This work presents numerical techniques to enforce continuity constraints on multi-patch surfaces for three distinct problem classes. The first involves structural analysis of thin shells that are described by general Kirchhoff-Love kinematics. Their governing equation is a vector-valued, fourth-order, nonlinear, partial differential equation (PDE) that requires at least $C^1$-continuity within a displacement-based finite element formulation. The second class are surface phase separations modeled by a phase field. Their governing equation is the Cahn-Hilliard equation - a scalar, fourth-order, nonlinear PDE - that can be coupled to the thin shell PDE. The third class are brittle fracture processes modeled by a phase field approach. In this work, these are described by a scalar, fourth-order, nonlinear PDE that is similar to the Cahn-Hilliard equation and is also coupled to the thin shell PDE. Using a direct finite element discretization, the two phase field equations also require at least a $C^1$-continuous formulation. Isogeometric surface discretizations - often composed of multiple patches - thus require constraints that enforce the $C^1$-continuity of displacement and phase field. For this, two numerical strategies are presented: For this, two numerical strategies are presented: A Lagrange multiplier formulation and a penalty method. The curvilinear shell model including the geometrical constraints is taken from Duong et al. (2017) and it is extended to model the coupled phase field problems on thin shells of Zimmermann et al. (2019) and Paul et al. (2020) on multi-patches. Their accuracy and convergence are illustrated by several numerical examples considering deforming shells, phase separations on evolving surfaces, and dynamic brittle fracture of thin shells.

翻译：这项工作为三个不同的问题类别提供了对多批量表面实施连续性限制的数值技术。 首先是对一般 Kirchhoff- love 运动学描述的薄壳进行结构分析。 其主导方程式是矢量值、 四级、 非线性、 部分差异方程式( PDE ), 需要至少 $C1 $- $- 连续性, 在基于迁移的限定元素配方中 。 第二类是用一个阶段字段模拟的表面级分离。 其主导方程式是 Cahn- Hilliard 方程式 - 直线性、 四级、 非线性PDE - 可以与薄性贝壳 PDE 相配合的薄壳壳壳。 第三个等方程式是用一个平面性、 四级、 非线性、 非线性、 四级、 非线性方程式 等方程式 。 等离子、 等式、 等式、 等式、 等式、 等式、 等式、 等式、 等式、 等式、 等式、 等式、 等式、 等式、 等式、 等式、 等式、 等式、 等式、 等式、 等式、 等式、 等式、 等式、 等式、 等式、 等式、 等式、 等式、 等式、 等式、 等式、 等式、 等式、 等式、 等式、 等式、 等式、 等式、 等式、 等式、 等式、 等式、 等式、 等式、 等式、 等式、 等式、 等式、 等式、 等式、 等式、 等式、 等式、 等式、 等式、 等式、 等式、 等式、 等式、 等式、 等式、 等式、 等式、 等式、 等式、 等式、 等式、 等式、 等式、 等式、 等式、 等式、 等式、 等式、 等式、 等式、