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Parameter estimation framework for fracture propagation problems under extreme mechanical loads
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This project was funded by the Croatian Science Foundation under the grant UIP-2020-02-6693. The research activities within the project were carried out from 1 February 2021 to 31 January 2026. The members of the research team gratefully acknowledge the support provided by the Croatian Science Foundation.
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Project Summary
Reliable prediction of the initiation and progressive failure in structures and materials represents one of the more challenging problems in computational mechanics. Although advanced nonlinear numerical models enable a detailed description of crack initiation and propagation, the reliability of their predictions largely depends on accurate knowledge of material parameters. A particular challenge is associated with parameters governing the post-peak material response, including strain localization and crack development.
The main objective of the FracID project was to develop a methodology for the reliable estimation of material parameters in fracture and crack propagation problems. The developed methodology is based on solving stochastic inverse problems, in which experimental or synthetic measurements are combined with advanced computational models within a Bayesian statistical framework.
Within the project, numerical models for describing strain localization and crack propagation were developed and enhanced, including models based on the Embedded Strong Discontinuity approach and discrete Lattice Element Models (LEM) for representing heterogeneous materials and their microstructure.
To efficiently solve computationally demanding stochastic inverse problems, Markov Chain Monte Carlo (MCMC) methods, Bayesian calibration, computationally efficient proxy/surrogate models, generalized Polynomial Chaos (gPC), as well as machine learning and artificial intelligence methods, were employed.
Team members
Assoc. Prof. Mijo Nikolić – Principal Investigator
Researcher in the fields of computational mechanics, numerical modelling of fracture and crack propagation, multiphysics problems in porous media, and stochastic inverse problems. Within the project, he led the development of methodologies for Bayesian parameter identification, as well as the development and application of numerical fracture models.
Dr. Matej Šodan – Doctoral Researcher
Within the project, he worked on the development of new numerical models for crack propagation and methodologies for stochastic parameter identification. His research focused particularly on models with embedded strong discontinuities, Bayesian parameter identification, and the application of proxy models and machine learning methods for the efficient solution of computationally demanding inverse problems.
Dr. Jadran Čarija – Postdoctoral Researcher
His research focused on the development of discrete Lattice Element Models (LEM) and algorithms for improved and efficient simulation of material failure in complex fracture problems under tensile and compressive loading.
Duje Pavić – Research Assistant
Research Assistant within the NPOO-DOK-2023-10-8014 project, involved in research on discrete Lattice Element Models and Bayesian identification of their parameters in two- and three-dimensional problems under mechanical and multiphysics loading conditions.
Assoc. Prof. Andjelka Stanić – Research Collaborator
Researcher in the fields of numerical modelling of crack propagation and stochastic problems. Within the project, she contributed to the development of methodologies for probabilistic parameter identification, with particular emphasis on dynamic fracture and crack propagation under high-rate dynamic loading.
Dr. Simona Dobrilla – Research Collaborator
She contributed to research on the Bayesian identification of mechanical properties of heterogeneous materials, combining experimental measurements, mesoscale numerical models, and probabilistic methods.
Assoc. Prof. Eduard Marenić – Research Collaborator
Researcher in the fields of computational mechanics and data-driven mechanics based on experimental data. Within the project, he contributed to the development and application of discrete fracture models.
Project results
The FracID project resulted in the development of a comprehensive computational framework for the stochastic identification of material parameters in nonlinear mechanics, strain localization, and material failure problems. The research conducted within the project encompassed the development of probabilistic parameter identification methods, new numerical fracture models, methods for the efficient solution of computationally demanding inverse problems, and their application to both synthetic and real experimental data [1–9].
Particular emphasis was placed on combining Bayes theorem with advanced numerical fracture models. The developed methodologies enable the identification of material parameters while simultaneously quantifying the associated uncertainties. To facilitate the application of probabilistic methods to computationally demanding strain localization and crack propagation problems, different approaches for reducing computational cost were investigated, including proxy models based on generalized Polynomial Chaos (gPC) and neural networks, as well as efficient algorithms for Bayesian parameter identification [1,4,7–9].
In parallel, numerical models for describing strain localization and fracture were developed and enhanced, including finite element models with embedded strong discontinuities [2,4–5] and discrete Lattice Element Models (LEM) for modelling heterogeneous materials and crack propagation [3,6,7,9]. The developed models and methodologies were applied to a wide range of problems, from academic numerical examples and synthetic measurements to real laboratory experiments and three-dimensional mesoscale models based on experimentally determined material geometries [1,5].
During the project, 9 scientific papers were published in international scientific journals [1–9]. The research results were presented at numerous international and national conferences and congresses, as well as through invited and keynote lectures.
