Embedded shell finite elements: Solid–shell interaction, surface locking, and application to image-based bio-structures

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In this article, we explore an embedded shell finite element method for the unfitted discretization of solid–shell interaction problems. Its core component is a variationally consistent approach that couples a shell discretization on the surface of an embedded solid domain to its unfitted discretization with hexahedral solid elements. Derived via an augmented Lagrangian formulation and the formal elimination of interface Lagrange multipliers, our method depends only on displacement variables, facilitated by a shift of the displacement-dependent traction vector entirely to the solid structure. We demonstrate that the weighted least squares term required for stability of the formulation triggers severe surface locking due to a mismatch in the polynomial spaces of the shell element and the embedding solid element. We show that reduced quadrature of the stabilization term that evaluates the kinematic constraint at the nodes of the embedded shell elements completely mitigates surface locking. For coarse discretizations, our variationally consistent method achieves superior accuracy with respect to a locking-free nodal penalty method. We illustrate the versatility of embedded shell finite elements for image-based analysis, including patient-specific stress prediction in a vertebra and local rind buckling in a plant structure.

Original languageEnglish (US)
Pages (from-to)298-326
Number of pages29
JournalComputer Methods in Applied Mechanics and Engineering
StatePublished - Jun 15 2018

Bibliographical note

Funding Information:
The first author (D. Schillinger) gratefully acknowledges support from the National Science Foundation via the NSF grant CISE-1565997 and the NSF CAREER Award No. 1651577 , and from the Minnesota Department of Agriculture under Grant No. 122130 . The second author (T. Gangwar) is partially supported by a Sommerfeld Fellowship awarded by the Department of Civil, Environmental, and Geo- Engineering at the University of Minnesota , which is gratefully acknowledged. The authors also acknowledge the Minnesota Supercomputing Institute (MSI) of the University of Minnesota for providing computing resources that have contributed to the research results reported within this paper ( https://www.msi.umn.edu/ ).

Publisher Copyright:
© 2018 Elsevier B.V.


  • Embedded shell finite elements
  • Reduced quadrature
  • Rotation-free shell formulation
  • Solid–shell interaction
  • Surface locking
  • Voxel finite elements


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