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.

Copyright © 2018 by ASME
Your Session has timed out. Please sign back in to continue.


Cocchetti, G. , Pagani, M. , and Perego, U. , 2013, “ Selective Mass Scaling and Critical Time-Step Estimate for Explicit Dynamics Analyses With Solid-Shell Elements,” Comput. Struct., 127, pp. 39–52. [CrossRef]
Noh, G. , and Bathe, K.-J. , 2013, “ An Explicit Time Integration Scheme for the Analysis of Wave Propagations,” Comput. Struct., 129, pp. 178–193. [CrossRef]
Frias, G. J. D. , Aquino, W. , Pierson, K. H. , Heinstein, M. W. , and Spencer, B. W. , 2014, “ A Multiscale Mass Scaling Approach for Explicit Time Integration Using Proper Orthogonal Decomposition,” Int. J. Numer. Methods Eng., 97(11), pp. 799–818. [CrossRef]
Gao, L. , and Calo, V. M. , 2014, “ Fast Isogeometric Solvers for Explicit Dynamics,” Comput. Methods Appl. Mech. Eng., 274, pp. 19–41. [CrossRef]
Chang, S.-Y. , 2010, “ A New Family of Explicit Methods for Linear Structural Dynamics,” Comput. Struct., 88(11–12), pp. 755–772. [CrossRef]
Jia, J. , 2014, Essentials of Applied Dynamic Analysis (Risk Engineering), Springer, Berlin. [CrossRef]
Olovsson, L. , Simonsson, K. , and Unosson, M. , 2005, “ Selective Mass Scaling for Explicit Finite Element Analyses,” Int. J. Numer. Methods Eng., 63(10), pp. 1436–1445. [CrossRef]
Ducobu, F. , Rivière-Lorphèvre, E. , and Filippi, E. , 2015, “ On the Introduction of Adaptive Mass Scaling in a Finite Element Model of ti6al4v Orthogonal Cutting,” Simul. Modell. Pract. Theory, 53, pp. 1–14. [CrossRef]
Paz, M. , 1984, “ Dynamic Condensation,” AIAA J., 22(5), pp. 724–727. [CrossRef]
Xiao, M. , Breitkopf, P. , Coelho, R. F. , Villon, P. , and Zhang, W. , 2014, “ Proper Orthogonal Decomposition With High Number of Linear Constraints for Aerodynamical Shape Optimization,” Appl. Math. Comput., 247, pp. 1096–1112.
Behzad, F. , Helenbrook, B. T. , and Ahmadi, G. , 2015, “ On the Sensitivity and Accuracy of Proper-Orthogonal-Decomposition-Based Reduced Order Models for Burgers Equation,” Comput. Fluids, 106, pp. 19–32. [CrossRef]
Chen, H. , Xu, M. , Hung, D. L. , and Zhuang, H. , 2014, “ Cycle-to-Cycle Variation Analysis of Early Flame Propagation in Engine Cylinder Using Proper Orthogonal Decomposition,” Exp. Therm. Fluid Sci., 58, pp. 48–55. [CrossRef]
Troshin, V. , Seifert, A. , Sidilkover, D. , and Tadmor, G. , 2016, “ Proper Orthogonal Decomposition of Flow-Field in Non-Stationary Geometry,” J. Comput. Phys., 311, pp. 329–337. [CrossRef]
Li, X. , Chen, X. , Hu, B. X. , and Navon, I. M. , 2013, “ Model Reduction of a Coupled Numerical Model Using Proper Orthogonal Decomposition,” J. Hydrol., 507, pp. 227–240. [CrossRef]
Mahapatra, P. S. , Chatterjee, S. , Mukhopadhyay, A. , Manna, N. K. , and Ghosh, K. , 2016, “ Proper Orthogonal Decomposition of Thermally-Induced Flow Structure in an Enclosure With Alternately Active Localized Heat Sources,” Int. J. Heat Mass Transfer, 94, pp. 373–379. [CrossRef]
Corigliano, A. , Dossi, M. , and Mariani, S. , 2015, “ Model Order Reduction and Domain Decomposition Strategies for the Solution of the Dynamic Elastic-Plastic Structural Problem,” Comput. Methods Appl. Mech. Eng., 290, pp. 127–155. [CrossRef]
