame rotating (and translating) with the element. Such methods are well suited to deformations where non-linearity arises from rigid body deformation but local strains are small (1-4%) and so suited to linear analysis. This thesis focuses on such small-strain, non-linear deformations. The key concept in small-strain corotational methods is the separation of deformation into its rigid body and elastic components. The elastic component then can be analyzed linearly. Assuming rigid translation is removed first, this separation can be viewed as a polar decomposition (F = vR) of the deformation gradient (F) into a rigid rotation (R) followed by a small, approximately linear, stretch (v). This stretch usually causes shear as well as pure stretch. Using linear algebra, Chapter 3 explains the EICR Method and Unified Small-Strain Corotational Formulation initially without, and then with, the projector operator, reflecting their historical development. Projectors are orthogonal projections which simplify the isolation of elastic deformation and improve element strain invariance to rigid body deformation. Turning to Lie theory, Chapter 4 summarizes and applies relevant Lie theory to explore rigid and elastic deformation, finite element methods in general, and the EICR Method in particular. Rigid body deformation from a Lie perspective is well represented in the literature which is summarized. A less developed but emerging area in differential geometry (notably, Marsden/Hughes [82]), elastic deformation is discussed thoroughly followed by various Lie aspects of finite element analysis. Finally, the EICR Method is explored using Lie theory. Given the available research, complexity of the area, and level of this thesis, this exploration is less developed than the earlier linear algebraic discussion, but offers a useful alternative perspective on corotational methods.