Conference Agenda

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 has been extended to nonlinear Koiter shells. 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 tangential-displacement and normal-normal-stress continuous (TDNNS) method has successfully been applied to linear Reissner-Mindlin plates leading to a shear locking free formulation.

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 Nedelec 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.

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.

We model the formation of wrinkles of an elastic substrate coated with a thin film. The elastic substrate is first stretched, then the film is attached to a part of the substrate boundary in the deformed state. Once the the external force is released, wrinkles form due to the stress mismatch between the two materials. The elastic substrate is modeled using a hyperelastic, homogeneous and isotropic material. The film is modeled using a geometrically exact Cosserat shell. The resulting deformation and microrotation $(varphi, R)$ are a minimizing pair of the combined energy functional

$$

J(varphi, R) = int_{Omega} W_textup{bulk}(nablavarphi) : dV + int_{Gamma_c}W_textup{coss}(nablavarphi_{|_{Gamma_c}}, R) : dS

$$

in the admissible set

begin{align*}

mathcal{A} = Big{&(varphi, R) in W^{1,q}(Omega, mathbb{R}^3) times H^1(Gamma_c,textup{SO(3)}) : Big| :

varphi textnormal{ is a deformation function, }& (varphi, R) textnormal{ fullfill the Dirichlet boundary conditions} Big}

end{align*}

with $q > 3$.

We discretize the problem using Lagrange finite elements for the substrate displacement. For the numerical treatment of the microrotation field, standard Lagrange finite elements cannot be used, as the microrotation field maps to the nonlinear manifold $textnormal{SO}(3)$. We present a generalization of Lagrange finite elements that is suitable for such manifold-valued functions: geometric finite elements.

The resulting finite element spaces are complete and invariant under isometries of the manifold. The best approximation error depends on the mesh size h. We prove the existence of solutions of the discrete coupled model. We compare two Newton-type methods to solve the resulting discrete problem: a Riemannian trust-region method and a Riemannian proximal Newton method.

Numerical experiments show that we can efficiently reproduce wrinkling patterns of coupled systems. Our approach works as well for more complex scenarios like multi-layer systems or systems involving various stress-free configurations.

In many different fields of engineering beam models play a significant role in the efficient simulation of slender structures. The most important model for large deformations is the so-called geometrically exact beam also often referred to as Simo-Reissner beam. The configuration manifold of the beam model is given by special Euclidian group as it describes the position of the centerline as well as the orientation of the beam's cross-section. The partial differential equations describing the behavior of the beam is usually solved with the help of the Finite Element Method (FEM). So it becomes necessary to discretize the special orthogonal group in a finite element sense.

A finite element discretization of the special orthogonal group is rather difficult as the special orthogonal group is not an abelian, additive group but a matrix group under multiplication. Though there exist parametrizations of the orthogonal group, which have an additive structure, they result in path-dependency. This can be overcome by discretizing the group directly by using so-called directors. The directors can be discretized additively, so in a classical finite element sense. This, however, leads to an increase in the number of degrees of freedom

if Lagrange multipliers are used to ensure the orthonormality of the directors. Further, this formulation does not conserve the structure of the manifold at every point of the discretization. A possible remedy could be a projection method via the polar decomposition, which is very costly in numerical terms.

The use of unit quaternions for the parametrization presents an interesting alternative. Even though unit quaternions have a complex mathematical structure, it can easily be ensured that their unit length is conserved after a finite element discretization by normalizing the discretized quaternions. This still allows for a classical additive discretization technique in a finite element sense.

In the literature, it is often shown that the Isogeometric Analysis (IGA) is advantageous over the classical FEM with Lagrangian elements, especially for dynamic problems. We thus apply the IGA to the quaternion formulation of the geometrically exact beam.