An important outcome of the project was also the training and development of early-career researchers. Within the project, doctoral researcher Matej Šodan completed and successfully defended his PhD thesis entitled Numerical model with embedded strong discontinuities and stochastic identification of parameters in material failure processes. In addition, one postdoctoral research fellowship (Jadran Čarija) was completed within the project.
Bayesian Methodology for the Identification of Fracture and Crack Propagation Parameters
One of the key outcomes of the FracID project was the development of a methodology for the probabilistic identification of material parameters governing fracture and crack propagation, based on the Bayesian approach. Unlike classical deterministic identification, in which a single optimal value is determined for each unknown parameter, the Bayesian approach enables the determination of their posterior probability distributions. In this way, the uncertainty associated with the identified parameter values is quantified simultaneously with their estimation.
Throughout the project, the methodology was developed, validated, and enhanced using various numerical and experimental examples, different fracture models, and problems of varying dimensionality and computational complexity [1,4,5,7–9]. Particular attention was devoted to problems involving a large number of unknown parameters, crack propagation under static and dynamic loading, and the development of efficient proxy models capable of supporting the large number of simulations required for probabilistic parameter identification.
In the paper by Dobrilla et al. [1], Bayesian identification was applied to determine the mesoscale mechanical parameters of mortar. The actual mesostructure of the laboratory specimen was reconstructed based on tomographic imaging, and its mechanical behaviour was represented using a three-dimensional discrete lattice model (3D Lattice Element Model) capable of simulating strain localization and crack propagation. Parameter identification was performed using the Generalized Ensemble Kalman Filter (EnGKF) method for a problem involving a total of 32 unknown parameters (tensile and shear yield limits, plasticity hardening parameters, tensile and shear strengths and fracture energies) describing different constituents of the mesostructure, including the cement paste, aggregates, and their interactions. The identified parameters included those governing localization and fracture, energy dissipation, as well as plastic behaviour and friction between the individual constituents [1].
Figure 1. a) 3D view of the reconstructed specimen, b) discrete lattice model, figure taken from [1]
A second research direction focused on coupling Bayesian parameter identification with fracture models based on embedded strong discontinuities. In the study by Šodan et al. [4], a methodology was developed for the identification of fracture parameters in a numerical model incorporating embedded strong discontinuities. The formulation of the model was developed based on the incompatible modes method and is presented in detail in [2].
Figure 2. a) dog bone specimen, b) FE mesh, [4]
Since Bayesian identification requires a very large number of numerical model evaluations, particular attention was devoted to the development of efficient proxy models, in which computationally demanding fracture simulations are replaced by significantly faster approximation models. For this purpose, different neural network models and models based on generalized Polynomial Chaos (gPC) were developed and systematically compared [4]. This resulted in an efficient computational framework that enables the practical implementation of probabilistic parameter identification in computationally demanding fracture problems [4].
Figure 3. Resulting posterior distributions for tensile strength and fracture energy obtained in dog bone tensile test [4]
The methodology was further extended to problems involving dynamic crack propagation. In the study by Stanić et al. [8], a probabilistic framework was developed for parameter identification in fracture problems subjected to dynamic loading, including impact loading. In such problems, crack initiation and propagation occur within a very short time interval, making the identification of material parameters a particularly challenging inverse problem. The developed methodology enabled the identification of key parameters governing the dynamic fracture process, including strength and fracture energy parameters, while simultaneously quantifying their probabilistic uncertainty [8].
Figure 4. Kalthoff test for dynamic crack propagation: a) horizontal displacement, b) vertical displacement, c) dissipated fracture energy; [8]
The methodology was further extended to multiphysics problems. In the study by Hadžalić et al. [9], a methodology for the stochastic identification of parameters was developed for a three-dimensional discrete lattice model for thermo-hydro-mechanical (THM) problems, in which heat and fluid transport processes are coupled with the mechanical response, strain localization, and material failure [9].
Figure 5. The progressive failure propagation under thermo-hydro-mechanical conditions [9]
New numerical fracture models
Within the project, numerical models were developed and enhanced for the simulation of strain localization, crack initiation and propagation, and failure of heterogeneous materials [2,3,6,7,9]. The development of advanced numerical models represented an important part of the project, as the reliability of material parameter identification directly depends on the reliability of the computational model and its ability to accurately represent the physical processes occurring during material damage and failure.
Figure 6. Mesostructural geometry and corresponding FE mesh; [2]
A new finite element model with an Embedded Strong Discontinuity was developed to describe strain localization and quasi-brittle fracture [2]. The model was based on an incompatible modes formulation, with the strong discontinuity embedded within the finite element [4,5].
