Proper Orthogonal Decomposition Framework for the Explicit Solution of Discrete Systems with Softening Response

[+] Author and Article Information
Chiara Ceccato

Dept. of Civil, Architectural and Environmental Eng., University of Padua, Padua, 35131, Italy

Xinwei Zhou

Engineering and Software System Solutions, Inc. (ES3), San Diego, CA 92101, USA

Daniele Pelessone

Engineering and Software System Solutions, Inc. (ES3), San Diego, CA 92101, USA

Gianluca Cusatis

Dept. of Civil and Environmental Eng., Northwestern University, Evanston, IL 60208, USA

1Corresponding author.

ASME doi:10.1115/1.4038967 History: Received November 27, 2017; Revised January 05, 2018


The application of explicit dynamics to simulate quasi-static events often becomes impractical in terms of computational cost. Different solutions have been investigated in the literature to decrease the simulation time and a family of interesting, increasingly adopted approaches, are the ones based on the Proper Orthogonal Decomposition (POD) as a model reduction technique. In this study, the algorithmic framework for the integration of the equation of motions through POD is proposed for discrete linear and nonlinear systems: a low dimensional approximation of the full order system is generated by the so called Proper Orthogonal Modes (POM), computed with snapshots from the full order simulation. Aiming to a predictive tool, the POMs are updated \emph{in itinere} alternating the integration in the complete system, for the snapshots collection, with the integration in the reduced system. The paper discusses details of the transition between the two systems and issues related to the application of essential and natural boundary conditions. Results show that, for one dimensional cases, just few modes are capable of excellent approximation of the solution, even in the case stress-strain softening behavior, allowing to conveniently increase the critical time step of the simulation without significant loss in accuracy. For more general three dimensional situations, the paper discusses the application of the developed algorithm to a discrete model formulated to simulate quasi-brittle materials characterized by a softening response. Efficiency and accuracy of the reduced order LDPM response are discussed with reference to both tensile and compressive loading conditions.

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