Lattice Element Model Elastic Behaviour in Solid Mechanics: Bayesian-Based Calibration and Fine-Tuning
DOI:
https://doi.org/10.13052/ejcm2642-2085.34343Keywords:
Lattice element model, Bayesian inference, elastic response, Parameter identification, Polynomial chaos methodAbstract
This work investigates the elastic behaviour of a mechanical discrete lattice element model based on Timoshenko beam elements. Due to the selected irregular meshes generated with Delaunay triangulation and one-dimensional elements, lattice models struggle to correctly simulate a wide range of elastic material responses, particularly in capturing lateral deformations associated with variations in the material’s Poisson ratio. To address this limitation, we introduce correction coefficients that modify the stiffness properties of the lattice beam elements, influencing the global mechanical behaviour of the lattice. A Bayesian stochastic identification framework is chosen to calibrate these coefficients using a set of standard mechanical tests, ensuring consistency with the proper continuum elastic response. The applicability of the identified lattice element stiffnesses is evaluated across different loading conditions and material properties. The methodology ensures a fine tuning of the model, accuracy in simulating mechanical deformations, and the basis for non-linear phenomena, crack initiation and its propagation across the lattice.
Downloads
References
A Ibrahimbegovic. (2009). Nonlinear solid mechanics: theoretical formulations and finite element solution methods (vol. 160). Heidelberg: Springer Science & Business Media.
M Nikolić, E Karavelić, A Ibrahimbegovic and P Miščević. (2018). Lattice element models and their peculiarities. Archives of Computational Methods in Engineering 25(3), 753–784.
A Lisjak and G Grasselli. (2014). A review of discrete modeling techniques for fracturing processes in discontinuous rock masses. Journal of Rock Mechanics and Geotechnical Engineering 6(4), 301–314.
J Oliver, A E Huespe and P J Sánchez. (2006). A comparative study on finite elements for capturing strong discontinuities: E-FEM vs X-FEM. Computer Methods in Applied Mechanics and Engineering 195(37–40), 4732–4752.
M Šodan, A Stanić and M Nikolić. (2024). Enhanced solid element model with embedded strong discontinuity for representation of mesoscale quasi-brittle failure. International Journal of Fracture 248(1), 1–25.
E Karavelić, M Nikolić, A Ibrahimbegovic and A Kurtović. (2019). Concrete meso-scale model with full set of 3D failure modes with random distribution of aggregate and cement phase. Part I: Formulation and numerical implementation. Computer Methods in Applied Mechanics and Engineering 344, 1051–1072.
A Hrennikoff. (1941). Solution of problems of elasticity by the framework method. Journal of Applied Mechanics 8(4), A169–A175.
M Ostoja-Starzewski. (2002). Lattice models in micromechanics. Applied Mechanics Reviews 55(1), 35–60.
E Schlangen and J G M Van Mier. (1992). Simple lattice model for numerical simulation of fracture of concrete materials and structures. Materials and Structures 25(9), 534–542.
E Schlangen and E J Garboczi. (1996). New method for simulating fracture using an elastically uniform random geometry lattice. International Journal of Engineering Science 34(10), 1131–1144.
G Cusatis, D Pelessone and A Mencarelli. (2011). Lattice Discrete Particle Model (LDPM) for failure behavior of concrete. I: Theory. Cement and Concrete Composites 33(9), 881–890.
J E Bolander Jr and S Saito. (1998). Fracture analyses using spring networks with random geometry. Engineering Fracture Mechanics 61(5–6), 569–591.
J E Bolander, J Eliáš, G Cusatis and K Nagai. (2021). Discrete mechanical models of concrete fracture. Engineering Fracture Mechanics 257, 108030.
J Čarija, E Marenić, T Jarak and M Nikolić. (2024). Discrete lattice element model for fracture propagation with improved elastic response. Applied Sciences 14(3), 1287.
M Nikolic, A Ibrahimbegovic and P Miscevic. (2015). Brittle and ductile failure of rocks: Embedded discontinuity approach for representing mode I and mode II failure mechanisms. International Journal for Numerical Methods in Engineering 102(8), 1507–1526.
D Asahina, K Ito, J E Houseworth, J T Birkholzer and J E Bolander. (2015). Simulating the Poisson effect in lattice models of elastic continua. Computers and Geotechnics 70, 60–67.
G F Zhao, Q Yin, A R Russell, Y Li, W Wu and Q Li. (2019). On the linear elastic responses of the 2D bonded discrete element model. International Journal for Numerical and Analytical Methods in Geomechanics 43(1), 166–182.
L Vaiani, A E Uva and A Boccaccio. (2025). Optimal lattice spring models derived from triangular and tetrahedral meshes. International Journal of Mechanical Sciences 110442.
D Xiu. (2010). Numerical methods for stochastic computations: a spectral method approach. Princeton: Princeton University Press.
