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

本后续研究建议关注的是分析先进的几何表示的固体，如修剪对象，采用等几何比例边界的方法。在项目的第一阶段开发的，这种方法使一个面向边界的固体建模，这是完全符合等几何范式。在缩放边界方法中，实体被分割成与其边界表面和缩放中心相关的部分。这在概念上不同于标准3D面片定义，其中假设三变量张量积表示。虽然在截面之间的界面上的位移插值仅是C0连续的，但相邻截面内的离散可以是高阶的，并且是协调的或非协调的。因此，我们寻求一种通用的方法来耦合相邻的部分，同时保持高阶连续的接口。在CAD中，实体是通过其边界曲面的定义来定义的。通常这些表面重叠，所有表面的核心代表实体。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着交点耦合。利用等几何比例边界法，我们的目标是一种方法，它提供了更高的连续性的位移近似的交点。基于连续性要求，推导出作用在交点上的自由度之间的关系。不同的方法来执行的连续性约束进行了讨论，如配置或砂浆的方法。此外，高阶耦合方法的推导意味着基函数的修改。为此，主-从框架的建议，允许在最小二乘设置的修改矩阵的推导。推导出的方法适用于不同类型的部分，如符合，分层和不符合部分。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。高阶连续性在结构动力学问题中特别有前途。该方法的扩展到时间相关的问题，使更高的连续性的空间维的评估，而微分代数扩展的α-方法形成的基础上的时间积分scheme.The方法有助于提供一个通用的方法，可以处理广泛的类的几何特征和复杂的多补丁星座的等几何分析。该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态问题提供准确和强大的数值分析框架。

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.