arXiv subjects

Felipe Galarce

Publications and source records attributed to Felipe Galarce.

17 recordsLinked to original sources

arXiv2026

Stress-divergence, Laplacian, and rotational forms of the incompressible Navier--Stokes equations with variable viscosity

In the Navier--Stokes equations, incompressibility allows rewriting the viscous term in various forms leading to distinct numerical properties and flow descriptions. Furthermore, models accounting for non-Newtonian, thermal or turbulent effects often break the constant-viscosity assumption, thereby producing additional consistency terms. In this context, the present work compares the classical symmetric-gradient diffusion term with more recent variable-viscosity generalizations of the Laplacian and rotational forms. We discuss, analyze and test their differences with respect to implementation, efficiency, numerical stability and outflow boundary conditions. With a focus on time-dependent flows, we consider second-order implicit-explicit (IMEX) temporal discretizations aimed at improving efficiency and numerical stability. Through a rigorous stability analysis, we show how selected explicit treatments can bypass algorithmic nonlinearities without inducing CFL conditions. Our numerical results highlight important differences between the three viscous formulations---especially in the presence of outflow boundaries, for which the generalized Laplacian form proves more suitable in diffusion-dominated regimes. %(as widely known for constant viscosity).

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arXiv2026

Dynamic Reduced-Order Data Assimilation from Sparse Velocity Measurements

We present a novel reduced-order data assimilation framework, termed Reduced-Order Dynamical Assimilation (RODAS), for reconstructing high-resolution, time-resolved flow fields from sparse velocity measurements. The method combines low-dimensional experimental observations with a physics-based parametric reduced-order model, enabling both spatial extrapolation beyond the measurement region and temporal super-resolution. The approach first identifies the dominant dynamics from sparse measurements through Dynamic Mode Decomposition (DMD), and subsequently reconstructs the corresponding full-order flow evolution by projecting the identified dynamics onto a parametric Proper Orthogonal Decomposition (POD) manifold generated from high-fidelity numerical simulations. Unlike conventional reduced-order data assimilation methods that estimate independent snapshots or treat time as an additional parameter, RODAS reconstructs an entire dynamical trajectory in a single inference step while naturally incorporating parametric variability. We assess the proposed methodology on vortex shedding behind a circular cylinder for both Newtonian and non-Newtonian (Carreau-Yasuda) fluids. Numerical experiments demonstrate accurate reconstruction of high-resolution velocity fields from localized, low-resolution measurements, achieving sub-percent reconstruction errors with sufficiently rich reduced bases, robust performance under severe temporal undersampling, and accurate prediction of engineering quantities of interest such as the drag coefficient. These results demonstrate that RODAS provides an efficient framework for real-time, physics-informed reconstruction of unsteady flows from sparse experimental data.

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arXiv2026

A Stress-Based Estimator for Pressure and Stress Recovery from Velocity Measurements

Non-invasive pressure field estimation from velocity measurements is a longstanding engineering problem. We propose, analyze, and test a pressure-recovery method that computes a full stress field from velocity measurements, and leaves the pressure estimation as a cheap post-processing step. The method relies on a stress-velocity first order formulation of the Navier-Stokes equations, and we show that the formulation accounts for deviations from incompressibility in the measured velocity field by construction. In addition, we theoretically establish the convergence of the finite element (FE) approximation scheme, the stability of the stress recovery with respect to finite-resolution velocity measurements, and then validate this theory numerically. Our results show that the proposed estimator is robust in convective flow regimes and remains accurate at reduced spatial resolution, improving upon state-of-the-art pressure-recovery strategies.

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arXiv2025

Improving performance estimation of a PCM-integrated solar chimney through reduced-order based data assimilation

This study evaluates a data assimilation framework based on reduced-order modeling (ROM-DA), complemented by a hybrid data-filling strategy, to reconstruct dynamic temperature fields in a phase-change-material (PCM) integrated solar chimney from limited temperature measurements. The goal is to enhance the estimation accuracy of the outlet airflow velocity. A regularized least-squares formulation is employed to estimate temperature distributions within an inclined solar chimney using RT-42 as the PCM. The methodology combines (i) a reduced-order model derived from high-fidelity finite-volume simulations of unsteady conjugate heat transfer with liquid-solid phase change and surface radiation, and (ii) three experimental datasets with 22, 135, and 203 measurement points. Missing data are reconstructed using a hybrid filling scheme based on boundary-layer and bicubic interpolations. The assimilated temperature fields are integrated into the thermally coupled forward solver to improve velocity predictions. Results show that the ROM-DA framework reconstructs the transient temperature fields in both the air and PCM domains with relative errors below 10 percent for sparse data and below 3 percent for expanded datasets. When applied to experimental measurements, the approach enhances the fidelity of temperature and velocity fields compared with the baseline model, reducing the outlet velocity RMS error by 20 percent. This represents the first application of a ROM-DA framework to a coupled multiphysics solar chimney with PCM integration, demonstrating its potential for near-real-time thermal state estimation and digital-twin development.

