Nonconforming Isogeometric Analysis For Geometrically Complex CAD Surfaces And Geometric Modeling

Iso-Geometric Analysis (IGA) is a new computational framework that uses regular NURBS patches for the geometric representation of a structure and can describe the displacement field of the structure. The biggest advantage of using IGA is its potential to fill the gap between the Computer Aided Design (CAD) and Computer-Aided Engineering (CAE). Traditional IGA requires that NURBS patches are conforming across the interfaces of subdomains, however, most CAD models are composed of nonconforming NURBS patches. While the NURBS patches may share geometrically identical interfaces, the control nets belonging to different patches may have no relationship to one another, thus, the NURBS functions are nonconforming across the patch interfaces. These flaws limit the further development of isogeometric analysis. This paper reports a method used to develop new spline theories, isogeometric analysis methods and the geometrical modeling techniques to analyze geometrically complex domains found in trimmed CAD models and or non-conforming CAD geometries.In this thesis, based on a brief introduction of isogeometric analysis, the theories, advan-tages and disadvantages of isogeometric analysis are introduced in detail. The highlights and contribution of this dissertation are summarized as follows:1. Nonconforming Isogeometric Analysis (NIGA) based on multi-point constraints and La-grange multiplier theory was proposed to address the nonconforming spline geometry. Weak Parametric Spline Space (WPSS) was introduced as the mathematical foundation of NIGA; The constrain conditions of weak continuity for geometric displacements and displacement functions are derived; A semi-definite equation system is obtained applying the Lagrange multiplier the-ory and the modified variational principle. FETI algorithm and the conjugate gradient iteration are employed to solve the semi-definite equation system. The present method does not require constructing analysis-suitable geometry.2. Nonconforming isogeometric analysis using mortar method was proposed. This method can deal with the nonconforming CAD geometries. A degrees of freedom reduced scheme was developed to assemble the stiffness matrix as simply as traditional finite clement method.3. NURBS-based FHTI algorithm was introduced, which is suitable for the parallel com-puting of isogeometric analysis. 4. The convergence of isogeometric analysis for trimmed NURBS geometries was inves-tigated. The results illustrate that isogeometric analysis for trimmed NURBS geometries is con-vergent. However, the convergence speed is not stable, which varies with the load conditions.5. One-step spatial unfolding algorithm was introduced to deal with geometrically com-plex CAD models. We also apply this method to mesh parameterizations.Compared with other methods for mesh parameterization, the new method can unfold a large number of meshes more rapidly. In addition, it can unfold the complex model step by step.6. Using the spatial unfolding algorithm, we reconstructed isogeometric analysis-suitable geometries. This method can construct the NURBS surface automatically.