arXiv subjects

Simone Deparis

Publications and source records attributed to Simone Deparis.

15 recordsLinked to original sources

arXiv2026

Can the perfect Swiss alphorn be designed? A combination of reduced basis method and machine learning for shape optimization

In this work, we investigate the shape optimization of a Swiss alphorn to achieve resonance frequencies as close as possible to prescribed target notes. We construct an accurate geometric model of the alphorn based on both experimental measurements and imaging techniques. The latter are used to reconstruct geometric information where direct measures are not available. The resulting 3D model is meshed using a fully structured approach. Then, we introduce a geometry-parametrized finite element (FE) formulation of the Helmholtz equation enabling the application of the Reduced Basis Method (RBM) to efficiently generate a large dataset of simulations, in which each set of geometric parameters corresponds to a specific set of resonance frequencies (notes). This dataset enables the training of machine learning (ML) models for forward prediction of resonance frequencies from geometric parameters, as well as inverse design, where geometries are estimated to achieve specified target notes.

math.NA↗

arXiv2025

Generalisation of the Total Linearisation Method to Three-dimensional Free-Surface Flows

An iterative Finite Element method predicated on a linearisation of the weak form around a reference configuration is derived for general, three-dimensional, free-surface flows, including systems with moving contact lines. The method is a rigorous generalisation of the Total Linearisation Method that was proposed by Kruyt et al. (1988) for two-dimensional flows with contact angles limited to $90^\circ$. In contrast to existing numerical methods for free-surface-flow problems, the present linearisation produces a weak form that is devoid of displacement degrees of freedom in the bulk, thus nearly halving the size of the linear system when compared to standard linearised methods. A novel preconditioner, whose implementation is made possible by the size reduction, is employed to solve the large resulting monolithic Jacobian systems with the Generalised Minimum Residual Method. The proposed method and the preconditioner are shown to be effective on two numerical examples of capillary flows, namely (i) the cylindrical die swell problem, solved both in 3D Cartesian coordinates and under the Ansatz of axisymmetry; and (ii) an enclosed 2D thermo-capillary problem. For the die swell, numerical results are validated by existing experimental results and prior simulations, and confirm both the extension to 3D and theoretical convergence rates. For the thermo-capillary problem, simulations verify earlier calculations. Additional simulations are also carried out for a new range of contact angles made possible by the extension.

physics.flu-dyn↗

arXiv2025

IMEX-RB: a self-adaptive implicit-explicit time integration scheme exploiting the reduced basis method

In this work, we introduce a self-adaptive implicit-explicit (IMEX) time integration scheme, named IMEX-RB, for the numerical integration of systems of ordinary differential equations (ODEs), arising from spatial discretizations of partial differential equations (PDEs) by finite difference methods. Leveraging the Reduced Basis (RB) method, at each timestep we project the high-fidelity problem onto a suitable low-dimensional subspace and integrate its dynamics implicitly. Following the IMEX paradigm, the resulting solution then serves as an educated guess within a full-order explicit step. Notably, compared to the canonical RB method, IMEX-RB neither requires a parametrization of the underlying PDE nor features an offline-online splitting, since the reduced subspace is built dynamically, exploiting the high-fidelity solution history. We present the first-order formulation of IMEX-RB, demonstrating and showcasing its convergence and stability properties. In particular, under appropriate conditions on the method's hyperparameters, IMEX-RB is unconditionally stable. The theoretical analysis is corroborated by numerical experiments performed on representative model problems in two and three dimensions. The results demonstrate that our approach can outperform conventional time integration schemes like backward Euler. Indeed, IMEX-RB yields high-fidelity accurate solutions, provided that its main hyperparameters - namely the reduced basis size and the stability tolerance - are suitably tuned. Moreover, IMEX-RB realizes computational gains over backward Euler for a range of timestep sizes above the forward Euler stability threshold.

math.NA↗

arXiv2025

Deformable registration and generative modelling of aortic anatomies by auto-decoders and neural ODEs

