Isogeometric continuity constraints for multi-patch shells governed by fourth-order deformation and phase field models
arXiv:2001.05964 · doi:10.1016/j.cma.2020.113219
Abstract
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 -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 -continuous formulation. Isogeometric surface discretizations - often composed of multiple patches - thus require constraints that enforce the -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.
arXiv commentIn this version, typos in the list of references were fixed, floating pictures in Chapter 6 were reduced, and the sign of the bending moments was changed (see text passages after Eqs. (63) and (71), footnotes 13-14 and 23, and Appendix A
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Cited by in corpus (5)
- A simple and efficient hybrid discretization approach to alleviate membrane locking in isogeometric thin shells
- A comparison of smooth basis constructions for isogeometric analysis
- A spatio-temporal adaptive phase-field fracture method
- An isogeometric finite element formulation for boundary and shell viscoelasticity based on a multiplicative surface deformation split
- A general isogeometric finite element formulation for rotation-free shells with in-plane bending of embedded fibers