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

Abstract

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.

An Adaptive Reduced-Dimensional Discrete Element Model for Dynamic Responses of Granular Materials With High-Frequency Noises

“…In DEM simulations, the governing equation is originated from the balance principle (Ceccato et al, 2018;Cundall and Strack, 1979;Kuhn and Mitchell, 1993;Wang and Sun, 2015. Here, we consider a simplified version of DEM in which the rotational degree of freedom is neglected.…”

“…are the eigenvectors ofC. However, computing POM and POV directly from the matrix C orC is still expensive and instead the singular value decomposition (SVD) is adopted (Ceccato et al, 2018;Chatterjee, 2000):…”

“…The ∆T here is the stable time step of reduced-order integration, and it is subjected to the relationship ∆T < 2/ω max , where ω max is computed from |Φ T m KΦ m −ω 2 Φ T m MΦ m | = 0, and Φ T m KΦ m is the stiffness matrix of the subspace (Ceccato et al, 2018). Thus, a larger stable time step is permissible for the integration of the reduced-order system.…”

“…Therefore, we present a predictive approach that updates the POM and the reduced-order system during the simulation automatically, and thus the online stage can be combined with the offline stage. The automatic-updating POD procedure Ceccato et al (2018) is demonstrated below. Assume T cycle is a presetting parameter of the time interval (or the number of time steps) that gives the duration of each updating cycle.…”

Homogenization coarse graining (HCG) of the lattice discrete particle model (LDPM) for the analysis of reinforced concrete structures

“…Furthermore, LDPM has shown superior capabilities in modeling concrete behavior under dynamic loading [50,52], Alkali Silica Reaction (ASR) deterioration [53,54,55], failure, and fracture of fiber-reinforced concrete [56,57,58]. Finally, LDPM was used successfully to simulate ultra-high performance [59,60], waterless concrete [61], fiber reinforced polymer (FRP) confined concrete [49], rocks [62,63], and to build reduced-order models of failure [64].…”