MOX Reports
The preprint collection of the Laboratory for Modeling and Scientific Computation MOX. It mainly contains works on numerical
analysis and mathematical modeling applied to engineering problems. MOX web site is mox.polimi.it
Found 1152 products
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28/2024 - 03/14/2024
Magri, M.; Riccobelli, D.
Modelling of initially stressed solids: structure of the energy density in the incompressible limit | Abstract | | This study addresses the modelling of elastic bodies, particularly when the relaxed configuration is unknown or non-existent. We adopt the theory of initially stressed materials, incorporating the deformation gradient and stress state of the reference configuration (initial stress tensor) into the response function. We show that for the theory to be applicable, the response function of the relaxed material is invertible up to an element of the material symmetry group. Additionally, we establish that commonly imposed constitutive restrictions, namely the initial stress compatibility condition and initial stress reference independence, naturally arise when assuming an initial stress generated solely from elastic distortion. The paper delves into modelling aspects concerning incompressible materials, showcasing the expressibility of strain energy density as a function of the deviatoric part of the initial stress tensor and the isochoric part of the deformation gradient. This not only reduces the number of independent invariants in the energy functional, but also enhances numerical robustness in finite element simulations. The findings of this research hold significant implications for modelling materials with initial stress, extending potential applications to areas such as mechanobiology, soft robotics, and 4D printing. |
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27/2024 - 03/07/2024
Antonietti, P.F.; Beirao da Veiga, L.; Botti, M.; Vacca, G.; Verani, M.
A Virtual Element method for non-Newtonian fluid flows | Abstract | | In this paper, we design and analyze a Virtual Element discretization for the steady motion of non-Newtonian, incompressible fluids. A specific stabilization, tailored to mimic the monotonicity and boundedness properties of the continuous operator, is introduced and theoretically investigated. The proposed method has several appealing features, including the exact enforcement of the divergence free condition and the possibility of making use of fully general polygonal meshes. A complete well-posedness and convergence analysis of the proposed method is presented under mild assumptions on the non-linear laws, encompassing common examples such as the Carreau–Yasuda model. Numerical experiments validating the theoretical bounds as well as demonstrating the practical capabilities of the proposed formulation are presented. |
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26/2024 - 03/06/2024
Bucelli, M.; Regazzoni, F.; Dede', L.; Quarteroni, A.
Robust radial basis function interpolation based on geodesic distance for the numerical coupling of multiphysics problems | Abstract | | Multiphysics simulations frequently require transferring solution fields between subproblems with non-matching spatial discretizations, typically using interpolation techniques. Standard methods are usually based on measuring the closeness between points by means of the Euclidean distance, which does not account for curvature, cuts, cavities or other non-trivial geometrical or topological features of the domain. This may lead to spurious oscillations in the interpolant in proximity to these features. To overcome this issue, we propose a modification to rescaled localized radial basis function (RL-RBF) interpolation to account for the geometry of the interpolation domain, by yielding conformity and fidelity to geometrical and topological features. The proposed method, referred to as RL-RBF-G, relies on measuring the geodesic distance between data points. RL-RBF-G removes spurious oscillations appearing in the RL-RBF interpolant, resulting in increased accuracy in domains with complex geometries. We demonstrate the effectiveness of RL-RBF-G interpolation through a convergence study in an idealized setting. Furthermore, we discuss the algorithmic aspects and the implementation of RL-RBF-G interpolation in a distributed-memory parallel framework, and present the results of a strong scalability test yielding nearly ideal results. Finally, we show the effectiveness of RL-RBF-G interpolation in multiphysics simulations by considering an application to a whole-heart cardiac electromecanics model. |
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25/2024 - 03/06/2024
Enrico Ballini e Luca Formaggia e Alessio Fumagalli e Anna Scotti e Paolo Zunino
Application of Deep Learning Reduced-Order Modeling for Single-Phase Flow in Faulted Porous Media | Abstract | | We apply reduced-order modeling (ROM) techniques to single-phase flow in faulted porous media, accounting for changing rock properties and fault geometry variations using a radial basis function mesh deformation method. This approach benefits from a mixed-dimensional framework that effectively manages the resulting non-conforming mesh. To streamline complex and repetitive calculations such as sensitivity analysis and solution of inverse problems, we utilize the Deep Learning Reduced Order Model (DL-ROM). This non-intrusive neural network-based technique is evaluated against the traditional Proper Orthogonal Decomposition (POD) method across various scenarios, demonstrating DL-ROM's capacity to expedite complex analyses with promising accuracy and efficiency. |
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22/2024 - 03/02/2024
Gatti, F.; de Falco, C.; Fois, M.; Formaggia, L.