B V Rosić, A Kučerová, J Sýkora, O Pajonk, A Litvinenko and H G Matthies. (2013). Parameter identification in a probabilistic setting. Engineering Structures 50, 179–196.
H G Matthies, E Zander, B V Rosić, A Litvinenko and O Pajonk. (2016). Inverse problems in a Bayesian setting. Computational Methods for Solids and Fluids: Multiscale Analysis, Probability Aspects and Model Reduction (245–286). Cham: Springer International Publishing.
B Rosić, J Sýkora, O Pajonk, A Kučerová and H G Matthies. (2016). Comparison of numerical approaches to Bayesian updating. Computational Methods for Solids and Fluids: Multiscale Analysis, Probability Aspects and Model Reduction (427–461). Cham: Springer International Publishing.
F Landi, F Marsili, N Friedman and P Croce. (2021). gPCE-based stochastic inverse methods: A benchmark study from a civil engineer’s perspective. Infrastructures 6(11), 158.
N Friedman, C Zoccarato, E Zander and H G Matthies. (2021). A worked-out example of surrogate-based bayesian parameter and field identification methods. Bayesian Methods for the Analysis of Engineering Systems; J Chiachio Ruano, M Chiachio Ruano, S Sankararaman, Eds. Boca Raton: CRC Press.
F Marsili, N Friedman, P Croce, P Formichi and F Landi. (2016). On Bayesian identification methods for the analysis of existing structures. Proceedings of the IABSE Congress Stockholm, Challenges in Design and Construction of an Innovative and Sustainable Built Environment, Stockholm, Sweden (21–23).
Y Huang, C Shao, B Wu, J L Beck and H Li. (2019). State-of-the-art review on Bayesian inference in structural system identification and damage assessment. Advances in Structural Engineering 22(6), 1329–1351.
M Å odan, A Urbanics, N Friedman, A Stanic and M Nikolić. (2025). Comparison of Machine Learning and gPC-based proxy solutions for an efficient Bayesian identification of fracture parameters. Computer Methods in Applied Mechanics and Engineering 436, 11768.
W K Hastings. (1970). Monte Carlo sampling methods using Markov chains and their applications. Biometrika Volume 57, Issue 1, 97–109.
A Saltelli, P Annoni, I Azzini, F Campolongo, M Ratto and S Tarantola. (2010). Variance based sensitivity analysis of model output. Design and estimator for the total sensitivity index. Computer Physics Communications 181(2), 259–270.
I M Sobol. (2001). Global sensitivity indices for nonlinear mathematical models and their Monte Carlo estimates. Mathematics and Computers in Simulation 55(1–3), 271–280.
M Nikolić. (2022). Discrete element model for the failure analysis of partially saturated porous media with propagating cracks represented with embedded strong discontinuities. Computer Methods in Applied Mechanics and Engineering 390, 114482.
A Stanic, M Nikolic, N Friedman and H G Matthies. (2021). Parameter identification in dynamic crack propagation. ECCOMAS MSF 2021 – 5th International Conference on Multi-scale Computational Methods for Solids and Fluids.
E Hadzalic, A Ibrahimbegovic and S Dolarevic. (2019). Theoretical formulation and seamless discrete approximation for localized failure of saturated poro-plastic structure interacting with reservoir. Computers & Structures 214, 73–93.
E Hadzalic, A Ibrahimbegovic and M Nikolic. (2018). Failure mechanisms in coupled poro-plastic medium. Coupled Systems Mechanics 7, 43–59.
E Hadzalic, A Ibrahimbegovic and S Dolarevic. (2020). 3D thermo-hydro-mechanical coupled discrete beam lattice model of saturated poro-plastic medium. Coupled Systems Mechanics 9(2), 125–145.
H Rana and A Ibrahimbegovic. (2025). Fine-scale model of concrete composite for long-term cycling transient loads incorporating nonlinear hardening and softening effects. Composite Structures, 119649.
H Rana and A Ibrahimbegovic. (2025). A hybrid physics-informed and data-driven approach for predicting the fatigue life of concrete using an energy-based fatigue model and machine learning. Computation 13(3), 61.
D Xenos, D Grégoire, S Morel and P Grassl. (2015). Calibration of nonlocal models for tensile fracture in quasi-brittle heterogeneous materials. Journal of the Mechanics and Physics of Solids 82, 48–60.
P J Green and R Sibson. (1978). Computing Dirichlet tessellations in the plane. The Computer Journal 21(2), 168–173.
S P Timoshenko. (1921). LXVI. On the correction for shear of the differential equation for transverse vibrations of prismatic bars. The London, Edinburgh, and Dublin Philosophical Magazine and Journal of Science 41(245), 744–746.
R L Taylor. (2014). FEAP, a finite element analysis program: Version 7.1 j user manual. Department of Civil and Environmental Engineering, University of California.
E Zander and J Vondrejc. (2016). A MATLAB/Octave toolbox for stochastic Galerkin methods. Github, San Francisco, CA.
