Dynamic Response of Nonlinear Problems with Large Rotations

Classical nonlinear beam theories have been studied independently by the research groups using the finite element method in Taiwan and Austria in the past years, each resulting in a mainstream of research papers published in reputed scientific journals. The former was mainly led by Prof. Y. B. Yang of National Taiwan University, and the latter by Prof. H. Irschik and Doz. J. Gerstmayr of the Johannes Kepler University Linz and of the Linz Center of Mechatronics, Austria. However, different approaches were adopted by the two groups. Both approaches have their own merits, but can be still improved to form a new and possibly better joint approach. The Austrian side has put much emphasis on the large displacement - finite deformation of beams based on the continuum mechanics virtual work principle, i.e., in terms of the 2nd Piola-Kirchhoff stress and Green strain. Such a continuum mechanics approach allows us to consider technically advanced material laws, which are based on generally nonlinear constitutive relations between stress and strain. The approach by Reissner (1972) based on structural mechanics, however, is considered to release some conceptual difficulties of defining the constitutive laws at the beam level. Among others, an absolute nodal coordinate formulation has been presented by the Austrian side, which allows inclusion of linearly elastic, hyperelastic or elasto-plastic materials in an easy way. At the present stage, the absolute nodal coordinate formulation suffers from a larger number of degrees of freedom and high computational costs. The formulation has been developed and tested for the analysis and simulation of dynamical problems of beams undergoing large rigid body motion and large deformations. In contrast, the Taiwan side has formulated their beam theory based on the updated Lagrangian formulation, with specific emphasis on application of the rigid body rule (Yang and Chiou 1987) in each stage of the analysis. In each incremental step of the analysis, division is made between the rigid rotation and natural deformation of each finite element, with procedures developed for dealing with each component. The nodal moments adopted for each element are the moments defined as stress resultants and the corresponding degrees of freedom are the rotations defined as the derivatives of transverse displacements. Such a pair of nodal quantities can be conveniently used by engineers working on design practice, but have the drawback of not being energetically conjugate (Reissner 1972). With the equilibrium of each joint defined in the deformed configuration, this approach has been extended to analysis of three-dimensional frames. In this project, both the Taiwan and Austrian approaches will be merged to obtain new beam finite elements for the dynamic analysis of beams and space frames involving large rotations. While the absolute nodal coordinate formulation up to now allows the inclusion of advanced material laws and dynamical effects, it is not efficient and shows low order convergence. Thus, the results of the Taiwan group will help to improve existing elements of the Austrian group and to develop efficient and high order convergent elements using e.g. the division into rigid body rotation and natural deformation. Existing numerical techniques based on corotational approaches of the Austrian group (Gerstmayr and Schöberl, 2006) will lead to synergies with the updated Lagrangian formulation of the Taiwan group. As a verification of the outcome of the joint project, existing benchmark problems will be evaluated and new benchmark problems will be established, along with some experimental testing of simple problems. As an essential part, the joint project will include exchange visits and joint workshops.

Aktuelle Herausforderungen der Modellierung und Simulation von Mehrkörpersystem sind insbesondere durch die Zusammenführung von CAD (Computer Aided Design) Daten und numerischen Simulationen, durch eine effiziente numerische Berechnung und Implementierung, sowie durch die korrekte mathematische Beschreibung des Materialverhaltens (Stoffgesetze) gegeben.Die taiwanesische und die österreichische Gruppe des vorliegenden Forschungsprojekts haben bereits im Vorfeld Mehrkörpersysteme mit verschiedenen Ansätzen studiert. Um die Vorteile der entstandenen Methoden und Theorien einander nutzbar zu machen, wurde der Schwerpunkt des vorliegenden Projektes auf die Zusammenarbeit der beiden Gruppen gelegt. Durch regelmäßige Projekttreffen (mindestens 1x pro Jahr) wurde der Ansatz der österreichischen Gruppe einer vollständig nichtlinearen Formulierung für strukturmechanische Elemente (Balken und Schalen) erweitert und ist nach und nach mit der Formulierung der taiwanesischen Gruppe verschmolzen.