A scalable well-balanced numerical scheme for a depth-integrated lava flow model | Abstract | | We propose a scalable well-balanced numerical method to efficiently solve a modified set of shallow water equations targeting the dynamics of lava flows. The governing equations are an extension of a depth-integrated model already available in the literature and proposed to model lava flows. Here, we consider the presence of vents that act as point sources in the mass and energy equations. Starting from a scheme developed in the framework of landslide simulation, we prove its capability to deal with lava flows. We show its excellent performances in terms of parallel scaling efficiency while maintaining good results in terms of accuracy. To verify the reliability of the proposed simulation tool, we first assess the accuracy and efficiency of the scheme on ideal scenarios. In particular, we investigate the well balancing property, we simulate benchmarks taken from the literature in the framework of lava flow simulations, and provide relevant scaling results for the parallel implementation of the method. Successively, we challenge the scheme on a real configuration taken from the available literature. |
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21/2024 - 02/26/2024
Caldana, M.; Antonietti P. F.; Dede' L.
Discovering Artificial Viscosity Models for Discontinuous Galerkin Approximation of Conservation Laws using Physics-Informed Machine Learning | Abstract | | Finite element-based high-order solvers of conservation laws offer large accuracy but face challenges near discontinuities due to the Gibbs phenomenon. Artificial viscosity is a popular and effective solution to this problem based on physical insight. In this work, we present a physics-informed machine learning algorithm to automate the discovery of artificial viscosity models in a non-supervised paradigm. The algorithm is inspired by reinforcement learning and trains a neural network acting cell-by-cell (the viscosity model) by minimizing a loss defined as the difference with respect to a reference solution thanks to automatic differentiation. This enables a dataset-free training procedure. We prove that the algorithm is effective by integrating it into a state-of-the-art Runge-Kutta discontinuous Galerkin solver. We showcase several numerical tests on scalar and vectorial problems, such as Burgers' and Euler's equations in one and two dimensions. Results demonstrate that the proposed approach trains a model that is able to outperform classical viscosity models. Moreover, we show that the learnt artificial viscosity model is able to generalize across different problems and parameters. |
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20/2024 - 02/22/2024
Torzoni, M.; Manzoni, A.; Mariani, S.
Structural health monitoring of civil structures: A diagnostic framework powered by deep metric learning | Abstract | | Recent advances in learning systems and sensor technology have enabled powerful strategies for autonomous data-driven damage detection in structural systems. This work proposes a novel method for the real-time localization of damage relying on a Siamese convolutional neural network. The method exploits a learnable mapping of raw vibration measurements onto a low-dimensional space, wherein damage locations can be easily identified. The mapping is learned in a supervised pairwise fashion exploiting labelled data, to induce a task-specific metric that allows to encode the damage position in the structure. Training data are generated through a reduced-order numerical model of the monitored structure. The damage position is then identified by performing a regression in the resulting low-dimensional features space. The proposed method does not require to define a-priori target classes and decision boundaries, thus requiring a limited amount of user-dependent assumptions. Results relevant to an L-shaped cantilever beam and a portal frame railway bridge demonstrate that the procedure can be effectively exploited for the purpose of damage localization. The method also proves to be insensitive to operational variability, measurement noise and modeling inaccuracies. |
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19/2024 - 02/22/2024
Torzoni, M.; Manzoni, A.; Mariani, S.
A multi-fidelity surrogate model for structural health monitoring exploiting model order reduction and artificial neural networks | Abstract | | Stochastic approaches to structural health monitoring (SHM) are often inevitably limited by computational constraints. For instance, for Markov chain Monte Carlo algorithms relying upon computationally expensive finite element models it is almost infeasible to sample the probability distribution of the structural state. To provide instead real-time procedures, this work proposes a non-intrusive surrogate modeling strategy, leveraging model order reduction and artificial neural networks. By relying upon a multi-fidelity (MF) framework, a composition of deep neural networks (DNNs) is devised to map damage and operational parameters onto time-dependent sensor recordings. Such an effective strategy is able to exploit datasets characterized by different fidelity levels without any prior assumption, allowing to blend a small high-fidelity (HF) dataset with a large low-fidelity (LF) dataset, ultimately alleviating the computational burden of supervised training while ensuring the accuracy of the approximated quantities of interest. The resulting surrogate model is made of an LF-DNN, which mimics sensor recordings in the undamaged condition, and of a long short-term memory HF-DNN, which adaptively refines the approximation with the effect of damage. An HF finite element model and an LF reduced order model are adopted offline to generate labeled training data of different fidelity, respectively in the presence or absence of a structural damage. Results relevant to an L-shaped cantilever beam and a portal frame railway bridge prove that the procedure efficiently provides remarkably accurate approximations, outperforming their single-fidelity counterparts. The capability of the MF-DNN to be exploited for SHM purposes is finally shown within an automated Bayesian procedure, aimed at updating the probability distribution of the structural state conditioned on sensor recordings, in the presence of operational variability and measurement noise. |
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