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

为了解决复杂工程结构的仿真需求，最近提出了旨在更紧密地集成产品开发的设计和分析阶段的方法。这种新的范例，广泛称为等几何分析（伊加）的目的是使用相同的基础功能，在计算机辅助设计（CAD）的离散化的解决方案领域。这允许在分析阶段中的所有细化阶段进行精确的（如在CAD中）几何表示。伊加的另一个重要优点是与标准有限元（FEM）相比，可以用更少的自由度获得更平滑，更精确的解决方案。 然而，数值积分的计算成本高于FEM，由于使用高次多项式和有理基，这需要更多的积分点或更复杂的积分schemes.In本项目中，我们计划研究和开发强大的配置方法伊加，克服了数值求积的需要，从而更快，更有效的实现。虽然样条配置已经是常微分方程研究中的一个既定工具，但在伊加的背景下，配置方法的发展仍处于起步阶段。拟议工作计划的一个主要目标是将已经为正交和样条配置开发的数学理论扩展到等几何范式。其他领域的考虑是误差估计和局部细化技术的发展，使用分层样条空间，允许进一步提高计算效率。此外，将并行开发适用于配置方法的参数化技术，目的是确定给定几何域的控制点和配置点的最佳位置。该项目还考虑了细分曲面生成的几何形状，这有助于设计具有高度平滑性的复杂对象。最后，所开发的方法将用于板和壳以及2D和3D连续体的分析。我们特别希望研究使用等几何配置壳动力学与应用服装和纺织品模拟。这将导致一个重要的工业领域的改进，例如通过简化新面料的开发或通过虚拟试穿模拟。

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.