This work introduces AD-SVFD, a deep learning model for the deformable registration of vascular shapes to a pre-defined reference and for the generation of synthetic anatomies. AD-SVFD operates by representing each geometry as a weighted point cloud and models ambient space deformations as solutions at unit time of ODEs, whose time-independent right-hand sides are expressed through artificial neural networks. The model parameters are optimized by minimizing the Chamfer Distance between the deformed and reference point clouds, while backward integration of the ODE defines the inverse transformation. A distinctive feature of AD-SVFD is its auto-decoder structure, that enables generalization across shape cohorts and favors efficient weight sharing. In particular, each anatomy is associated with a low-dimensional code that acts as a self-conditioning field and that is jointly optimized with the network parameters during training. At inference, only the latent codes are fine-tuned, substantially reducing computational overheads. Furthermore, the use of implicit shape representations enables generative applications: new anatomies can be synthesized by suitably sampling from the latent space and applying the corresponding inverse transformations to the reference geometry. Numerical experiments, conducted on healthy aortic anatomies, showcase the high-quality results of AD-SVFD, which yields extremely accurate approximations at competitive computational costs.

cs.CV↗

arXiv2025

Model order reduction of hemodynamics by space-time reduced basis and reduced fluid-structure interaction

In this work, we apply the space-time Galerkin reduced basis (ST-GRB) method to a reduced fluid-structure interaction model, for the numerical simulation of hemodynamics in arteries. In essence, ST-GRB extends the classical reduced basis (RB) method, exploiting a data-driven low-dimensional linear encoding of the temporal dynamics to further cut the computational costs. The current investigation brings forth two key enhancements, compared to previous works on the topic. On the one side, we model blood flow through the Navier-Stokes equations, hence accounting for convection. In this regard, we implement a hyper-reduction scheme, based on approximate space-time reduced affine decompositions, to deal with nonlinearities effectively. On the other side, we move beyond the constraint of modelling blood vessels as rigid structures, acknowledging the importance of elasticity for the accurate simulation of complex blood flow patterns. To limit computational complexity, we adopt the Coupled Momentum model, incorporating the effect of wall compliance in the fluid's equations through a generalized Robin boundary condition. In particular, we propose an efficient strategy for handling the spatio-temporal projection of the structural displacement, which ultimately configures as a by-product. The performances of ST-GRB are assessed in three different numerical experiments. The results confirm that the proposed approach can outperform the classical RB method, yielding precise approximations of high-fidelity solutions at more convenient costs. However, the computational gains of ST-GRB vanish if the number of retained temporal modes is too large, which occurs either when complex dynamics arise or if very precise solutions are sought.

math.NA↗

arXiv2025

A spline-based hexahedral mesh generator for patient-specific coronary arteries

This paper presents a spline-based hexahedral mesh generator for tubular geometries commonly encountered in haemodynamics studies, in particular coronary arteries. We focus on techniques for accurately meshing vessels with stenoses and aneurysms, as well as non-planar bifurcations. Our approach incorporates several innovations, including a spline-based description of the vessel geometry in both the radial and the longitudinal directions, the use of Hermite curves for modeling non-planar bifurcations, and a generalization to non-planar n intersecting branches. This method eliminates the need for a concrete vessel surface, grid smoothing, and other post-processing. A technique to generate grids with boundary layers is also presented. We validate the generated meshes using commonly employed quality indices, compare them against state-of-the-art mesh generators and apply our method to complex coronary trees. Finally, we present finite element fluid flow simulations with physiological boundary conditions. To validate the proposed framework, a wall-shear-stress-based convergence test and computations of haemodynamic indices are also presented.

math.NA↗

arXiv2024

Mimetic finite difference schemes for transport operators with divergence-free advective field and applications to plasma physics

In wave propagation problems, finite difference methods implemented on staggered grids are commonly used to avoid checkerboard patterns and to improve accuracy in the approximation of short-wavelength components of the solutions. In this study, we develop a mimetic finite difference (MFD) method on staggered grids for transport operators with divergence-free advective field that is proven to be energy-preserving in wave problems. This method mimics some characteristics of the summation-by-parts (SBP) operators framework, in particular it preserves the divergence theorem at the discrete level. Its design is intended to be versatile and applicable to wave problems characterized by a divergence-free velocity. As an application, we consider the electrostatic shear Alfv\'en waves (SAWs), appearing in the modeling of plasmas. These waves are solved in a magnetic field configuration recalling that of a tokamak device. The study of the generalized eigenvalue problem associated with the SAWs shows the energy conservation of the discretization scheme, demonstrating the stability of the numerical solution.

