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A mixed-dimensional model for the electrostatic problem on coupled domains

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Journal of Computational Physics 2025年 534卷

作者： Crippa, Beatrice Scotti, Anna Villa, Andrea MOX-Laboratory for Modeling and Scientific Computing Department of Mathematics Politecnico di Milano Piazza Leonardo da Vinci 32 Milan20133 Italy Via R. Rubattino 54 Milan20134 Italy

We derive a mixed-dimensional 3D-1D formulation of the electrostatic equation in two domains with different dielectric constants to compute, with an affordable computational cost, the electric field and potential in the relevant case of thin inclusions in a larger 3D domain. The numerical solution is obtained by Mixed Finite Elements for the 3D problem and Finite Elements on the 1D domain. We analyze some test cases with simple geometries to validate the proposed approach against analytical solutions, and perform comparisons with the fully resolved 3D problem. We treat the case where ramifications are present in the one-dimensional domain and show some results on the geometry of an electrical treeing, a ramified structure that propagates in insulators causing their failure. © 2025 The Authors

关键词： Mixed-dimensions 3d-1d coupling Multiscale Electrical treeing

Defining myocardial fiber bundle architecture in atrial digital twins

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Computers in Biology and Medicine 2025年 188卷 109774-109774页

作者： Piersanti, Roberto Bradley, Ryan Ali, Syed Yusuf Quarteroni, Alfio Dede’, Luca Trayanova, Natalia A. MOX - Laboratory of Modeling and Scientific Computing Dipartimento di Matematica Politecnico di Milano Milano Italy ADVANCE - Alliance for Cardiovascular Diagnostic and Treatment Innovation Johns Hopkins University Baltimore United States Research Computing Lehigh University BethlehemPA United States Department of Biomedical Engineering Johns Hopkins University Baltimore United States Mathematics Institute École Polytechnique Fédérale de Lausanne Lausanne Switzerland

A key component in developing atrial digital twins (ADT) - virtual representations of patients’ atria — is the accurate prescription of myocardial fibers which are essential for the tissue characterization. Due to the difficulty of reconstructing atrial fibers from medical imaging, a widely used strategy for fiber generation in ADT relies on mathematical models. Existing methodologies utilize semi-automatic approaches, are tailored to specific morphologies, and lack rigorous validation against imaging fiber data. In this study, we introduce a novel atrial Laplace–Dirichlet-Rule-Based Method (LDRBM) for prescribing highly detailed myofiber orientations and providing robust regional annotation in bi-atrial morphologies of any complexity. The robustness of our approach is verified in eight extremely detailed bi-atrial geometries, derived from a sub-millimeter Diffusion-Tensor-Magnetic-Resonance Imaging (DTMRI) human atrial fiber dataset. We validate the LDRBM by quantitatively recreating each of the DTMRI fiber architectures: a comprehensive comparison with DTMRI ground truth data is conducted, investigating differences between electrophysiology (EP) simulations provided by either LDRBM and DTMRI fibers. Finally, we demonstrate that the novel LDRBM outperforms current state-of-the-art (LDRBMs and Universal Atrial Coordinates) fiber models, confirming the exceptional accuracy of our methodology and the critical importance of incorporating detailed fiber orientations in EP simulations. Ultimately, this work represents a fundamental step towards the development of physics-based digital twins of the human atria, establishing a new standard for prescribing fibers in ADT. © 2025 The Authors

关键词： Magnetic resonance imaging

Schwarz methods for a preconditioned WOPSIP method for elliptic problems

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Computational Methods in Applied Mathematics 2012年第3期12卷 241-272页

作者： Antonietti, Paola F. De Dios, Blanca Ayuso Brenner, Susanne C. Sung, Li-Yeng MOX-Laboratory for Modeling and Scientific Computing Dipartimento di Matematica Politecnico di Milano Piazza Leondardo da Vinci 32 20133 Milano Italy Spain Department of Mathematics Center for Computation and Technology Louisiana State University Baton Rouge LA 70803 United States

We propose and analyze several two-level non-overlapping Schwarz methods for a preconditioned weakly over-penalized symmetric interior penalty (WOPSIP) discretization of a second order boundary value problem. We show that the preconditioners are scalable and that the condition number of the resulting preconditioned linear systems of equations is independent of the penalty parameter and is of order Hh-1, where H and h represent the mesh sizes of the coarse and fine partitions, respectively. Numerical experiments that illustrate the performance of the proposed two-level Schwarz methods are also presented. © 2012 Institute of Mathematics, NAS of Belarus.

关键词： Numerical methods

Mixed-primal Discontinuous Galerkin approximation of flows in fractured porous media on polygonal and polyhedral grids

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Mixed-primal Discontinuous Galerkin approximation of flows i...

