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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

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. © 2024, CC BY.

关键词： Electric fields

Polytopal discontinuous Galerkin methods for low-frequency poroelasticity coupled to unsteady Stokes flow

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arXiv 2025年

作者： Botti, Michele Fumagalli, Ivan Mazzieri, Ilario MOX Laboratory for Modeling and Scientific Computing Dipartimento di Matematica Politecnico di Milano Piazza Leonardo da Vinci 32 MilanoI-20133 Italy

We focus on the numerical analysis of a polygonal discontinuous Galerkin scheme for the simulation of the exchange of fluid between a deformable saturated poroelastic structure and an adjacent free-flow channel. We specifically address wave phenomena described by the low-frequency Biot model in the poroelastic region and unsteady Stokes flow in the open channel, possibly an isolated cavity or a connected fracture system. The coupling at the interface between the two regions is realized by means of transmission conditions expressing conservation laws. The spatial discretization hinges on the weak form of the two-displacement poroelasticity system and a stress formulation of the Stokes equation with weakly imposed symmetry. We present a complete stability analysis for the proposed semi-discrete formulation and derive a-priori hp-error *** Codes 65M12, 65M60 Copyright © 2025, The Authors. All rights reserved.

关键词： Galerkin methods

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)

A polytopal discontinuous Galerkin method for the pseudo-stress formulation of the unsteady Stokes problem

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arXiv 2024年

作者： Antonietti, Paola F. Botti, Michele Cancrini, Alessandra Mazzieri, Ilario MOX Laboratory for Modeling and Scientific Computing Dipartimento di Matematica Politecnico di Milano Piazza Leonardo da Vinci 32 MilanoI-20133 Italy

This work aims to construct and analyze a discontinuous Galerkin method on polytopal grids (PolydG) to solve the pseudo-stress formulation of the unsteady Stokes problem. The pseudo-stress variable is introduced due to the growing interest in non-Newtonian flows and coupled interface problems, where stress assumes a fundamental role. The space-time discretization of the problem is achieved by combining the PolydG approach with the implicit θ-method time integration scheme. For both the semi- and fully-discrete problems we present a detailed stability analysis. Moreover, we derive convergence estimates for the fully discrete space-time discretization. A set of verification tests is presented to verify the theoretical estimates and the application of the method to cases of engineering *** Codes 65M12, 65M60 Copyright © 2024, The Authors. All rights reserved.

关键词： Non Newtonian flow

Shape-informed surrogate models based on signed distance function domain encoding

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arXiv 2024年

作者： Zhang, Linying Pagani, Stefano Zhang, Jun Regazzoni, Francesco MOX Laboratory of Modeling and Scientific Computing Dipartimento di Matematica Politecnico di Milano Milano Italy School of Aeronautic Science and Engineering Beihang University Beijing China

We propose a non-intrusive method to build surrogate models that approximate the solution of parameterized partial differential equations (PDEs), capable of taking into account the dependence of the solution on the shape of the computational domain. Our approach is based on the combination of two neural networks (NNs). The first NN, conditioned on a latent code, provides an implicit representation of geometry variability through signed distance functions. This automated shape encoding technique generates compact, low-dimensional representations of geometries within a latent space, without requiring the explicit construction of an encoder. The second NN reconstructs the output physical fields independently for each spatial point, thus avoiding the computational burden typically associated with high-dimensional discretizations like computational meshes. Furthermore, we show that accuracy in geometrical characterization can be further enhanced by employing Fourier feature mapping as input feature of the NN. The meshless nature of the proposed method, combined with the dimensionality reduction achieved through automatic feature extraction in latent space, makes it highly flexible and computationally efficient. This strategy eliminates the need for manual intervention in extracting geometric parameters, and can even be applied in cases where geometries undergo changes in their topology. Numerical tests in the field of fluid dynamics and solid mechanics demonstrate the effectiveness of the proposed method in accurately predict the solution of PDEs in domains of arbitrary shape. Remarkably, the results show that it achieves accuracy comparable to the best-case scenarios where an explicit parametrization of the computational domain is available. © 2024, CC BY-SA.