Figure 7. Meso-structural crack propagation; [2]
In parallel, discrete Lattice Element Models (LEM) were developed and enhanced for modelling heterogeneous materials and crack propagation processes [3,6,7,9]. Particular attention was devoted to improving their elastic response and overcoming the limitations of classical lattice formulations [3], as well as to the calibration and accurate determination of their parameters using Bayesian methods [7].
Figure 8. The development of improved discrete lattice models (Lattice Element Method); [3]
During the project, two complementary approaches to fracture modelling were developed: a continuum approach based on finite elements with embedded strong discontinuities [2,4,5] and a discrete approach based on Lattice Element Models [1,3,6,7,9]. Both approaches were successfully coupled with the probabilistic parameter identification methodologies developed within the project.
Figure 9. Calibration and identification of correction coefficients in discrete lattice models [7]
Efficient Solution of Computationally Demanding Stochastic Problems
One of the key challenges in Bayesian parameter identification is the need for a very large number of repeated numerical simulations. When an individual simulation involves nonlinear material behaviour, strain localization, or crack propagation, the direct application of MCMC and other probabilistic algorithms may require substantial computational resources [1,4,8]. To perform a large number of numerical simulations and enable parallel computations, the computational resources of the CIGLA cluster at the Faculty of Civil Engineering, Architecture and Geodesy and the STINA cluster at the University of Split were utilized.
In addition to the use of high-performance computing clusters, an important research direction focused on the development of proxy or surrogate models, in which a computationally demanding numerical model is replaced by a significantly faster approximation of its response. For this purpose, models based on generalized Polynomial Chaos (gPC), as well as various neural network architectures and machine learning methods, were employed [4].
Figure 10. Utilization of artificial neural networks for construction of efficient proxy models [4]
Application to Numerical and Experimental Problems
The developed methodology is not limited to a specific type of material, numerical model, or experimental procedure, but represents a general methodology for parameter identification in complex nonlinear mechanic’s problems, with applications ranging from academic numerical examples and synthetic measurements to real laboratory experiments and complex mesoscale models of heterogeneous materials [1,4,5,8].
The methodology was further applied to real experimental data and non-standard laboratory specimens [1,5]. In the study by Šodan et al. [5], Bayesian parameter identification was investigated using non-standard laboratory specimens.
Figure 11. Failure mechanism in non-standard dog bone sample [5]
A particularly complex application was carried out at the mesoscale level of heterogeneous materials [1]. The actual internal structure of the specimen was obtained using tomographic imaging, and the reconstructed geometry was used to develop a three-dimensional discrete lattice model. The numerical model was coupled with experimental data obtained from laboratory testing of the specimen, followed by probabilistic identification of a large number of parameters describing the different material constituents [1].
The results of the conducted research demonstrate the potential for integrating numerical and experimental data, advanced fracture models, Bayesian statistics, and artificial intelligence methods within a unified framework for material parameter identification and uncertainty quantification [1,4,5,8,9].
Presentations at Conferences, Congresses, and Scientific Meetings
The research results arising from the FracID project were continuously presented by members of the research team to the scientific and professional community through presentations at numerous international and national conferences, congresses, and scientific meetings.
The project results were presented at the following events:
- ECCOMAS MSF 2021, Split
- COMPLAS 2021, Barcelona
- ECCOMAS Congress 2022, Oslo
- ESMC 2022, Galway
- ICCSM 2022, Pula
- UNCECOMP 2023, Atena
- HDM Susret 2023, Split
- ECCOMAS MSF 2023, Sarajevo
- MORTech 2023, Pariz
- ECCOMAS Congress 2024, Lisabon
- CEACM Synergy 2024, Prag
- ECCOMAS MSF 2025, Split
- ICCSM 2025, Vodice
Through participation in these events, the project results were disseminated to the international research community in the fields of computational mechanics, fracture mechanics, stochastic modelling, parameter identification, and the application of artificial intelligence methods in mechanics.
Figure 12. Organizing Int. Conference ECCOMAS MSF 2025 in Split, presentations at Int. Conferences: dr.sc. Jadran Čarija, dr.sc. Matej Šodan, mag.ing. Duje Pavić
Invited and Keynote Lectures
The research results achieved within the project were also presented through a series of invited and keynote lectures, further contributing to the international and national dissemination of the project results.
Invited and keynote lectures presenting results from the project were delivered at the following events:
- ECCOMAS MSF 2021
- ECCOMAS MSF 2023
- ECCOMAS MSF 2025
- Invited lecture Croatian Society of Mechanics
- Invited lecture 1st Symposium on application of artificial intelligence in mechanics.
