Tangential differential calculus for curved, linear kirchhoff beams with systematic convergence studies

We propose a reformulation of linear Kirchhoff beams in two dimensions based on the tangential differential calculus (TDC). The rotation-free formulation of the Kirchhoff beam is classically based on curvilinear coordinates. However, for general applications in engineering and sciences that take place on curved geometries embedded in a higher-dimensional space, the tangential differential calculus enables a formulation independent of curvilinear coordinates and, hence, is suitable also for implicitly defined geometries. The geometry and differential operators are formulated in global Cartesian coordinates related to the embedding space. Isogeometric analysis (IGA) is employed for the generation of shape functions in the numerical analysis because Kirchhoff kinematics require C₁-continuous shape functions. The boundary conditions are enforced using Lagrange multipliers. We emphasize systematic convergence studies for established and new test cases by investigating residual errors. Therefore, the approximated solution obtained by the FEM is inserted into the strong form of the governing equations in a post-processing step. The error is then integrated over the domain in an L₂-sense. For sufficiently smooth physical fields, higher-order convergence rates in the residual errors are achieved. For classical benchmark test cases with known analytical solutions, we also confirm optimal convergence rates in the displacements.