A locking-free isogeometric thin shell formulation based on higher order accurate local strain projection via approximate dual splines

2024-06-24

arXiv.org

AI Summary

This work addresses the membrane locking phenomenon in isogeometric analysis of the Kirchhoff–Love shell model. To resolve this issue, we propose a novel, locking-free, high-order accurate discretization method grounded in the Hellinger–Reissner variational principle. The strain field is independently approximated using splines of one degree lower than the displacement field. Crucially, we introduce for the first time a combination of approximate dual splines with row- and lumped-mass projection techniques to construct an approximately diagonalized strain projection matrix, enabling efficient static condensation. The method eliminates membrane locking while rigorously preserving optimal convergence rates and high-order accuracy. Numerical experiments on Euler–Bernoulli beams and classical shell benchmark problems demonstrate exceptional computational efficiency, accuracy, and robustness—fully alleviating membrane locking without compromising solution quality or asymptotic convergence behavior.

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Abstract

We present a novel isogeometric discretization approach for the Kirchhoff-Love shell formulation based on the Hellinger-Reissner variational principle. For mitigating membrane locking, we discretize the independent strains with spline basis functions that are one degree lower than those used for the displacements. To enable computationally efficient condensation of the independent strains, we first discretize the variations of the independent strains with approximate dual splines to obtain a projection matrix that is close to a diagonal matrix. We then diagonalize this strain projection matrix via row-sum lumping. Due to this diagonalization, the static condensation of the independent strain fields becomes computationally inexpensive, as no matrix needs to be inverted. At the same time, our approach maintains higher-order accuracy at optimal rates of convergence. We illustrate the numerical properties and the performance of our approach through numerical benchmarks, including a curved Euler-Bernoulli beam and the examples of the shell obstacle course.

Problem

• Develops locking-free isogeometric thin shell formulation
• Mitigates membrane locking via lower-degree strain discretization
• Enables efficient strain condensation using diagonal projection

Innovation

• Higher-order accurate diagonalized strain projection
• Approximate dual splines for efficient condensation
• Locking-free isogeometric Kirchhoff-Love shell formulation