Isogeometric Collocation Methods with Structured Constraints for PDE-based and Field-aligned Analysis Suitable Parameterization

Isogeometric analysis (IGA) is an active field of research for integrated design and analysis of engineering objects using the same set of basis functions for computer aided design (CAD). However, the present CAD industry is not entirely compatible with the concept of isogeometric analysis and not all CAD models can be directly used in isogeometric analysis. One often needs to construct an analysis suitable parameterization of the computational domain from a boundary representation (B-rep) model for isogeometric analysis. Construction of spline surfaces or volumetric spline models with given boundaries is one of the classical problems in computer aided geometric design. In connection with analysis suitable parameterization for IGA, the topic re-attracted much interest in recent years. It is still a challenging problem especially for parameterization from a complex input boundary representation. This project investigates alternative methods for analysis suitable parameterization of computational domains from input boundary representations using isogeometric collocation methods with structured constraints for achieving desired parameterization. The methods are based on isogeometric solutions of partial differential equations (PDEs) with boundary conditions being the input boundary representation plus structured constraints that can be specified either over the boundary or inside the computational domain in meeting desired properties. While the resulting parameterization will be for isogeometric analysis, the PDEs will also be solved using one of the general methodologies of isogeometric analysis, namely isogeometric collocation methods. Additional structured constraints will be designed and implemented in the system of collocation equations of the PDEs for achieving desired properties of the resulting parameterization. The project will also investigate field-aligned parameterization by applying structured constraints based on extracted structural shape information of the input boundary and a posterior analysis of solution error distribution. Main areas of research in the project include (1) the development of a general framework on isogeometric collocation methods for parameterization of computational domains using NURBS and subdivision schemes, (2) design and analysis of structured constraints for achieving desired properties of resulting parameterization, (3) field-aligned dynamic parameterization with desired distribution of degrees of freedom for application specific solutions, and (4) application of the resulting parameterization in thermal analysis and simulation. The proposed research will contribute to isogeometric analysis with added capability and flexibility in handling a wider range of engineering objects and solutions to several key research issues of isogeometric analysis. It will also contribute to future CAD technologies compatible with isogeometric analysis.