TY - JOUR

T1 - Long-time principal geodesic analysis in director-based dynamics of hybrid mechanical systems

AU - Gebhardt, Cristian G.

AU - Schubert, Jenny

AU - Steinbach, Marc C.

N1 - Funding Information: The second and third authors are funded by the Deutsche Forschungsgemeinschaft (DFG, German Research Foundation) – SFB1463 – 434502799. The authors acknowledge as well the University of Bergen for the open access funding. Finally the authors would like to thank two anonymous referees for their careful reading and constructive feedback.

PY - 2023/7

Y1 - 2023/7

N2 - In this article, we investigate an extended version of principal geodesic analysis for the unit sphere S2 and the special orthogonal group SO(3). In contrast to prior work, we address the construction of long-time smooth lifts of possibly non-localized data across branches of the respective logarithm maps. To this end, we pay special attention to certain critical numerical aspects such as singularities and their consequences on the numerical accuracy. Moreover, we apply principal geodesic analysis to investigate the behavior of several mechanical systems that are very rich in dynamics. The examples chosen are computationally modeled by employing a director-based formulation for rigid and flexible mechanical systems. Such a formulation allows to investigate our algorithms in a direct manner while avoiding the introduction of additional sources of error that are unrelated to principal geodesic analysis. Finally, we test our numerical machinery with the examples and, at the same time, we gain deeper insight into their dynamical behavior.

AB - In this article, we investigate an extended version of principal geodesic analysis for the unit sphere S2 and the special orthogonal group SO(3). In contrast to prior work, we address the construction of long-time smooth lifts of possibly non-localized data across branches of the respective logarithm maps. To this end, we pay special attention to certain critical numerical aspects such as singularities and their consequences on the numerical accuracy. Moreover, we apply principal geodesic analysis to investigate the behavior of several mechanical systems that are very rich in dynamics. The examples chosen are computationally modeled by employing a director-based formulation for rigid and flexible mechanical systems. Such a formulation allows to investigate our algorithms in a direct manner while avoiding the introduction of additional sources of error that are unrelated to principal geodesic analysis. Finally, we test our numerical machinery with the examples and, at the same time, we gain deeper insight into their dynamical behavior.

KW - Director-based dynamics of mechanical systems

KW - Long-time principal geodesic analysis

KW - Singularities and numerical accuracy

KW - Special orthogonal group SO(3)

KW - Unit sphere S

UR - http://www.scopus.com/inward/record.url?scp=85151300724&partnerID=8YFLogxK

U2 - 10.1016/j.cnsns.2023.107240

DO - 10.1016/j.cnsns.2023.107240

M3 - Article

AN - SCOPUS:85151300724

VL - 122

JO - Communications in Nonlinear Science and Numerical Simulation

JF - Communications in Nonlinear Science and Numerical Simulation

SN - 1007-5704

M1 - 107240

ER -