关键词： Mesh generation

A space-time discontinuous Galerkin method for coupled poroelasticity-elasticity problems

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arXiv 2023年

作者： Antonietti, Paola F. Botti, Michele Mazzieri, Ilario MOX Laboratory for Modeling and Scientific Computing Dipartimento di Matematica Politecnico di Milano Piazza Leonardo da Vinci 32 MilanoI-20133 Italy

This work is concerned with the analysis of a space-time finite element discontinuous Galerkin method on polytopal meshes (XT-PolydG) for the numerical discretization of wave propagation in coupled poroelastic-elastic media. The mathematical model consists of the low-frequency Biot’s equations in the poroelastic medium and the elastodynamics equation for the elastic one. To realize the coupling, suitable transmission conditions on the interface between the two domains are (weakly) embedded in the formulation. The proposed PolydG discretization in space is then coupled with a dG time integration scheme, resulting in a full space-time dG discretization. We present the stability analysis for both the continuous and the semidiscrete formulations, and we derive error estimates for the semidiscrete formulation in a suitable energy norm. The method is applied to a wide set of numerical test cases to verify the theoretical bounds. Examples of physical interest are also presented to investigate the capability of the proposed method in relevant geophysical *** Codes 65M12, 65M60 Copyright © 2023, The Authors. All rights reserved.

关键词： Multiphysics

A hybrid finite volume - spectral element method for aeroacoustic problems

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arXiv 2023年

作者： Artoni, Alberto Antonietti, Paola F. Mazzieri, Ilario Parolini, Nicola Rocchi, Daniele MOX-Laboratory for Modeling and Scientific Computing Department of Mathematics Politecnico di Milano Milan20133 Italy Department of Mechanical Engineering Politecnico di Milano Milan20156 Italy

We propose a hybrid Finite Volume (FV) - Spectral Element Method (SEM) for modelling aeroacoustic phenomena based on the Lighthill’s acoustic analogy. First the fluid solution is computed employing a FV method. Then, the sound source term is projected onto the acoustic grid and the inhomogeneous Lighthill’s wave equation is solved employing the SEM. The novel projection method computes offline the intersections between the acoustic and the fluid grids in order to preserve the accuracy. The proposed intersection algorithm is shown to be robust, scalable and able to efficiently compute the geometric intersection of arbitrary polyhedral elements. We then analyse the properties of the projection error, showing that if the fluid grid is fine enough we are able to exploit the accuracy of the acoustic solver and we numerically assess the obtained theoretical estimates. Finally, we address two relevant aeroacoustic benchmarks, namely the corotating vortex pair and the noise induced by a laminar flow around a squared cylinder, to demonstrate in practice the effectiveness of the projection method when dealing with high order solvers. The flow computations are performed with OpenFOAM [50], an open-source finite volume library, while the inhomogeneous Lighthill’s wave equation is solved with SPEED [34], an open-source spectral element library. Copyright © 2023, The Authors. All rights reserved.

关键词： Wave equations

Coupled Eikonal problems to model cardiac reentries in Purkinje network and myocardium

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arXiv 2024年

作者： Brunati, Samuele Bucelli, Michele Piersanti, Roberto Dede, Luca Vergara, Christian MOX Laboratory of Modeling and Scientific Computing Dipartimento di Matematica Politecnico di Milano Milano Italy LaBS Laboratory of Biological Structure Mechanics Dipartimento di Chimica Materiali e Ingegneria Chimica Politecnico di Milano Milano Italy

We propose a novel partitioned scheme based on Eikonal equations to model the coupled propagation of the electrical signal in the His-Purkinje system and in the myocardium for cardiac electrophysiology. This scheme allows, for the first time in Eikonal-based modeling, to capture all possible signal reentries between the Purkinje network and the cardiac muscle that may occur under pathological conditions. As part of the proposed scheme, we introduce a new pseudo-time method for the Eikonal-diffusion problem in the myocardium, to correctly enforce electrical stimuli coming from the Purkinje network. We test our approach by performing numerical simulations of cardiac electrophysiology in a real biventricular geometry, under both pathological and therapeutic conditions, to demonstrate its flexibility, robustness, and accuracy. © 2024, CC BY.

关键词： Heart

Discontinuous Galerkin approximation of the fully-coupled thermo-poroelastic problem

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arXiv 2022年

作者： Bonetti, Stefano Botti, Michele Antonietti, Paola F. MOX-Laboratory for Modeling and Scientific Computing Dipartimento di Matematica Politecnico di Milano Piazza Leondardo da Vinci 32 Milano20133 Italy

We present and analyze a discontinuous Galerkin method for the numerical modelling of the fully-coupled quasi-static thermo-poroelastic problem. In particular, for the space discretization we introduce a discontinuous Galerkin method over polygonal and polyhedral grids and we present the stability analysis via two different approaches: first exploiting the Poincarè's inequality and second using the generalized inf-sup condition. Error estimates are derived for the resulting semi-discrete formulation in a suitable mesh dependent energy norm. Numerical simulations are presented in order to validate the theoretical analysis and to show the application of the model to a realistic case test. Copyright © 2022, The Authors. All rights reserved.

关键词： Galerkin methods