Selected space-time integration strategies for linear and nonlinear frame structures

• Ausgewählte Raum-Zeit-Integrationsstrategien für lineare und nichtlineare Rahmenstrukturen

Shirafkan, Nima; Markert, Bernd (Thesis advisor); de Saxcé, Géry (Thesis advisor)

Aachen : RWTH Aachen University (2022, 2023)
Dissertation / PhD Thesis

Dissertation, Rheinisch-Westfälische Technische Hochschule Aachen, 2022

Abstract

Evaluating the response history of engineering structures subjected to external force functions constitutes a key issue in structural mechanics. Classical solution techniques entail step-by-step time integration schemes and reveal several drawbacks. In order to save calculation time and make solution strategies more efficient, several approximation schemes and model order reduction techniques have been proposed and applied to a wide range of structural problems. In this thesis, a new space-time approach is presented and applied to solve and analyse the nonlinear and linear problems using the Proper Generalized Decomposition (PGD).Regarding nonlinear quasi static problems, the equation of equilibrium is transformed into its space-time equivalent. To find the solution ansatz for structural problems, the set of equations of motion is also transformed into its space-time equivalent, using a space-time formulation of the direct integration ansatz such as the Newmark, the Finite Difference Method, and the Euler scheme. In this way, the mechanical system is then represented by one algebraic equation only. The solution, dependent on space and time, is built by iteratively summing up enrichments until convergence is achieved. Each enrichment step is represented by the dyadic product of spatial and temporal functions, which are called the spatial and temporal modes. The space-time formulation in nonlinear quasi-static and linear dynamic problems is projected in two coupled space and time equations, which are solved simultaneously using the fixed point algorithm. The evaluation of the spatial and temporal modes within the proposed iterative scheme is shown and investigated during convergence and interpreted by comparing them to the set of lower modes of vibration. Furthermore, the results of the proposed space-time formulation are compared with the conventional step-by-step algorithm regarding accuracy and efficiency. The new method is demonstrated using several numerical examples, presenting not only the excellent convergence behavior and the numerical efficiency but also the limits of the proposed approach. To this aim, four different 2D beam and frame structures subjected to the transient and harmonic ground excitations and impact forces are investigated and the solution of these linear dynamic problems is compared with conventional direct integration schemes. In addition, an elastoplastic frame structure subjected to an external cyclic force history is analysed and the spatial and temporal modes are evaluated and their behavior is mechanically interpreted. It is shown that inelastic deformation can be directly located by the evolution of the temporal functions. However, the corresponding spatial modes show a high correlation containing global elastic and plastic deformation patterns that can be interpreted by an orthonormal basis representation using the singular value decomposition.