math.NA↗

arXiv2022

Space-time reduced basis methods for parametrized unsteady Stokes equations

In this work, we analyse space-time reduced basis methods for the efficient numerical simulation of hemodynamics in arteries. The classical formulation of the reduced basis (RB) method features dimensionality reduction in space, while finite differences schemes are employed for the time integration of the resulting ordinary differential equation (ODE). Space-time reduced basis (ST-RB) methods extend the dimensionality reduction paradigm to the temporal dimension, projecting the full-order problem onto a low-dimensional spatio-temporal subspace. Our goal is to investigate the application of ST-RB methods to the unsteady incompressible Stokes equations, with a particular focus on stability. High-fidelity simulations are performed using the Finite Element (FE) method and BDF2 as time marching scheme. We consider two different ST-RB methods. In the first one - called ST-GRB - space-time model order reduction is achieved by means of a Galerkin projection; a spatio-temporal velocity basis enrichment procedure is introduced to guarantee stability. The second method - called ST-PGRB - is characterized by a Petrov--Galerkin projection, stemming from a suitable minimization of the FOM residual, that allows to automatically attain stability. The classical RB method - denoted as SRB-TFO - serves as a baseline for the theoretical development. Numerical tests have been conducted on an idealized symmetric bifurcation geometry and on the patient-specific one of a femoropopliteal bypass. The results show that both ST-RB methods provide accurate approximations of the high-fidelity solutions, while considerably reducing the computational cost. In particular, the ST-PGRB method exhibits the best performance, as it features a better computational efficiency while retaining accuracies in accordance with theoretical expectations.

math.NA↗

arXiv2021

DeepBND: a Machine Learning approach to enhance Multiscale Solid Mechanics

Effective properties of materials with random heterogeneous structures are typically determined by homogenising the mechanical quantity of interest in a window of observation. The entire problem setting encompasses the solution of a local PDE and some averaging formula for the quantity of interest in such domain. There are relatively standard methods in the literature to completely determine the formulation except for two choices: i) the local domain itself and the ii) boundary conditions. Hence, the modelling errors are governed by the quality of these two choices. The choice i) relates to the degree of representativeness of a microscale sample, i.e., it is essentially a statistical characteristic. Naturally, its reliability is higher as the size of the observation window becomes larger and/or the number of samples increases. On the other hand, excepting few special cases there is no automatic guideline to handle ii). Although it is known that the overall effect of boundary condition becomes less important with the size of the microscale domain, the computational cost to simulate such large problem several times might be prohibitive even for relatively small accuracy requirements. Here we introduce a machine learning procedure to select most suitable boundary conditions for multiscale problems, particularly those arising in solid mechanics. We propose the combination Reduced-Order Models and Deep Neural Networks in an offline phase, whilst the online phase consists in the very same homogenisation procedure plus one (cheap) evaluation of the trained model for boundary conditions. Hence, the method allows an implementation with minimal changes in existing codes and the use of relatively small domains without losing accuracy, which reduces the computational cost by several orders of magnitude.

math.NA↗

arXiv2020

Model order reduction of flow based on a modular geometrical approximation of blood vessels

We are interested in a reduced order method for the efficient simulation of blood flow in arteries. The blood dynamics is modeled by means of the incompressible Navier-Stokes equations. Our algorithm is based on an approximated domain-decomposition of the target geometry into a number of subdomains obtained from the parametrized deformation of geometrical building blocks (e.g. straight tubes and model bifurcations). On each of these building blocks, we build a set of spectral functions by proper orthogonal decomposition of a large number of snapshots of finite element solutions (offline phase). The global solution of the Navier-Stokes equations on a target geometry is then found by coupling linear combinations of these local basis functions by means of spectral Lagrange multipliers (online phase). Being that the number of reduced degrees of freedom is considerably smaller than their finite element counterpart, this approach allows us to significantly decrease the size of the linear system to be solved in each iteration of the Newton-Raphson algorithm. We achieve large speedups with respect to the full order simulation (in our numerical experiments, the gain is at least of one order of magnitude and grows inversely with respect to the reduced basis size), whilst still retaining satisfactory accuracy for most cardiovascular simulations.

math.NA↗

arXiv2019

Data driven approximation of parametrized PDEs by Reduced Basis and Neural Networks

We are interested in the approximation of partial differential equations with a data-driven approach based on the reduced basis method and machine learning. We suppose that the phenomenon of interest can be modeled by a parametrized partial differential equation, but that the value of the physical parameters is unknown or difficult to be directly measured. Our method allows to estimate fields of interest, for instance temperature of a sample of material or velocity of a fluid, given data at a handful of points in the domain. We propose to accomplish this task with a neural network embedding a reduced basis solver as exotic activation function in the last layer. The reduced basis solver accounts for the underlying physical phenomenonon and it is constructed from snapshots obtained from randomly selected values of the physical parameters during an expensive offline phase. The same full order solutions are then employed for the training of the neural network. As a matter of fact, the chosen architecture resembles an asymmetric autoencoder in which the decoder is the reduced basis solver and as such it does not contain trainable parameters. The resulting latent space of our autoencoder includes parameter-dependent quantities feeding the reduced basis solver, which -- depending on the considered partial differential equation -- are the values of the physical parameters themselves or the affine decomposition coefficients of the differential operators.

