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Research Papers

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

[+] Author and Article Information
Chiara Ceccato

Department of Civil, Architectural
and Environmental Engineering,
University of Padua,
Padua 35131, Italy

Xinwei Zhou, Daniele Pelessone

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

Gianluca Cusatis

Department of Civil and Environmental Engineering,
Northwestern University,
Evanston, IL 60208
e-mail: g-cusatis@northwestern.edu

1Corresponding author.

Contributed by the Applied Mechanics Division of ASME for publication in the JOURNAL OF APPLIED MECHANICS. Manuscript received November 27, 2017; final manuscript received January 5, 2018; published online March 7, 2018. Editor: Yonggang Huang.

J. Appl. Mech 85(5), 051004 (Mar 07, 2018) (13 pages) Paper No: JAM-17-1655; 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 (POMs), computed with snapshots from the full order simulation. Aiming to a predictive tool, the POMs are updated 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 (BCs). Results show that, for one-dimensional (1D) cases, just few modes are capable of excellent approximation of the solution, even in the case of 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 (3D) situations, the paper discusses the application of the developed algorithm to a discrete model called lattice discrete particle model (LDPM) 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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Figures

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Fig. 2

One-dimensional model of a dynamic discrete system

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Fig. 1

(a) Time step scheme and (b) and (c) definition of time steps in the transition from complete to reduced integration (b) and from reduced to complete integration (c)

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Fig. 4

Force versus displacement curves for the linear material and zoom in for (a) 1 mode, (b) 3 modes, and (c) 5 modes (51DOFs)

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Fig. 3

Shape of the first six POD modes

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Fig. 5

Force versus displacement curves for the softening material (a) without modes update, (b) with modes update and 1 mode, (c) and (d) with modes update and 2 modes, (e) with 12 automatic updates, and (f) with 8 automatic updates (51 DOFs)

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Fig. 6

(a) Shape of the first POM and displacements distribution along the system (b) after 1.0 ms (δ = 0.01 mm applied displacement) and (c) after 10 ms (δ = 0.1 mm applied displacement) using only the first POM in the reduced integration, (d) shape of the second POM and displacements distribution along the 1D system (b) after 1.0 ms (δ = 0.011 mm applied displacement), and (e) after 10 ms (δ = 0.1 mm applied displacement) using the first and second POMs in the reduced integration

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Fig. 7

Force distribution trend along the 1D system at the beginning of softening (δ = 0.025 mm applied displacement) for different numbers of POMs and snapshots (out of 600)

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Fig. 8

Mass scaling and POD for elastic response: (a) ΔtR = 50 Δt (mass scaling only), (b) ΔtR = 1000 dt (mass scaling only), (c) and (d) ΔtR = 50 dt, and (e) and (f) ΔtR = 1000 Δt

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Fig. 9

Load versus displacement curves for mass scaling and POD: (a) ΔtR = 10 Δt, (b) ΔtR = 25 Δt, and (c) ΔtR = 50 Δt (51 DOFs, softening material)

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Fig. 10

(a) LDPM polyhedral particle enclosing spherical aggregate pieces; (b) typical LDPM tetrahedron connecting four adjacent aggregates and its associated tessellation; and (c) tetrahedron portion associated with aggregate I

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Fig. 11

Dogbone shaped specimen for direct tension test: (a) LDPM geometry and (b) fracture pattern

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Fig. 12

Load versus displacement curves from the fully explicit simulation and the POD algorithm (a) with different numbers of POMs and fixed number of snapshots (1000), (b) with different numbers of snapshots and fixed number of POMs (900), and (c) with automatic updates (2 POM and 10 snapshots)

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Fig. 13

Load versus displacement curves for the concrete cylinder under unconfined and confined compression, from the fully explicit simulation and the POD algorithm: (a) unconfined compression simulations with increasing number of POMs (500 snapshots), (b) with increasing number of snapshots (100 POMs), (c) with automatic updates (2 POMs, 10 snapshots), (d) confined compression simulations with increasing number of POMs (500 snapshots), (e) with increasing number of snapshots (100 POMs), and (f) with automatic updates (2 POMs, 10 snapshots)

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