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Isogeometric continuity constraints for multi-patch shells governed by fourth-order deformation and phase field models. (English) Zbl 1506.74429

Summary: 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: A Lagrange multiplier formulation and a penalty method. The curvilinear shell model including the geometrical constraints is taken from [the third author et al., ibid. 316, 43–83 (2017; Zbl 1439.74409)] and it is extended to model the coupled phase field problems on thin shells of the second author et al. [ibid. 351, 441–477 (2019; Zbl 1441.74286)] and the first author et al. [Comput. Mech. 65, No. 4, 1039–1062 (2020; Zbl 1464.74150)] 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.

MSC:

74S05 Finite element methods applied to problems in solid mechanics
65N30 Finite element, Rayleigh-Ritz and Galerkin methods for boundary value problems involving PDEs
65D07 Numerical computation using splines
74K25 Shells
74S22 Isogeometric methods applied to problems in solid mechanics
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References:

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