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arXiv2025

Accelerated implicitization: Robust fixed-point iterations arising from an explicit scheme

This work proposes a general strategy for solving possibly nonlinear problems arising from implicit time discretizations as a sequence of explicit solutions. The resulting sequence may exhibit instabilities similar to those of the base explicit scheme, which can be mitigated through Anderson acceleration. The approach uses explicit fixed-point subiterations for nonlinear problems, combined with Anderson acceleration to improve convergence and computational efficiency. Its usability and scalability are verified on three nonlinear differential equations. An error analysis is presented to establish the expected properties of the proposed strategy for both time and space-time formulations. Several examples illustrate the simplicity of the implementation and reveal the influence of parameter choices. The method proves simple to implement and performs well across a range of problems, particularly when matrix assembly is expensive or a good preconditioner for the implicit system is unavailable, such as in highly convective fluid flows. This work formalizes the delay of implicit terms in time discretization, provides a concise error analysis, and enhances the approach using Anderson acceleration. The results are encouraging and well supported by existing theory, laying the groundwork for further research.

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arXiv2025

Estimation of Hemodynamic Parameters via Physics Informed Neural Networks including Hematocrit Dependent Rheology

Physics-Informed Neural Networks (PINNs) show significant potential for solving inverse problems, especially when observations are limited and sparse, provided that the relevant physical equations are known. We use PINNs to estimate smooth velocity and pressure fields from synthetic 4D flow Magnetic Resonance Imaging (MRI) data. We analyze five non-Newtonian dynamic 3D blood flow cases within a realistic aortic model, covering a range of hematocrit values from anemic to polycythemic conditions. To enhance state estimation results, we consider various design and training techniques for PINNs, including adaptive loss balancing, curriculum training, and a realistic measurement operator. Regarding blood rheology, the PINN approach accurately estimates viscosity globally and locally under peak systolic conditions. It also provides a clear pattern recognition for diastolic stages. Regarding mass conservation, PINN estimations effectively reproduce the bifurcation of flow through the different branches of the aorta, demonstrate an excellent representation of the non-slip conditions at the walls, and accurately estimate pressure drops with relative errors below the 5% in the whole pressure field. We test our pressure drop estimations against the state of the art Virtual Work Energy Relative Pressure (vWERP) estimator, and we observe how our results outperform vWERP in terms of both accuracy and time resolution. Additionally, we find that the best results are achieved by computing the velocity field using the PINN, which is then integrated into the vWERP framework, leading to time super-sampled and high-order approximations, with a clinically admissible accuracy.

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arXiv2025

Data assimilation performed with robust shape registration and graph neural networks: application to aortic coarctation

Image-based, patient-specific modelling of hemodynamics can improve diagnostic capabilities and provide complementary insights to better understand the hemodynamic treatment outcomes. However, computational fluid dynamics simulations remain relatively costly in a clinical context. Moreover, projection-based reduced-order models and purely data-driven surrogate models struggle due to the high variability of anatomical shapes in a population. A possible solution is shape registration: a reference template geometry is designed from a cohort of available geometries, which can then be diffeomorphically mapped onto it. This provides a natural encoding that can be exploited by machine learning architectures and, at the same time, a reference computational domain in which efficient dimension-reduction strategies can be performed. We compare state-of-the-art graph neural network models with recent data assimilation strategies for the prediction of physical quantities and clinically relevant biomarkers in the context of aortic coarctation.

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arXiv2025

A parametric study of the pipeline hammer phenomenon in plastic Bingham slurry flows using the finite element method

We conduct a numerical study of the transient phenomenon in pipelines transporting plastic Bingham slurry flows, using a lowest-order finite element method (FEM). While most pipeline hammer studies focus on Newtonian fluids, the transient dynamics in Bingham fluids remains elusive and poorly afforded, despite their significant industrial impact, particularly in mining. A detailed parametric study assesses the effects of the slurry yield stress and the valve closure times on both pressure and velocity distributions along the pipeline, using an adaptive friction model to account for turbulent slurries. Results reveal that yield stress enhances flow resistance and accelerates pressure peak attenuation, underscoring the damping role of Bingham rheology compared to Newtonian flows. These insights emphasize the need for advanced FEM-based schemes in non-Newtonian shockwave modeling, with implications for industrial pipeline design and operational safety.

