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.

中文翻译：

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由四阶变形和相场模型控制的多面片壳的等几何连续性约束

这项工作提出了数值技术，以对三个不同问题类别的多面片表面实施连续性约束。第一个涉及由一般 Kirchhoff-Love 运动学描述的薄壳的结构分析。他们的控制方程是矢量值的四阶非线性偏微分方程 (PDE)，在基于位移的有限元公式中至少需要 $C^1$-连续性。第二类是由相场建模的表面相分离。他们的控制方程是 Cahn-Hilliard 方程——一个标量、四阶、非线性 PDE——可以耦合到薄壳 PDE。第三类是通过相场方法模拟的脆性断裂过程。在这项工作中，这些由标量、四阶、类似于 Cahn-Hilliard 方程的非线性 PDE，也耦合到薄壳 PDE。使用直接有限元离散化，两相场方程还需要至少一个 $C^1$-连续公式。等几何表面离散化 - 通常由多个补丁组成 - 因此需要强制执行 $C^1$-位移和相场连续性的约束。为此，提出了两种数值策略：为此，提出了两种数值策略：拉格朗日乘数公式和惩罚方法。包括几何约束的曲线壳模型取自 Duong 等人。(2017) 并扩展到对 Zimmermann 等人的薄壳上的耦合相场问题进行建模。(2019) 和保罗等人。(2020) 关于多补丁。