Mariani, R. , and Dessi, D. , 2012, “ Analysis of the Global Bending Modes of a Floating Structure Using the Proper Orthogonal Decomposition,” J. Fluids Struct., 28, pp. 115–134. [CrossRef]
Azam, S. E. , and Mariani, S. , 2013, “ Investigation of Computational and Accuracy Issues in Pod-Based Reduced Order Modeling of Dynamic Structural Systems,” Eng. Struct., 54, pp. 150–167. [CrossRef]
Aubry, N. , 1991, “ On the Hidden Beauty of the Proper Orthogonal Decomposition,” Theor. Comput. Fluid Dyn., 2(5), pp. 339–352. [CrossRef]
Berkooz, G., Holmes, P., and Lumley, J. L., 1993, “The Proper Orthogonal Decomposition in the Analysis of Turbulent Flows,” Annu. Rev. Fluid Mech., 25(1), pp. 539–575.
Kunisch, K. , and Volkwein, S. , 1999, “ Control of the Burgers Equation by a Reduced-Order Approach Using Proper Orthogonal Decomposition,” J. Optim. Theory Appl., 102(2), pp. 345–371. [CrossRef]
Kerfriden, P. , Goury, O. , Rabczuk, T. , and Bordas, S. , 2013, “ A Partitioned Model Order Reduction Approach to Rationalise Computational Expenses in Nonlinear Fracture Mechanics,” Comput. Methods Appl. Mech. Eng., 256, pp. 169–188. [CrossRef] [PubMed]
Radermacher, A. , and Reese, S. , 2016, “ POD-Based Model Reduction With Empirical Interpolation Applied to Nonlinear Elasticity,” Int. J. Numer. Methods Eng., 107(6), pp. 477–495. [CrossRef]
Rapún, M.-L. , Terragni, F. , and Vega, J. M. , 2015, “ Adaptive Pod-Based Low-Dimensional Modeling Supported by Residual Estimates,” Int. J. Numer. Methods Eng., 104(9), pp. 844–868. [CrossRef]
Wang, Z. , McBee, B. , and Iliescu, T. , 2016, “ Approximate Partitioned Method of Snapshots for POD,” J. Comput. Appl. Math., 307, pp. 374–384. [CrossRef]
Peng, L. , and Mohseni, K. , 2016, “ Nonlinear Model Reduction Via a Locally Weighted POD Method,” Int. J. Numer. Methods Eng., 106(5), pp. 372–396. [CrossRef]
Cusatis, G. , Pelessone, D. , and Mencarelli, A. , 2011, “ Lattice Discrete Particle Model (LDPM) for Failure Behavior of Concrete. I: Theory,” Cem. Concrete Compos., 33(9), pp. 881–890. [CrossRef]
Smith, J. , Cusatis, G. , Pelessone, D. , Landis, E. , O'Daniel, J. , and Baylot, J. , 2014, “ Discrete Modeling of Ultra-High-Performance Concrete With Application to Projectile Penetration,” Int. J. Impact Eng., 65, pp. 13–32. [CrossRef]
Wan, L. , Wendner, R. , and Cusatis, G. , 2016, “ A Novel Material for In Situ Construction on Mars: Experiments and Numerical Simulations,” Constr. Build. Mater., 120, pp. 222–231. [CrossRef]
Wan, L. , Wendner, R. , Liang, B. , and Cusatis, G. , 2016, “ Analysis of the Behavior of Ultra High Performance Concrete at Early Age,” Cem. Concr. Compos., 74, pp. 120–135. [CrossRef]
Li, W. , Rezakhani, R. , Jin, C. , Zhou, X. , and Cusatis, G. , 2017, “ A Multiscale Framework for the Simulation of the Anisotropic Mechanical Behavior of Shale,” Int. J. Numer. Anal. Methods Geomech., 41(14), pp. 1494–1522.
Cusatis, G. , and Zhou, X. , 2014, “ High-Order Microplane Theory for Quasi-Brittle Materials With Multiple Characteristic Lengths,” J. Eng. Mech., 140(7), p. 04014046.
Rezakhani, R. , and Cusatis, G. , 2016, “ Asymptotic Expansion Homogenization of Discrete Fine-Scale Models With Rotational Degrees of Freedom for the Simulation of Quasi-Brittle Materials,” J. Mech. Phys. Solids, 88, pp. 320–345. [CrossRef]