physics.flu-dyn↗

arXiv2024

A fast food-freezing temperature estimation framework using optimally located sensors

This article presents and assesses a framework for estimating temperature fields in real time for food-freezing applications, significantly reducing computational load while ensuring accurate temperature monitoring, which represents a promising technological tool for optimizing and controlling food engineering processes. The strategy is based on (i) a mathematical model of a convection-dominated problem coupling thermal convection and turbulence, and (ii) a least-squares approach for solving the inverse data assimilation problem, regularized by projecting the governing dynamics onto a reduced-order model (ROM). The unsteady freezing process considers a salmon slice in a freezer cabinet, modeled with temperature-dependent thermophysical properties. The forward problem is approximated using a third-order WENO finite volume solver, including an optimized second-order backward scheme for time discretization. We employ our data assimilation framework to reconstruct the temperature field based on a limited number of sensors and to estimate temperature distributions within frozen food. Sensor placement is optimized using a novel greedy algorithm, which maximizes the observability of the reduced-order dynamics for a fixed set of sensors. The proposed approach allows efficient extrapolation from external sensor measurements to the internal temperature of the food under realistic turbulent flow conditions, which is crucial for maintaining food quality.

math.NA↗

arXiv2024

Fully consistent lowest-order finite element methods for generalised Stokes flows with variable viscosity

Variable viscosity arises in many flow scenarios, often imposing numerical challenges. Yet, discretisation methods designed specifically for non-constant viscosity are few, and their analysis is even scarcer. In finite element methods for incompressible flows, the most popular approach to allow equal-order velocity-pressure interpolation are residual-based stabilisations. For low-order elements, however, the viscous part of that residual cannot be approximated, often compromising accuracy. Assuming slightly more regularity on the viscosity field, we can construct stabilisation methods that fully approximate the residual, regardless of the polynomial order of the finite element spaces. This work analyses two variants of this fully consistent approach, with the generalised Stokes system as a model problem. We prove unique solvability and derive expressions for the stabilisation parameter, generalising some classical results for constant viscosity. Numerical results illustrate how our method completely eliminates the spurious pressure boundary layers typically induced by low-order PSPG-like stabilisations.

math.NA↗

arXiv2024

A stabilized total pressure-formulation of the Biot's poroelasticity equations in frequency domain: numerical analysis and applications

This work focuses on the numerical solution of the dynamics of a poroelastic material in the frequency domain. We provide a detailed stability analysis based on the application of the Fredholm alternative in the continuous case, considering a total pressure formulation of the Biot's equations. In the discrete setting, we propose a stabilized equal order finite element method complemented by an additional pressure stabilization to enhance the robustness of the numerical scheme with respect to the fluid permeability. Utilizing the Fredholm alternative, we extend the well-posedness results to the discrete setting, obtaining theoretical optimal convergence for the case of linear finite elements. We present different numerical experiments to validate the proposed method. First, we consider model problems with known analytic solutions in two and three dimensions. As next, we show that the method is robust for a wide range of permeabilities, including the case of discontinuous coefficients. Lastly, we show the application for the simulation of brain elastography on a realistic brain geometry obtained from medical imaging.

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arXiv2024

Enhancing Hemodynamic Parameter Estimations: Nonlinear Blood Behavior in 4D Flow MRI

Hemodynamic parameters are often estimated assuming a constant Newtonian viscosity, even though blood exhibits shear-thinning behavior. This article investigates the influence of blood rheology and hematocrit (Hct) percentage on the estimation of Wall Shear Stress (WSS), rate of viscous Energy Loss ($\dot{E}_L$) at different points in the cardiac cycle, and the Oscillatory Shear Index (OSI). We focus on a hematocrit-dependent power-law non-Newtonian model, considering a wide range of Hct values at physiological temperature, with rheological parameters obtained from previously reported experimental data. In all cases, we systematically compared WSS, $\dot{E}_L$, and OSI using both Newtonian and power-law models, underscoring the crucial role of blood rheology in accurately assessing cardiovascular diseases. Our results show that, in in-silico experiments, differences in WSS and $\dot{E}_L$ across a wide range of Hct values can reach as high as 190\% and 113\% at systole, and as low as -72\% and -74\% at diastole, respectively. In in-vivo data, differences in WSS and $\dot{E}_L$ can reach up to -45\% and -60\% at systole, and range from -69\% to 73\% at diastole. This study enhances our understanding of the impact of blood rheology on hemodynamic parameter estimations using both in-silico and in-vivo aortic 4D Flow MRI data.

