Scaled boundary isogeometric analysis with advanced features for trimmed objects, higher order continuity, and structural dynamics

这一后续研究方案涉及通过使用等距缩放边界方法来分析实体的高级几何表示，例如修剪对象。该方法是在该项目的第一阶段开发的，能够对固体进行以边界为导向的建模，这完全符合等几何范例。在缩放边界方法中，实体被分割成相对于其边界曲面和缩放中心的部分。这在概念上与标准的3D面片定义不同，在标准3D面片定义中，假设了三变量张量积表示。当截面间的位移插值仅为C0连续时，相邻截面内的离散化可以是高阶的，可以是协调的，也可以是非协调的。因此，我们寻求一种通用的方法来耦合相邻的截面，同时保持界面上的高阶连续性。在CAD中，实体是通过定义它们的边界曲面来定义的。通常，这些曲面重叠，并且所有曲面的核心表示实体。此处可能出现的问题会影响相邻曲面的自由度。由于缺少共享控制点，这些自由度可能不会沿交叉点耦合。利用等距尺度边界方法，我们的目标是为交叉点上的位移逼近提供一种更高连续性的方法。基于连续性要求，推导了作用在交点上的自由度之间的关系。讨论了实施连续性约束的不同方法，如搭配方法或迫击炮方法。此外，高阶耦合方法的推导意味着基函数的修改。为此，提出了一种主从框架，该框架允许在最小二乘设置下推导修改矩阵。该方法适用于不同类型的截面，如一致性截面、层次型截面和非一致性截面。CK连续公式的一个优点在于减少了结构动力学中存在的光学分支，例如有限元分析。高阶连续性特别适用于结构动力学问题。将该方法扩展到与时间相关的问题，可以评估空间维度的更高连续性，而α方法的微分代数扩展形成了时间积分方案的基础。该方法有助于提供一种通用的方法，可以处理广泛的几何特征和复杂的多面片星座。这个项目的另一个目的是利用增加连续性的好处，以便为固体力学中的动力学问题提供一个准确和健壮的数值分析框架。

This follow-up research proposal is concerned with the analysis of advanced geometry representations of solids, such as trimmed objects, by employing the isogeometric scaled boundary approach. Developed in the first phase of the project, this approach enables a boundary oriented modelling of solids, which is in full accordance with the isogeometric paradigm. In the scaled boundary approach, the solid is split into sections in relation to its boundary surfaces and the scaling centre. This is conceptually different to the standard 3D-patch definition where a tri-variate tensor product representation is assumed. While the displacement interpolation on the interface between the sections is only C0-continuous, the discretization within adjacent sections can be of higher order and conforming or non-conforming. Therefore, we seek for a general method to couple adjacent sections while preserving higher-order continuity on the interface. In CAD, solids are defined through the definition of their bounded surfaces. Commonly these surfaces overlap and the kernel of all surfaces represents the solid. An issue that may occur here affects the degrees of freedom of adjacent surfaces. Due to the lack of shared control points, these degrees of freedom might not be coupled along the intersection. Using the isogeometric scaled boundary approach, we are aiming for a method which provides higher continuity for the displacement approximation on the intersections. Based on the continuity requirement, a relation is derived between the degrees of freedom acting on the intersection. Different approaches for the enforcement of the continuity constraint are discussed, such as a collocation or the mortar approach. Furthermore, the derivation of a higher-order coupling approach implies a modification of the basis functions. To this end, a master-slave framework is suggested that allows the derivation of a modification matrix in a least square setting. The derived methodology is applied to different types of sections such as conforming, hierarchical and non-conforming sections. One advantage of the Ck-continuous formulation lies in the reduction of optical branches present in structural dynamics, e.g. the finite element analysis. Higher-order continuity is in particular promising for problems in structural dynamics. The extension of the methodology to time-dependent problems enables the assessment of higher-continuity in the spatial dimension while the differential-algebraic extension of the α-method forms the basis for the time integration scheme.The methodology contributes towards providing a general approach for isogeometric analysis that can handle a wide class of geometric features and complex multi-patch constellations. A further aim of this project is to exploit the benefits of increased continuity in order to provide an accurate and robust numerical analysis framework for dynamic problems in solid mechanics.