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 1251 products
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11/2024 - 01/29/2024
Antonietti, P.F.; Corti, M.
Numerical modelling of protein misfolding in neurodegenerative diseases: a computational study | Abstract | | The spreading of misfolded proteins is a known hallmark in some neurodegenerative diseases, known as proteinopathies. A significant example is the tau protein, associated with many pathologies, such as Alzheimer's. In this work, we discuss and compare two different models for the mathematical modelling of protein misfolding, namely the heterodimer model and the Fisher-Kolmogorov model, as well as their numerical discretizations. We introduce a discontinuous Galerkin method on polygonal and polyhedral grids for space discretization to accurately simulate the wavefronts typically observed in the prionic spreading. Starting from the semidiscrete formulations, we use a Crank-Nicolson scheme to advance in time. Finally, we simulate the spreading of the misfolded tau protein in a two-dimensional brain slice in the sagittal plane with a polygonal agglomerated grid. The simulation is performed using both the presented models, and we compare the results and the differences deriving from the modelling choices. |
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10/2024 - 01/25/2024
Capuano E.; Regazzoni F.; Maines M.; Fornara S.; Locatelli V.; Catanzariti D.; Stella S.; Nobile F.; Del Greco M.; Vergara C.
Personalized Computational Electro-mechanics Simulations to Optimize Cardiac Resynchronization Therapy | Abstract | | In this study, we present a computational framework designed to evaluate virtual scenarios of Cardiac Resynchronization Therapy (CRT) and compare their effectiveness based on relevant clinical biomarkers. Our approach involves electro-mechanical numerical simulations calibrated, for patients with left bundle branch block, using data from Electro-Anatomical Mapping System (EAMS) measures, as well as ventricular pressures and volumes, both obtained pre-implantation. We validate the calibration by using EAMS data coming from right pacing conditions. Three patients with fibrosis and three without are considered to explore various conditions.
Our virtual scenarios consist of personalized numerical experiments, incorporating different positions of the left electrode along reconstructed epicardial veins; different locations of the right electrode; different ventriculo-ventricular delays. The aim is to offer a comprehensive tool capable of optimizing CRT efficiency for individual patients, by providing preliminary answers on optimal electrode placement and delay. |
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09/2024 - 01/25/2024
Leimer Saglio, C. B.: Pagani, S.; Corti, M.; Antonietti, P. F.
A high-order discontinuous Galerkin method for the numerical modeling of epileptic seizures | Abstract | | Epilepsy is a clinical neurological disorder characterized by recurrent and spontaneous seizures consisting of abnormal high-frequency electrical activity in the brain.
In this condition, the transmembrane potential dynamics are characterized by rapid and sharp wavefronts traveling along the heterogeneous and anisotropic conduction pathways of the brain.
This work employs the monodomain model, coupled with specific neuronal ionic models characterizing ion concentration dynamics, to mathematically describe brain tissue electrophysiology in grey and white matter at the organ scale. This multiscale model is discretized in space with the high-order discontinuous Galerkin method on polygonal and polyhedral grids (PolyDG) and advanced in time with a Crank-Nicolson scheme. This ensures, on the one hand, efficient and accurate simulations of the high-frequency electrical activity that is responsible for epileptic seizure and, on the other hand, keeps reasonably low the computational costs by a suitable combination of high-order approximations and agglomerated polytopal meshes.
We numerically investigate synthetic test cases on a two-dimensional heterogeneous squared domain discretized with a polygonal grid, and on a two-dimensional brainstem in a sagittal plane with an agglomerated polygonal grid that takes full advantage of the flexibility of the PolyDG approximation of the semidiscrete formulation. Finally, we provide a theoretical analysis of stability and an a-priori convergence analysis for a simplified mathematical problem. |
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05/2024 - 01/24/2024
Conti, P.; Gobat, G.; Fresca, S.; Manzoni, A.; Frangi, A.
Reduced order modeling of parametrized systems through autoencoders and SINDy approach: continuation of periodic solutions | Abstract | | Highly accurate simulations of complex phenomena governed by partial differential equations (PDEs) typically require intrusive methods and entail expensive computational costs, which might become prohibitive when approximating steady-state solutions of PDEs for multiple combinations of control parameters and initial conditions. Therefore, constructing efficient reduced order models (ROMs) that enable accurate but fast predictions, while retaining the dynamical characteristics of the physical phenomenon as parameters vary, is of paramount importance. In this work, a data-driven, non-intrusive framework which combines ROM construction with reduced dynamics identification, is presented. Starting from a limited amount of full order solutions, the proposed approach leverages autoencoder neural networks with parametric sparse identification of nonlinear dynamics (SINDy) to construct a low-dimensional dynamical model. This model can be queried to efficiently compute full-time solutions at new parameter instances, as well as directly fed to continuation algorithms. These aim at tracking the evolution of periodic steady-state responses as functions of system parameters, avoiding the computation of the transient phase, and allowing to detect instabilities and bifurcations. Featuring an explicit and parametrized modeling of the reduced dynamics, the proposed data-driven framework presents remarkable capabilities to generalize with respect to both time and parameters. Applications to structural mechanics and fluid dynamics problems illustrate the effectiveness and accuracy of the proposed method. |
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06/2024 - 01/24/2024
Antonietti, P.F., Bonetti, S., Botti, M., Corti, M., Fumagalli, I., Mazzieri, I.