Research Workshops
Two research workshops were organized within the project. In October 2022, a Research Workshop was held at the Faculty of Civil Engineering, Architecture and Geodesy in Split, bringing together project team members, invited speakers, and researchers. A total of 12 presentations were delivered, covering project results, new research topics, and research directions relevant to the project.
In December 2022, a research workshop was held at the Institut Clément Ader (Institut National des Sciences Appliquées de Toulouse), focusing on the development of models based on large datasets and data-driven approaches. The workshop was hosted by Assoc. Prof. Eduard Marenić.
Figure 13. Research Workshop, October 2022., Faculty of Civil Engineering, Architecture and Geodesy Split
Final Project Conference
The Final Conference of the FracID project was held in January 2026 under the thematic title “From Physics-Based Models to Bayesian and Machine-Learning Frameworks”, where the main research results achieved within the project were presented. The conference featured presentations by renowned researchers in the fields of computational and experimental mechanics, machine learning, and stochastic methods.
Figure 14. The final project conference agenda
Figure 15. The final conference
International Collaboration
The FracID project enabled the development and strengthening of international collaboration with several leading research groups in the fields of computational mechanics, stochastic modelling, and fracture mechanics.
In particular, close collaborations were established with the research teams of:
- Dr. Noemi Friedman, HUN-REN SZTAKI, Hungary
- Prof. Adnan Ibrahimbegović, Université de Technologie de Compiègne / Sorbonne Universités, France
- Prof. Hermann Matthies, Technische Universität Braunschweig, Germany
- Prof. Bojana Rosić, University of Twente, the Netherlands
These collaborations resulted in joint research activities, scientific publications, researcher exchanges, and the development of new research directions that continue beyond the completion of the project.
Journal papers
During the project, 9 scientific papers were published in international journals:
[1] S. Dobrilla, M. Lunardelli, M. Nikolić, D. Lowke, B. Rosić, Bayesian inference of mesoscale mechanical properties of mortar using experimental data from a double shear test, Computer Methods in Applied Mechanics and Engineering, 409, 115964, 2023.
[2] M. Šodan, A. Stanić, M. Nikolić, Enhanced solid element model with embedded strong discontinuity for representation of mesoscale quasi-brittle failure, International Journal of Fracture, 248, 1–25, 2024.
[3] J. Čarija, E. Marenić, T. Jarak, M. Nikolić, Discrete Lattice Element Model for Fracture Propagation with Improved Elastic Response, Applied Sciences, 14(3), 1287, 2024.
[4] M. Šodan, A. Urbanics, N. Friedman, A. Stanić, M. Nikolić, 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, 117686, 2025.
[5] M. Šodan, V. Divić, N. Friedman, M. Nikolić, Investigating Bayesian Parameter Identification Using Non-Standard Laboratory Specimens, Applied Sciences, 15(11), 6194, 2025.
[6] V. Tojaga, M. Nikolić, M. Denzel, J. Ulloa, A. Ibrahimbegović, M. Evertsson, A. Bilock, T. Saksala, J. Quist, Advances in discrete element modeling of rock fracture for next-generation comminution models, Computational Particle Mechanics, 12, 4431–4449, 2025.
[7] D. Pavić, N. Friedman, H. Matthies, M. Nikolić, Lattice Element Model Elastic Behaviour in Solid Mechanics: Bayesian-Based Calibration and Fine Tuning, European Journal of Computational Mechanics, 34(3–4), 241–274, 2025.
[8] A. Stanić, M. Nikolić, N. Friedman, H. Matthies, Probabilistic Identification of Parameters in Dynamic Fracture Propagation, International Journal for Numerical Methods in Engineering, 127(4), e70282, 2026.
[9] E. Hadžalić, D. Pavić, Z. Rizvi, A. Ibrahimbegović, M. Nikolić, A Three-Dimensional Thermo-Hydro-Mechanical Discrete Lattice Element Framework for Localized Failure Enhanced with Bayesian Parameter Calibration, Computers and Geotechnics, 2026.
A complete list of publications, including presentations at conferences, congresses, and scientific meetings, is available at the https://www.croris.hr/projekti/projekt/6257
Final Remarks
The results of the FracID project demonstrated that combining advanced computational mechanics models, Bayesian statistics, experimental measurements, and artificial intelligence methods can provide an efficient framework for the reliable identification of parameters in complex materials.
Starting from the initial problem of fracture parameter identification, the project evolved into a broader methodological framework for solving stochastic inverse problems in computational mechanics. The developed methods can be applied to a wide range of materials and structural problems and provide a foundation for further research in probabilistic modelling, computational mechanics, and the application of artificial intelligence methods to the identification of material behaviour.