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arXiv2023

Bias and Multiscale Correction Methods for Variational State Estimation

Data assimilation performance can be significantly impacted by biased noise in observations, altering the signal magnitude and introducing fast oscillations or discontinuities when the system lacks smoothness. To mitigate these issues, this paper employ variational state estimation using the so-called parametrized-background data-weak method. This approach relies on a background manifold parametrized by a set of constraints, enabling the state estimation by solving a minimization problem on a reduced-order background model, subject to constraints imposed by the input measurements. The proposed formulation incorporates a novel bias correction mechanism and a manifold decomposition that handles rapid oscillations by treating them as slow-decaying modes based on a two-scale splitting of the classical reconstruction algorithm. The method is validated in different examples, including the assimilation of biased synthetic data, discontinuous signals, and Doppler ultrasound data obtained from experimental measurements.

math.NA↗

arXiv2022

Displacement and pressure reconstruction from magnetic resonance elastography images: application to an in silico brain model

Magnetic resonance elastography is a motion-sensitive image modality that allows to measure in vivo tissue displacement fields in response to mechanical excitations. This paper investigates a data assimilation approach for reconstructing tissue displacement and pressure fields in an in silico brain model from partial elastography data. The data assimilation is based on a parametrized-background data weak methodology, in which the state of the physical system -- tissue displacements and pressure fields -- is reconstructed from the available data assuming an underlying poroelastic biomechanics model. For this purpose, a physics-informed manifold is built by sampling the space of parameters describing the tissue model close to their physiological ranges to simulate the corresponding poroelastic problem, and computing a reduced basis via Proper Orthogonal Decomposition. Displacements and pressure reconstruction is sought in a reduced space after solving a minimization problem that encompasses both the structure of the reduced-order model and the available measurements. The proposed pipeline is validated using synthetic data obtained after simulating the poroelastic mechanics of a physiological brain. The numerical experiments demonstrate that the framework can exhibit accurate joint reconstructions of both displacement and pressure fields. The methodology can be formulated for an arbitrary resolution of available displacement data from pertinent images.

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arXiv2021

State Estimation with Model Reduction and Shape Variability. Application to biomedical problems

We develop a mathematical and numerical framework to solve state estimation problems for applications that present variations in the shape of the spatial domain. This situation arises typically in a biomedical context where inverse problems are posed on certain organs or portions of the body which inevitably involve morphological variations. If one wants to provide fast reconstruction methods, the algorithms must take into account the geometric variability. We develop and analyze a method which allows to take this variability into account without needing any a priori knowledge on a parametrization of the geometrical variations. For this, we rely on morphometric techniques involving Multidimensional Scaling, and couple them with reconstruction algorithms that make use of reduced model spaces pre-computed on a database of geometries. We prove the potential of the method on a synthetic test problem inspired from the reconstruction of blood flows and quantities of medical interest with Doppler ultrasound imaging.

math.NA↗

arXiv2021

Reconstructing Haemodynamics Quantities of Interest from Doppler Ultrasound Imaging

The present contribution deals with the estimation of haemodynamics Quantities of Interest by exploiting Ultrasound Doppler measurements. A fast method is proposed, based on the PBDW method. Several methodological contributions are described: a sub-manifold partitioning is introduced to improve the reduced-order approximation, two different ways to estimate the pressure drop are compared, and an error estimation is derived. A test-case on a realistic common carotid geometry is presented, showing that the proposed approach is promising in view of realistic applications.

math.NA↗

arXiv2020

Fast reconstruction of 3D blood flows from Doppler ultrasound images and reduced models

This paper deals with the problem of building fast and reliable 3D reconstruction methods for blood flows for which partial information is given by Doppler ultrasound measurements. This task is of interest in medicine since it could enrich the available information used in the diagnosis of certain diseases which is currently based essentially on the measurements coming from ultrasound devices. The fast reconstruction of the full flow can be performed with state estimation methods that have been introduced in recent years and that involve reduced order models. One simple and efficient strategy is the so-called Parametrized Background Data-Weak approach (PBDW). It is a linear mapping that consists in a least squares fit between the measurement data and a linear reduced model to which a certain correction term is added. However, in the original approach, the reduced model is built a priori and independently of the reconstruction task (typically with a proper orthogonal decomposition or a greedy algorithm). In this paper, we investigate the construction of other reduced spaces which are built to be better adapted to the reconstruction task and which result in mappings that are sometimes nonlinear. We compare the performance of the different algorithms on numerical experiments involving synthetic Doppler measurements. The results illustrate the superiority of the proposed alternatives to the classical linear PBDW approach.

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