lymph: discontinuous poLYtopal methods for Multi-PHysics differential problems | Abstract | | We present the library lymph for the finite element numerical discretization of coupled multi-physics problems. lymph is a Matlab library for the discretization of partial differential equations based on high-order discontinuous Galerkin methods on polytopal grids (PolyDG) for spatial discretization coupled with suitable finite-difference time marching schemes. The objective of the paper is to introduce the library by describing it in terms of installation, input/output data, and code structure, highlighting -- when necessary -- key implementation aspects related to the method. A user guide, proceeding step-by-step in the implementation and solution of a Poisson problem, is also provided. In the last part of the paper, we show the results obtained for several differential problems, namely the Poisson problem, the heat equation, and the elastodynamics system. Through these examples, we show the convergence properties and highlight some of the main features of the proposed method, i.e. geometric flexibility, high-order accuracy, and robustness with respect to heterogeneous physical parameters. |
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04/2024 - 01/22/2024
Torzoni, M.; Tezzele, M.; Mariani, S.; Manzoni, A.; Willcox, K.E.
A digital twin framework for civil engineering structures | Abstract | | The digital twin concept represents an appealing opportunity to advance condition-based and predictive maintenance paradigms for civil engineering systems, thus allowing reduced lifecycle costs, increased system safety, and increased system availability. This work proposes a predictive digital twin approach to the health monitoring, maintenance, and management planning of civil engineering structures. The asset-twin coupled dynamical system is encoded employing a probabilistic graphical model, which allows all relevant sources of uncertainty to be taken into account. In particular, the time-repeating observations-to-decisions flow is modeled using a dynamic Bayesian network. Real-time structural health diagnostics are provided by assimilating sensed data with deep learning models. The digital twin state is continually updated in a sequential Bayesian inference fashion. This is then exploited to inform the optimal planning of maintenance and management actions within a dynamic decision-making framework. A preliminary offline phase involves the population of training datasets through a reduced-order numerical model and the computation of a health-dependent control policy. The strategy is assessed on two synthetic case studies, involving a cantilever beam and a railway bridge, demonstrating the dynamic decision-making capabilities of health-aware digital twins. |
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03/2024 - 01/16/2024
Ciaramella, G.; Gander, M.J.; Vanzan, T.
A gentle introduction to interpolation on the Grassmann manifold | Abstract | | These notes originated from the authors’ effort while studying interpolation techniques on the Grassmann manifold. This has been a hot topic recently since it is an important tool in parametric reduced order modelling. Fortunately, there is an extensive literature available with seminal contributions both from the engineering and mathematical communities. More generally, the development of numerical methods involving manifolds is a very active research area.
Given all literature about interpolation on Grassmann manifold, the reader may immediately ask the following question: is there any need for an additional introductory manuscript? It is the authors’ belief that this is actually the case. The aim of these notes is to precisely fill a gap in the literature, by providing a reference which gently introduces numerical analysts to the very interesting research topic of interpolation on the Grassmann manifold. Indeed, on the one hand, the engineering literature often does not provide the necessary mathematical details needed by a numerical analyst to understand the subject and to solidly build new computational algorithms. On the other hand, manuscripts from the mathematical community, despite being seminal references, tend to be overwhelming in terms of details, and mathematically concepts that are often not familiar to numerical analysts approaching the topic for the first time. These notes are meant to be a first very gentle introduction to these numerical methods, before approaching the more organic references. Further, the notes are self-contained concerning the derivation of geodesics, the algorithms to compute the exponential and logarithmic maps, and interpolation algorithms on the Grassmann manifold. These mathematical results are all well-known, but the original proofs are scattered across several manu scripts, often using different notations and level of detail, so that their study may not be immediate. |
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02/2024 - 01/13/2024
Parolini, N.; Poiatti, A.; Vene', J.; Verani, M.
Structure-preserving neural networks in data-driven rheological models | Abstract | | In this paper we address the importance and the impact of employing structure preserving neural networks as surrogate of the analytical physics-based models typically employed to describe the rheology of non-Newtonian fluids in Stokes flows. In particular, we propose and test on real-world scenarios a novel strategy to build data-driven rheological models based on the use of Input-Output Convex Neural Networks (ICNNs), a special class of feedforward neural network scalar valued functions that are convex with respect to their inputs. Moreover, we show, through a detailed campaign of numerical experiments, that the use of ICNNs is of paramount importance to guarantee the well-posedness of the associated non-Newtonian Stokes differential problem. Finally, building upon a novel perturbation result for non-Newtonian Stokes problems, we study the impact of our data-driven ICNN based rheological model on the accuracy of the finite element approximation. |
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