Exploiting lower dimensional models for efficient three-dimensional finite element analysis.

摘要

There are two broad strategies for 3-D structural analysis today: (I) classical or simplified modeling, and (2) 3-D modeling. Classical modeling is applicable when the underlying geometry and physics exhibits special properties. For example, 2-D plane-stress, 1-D beam, 2d plate / shell or 2-D axi-symmetric approximations fall under this category. Such simplified models are computationally efficient, but with increased geometric complexity justification and automated construction of reduced models is cumbersome. Moreover, the inability of these reduced models to capture 3-D stress limits their use to certain applications.;If the model cannot be simplified, or when stress prediction is critical, one must use a 3-D method, among which finite element analysis (FEA) is most popular. However, 3-D FEA typically results in a large system of equations that can be computationally challenging to solve.;The main contribution of this thesis is a novel framework that integrates classical methods with 3-D FEA, for faster and accurate iterative analysis. An implementation of the framework for thin structure analysis is presented here. The proposed framework uses standard 3-D finite element formulation for assembling the linear systems of equations over the full 3-D solid. However, these equations are solved with the help of reduced beam or plate models. This is achieved by combining the multi-grid idea with recently developed dual-representation method.;In the proposed scheme, the multi-grid setup provides the necessary framework for bi-directional sweeping, where high-frequency components of displacement formulation are smoothed out using 3-D ETA. and low-frequency components are captured via reduced models. Finally, dual-representation provides an accurate and easily computable reduced dimension stiffness matrix, based on actual 3-D geometry, accounting for all 3-D features.;Numerical examples are presented to illustrate that the proposed method not only leads to significant computational gains over pre-conditioned iterative methods. but also has the flexibility to incorporate existing pre-conditioners within its framework Since 3-D FEA forms the basis of the method, it can accurately capture stress concentration as well.