The Hellan–Herrmann–Johnson and TDNNS method for nonlinear Koiter and Naghdi shells
Date:
The slides can be found here.
Abstract:
The development of effective and locking free shell elements is intensive topic of research since several decades. Recently, the Hellan–Herrmann–Johnson (HHJ) method for linear Kirchhoff–Love plates [M. Comodi, The Hellan-Herrmann-Johnson method: some new error estimates and postprocessing, Mathematics of Computation, 52 (1989), 17–29] has been extended to nonlinear Koiter shells [M. Neunteufel, J. Schöberl, The Hellan–Herrmann–Johnson method for nonlinear shells, Computers & Structures, 225 (2019), 106109]. Therein, the bending moment tensor is introduced as additional unknown to rewrite the fourth order as a second order mixed saddle point problem circumventing the necessity of C1-conforming finite elements. Via hybridization techniques the saddle point translates into a minimization problem again.
The TDNNS method has successfully been applied to Reissner–Mindlin plates leading to a shear locking free formulation [A. Pechstein and J. Schöberl, The TDNNS method for Reissner-Mindlin plates, J. Numer. Math. (2017) 137, pp. 713-740.].
In this talk we present a shear locking free extension of the TDNNS method from linear Reissner–Mindlin plates to nonlinear Naghdi shells by means of a hierarchical approach. Therefore, the HHJ method for Koiter shells is enriched with shearing degrees of freedom, discretized by H(curl)-conforming Nédélec elements. We discuss the small-strain regime leading to the HHJ and TDNNS method for linear Koiter and Naghdi shells. We show how the so-called Regge interpolant can be used in all methods to avoid membrane locking by inserting into the membrane energy term [M. Neunteufel, J. Schöberl, Avoiding membrane locking with Regge interpolation, Comput. Methods Appl. Mech. Eng., 373 (2021), 113524].
Several benchmark examples, implemented in the open-source finite element software NGSolve (www.ngsolve.org), are presented to demonstrate the excellent performance of the proposed shell elements.