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A consistent mixed-dimensional coupling approach for 1D Cosserat beams and 2D surfaces in 3D space

Ivo Steinbrecher, Nora Hagmeyer, Christoph Meier, Alexander Popp

TL;DR

This work delivers a rigorous 1D-2D mixed-dimensional coupling framework for linking geometrically exact Cosserat beams to 2D surfaces in 3D, applicable to both solid and shell surfaces. It introduces a fully consistent BTS-FULL scheme that couples beam centerline positions and cross-section orientations via mortar-type constraints, with a surface-triad construction to support rotational coupling. A key contribution is proving exact linear and angular momentum conservation for the consistent BTS-POS-CONS variant, even with nonzero surface normal distance, and demonstrating via numerical examples (including a vascular-stent-like FSI scenario) that consistency is essential for physical accuracy. The approach enables efficient, accurate simulations of complex beam-surfaces in multi-physics contexts and is extendable to surface-to-surface mesh tying problems where non-matching interfaces occur.

Abstract

The present article proposes a novel computational method for coupling arbitrarily curved 1D fibers with a 2D surface as defined, e.g., by the 2D surfaces of a 3D solid body or by 2D shell formulations. The fibers are modeled as 1D Cosserat continua (beams) with six local degrees of freedom, three positional and three rotational ones. A kinematically consistent 1D-2D coupling scheme for this problem type is proposed considering the positional and rotational degrees of freedom along the beams. The positional degrees of freedom are coupled by enforcing a constant normal distance between a point on the beam centerline and a corresponding point on the surface. This strategy requires a consistent description of the surface normal vector field to guarantee fundamental mechanical properties such as conservation of angular momentum. Coupling of the rotational degrees of freedom of the beams and a suitable rotation tensor representing the local orientation within a solid volume has been considered in a previous contribution. In the present work, this coupling approach will be extended by constructing rotation tensors that are representative of local surface orientations. Several numerical examples demonstrate the consistency, robustness and accuracy of the proposed method. To showcase its applicability to multi-physics systems of practical relevance, the fluid-structure interaction example of a vascular stent is presented.

A consistent mixed-dimensional coupling approach for 1D Cosserat beams and 2D surfaces in 3D space

TL;DR

This work delivers a rigorous 1D-2D mixed-dimensional coupling framework for linking geometrically exact Cosserat beams to 2D surfaces in 3D, applicable to both solid and shell surfaces. It introduces a fully consistent BTS-FULL scheme that couples beam centerline positions and cross-section orientations via mortar-type constraints, with a surface-triad construction to support rotational coupling. A key contribution is proving exact linear and angular momentum conservation for the consistent BTS-POS-CONS variant, even with nonzero surface normal distance, and demonstrating via numerical examples (including a vascular-stent-like FSI scenario) that consistency is essential for physical accuracy. The approach enables efficient, accurate simulations of complex beam-surfaces in multi-physics contexts and is extendable to surface-to-surface mesh tying problems where non-matching interfaces occur.

Abstract

The present article proposes a novel computational method for coupling arbitrarily curved 1D fibers with a 2D surface as defined, e.g., by the 2D surfaces of a 3D solid body or by 2D shell formulations. The fibers are modeled as 1D Cosserat continua (beams) with six local degrees of freedom, three positional and three rotational ones. A kinematically consistent 1D-2D coupling scheme for this problem type is proposed considering the positional and rotational degrees of freedom along the beams. The positional degrees of freedom are coupled by enforcing a constant normal distance between a point on the beam centerline and a corresponding point on the surface. This strategy requires a consistent description of the surface normal vector field to guarantee fundamental mechanical properties such as conservation of angular momentum. Coupling of the rotational degrees of freedom of the beams and a suitable rotation tensor representing the local orientation within a solid volume has been considered in a previous contribution. In the present work, this coupling approach will be extended by constructing rotation tensors that are representative of local surface orientations. Several numerical examples demonstrate the consistency, robustness and accuracy of the proposed method. To showcase its applicability to multi-physics systems of practical relevance, the fluid-structure interaction example of a vascular stent is presented.
Paper Structure (36 sections, 62 equations, 27 figures, 3 tables)

This paper contains 36 sections, 62 equations, 27 figures, 3 tables.

Table of Contents

  1. Introduction
  2. Problem formulation
  3. Finite rotations
  4. Structure formulation
  5. Solid formulation
  6. Kirchhoff--Love shell formulation
  7. Geometrically exact beam theory
  8. Beam-to-surface coupling (BTS-FULL)
  9. Closest point projection
  10. Positional beam-to-surface coupling (BTS-POS)
  11. Consistent positional coupling (BTS-POS-CONS)
  12. Forced reference configuration coupling (BTS-POS-REF)
  13. Displacement coupling (BTS-POS-DISP)
  14. Rotational beam-to-surface coupling (BTS-ROT)
  15. Surface triad field
  16. ...and 21 more sections

Figures (27)

  • Figure 1: Types of considered beam-to-surface scenarios in this work, \ref{['fig:introduction:problem_types:solid']} the beam is coupled to the 2D surface of a classical 3D Boltzmann continua (beam-to-solid-surface scenario) and \ref{['fig:introduction:problem_types:shell']} the beam is coupled to a reduced dimensional 2D shell formulation (beam-to-shell-surface scenario).
  • Figure 2: \ref{['fig:introduction:kelvin_flamant:kelvin']} The Kelvin problem of an embedded line load acting on an infinite solid and \ref{['fig:introduction:kelvin_flamant:flamant']} the Flamant problem of a line load acting on an infinite solid half space.
  • Figure 3: Notation for the finite deformation BTS-FULL coupling problem.
  • Figure 4: Illustration of possible beam-to-surface coupling problems. For the beam-to-solid-surface scenario: \ref{['fig:problem:btss_possible:curved_wo_offset']} beam centerline on a surface, \ref{['fig:problem:btss_possible:curved']} beam centerline offset by the cross-section radius in surface normal direction, and \ref{['fig:problem:btss_possible:flipped']} a general non-matching case. For the beam-to-shell-surface scenario: \ref{['fig:problem:btss_possible:curved_wo_offset_shell']} beam centerline on a shell midsurface, \ref{['fig:problem:btss_possible:curved_shell']} beam centerline offset by the cross-section radius and half of the shell thickness in shell midsurface normal direction, and \ref{['fig:problem:btss_possible:flipped_shell']} a general non-matching case.
  • Figure 5: Illustration of the different positional beam-to-surface (BTS-POS) coupling variants. \ref{['fig:problem:trans:cons']} Consistent positional coupling (BTS-POS-CONS) via the surface normal vector, \ref{['fig:problem:trans:ref']} forced reference configuration coupling (BTS-POS-REF) by forcing beam centerline points to lie on the surface and \ref{['fig:problem:trans:disp']} displacement coupling (BTS-POS-DISP), where the displacement of beam centerline and surface are coupled. The surface-to-surface equivalents of the BTS-POS-REF and BTS-POS-DISP variants are commonly used in classical surface-to-surface mesh tying problems Puso2004.
  • ...and 22 more figures

Theorems & Definitions (4)

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