Certified reduced basis method for variational inequalities

• Zertifizierte Reduzierte-Basis-Methode für Variationsungleichungen

Zhang, Zhenying; Veroy-Grepl, Karen Paula (Thesis advisor); Herty, Michael Matthias (Thesis advisor)

Aachen (2016)
Dissertation / PhD Thesis

Dissertation, RWTH Aachen University, 2016

Abstract

In this work, we present certified reduced basis methods for variational inequalities of the first kind. The proposed approaches allow us to compute real-time solutions online efficiently with notably shaper error bounds.The reduced basis method was developed over the last decades for rapid and reliable model order reduction for parametrized partial differential equations (PDEs). This method is designed to approximate high dimensional finite element problems through low dimensional surrogates for many-query PDE solutions and real-time contexts of optimization, parameter estimation, and control. One of the many merits of the reduced basis method is the significantly low computational cost, which permits rapid simulation, whereas the rigorous error estimators provide quantified quality control with reference to the high dimensional finite element solutions. The method therefore enables scientists and engineers to make real-time decisions based on reliable information.Based on previous research, this work proposes new reduced basis approaches and improves the a posteriori error estimators with regard to both the computational costs and the sharpness of the error bounds. This is achieved as follows: First, we present abstract elliptic variational inequalities in different forms and their finite element approximations. Second, we summarize the existing approach for variational inequalities and propose new reduced basis approaches and the associated error estimators. Third, we validate discussed approaches against various numerical models for comparison. Lastly, we extend proposed approaches to parabolic variational inequalities.Motivated by applications in the field of solid mechanics, the proposed approaches are applied to several contact problems. Providing comparison between the proposed approaches and the existing one, numerical results illustrate several advantages of the proposed approaches in practice.

Institutions

• Aachen Institute for Advanced Study in Computational Engineering Science [080003]
• High-Performance Computing of Engineering Models Teaching and Research Area [316320]
• Lehr- und Forschungsgebiet Mathematik [114620]
• Department of Mathematics [110000]