Barbosa, R. , and Ghaboussi, J. , 1992, “ Discrete Finite Element Methods,” Eng. Comput., 9(2), pp. 253–266. [CrossRef]
Ji, S. , Di, S. , and Long, X. , 2017, “ DEM Simulation of Uniaxial Compressive and Flexural Strength of Sea Ice: Parametric Study,” J. Eng. Mech., 143(1), p. C4016010.
Cundall, P. A. , and Strack, O. D. L. , 1979, “ A Discrete Numerical Model for Granular Assemblies,” Géotechnique, 29(1), pp. 47–65. [CrossRef]
Proctor, E. A. , Ding, F. , and Dokholyan, N. V. , 2011, “ Discrete Molecular Dynamics,” Wiley Interdiscip. Rev.: Comput. Mol. Sci., 1(1), pp. 80–92. [CrossRef]
Rapaport, D. C. , Blumberg, R. L. , McKay, S. R. , and Christian, W. , 1996, “ The Art of Molecular Dynamics Simulation,” Comput. Phys., 10(5), pp. 456–456. [CrossRef]
Courant, R. , Friedrichs, K. , and Lewy, H. , 1976, “ On the Partial Difference Equations of Mathematical Physics,” IBM J., 11, pp. 215–234. [CrossRef]
Belytschko, T. , Liu, W. K. , and Moran, B. , 2000, Nonlinear Finite Elements for Continua and Structures, Wiley, Chichester, UK.
Linang, Y. , Lee, H. , Lim, S. , Lin, W. , Lee, K. , and Wu, C. , 2002, “ Proper Orthogonal Decomposition and Its Applications—Part I: Theory,” J. Sound Vib., 252(3), pp. 527–544. [CrossRef]
Chatterjee, A. , 2000, “ An Introduction to the Proper Orthogonal Decomposition,” Curr. Sci., 78(7), pp. 808–817. http://www.jstor.org/stable/24103957
Antoulas, A. C. , 2005, Approximation of Large-Scale Dynamical Systems, Vol. 6, SIAM, Philadelphia, PA. [CrossRef]
Kalashnikova, I. , and Barone, M. F. , 2012, “ Efficient Non-Linear Proper Orthogonal Decomposition/Galerkin Reduced Order Models With Stable Penalty Enforcement of Boundary Conditions,” Int. J. Numer. Methods Eng., 90(11), pp. 1337–1362. [CrossRef]
Cusatis, G. , Mencarelli, A. , Pelessone, D. , and Baylot, J. T. , 2011, “ Lattice Discrete Particle Model (LDPM) for Failure Behavior of Concrete. II: Calibration and Validation,” Cem. Concr. Compos., 33(9), pp. 891–905. [CrossRef]
Lale, E. , Zhou, X. , and Cusatis, G. , 2015, “ Isogeometric Implementation of High Order Microplane Model for the Simulation of High Order Elasticity, Softening, and Localization,” ASME J. Appl. Mech., 84(1), pp. 3523–3545.
Pelessone, D. , 2015, “MARS, Modeling and Analysis of the Response of Structures” User's Manual, Engineering and Software System Solutions, Inc., San Diego, CA.
Smith, J. , and Cusatis, G. , 2016, “ Numerical Analysis of Projectile Penetration and Perforation of Plain and Fiber Reinforced Concrete Slabs,” Int. J. Numer. Anal. Methods Geomech., 41(3), pp. 315–337.
Alnaggar, M. , Cusatis, G. , and Di Luzio, G. , 2013, “ Lattice Discrete Particle Modeling (LDPM) of Alkali Silica Reaction (ASR) Deterioration of Concrete Structures,” Cem. Concr. Compos., 41, pp. 45–59. [CrossRef]
Alnaggar, M. , Di Luzio, G. , and Cusatis, G. , 2017, “ Modeling Time-Dependent Behavior of Concrete Affected by Alkali Silica Reaction in Variable Environmental Conditions,” Materials, 10(5), p. 471. [CrossRef]
Alnaggar, M. , Liu, M. , Qu, L. , and Cusatis, G. , 2015, “ Lattice Discrete Particle Modeling of Acoustic Nonlinearity Change in Accelerated Alkali Silica Reaction (Asr) Tests,” Mater. Struct. J., 49(9), pp. 3523–3545. [CrossRef]