math.NA↗

arXiv2018

Coupling non-conforming discretizations of PDEs by spectral approximation of the Lagrange multiplier space

This work focuses on the development of a non-conforming domain decomposition method for the approximation of PDEs based on weakly imposed transmission conditions: the continuity of the global solution is enforced by a discrete number of Lagrange multipliers defined over the interfaces of adjacent subdomains. The method falls into the class of primal hybrid methods, which also include the well-known mortar method. Differently from the mortar method, we discretize the space of basis functions on the interface by spectral approximation independently of the discretization of the two adjacent domains; one of the possible choices is to approximate the interface variational space by Fourier basis functions. As we show in the numerical simulations, our approach is well-suited for the solution of problems with non-conforming meshes or with finite element basis functions with different polynomial degrees in each subdomain. Another application of the method that still needs to be investigated is the coupling of solutions obtained from otherwise incompatible methods, such as the finite element method, the spectral element method or isogeometric analysis.

math.NA↗

arXiv2018

Reduced order models for fluid-structure interaction problems with applications in haemodynamics

This paper deals with fast simulations of the haemodynamics in large arteries by considering a reduced model of the associated fluid-structure interaction problem, which in turn allows an additional reduction in terms of the numerical discretisation. The resulting method is both accurate and computationally cheap. This goal is achieved by means of two levels of reduction: first, we describe the model equations with a reduced mathematical formulation which allows to write the fluid-structure interaction problem as a Navier-Stokes system with non-standard boundary conditions; second, we employ numerical reduction techniques to further and drastically lower the computational costs. The numerical reduction is obtained coupling two well-known techniques: the proper orthogonal decomposition and the reduced basis method, in particular the greedy algorithm. We start by reducing the numerical dimension of the problem at hand with a proper orthogonal decomposition and we measure the system energy with specific norms; this allows to take into account the different orders of magnitude of the state variables, the velocity and the pressure. Then, we introduce a strategy based on a greedy procedure which aims at enriching the reduced discretization space with low offline computational costs. As application, we consider a realistic haemodynamics problem with a perturbation in the boundary conditions and we show the good performances of the reduction techniques presented in the paper. The gains obtained in term of CPU time are of three orders of magnitude.

math.NA↗

arXiv2017

The LifeV library: engineering mathematics beyond the proof of concept

LifeV is a library for the finite element (FE) solution of partial differential equations in one, two, and three dimensions. It is written in C++ and designed to run on diverse parallel architectures, including cloud and high performance computing facilities. In spite of its academic research nature, meaning a library for the development and testing of new methods, one distinguishing feature of LifeV is its use on real world problems and it is intended to provide a tool for many engineering applications. It has been actually used in computational hemodynamics, including cardiac mechanics and fluid-structure interaction problems, in porous media, ice sheets dynamics for both forward and inverse problems. In this paper we give a short overview of the features of LifeV and its coding paradigms on simple problems. The main focus is on the parallel environment which is mainly driven by domain decomposition methods and based on external libraries such as MPI, the Trilinos project, HDF5 and ParMetis. Dedicated to the memory of Fausto Saleri.

math.NA↗

arXiv2017

Simulation of left ventricle fluid dynamics with mitral regurgitation from magnetic resonance images with fictitious elastic structure regularization

Computer modeling can provide quantitative insight into cardiac fluid dynamics phenomena that are not evident from standard imaging tools. We propose a new approach to modeling left ventricle fluid dynamics based on an image-driven model-based description of ventricular motion. In this approach, the end-diastolic geometry and time-dependent deformation of the left ventricle cavity are obtained from cardiac magnetic resonance images and a fictitious elastic structure is used to impose the contractile behavior of the left ventricle. This allows seamless treatment of the isovolumic phases. Besides the ventricular motion, the intracavitary fluid dynamics is controlled by the mitral valve. Three different mitral valve models are included in the simulation: an idealized diode (with or without regurgitation) and a lumped parameter model accounting for the opening dynamics of the valve and including regurgitation.

physics.med-ph↗