90th Annual Meeting of the International Association of Applied Mathematics and Mechanics, GAMM 2019

作者： Facciolà, Chiara Antonietti, Paola Francesca Verani, Marco MOX – Laboratory for Modeling and Scientific Computing Dipartimento di Matematica Politecnico di Milano Piazza Leonardo da Vin ci 32 Milano20133 Italy

We propose a formulation based on discontinuous Galerkin methods on polygonal/polyhedral grids for the simulation of flows in fractured porous media. We adopt a model for single-phase flows where the fracture is modelled as a (d − 1)-dimensional interface in a d-dimensional bulk domain and the flow is governed by the Darcy’s law in both the bulk and the fracture. The two problems are then coupled through physically consistent conditions. We focus on the numerical approximation of the coupled bulk-fracture problem, discretizing the bulk problem in mixed form and the fracture problem in primal form. We present an priori h-and p-version error estimate in a suitable (mesh-dependent) energy norm and numerical tests assessing it. © 2019 Wiley-VCH Verlag GmbH & Co. KGaA, Weinheim.

关键词： Fracture

On mathematical and numerical modelling of multiphysics wave propagation with polytopal discontinuous galerkin methods

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arXiv 2021年

作者： Antonietti, Paola F. Botti, Michele Mazzieri, Ilario MOX-Laboratory for Modeling and Scientific Computing Department of Mathematics Politecnico di Milano Italy

In this work we review discontinuous Galerkin finite element methods on polytopal grids (PolydG) for the numerical simulation of multiphysics wave propagation phenomena in heterogeneous media. In particular, we address wave phenomena in elastic, poro-elastic, and poro-elasto-acoustic materials. Wave propagation is modeled by using either the elastodynamics equation in the elastic domain, the acoustics equations in the acoustic domain and the low-frequency Biot’s equations in the poroelastic one. The coupling between different models is realized by means of (physically consistent) transmission conditions, weakly imposed at the interface between the subdomains. For all models configuration, we introduce and analyse the PolydG semi-discrete formulation, which is then coupled with suitable time marching schemes. For the semi-discrete problem, we present the stability analysis and derive a-priori error estimates in a suitable energy norm. A wide set of verification tests with manufactured solutions are presented in order to validate the error analysis. Examples of physical interest are also shown to demonstrate the capability of the proposed *** Codes 35L05, 65M12, 65M60, 74F10 Copyright © 2021, The Authors. All rights reserved.

关键词： Galerkin methods

Two-Level Schwarz Preconditioners for Super Penalty Discontinuous Galerkin Methods

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Communications in Computational Physics 2009年第2期5卷 398-412页

作者： Paola F.Antonietti Blanca Ayuso MOX–Laboratory for Modeling and Scientific Computing Dipartimento di Matematica”F.Brioschi”Politecnico di Milanovia Bonardi 920133 MilanoItaly Departamento de Matematicas Universidad Autonoma de MadridCampus de CantoblancoCtra.de Colmenar Viejo28049 MadridSpain

We extend the construction and analysis of the non-overlapping Schwarz preconditioners proposed in[2,3]to the(non-consistent)super penalty discontinuous Galerkin methods introduced in[5]and[8].We show that the resulting preconditioners are scalable,and we provide the convergence *** also present numerical experiments confirming the sharpness of the theoretical results.

关键词： Schwarz preconditioners super penalty discontinuous Galerkin methods

Numerical Solution of linear drift-diffusion and pure drift equations on one-dimensional graphs

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arXiv 2024年

作者： Crippa, Beatrice Scotti, Anna Villa, Andrea MOX-Laboratory for Modeling and Scientific Computing Department of Mathematics Politecnico di Milano Milan20133 Italy Milano20134 Italy

We propose numerical schemes for the approximate solution of problems defined on the edges of a one-dimensional graph. In particular, we consider linear transport and a drift-diffusion equations, and discretize them by extending Finite Volume schemes with upwind flux to domains presenting bifurcation nodes with an arbitrary number of incoming and outgoing edges, and implicit time discretization. We show that the discrete problems admit positive unique solutions, and we test the methods on the intricate geometry of an electrical treeing. © 2024, CC BY.

关键词： Partial differential equations

A mixed-dimensional model for the electrostatic problem on coupled domains

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arXiv 2024年

作者： Crippa, Beatrice Scotti, Anna Villa, Andrea MOX-Laboratory for Modeling and Scientific Computing Department of Mathematics Politecnico di Milano Milan20133 Italy Milano20134 Italy

关键词： Electric fields

The lifex library version 2.0

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arXiv 2024年

作者： Bucelli, Michele MOX Laboratory of Modeling and Scientific Computing Dipartimento di Matematica Politecnico di Milano Piazza Leonardo da Vinci 32 Milano20133 Italy

This article presents updates to lifex [Africa, SoftwareX (2022)], a C++ library for high-performance finite element simulations of multiphysics, multiscale and multidomain problems. In this release, we introduce an additional intergrid transfer method for non-matching multiphysics coupling on the same domain, significantly optimize nearest-neighbor point searches and interface coupling utilities, extend the support for 2D and mixed-dimensional problems, and provide improved facilities for input/output and simulation serialization and restart. These advancements also propagate to the previously released modules of lifex specifically designed for cardiac modeling and simulation, namely lifex-fiber [Africa et al., BMC Bioinformatics (2023)], lifex-ep [Africa et al., BMC Bioinformatics (2023)] and lifex-cfd [Africa et al., Computer Physics Communications (2024)]. The changes introduced in this release aim at consolidating lifex’s position as a valuable and versatile tool for the simulation of multiphysics systems. © 2024, CC BY.

关键词： C++ (programming language)