Ceccato, C. , Salviato, M. , Pellegrino, C. , and Cusatis, G. , 2017, “ Simulation of Concrete Failure and Fiber Reinforced Polymer Fracture in Confined Columns With Different Cross Sectional Shape,” Int. J. Solids Struct., 108, pp. 216–229. [CrossRef]
Schauffert, E. A. , and Cusatis, G. , 2011, “ Lattice Discrete Particle Model for Fiber-Reinforced Concrete. I: Theory,” J. Eng. Mech., 138(7), pp. 826–833. [CrossRef]
Schauffert, E. A. , Cusatis, G. , Pelessone, D. , O'Daniel, J. L. , and Baylot, J. T. , 2012, “ Lattice Discrete Particle Model for Fiber-Reinforced Concrete. II: Tensile Fracture and Multiaxial Loading Behavior,” J. Eng. Mech., 138(7), pp. 834–841. [CrossRef]
Jin, C. , Buratti, N. , Stacchini, M. , Savoia, M. , and Cusatis, G. , 2016, “ Lattice Discrete Particle Modeling of Fiber Reinforced Concrete: Experiments and Simulations,” Eur. J. Mech.−A/Solids, 57, pp. 85–107. [CrossRef]
Wan, L., Wendner, R., and Cusatis, G., 2015, “A Hygro-Thermo-Chemo-Mechanical Model for the Simulation of Early Age Behavior of Ultra High Performance Concrete,” 10th International Conference on Mechanics and Physics of Creep, Shrinkage, and Durability of Concrete and Concrete Structures (CONCREEP), Sept. 19–20, Vienna, Austria. https://ascelibrary.org/doi/abs/10.1061/9780784479346.020
Wan-Wendner, L. , Wan-Wendner, R. , and Cusatis, G. , 2018, “ Age-Dependent Size Effect and Fracture Characteristics of Ultra-High Performance Concrete,” Cem. Concr. Compos., 85(Suppl. C), pp. 67–82. [CrossRef]
Ashari, S. E. , Buscarnera, G. , and Cusatis, G. , 2017, “ A Lattice Discrete Particle Model for Pressure-Dependent Inelasticity in Granular Rocks,” Int. J. Rock Mech. Min. Sci., 91, pp. 49–58.
Li, W., Cusatis, G., and Jin, C., 2016, “Integrated Experimental and Computational Characterization of Shale at Multiple Length Scales,” New Frontiers in Oil and Gas Exploration, Springer, Berlin.


Grahic Jump Location
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)

Grahic Jump Location
Fig. 2

One-dimensional model of a dynamic discrete system

Grahic Jump Location
Fig. 3

Shape of the first six POD modes

Grahic Jump Location
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)

Grahic Jump Location
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)

Grahic Jump Location
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

Grahic Jump Location
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)

Grahic Jump Location
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

Grahic Jump Location
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)

Grahic Jump Location
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

Grahic Jump Location
Fig. 11

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

Grahic Jump Location
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)

Grahic Jump Location
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)



Some tools below are only available to our subscribers or users with an online account.

Related Content

Customize your page view by dragging and repositioning the boxes below.

Related Journal Articles
Related eBook Content
Topic Collections

Sorry! You do not have access to this content. For assistance or to subscribe, please contact us:

  • TELEPHONE: 1-800-843-2763 (Toll-free in the USA)
  • EMAIL: asmedigitalcollection@asme.org
Sign In