Research on key issues of fast isogeometric collocation methods for complex models

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In order to address the simulation requirements of complex engineering structures, methods that aim to more closely integrate the design and analysis stages of product development have been recently proposed. This new paradigm, widely known as isogeometric analysis (IGA) aims to use the same basis functions employed in Computer Aided Design (CAD) for the discretization of the solution field. This allows for an exact (as in CAD) geometry representation at all stages of refinement in the analysis phase. Another important advantage of IGA is the possibility for smoother, more accurate solutions obtained with significantly fewer degrees of freedom compared to standard finite elements (FEM). Nevertheless the computational costs related to numerical integration are higher than in FEM due to the usage of higher degree polynomial and rational bases, which require more quadrature points or more complex integration schemes.In this project, we plan to investigate and develop robust collocation methods for IGA, which overcome the need for numerical quadrature, resulting in much faster, more efficient implementations. While spline collocation has been an established tool in the study of ordinary differential equations, the development of collocated methods in the context of IGA is still in its infancy. A main goal of the proposed work plan is to extend the mathematical theory already developed for orthogonal and spline collocation to the isogeometric paradigm. Other areas of consideration are the development of error estimation and local refinement techniques using hierarchical spline spaces, allowing for further improvement in computational efficiency. Furthermore, parametrization techniques which are suitable for collocation methods will be developed in parallel, with the aim to determine the optimal location of control and collocation points for a given geometric domain. The proposed project also considers geometries generated by subdivision surfaces, which facilitate the design of complex objects with a high degree of smoothness. Finally, the developed methods will be used in the analysis of plates and shells as well as 2D and 3D continua. We particularly wish to study the use of isogeometric collocation for shell dynamics with application to clothing and textile simulations. This would lead to improvements to an important industrial area, for example by easing the development of new fabrics or through virtual try-on simulations.