3D-shell element technology, nonlinear poisson stiffening and data-integrated time step estimation

Abstract

This thesis deals with three aspects of the finite element method: the development of three-dimensional shell elements, the explanation of a previously largely unknown stiffening effect, and a data-integrated approach for estimating the critical time step size in simulations with explicit time integration. A family of three-dimensional finite shell elements is developed by describing the velocity field as a polynomial in the out-of-plane coordinate while discretizing the shell plane with bilinear Lagrange polynomials. The elements use reduced integration in combination with an hourglass control. Numerical examples show that the developed shell elements with a quadratic or cubic velocity field in the out-of-plane coordinate yield more accurate results compared to elements with linear velocity field or elements based on the Reissner-Mindlin shell model. The thesis furthermore investigates and explains the nonlinear Poisson stiffening effect, a previously largely unknown stiffening effect. This effect is a nonlinear variant of Poisson thickness locking. Its causes and symptoms as well as mitigation strategies are discussed. Additionally, its relevance is demonstrated with a series of numerical examples. For explicit dynamic simulations, the thesis introduces a data-integrated approach to estimate the critical time step size. A time step estimator based on a neural network is developed for quadrilateral two-dimensional solid elements and shell elements with a linear velocity field in the out-of-plane coordinate. An update algorithm enhances computational efficiency, making this time step estimation method applicable in practical simulations.